NON COMMUTATIVE
GEOMETRY
A. CONNES
IuNtitut des Hautes Etudes Scientifiques
35, route de Chartres
91440 - Вш-es-sur-Yvette (France)
Octobre 1993
IHES/M/93/54

NON COMMUTATIVE
GEOMETRY
A. CONNES
This book is the English translation of the French book "Geometrie non commutative" published by
Intereditions Paris A990). The translation will be published by Academic Press. After the initial translation
of S.K. Berberian, a considerable amount of rewriting and many additions have been done, tripling the size
of the original manuscript. In particular the present text contains several unpublished results. My thanks
go above all to Cecile whose patience and care for the manuscript have been essential in its completion.
This second version of the book greatly benefited of the important modifications suggested by M. Riefiel, D.
Sullivan, J.L. Loday, J. Lott, J. Bellissard, P. Cohen, R. Coquereaux, J. Dixmier, M. Karoubi, P. Kree, H.
Bacry, P. de la Harpe, A. Hof.
Institut des Hautes Etudes Scientifiques
35, route de Chartres
91440 - Bures-sur-Yvette (France)
Octobre 1993
IHES/M/93/54

TABLE OF CONTENTS
Introduction
Chapter I. Non commutative spaces and measure theory.
1.1. Heisenberg and the non commutative algebra of physical quantities associated to a microscopic system.
1.2. Statistical state of a microscopic system and quantum statistical mechanics.
1.3. Modular theory and the classification of factors.
1.4. Geometric examples of von Neumann algebras: Measure theory of non commutative spaces.
1.5. The index theorem for measured foliations.
Appendix A. Transverse measures and averaging sequences.
Appendix B. Abstract transverse measure theory,
Appendix C. Non commutative spaces aud set theory.
Chapter II. Topology and A*-theory.
11.1. C*-algebras and their К theory.
11.2. Elementary examples of quotient spaces.
11.3. The space of Penrose tilings.
11.4. The dual space of discrete groups.
11.5. The tangent groupoid of a manifold.
П.6. Wrong way functorialit.y as a deformation.
11.7. The orbit space of a group action.
11.8. The leaf space of a foliation.
11.9. The longitudinal index theorem.
11.10. The analytic assembly map and Lie groups.
Appendix А. C* modules and strong Morita equivalence of C* algebras.
Appendix В. Е theory and deformations.
Appendix C. Crossed products of C"-algebras.
Appendix D. Penrose tilings.
Chapter III. Cyclic cohomology and differential geometry.
111.1. Cyclic cohomology.
111.2. Examples of computation of HC"(A); manifolds, group rings and crossed products.
Ш.З. Pairing of cyclic cohomology with К theory.
Ш.4. The higher index theorem for covering spaces.
111.5. The Novikov conjecture for hyperbolic groups.
111.6. The Godbillon Vey invariant.
111.7. The transverse fundamental class for foliations and geometric corollaries.
1

Appendix A. The cyclic category Л.
Appendix B. Locally convex algebras.
Appendix C. Stability under holomorphic functional calculus.
Chapter IV. Quantized differential calculus.
IV.1. Quantized differential calculus and cyclic cohomology.
IV.2. The Dixmier trace and the residue formula.
IV.3. Quantized calculus in one real variable and fractals.
IV.4. Conformal manifolds.
IV.5. Rank one discrete groups.
IV.6. The Quantum Hall effect and elliptic theory on the non commutative torus.
IV.7. Entire cyclic cohomology.
IV.8. The Chern character of 0-summable Fredholm modules.
IV.9. 0-summable K cycles, discrete groups and Quantum field theory.
Appendix A. Kasparov's Invariant theory.
Appendix B. Real and complex interpolation of Banach spaces.
Appendix C. Normed ideals of operators
Appendix D. Deformations of algebras and cyclic cohomology.
Chapter V. Operator algebras.
V.I. The papers of Murray and von Neumann.
V.2. Representations of C"-algebras.
V.3. The algebraic framework for noncommutat.ive integration and the theory of weights.
V.4. The factors of Powers, Araki and Woods, and of Krieger.
V.5. The Radon-Nikodym theorem and factors of type lllj.
V.6. Noncommutative ergoclic theory.
V.7. Amenable von Neumann algebras.
V.8. The flow of weights.
V.9. The classification of amenable factors.
V.10. Subfactors of type Hi factors.
V.ll. Hecke algebras, type III factors and statistical theory of prime numbers.
Appendix A. Crossed products of von Neumann algebras.
Appendix B. Correspondences.
Chapter VI. The metric aspect of non commutative geometry.
VI.1. Riemannian geometry and the Dirac operator.
VI.2. Positivity in Hochschild cohoniology and inequalities for the Yang Mills action.
VI.3. Product of continuum by discrete and the symmetry breaking mechanism.
VI.4. The notion of manifold in non commutative geometry.
VI.5. The standard U{\) x SUB) x SV(Z) model.
References
INTRODUCTION
The correspondence between geometric spaces and commutative algebras is a familiar and basic idea of
algebraic geometry. The purpose of this book is to extend this correspondence to the non commutative
case in the framework of real analysis. The theory, called non commutative geometry rests on two essential
points:
1. The existence of many natural spaces for which the classical set theoretic tools of analysis, such as measure
theory, topology, calculus and metric ideas lose their pertinence but which correspond very naturally to a
non commutative algebra. Such spaces arise both in mathematics and in quantum physics and we shall
discuss them in more detail below, examples include:
a) The space of Penrose tilings
b) The space of leaves of a foliation
c) The space of irreducible unitary representations of a discrete group
d) The phase space in quantum mechanics
e) The Brillouin zone in the quantum Hall effect.
f) Space time.
Moreover even for classical spaces, which correspond to a commutative algebra, the new point of view will
give new tools and results, for instance for the Julia sets of it.era.tion theory.
2. The extension of the classical tools: measure theory, topology, differential calculus and Riemannian
geometry, to the non commutative situation. This extension involves of course an algebraic reformulation
of the above tools but passing from the commutative to the non commutative case is never straightforward.
On the one hand completely new phenomena arise in the non commutative case, such as the existence of
a canonical time evolution for a non commutative measure space. On the other hand the constraint of
developing the theory in the non commutative framework leads to a new point of view and new tools even in
the commutative case, such as cyclic cohomology and also the quantized differential calculus which, unlike
the theory of distributions, is perfectly adapted to products and gives meaning and uses expressions like
f f(Z)\dZ\p where Z is non differentiable (and p not necessarily an integer).
Let us now discuss in more detail the extension of the classical tools of analysis to the non commutative
case.
A. Measure Theory (Chapters 1 and 5)
It has long been known to operator algebraists that the theory of von Neumann algebras and weights
constitutes a far reaching generalization of classical measure theory. Given a countably generated measure
space X the linear space of square integrable (classes of) measurable functions on Л* forms a Hilbert space. Ц
is one of the great virtues of the Lehesgue theory that every element of the latter Hilbert space is represented
by a measurable function, a fact which easily implies the Radon Nikodym theorem for instance. There is,
up to isomorphism, only one Hilbert space with a countable basis ami in the above construction the original
2
3

measure space is encoded by the representation (by multiplication operators) of its algebra of bounded
measurable functions. This algebra turns out to be the prototype of a commutative von Neumann algebra,
which is dual to an (essentially unique) measure space Л*.
In general aconstruction of a Hilbert space (with a countable basis) provides one with specific automorphisms
(unitary operators) of that space. The algebra of operators in the Hilbert space which commute with these
particular automorphisms is a von Neumann algebra and all von Neumann algebras are obtained in that
manner. The theory of not necessarily commutative von Neumann algebras was initiated by Murray and
von Neumann and is considerably more difficult than the commutative case.
The center of a von Neumann algebra is a commutative von Neumann algebra and, as such, dual to an
(essentially unique) measure space. The general case thus decomposes over the center as a direct integral of
the so called factors, i.e. von Neumann algebras with trivial center.
In increasing degree of complexity the factors were initially classified by Murray and von Neumann into three
types, I, II, III.
The type I factors and more generally the type I von Neumann algebras, (i.e. direct integrals^of type I
factors) are isomorphic to commutants of commviaiivt von Neumann algebras. Thus up to the notion of
multiplicity they correspond to classical measure theory.
The type II factors exhibit a completely new phenomenon, thai, of continuous dimension. Thus, whereas a
type I factor corresponds to the geometry of lines, planes, ..., ^--dimensional complex subspaces of a given
Hilbert space, the subspaces that belong to a type II factor arc no longer classified by a dimension which
is an integer but by a dimension which is a positive real number and will span a continuum of values (an
interval). Moreover crucial properties such as the equality:
dim(?A
dim(? V F) = dimfC) + rlini(F)
remain true in this continuous geometry (if Л F is tin; intersection of the subspaces and EVF the closure
of the linear span of E and F).
The type III factors are those which remain after the type I and type II cases have been considered. They
appear at first sight as singular and untractable. Relying on Tomita's theory of modular Hilbert algebras and
on the earlier work of Powers, Araki Woods snd Krieger, 1 showed in my thesis that type III is subdivided
into types Шд, A € [0,1] and that factors of type 1ПЛ, А ф 1, can be reconstructed uniquely as a crossed
product of a type II von Neumann algebra by an automorphism contracting the trace. This result was then
extended by M. Takesaki to cover the IHi case as well, using a one parameter group of automorphisms
instead of a single automorphism. These results thus reduce l.he understanding of type III factors to that
of type II and their automorphisms, a task which was completed in the hyperfinite case and culminates in
the complete classification of hyperfinite von Neumann algebras exposed briefly in chapter I section 3 and in
great detail in chapter V. The reduction from type III to type \\ has some resemblance with the reduction
of arbitrary locally compact groups to unimodular ones by a semidirect product construction. There is one
essential difference however which is that, the range of the module which is a closed subgroup of R+ in the
locally compact group case has to be replaced for type lllo factors by an ergodic action of R+: the flow of
weights of the type III factor. This flow is an invariant of the factor and can, by Krieger's theorem (chapter
V) be any ergodic flow, thus exhibiting an intrinsic relation between type III factors and ergodic theory and
bringing support to the ideas of G. Mackey on virtual subgroups. Indeed in Mackey's terminology a virtual
subgroup of R+ corresponds exactly to an ergodic action of H'+. Since general von Neumann algebras have
such an unexpected and powerful structure theory it. is natural to look for their occurence in more common
parts of mathematics and to start using them as tools. After some earlier work by Singer, Coburn Douglas
and Schaeffer, and by Shubin (whose work is the first, application of type II technic|ues to the spectral theory
of operators), a decisive step in this direction was taken up by M.F. Atiyah and I. Singer. They showed
that the type II von Neumann algebra generated by the regular representation of a discrete group (already
considered by Murray and von Neumann) provides, thanks to the continuous dimensions, the necessary tool
to measure the multiplicity of the kernel of an invariant elliptic differential operator on a Galois covering
space. Moreover they showed that the type II index on the covering equals the ordinary (type I) index on the
quotient manifold. Atiyah then went on, with W. ScJimid to show the power of this result in the geometric
construction of discrete series representations of semisimple Lie groups.
Motivated by this result and by the second construction of factors by Murray and von Neumann, namely the
group measure space construction, I then showed that a foliated manifold gives rise in a canonical manner
to a von Neumann algebra (chapter I section 4). A general element of this algebra is just a random operator
in the L2 space of the generic leaf of the foliation and can thus be seen as an operator valued function on the
badly behaved leaf space A' of the foliation. As in the case of covering spaces the generic leaf is in general
non compact even if the ambient manifold is compact. A notable first difference with the case of discrete
groups and covering spaces is that in general the von Neumann algebra of a foliation is not of type II. In
fact every type can occur and for instance very standard foliations such as the Anosov foliation of the unit
sphere bundle of a compact Riemann surface of genus > 1 give a factor of type III,. This allows to illustrate
by concrete geometric examples the meaning of type I, type II, type IIIj ... and we shall do that as early as
section 4 of chapter I. Geometrically the reduction from type III to type II amounts to the replacement of
the initial non commutative space by the total space of an R^J. principal bundle over it.
We shall see much later (chapter 3 section 6) the deep relation between the flow of weights and the Godbillon
Vey class for codimension one foliations.
The second notable difference is in the formulation of the index theorem, which as in Atiyah's case uses the.
type II continuous dimensions as the key tool. For foliations one needs first to realize that the type II. Radon
measures, i.e. the traces on the C*-algebra of the foliation (cf. below) correspond exactly to the holonomy
invariant measures. Such measures are characterized (cf. chapter I section 5) by a de Rham current: the
Ruelle Sullivan current, and the index formula for the type II index of a longitudinal elliptic differential
operator now involves the homology class of the Ruelle Sullivan current. Unlike the case of covering spaces
this homology class is in general not even rational, the continuous dimensions involved can now assume
arbitrary real values and the index is not. related to an integer valued index. In the case of measured
foliations the continuous dimensions aquire a very clear geometric meaning. First, a general projection
belonging to the von Neumann algebra of the foliation yields a random Hilbert space: i.e. a measurable
bundle of Hilbert spaces over the badly behaved space Л' of leaves of the foliation. Next, any such random
Hilbert space is isomorphic to one associated to a transversal as follows: the transversal intersected with a
generic leaf yields a countable set, the fiber over the leaf is then the Hilbert space with basis this countable
set. Finally in the above isomorphism, the transverse measure of the transversal is independent of any
choices and gives the continuous dimension of the original projection. One can then formulate the index
theorem independently of von Neumann algebras, which we do in section 5 of chapter I. In simple cases
where the ergodic theorem applies, one recovers the transverse measure of a transversal as the density of
the corresponding discrete subset of the generic leaf, i.e. as the limit of the number of points of this subset
over increasingly large volumes. Thus the Murray and von Neumann continuous dimensions bear the same
relation with ordinary dimensions as (continuous) densities bear to the counting of finite sets.
In order to get some intuitive idea of what the generic leaf of a foliation can be like, as well as its space X of
leaves (at this level of measure theory) one can consider the space of Penrose tilings of the plane (or Penrose
universes). After the discovery of aperiodic tilings of the plane, the number of required tiles was gradually
reduced to two. Given the two basic tiles: the Penrose kites and darts of figure 1, one can tile the plane with
these two tiles (with the matching condition of the colors of the vertices) but no such tiling is periodic. Two
tilings are called identical if they are deduced from each other by an isometry of the plane. An example of
non identical tilings is given by the cartwheel tiling of figure 2 and the star tiling of figure 3. The set X of
all non identical tilings of the plane by the above two tiles is a very strange set because of the following:
"Every finite patch of tiles in a tiling by kites and darts does occur, and infinitely many times, in any
other tiling by the same tiles".
Fig. 1. Kites and Darts
5

Fig.2. Cartwheel tiling ([Gru-S])
Fig..!. Star tiling ([Gru-S])
This means that it is impossible to decide locally with which tiling one is dealing. Any pair of tilings such as
those of figures 2 and 3 can be matched on arbitrarily large patches. When analyzed with classical tools the
space X appears as pathological, and the usual tools of measure theory or topology give poor results: the only
natural topology on the set X is the coarse topology with 0 and X as the only open sets and a similar result
holds in measure theory. In fact one can go even further and show that the effective cardinality of the set X is
strictly larger than the continuum (cf. appendix С of chapter I). The natural first reaction in front of such a
space X is to declare it as pathological. Our thesis in this book is that X only looks pathological because one
tries to understand it with classical tools whereas even as a set, X is inherently of quantum mechanical nature.
What we mean by this, and we shall fully justify it in chapters I anil II, is that all the pathological properties
of X disappear if we analyse it using, instead of the usual complex valued functions, ^-number or operator
valued functions. Equivalently one can write (in many ways) the space X as a quotient of an ordinary well
behaved space by an equivalence relation. For instance the space У of pairs (tiling, tile belonging to the
tiling), is well behaved and possesses an obvious equivalence relation with quotient X. One could also use
in a similar manner the group of isometries of the plane. All these constructions yield equivalence relations
or better groupoids or preequivalence relation in the sense of Grothendieck. The non commutative algebra is
then the convolution algebra of the groiipoid. It is fashionable among mathematicians to despise groupoids
and consider that only groups have an authentic mathrnmlical status, probably because of the pejorative
suffix oid. To remove this prejudice we start chapter [ by Meisenberg's discovery of quantum mechanics.
The reader will hopefully realise there how the experimental results of spectroscopy forced Heisenberg to
replace the classical frequency group of the system by i.lie groupokl of quantum transitions. Imitating for
this groupoid the construction of a group convolution algebra lleisonberg rediscovered matrix multiplication
and invented quantum mechanics.
In the case of the space X of Penrose tilings the non commutative C*-algebra which replaces the commutative
C*-algebra of continuous functions on ,Y has trivial center and a unique trace. The corresponding factor
of type II which describes X from the measure theory point of view has a natural subfactor of Jones index
Bcost/5J (cf. chapter 5 section 10).
Before we pass to the natural extension of topology to the non commutative spaces we need to describe its
role in non commutative measure theory. One crucial use of the (local) compactness of an ordinary space
X is the Riesz representation theorem. It extends the construction of the Lebesgue integral, starting from
an arbitrary positive linear form on the algebra of continuous functions over the space X. This theorem
extends as follows to the non commutative case. First, the involutive algebras (over C) of complex valued
functions over compact spaces are, by Gelfand's fundamental result, exactly the commutative C*-algebras
with unit. Moreover this establishes a perfect duality between the categories of compact (resp. locally
compact) spaces and continuous (resp. proper and continuous) maps and the category of unital C"-algebras
and * homomorphisms (resp. not necessarily unital).
The algebraic definition of a C-algebra turns out to be remarkably simple and to make no use of commuta-
tivity. One is used to introduce C"*-algebras as involntive Banach algebras for which the following equality
holds:
for any element x. But this hides an absolutely crucial feature by letting one believe that, as in a Banach
algebra, there is freedom in the choice of the norm. In fact il' an involutive algebra is a C*-algebra it is so
for a unique norm, given for any x by the equality:
||:r|j = (Spectral radius of x*x) .
The general tools of functional analysis show that C"-algebras constitute the natural framework for non
commutative Radon measure theory. Thus for instance the set of elements of a C""-algebra which are of
the form x*x constitute a closed convex cone, the cone of positive elements of the C-algebra. Any element
of the dual cone, i.e. any positive linear form yields by the Gelfand Naimark Segal construction a Hilbert
space representation of the C*-algelira. This bridges the gap with the. non commutative measure theory,
i.e. von Neumann algebras. Indeed the positive linear form extends by continuity to the von Neumann

algebra generated in the above Hilbert space representation. Moreover the remarkable up down theorem of
G. Pedersen asserts that any selfadjoint element of the von Neumann algebra is obtained by monotone limits
(iterated twice) of "continuous functions" i.e. of elements of I.he C-algebra.
This construction of measures from positive linear forms on C-algebras plays an important role in the
case of foliated manifolds where as we already mentioned the traces on the C"-algebra of the foliation
correspond exactly to the holoiiomy invariant transverse measures. It also plays a crucial role in quantum
statistical mechanics in formulating what a state of the system is, in the thermodynamic limit: exactly a
positive normalized linear form oil the C-algebra of observables. The equilibrium or Gibbs states are then
characterized by the Kubo Martin Schwinger condition as unveiled by [[aag Hngenholtz and Winnink (cf.
chapter I section 2). The relation between this "KMS" condition and the modular operator of Tomita's
Hilbert algebras, discovered by M. Takesaki and M. Winnink remains one of the deepest points of contact
between physics and pure mathematics.
B. Topology and A' theory (Chapter 2)
In the above use of C-algebras as a tool to construct measures (in the commutative or non commutative
cases) the fine topologica] features of the spares under consideration are not relevant and do not show up.
However by Gelfand's theoreiii any homcomorphism invariant of a compact space Л' is an algebraic invariant
of the C*-algebra С(Л') of continuous functions on X so that, one should be able to recover it, purely
algebraically.
The first invariant for which this was done is the Atiyah-llirzebiiicli topological Л' theory. Indeed the abelian
group /C(A') generated by stable isomorphism classes of complex vector bundles over the compact topological
space Л' has a very simple and natural description in terms of the C-algebra C(X). By a result of Serre
and Swan, it is the abelian group generated by .stable isomorphism classes of finite projective modules over
C(X), a purely algebraic notion, which moreover makes no use of the commutativity of C(X). The key
result of topological A' theory is the periodicity theorem of R. Bott. The original proof of R. Bott relied on
Morse theory applied to loop spaces of Lie groups. Thanks to the work of Atiyah and Bott, the result once
formulated in the algebraic context has a very simple proof and holds for any (not necessarily commutative)
Banach algebra and in particular for C-algebras. The second invariant which was naturally extended to the
non commutative and algebraic framework is A' homology which appeared under the influence of the work
of Atiyah and Singer on the index theorem for elliptic operators on a compact manifold and was developed
by Atiyah, Kasparov, Brown Douglas and Fillmore. The pseiulodiirerontial calculus on a compact manifold,
as used in the proof of the index theorem, gives rise to a short exact sequence of C-algebras which encodes
in an algebraic way the information given by the index map. The last term of the exact sequence is the
commutative algebra of continuous functions on the unit sphere of the cotangent space of the manifold and
its A' theory group is the natural receptacle for the symbol of the elliptic operator. The first term of the exact
sequence is a non commutative C"-algebra, independent of I lie context: the algebra of compact operators
in a Hilbert space with a countable basis. This unique C-algehra is called the clcmcniarg C"-algebra and
is the only separable infinite dimensional C'"-algebra which admits (up to unitary equivalence) only one
irreducible unitary representation. Its A' theory group is equal to the additive group of relative integers
and is the natural receptacle for indices of Fredholm operators. The connecting map of the exact sequence
of pseudodifferential calculus is the index map of Atiyah and Singer. In their work on extension theory,
L. Brown, R. Douglas and P. Fillmore showed how to associate to any compact space X an abelian group
classifying short exact sequences of the above type, called extensions of С(Л') by the elementary C*-algebra.
They proved that the obtained invariant is /\' hoinology, i.e. the homology theory associated by duality
to A' theory in the odd case. The even case of that theory was obtained by M.F. Atiyah and Kasparov.
The resulting theory was then extended to the non commutative case, i.e. with С(Л') replaced by a non
commutative C*-a]gebra thanks to the remarkable generalisation by D. Voiculescu of the Weyl von Neumann
perturbation theorein. This allowed to prove that classes of extensions (by the elementary C"-algebra) form
a group, provided the original C-algebra is nuclear. The class of nuclear C""-algebras was introduced by M.
Takesaki in his work on tensor products of C-algebras. Л C-algehra is nuclear if and only if its associated
von Neumann algebra is hyperfiuite. in any unitary representation. This class of C""-a]gebras covers many,
though not all, interesting examples.
In the meantime considerable progress had been made by topologists concerning the use, in surgery theory
of non simply connected manifolds M, of algebraic invariants of the group ring of the fundamental group.
In 1965, S. Novikov conjectured the homotopy invariance of numbers of the form (Lx,[M]) where L is the
characteristic class of the Hirzebrucli signature theorem, and x a product of one dimensional cohomology
classes. His conjecture was proved independently by Farell-IIsiang and Kasparov using geometric methods.
The higher signatures of manifolds" M with fundamental group Г are the numbers of the form (L V"*(y), [Af])
where у is a cohomology class on ВГ and ф : M — ВГ the classifying map. The original conjecture of
Novikov is the special case Г abelian. In 1970 Miscenko constructed the equivariant signature of non simpljr
connected manifolds as an element of the Wall group of the group ring. He proved that this signature is a
homotopy invariant of the manifold. Lusztig gave a new proof of the result of Farell-Hsiang and Kasparov
based on families of elliptic operators and extended it to the symplectic groups. Following this line, and
by a crucial use of C->algebras, Miscenko was able to prove the homotopy invariance of higher signatures
under the assumption that the classifying space ВГ of the fundamental group Г is a compact manifold with
a Riemannian metric of non positive curvature. The reason why C-algebras play a key role at this point is
the following. The Wall group of an involutive algebra (such as a group ring) classifies hermitian quadratic
forms over that algebra and is in general far more difficult to compute than the A' group which classifies
finite projective modules. However, as shown by Gelfand and Miscenko, the two groups coincide when the
involutive algebra is a C-algebra. This equality does not hold for general Banach involutive algebras. As the
group ring of a discrete group can be completed canonically to а С -algebra one thus obtains the equivariant
signature of a non simply connected manifold as an element, of the Л' group of this C-algebra.
Miscenko then went on and used the dual theory, namely A homology, in the guise of Fredholm representa-
representations of the fundamental group, to obtain numerical invariants by pairing with A* theory.
Thus the К theory of the highly non commutative C-algebra of the fundamental group played a key role
in the solution of a classical problem in the theory of non simply connected manifolds.
The К theory of C*-algebras, the extension theory of Brown, Douglas, Fillmore, the Ell theory of Atiyah and
the Fredholm representations of Miscenko are all special cases of the bivariant A'A' theory of Kasparov. With
his theory, whose main tool is the intersection product, Kasparov proved the homotopy invariance of the
extension theory and solved the Novikov conjecture for discrete subgroups of arbitrary Lie groups. Together
with the breakthrough of Pimsner and Voicnlescu in the computation of A' groiips of crossed products the
Kasparov theory played a decisive role in the understanding of К groups of non commutative C-algebras.
Ill chapter II we shall use a variant of Kasparov's theory, the deformation theory (chapter II appendix B) due
to N. Higson and myself which succeeds in defining the intersection product in full generality for extensions.
It also has the advantage of having an easily defined product, (cf. appendix B) and of being exact in full
generality. Nevertheless we shall need A'A' theory later and have gathered its main results in appendix A of
chapter IV.
The importance of the К group as an invariant of non commutative C-algebras was already clear for the
work of Bratteli and Elliott on the classification of C-algebras which are inductive limits of finite dimensional
algebras. They showed that the pointed A' group, with its natural order is a complete invariant in the above
class of algebras, and Effros Chen and Ilandelman completely characterized the pointed ordered groups thus
obtained. As a simple example where this applies consider (as above) the space Л' of Penrose tilings and the
non commutative C-algebra which replaces in this singular example, the algebra C(X). This C-algebra
turns out to be an inductive limit of finite dimensional algebras and the computation of its К group (chapter
II section 3) gives the abelian group Z2 ordered by the half plane determined by the golden ratio. The space
X is thus well understood as a  dimensional" non commutative space.
In order to be able to apply the above tools of non commutative topology to spaces such as the space of
leaves of a foliation we need to describe more carefully how the topology of such spaces give rise to a non
commutative C-algebra.
This is done in great detail in chapter II with a lot of examples. The general principle is, instead of taking
the quotient of a space by an equivalence relation, to retain this equivalence relation as the basic information.
An important intermediate notion which emerges is lliat of smooth gronpoid. It plays the same role as the
preequivalence relations of Grothendieck. The same non commutative space can be presented by several
9

equivalent smooth groupoids. The corresponding C*-algebras are then strongly Morita equivalent in the
sense of M. Rieffel and have consequently the same A" theory invariants (cf. chapter II appendix A). Even in
the context of ordinary manifolds smooth groupoids are quite pertinent. To illustrate this point we give in
chapter II section 5 a proof of the Atiyali Singer index theorem. It follows directly from the construction of the
tangent groupoid of an arbitrary manifold, a geometric construction which encodes the naive interpretation
of a tangent vector as a pair of points whose distance is со lparable to a given infinitesimal number e.
Smooth groupoids contain manifolds, Lie groups and discrete groups as special cases. The smooth groupoid
which corresponds to the space of leaves of a foliation is the holoiiomy groupoid or graph of the foliation,
first considered by R. Thom. The smooth groupoid which corresponds to the quotient of a manifold by a
group of diffeomorphisms is the semidirect product of the manifold by the action of the group.
Smooth groupoids are special cases of locally compact groupoids and i. Renault has shown how to associate
a C'-algebra to the latter. The advantage of smoothness however is that, as in the original case of foliations,
the use of ^ densities removes any artificial choices in the construction.
We define the Л' theory of spaces such as the space of leaves of a foliation, ts the A" theory of the associated
C'-algebra, i.e. here the convolution algebra of the holononiy groupoid. In the special case of fibrations the
leaf space is a manifold and the above definition of its A" theory coincides with the usual one. The first role of
the К group is as an invariant of the leaf space, unafWted by modifications of the foliation such as leafwise
surgery, which do not modify the space of leaves. Thus for instance for the Kronecker foliations dy = 9dx of
the two torus one gets back the class of 0 modulo PSLB,2) from the Л' theory of the leaf space. But the
main role of the К group is as a receptacle for the index of families. Given a space Л" such as the leaf space of
a foliation, an example of a family (Dt)ce\ of elliptic operators parametrized by ,Y is given by a longitudinal
elliptic differential operator D on the ambient manifold: It restricts to each leaf t as an elliptic operator:
Dt- In terms of the holononiy groupoid of the foliation this provides us with a functor to the category of
manifolds and elliptic operators. The general notion of family is defined in these terms. It is a general
principle of great relevance that the analytical index of such families (О,)*ел' makes sense as an element of
the A* group of the parameter space (defined as above through the associated C*-algebra). Thus the index
of a longitudinal elliptic operator now makes sense, irrespective of the existence of a transverse measure, as
an element of the A* group of the leaf space. Similarly the index of an invariant elliptic operator on a Galois
covering of a manifold, makes sense as an element of the A" group of the С-algebra of the covering group.
Moreover the invariance properties, such as the homolopy invariauie of the eqnivariant signature (due to
Miscenko and Kasparov', or the vanishing of the Dirac index in the presence of positive scalar curvature (due
to Gromov Lawson Rosenberg) do hold at the level of the /\' group.
The obvious problem then is to compute these Л' groups in geometric t.erms for the above spaces. Motivated
by the case of foliations where closed transversals give idenipotents of the C*-algebra, I was lei with P.
Baum and Skandalis to the construction, in the above generality of smooth groupoids G, of a geometrically
defined group: A'*op(G) and of an adciitive map \t from this group to the К group of the C-algebra. So
far each new computation of A groups confirms the validity of the general conjecture according to which
the map ji is an isomorphism. Roughly speaking the injectivity of /< is a generalized forn: of the Novikov
higher signature conjecture, while the surjectivil.y of // is a general form of the Selberg principle on the
vanishing of orbital integrals of non elliptic elements in the theory of semi simple Lie groups. It also has
deep connections with the zero divisor conjecture, of discrete group theory. Chapter II contains a detailed
account of the construction of the geometric group, of the map /* and their properties. Besides the cases of
discrete groups, quotients of manifolds by group actions, and foliations, we also treat carefully the case of
Lie groups where the conjecture, intimately related to the work of Atiyah and Schmid has been proved, in
the semisimple case, by A. Wassermann.
The general problem of injeetivity of the analytic assembly map ;i is an important reason for developing the
analogue of de Rhani homology for the above spaces which is done in chapter 3.
С Cyclic cohomology (Chapter 3)
I discovered cyclic cohomology and the spectral sequence relating it. to Ilochschild cohomology in 1981
([Coi4]). My original motivation came from the trace formulae of Helton-Howe and Carey-Pincus for op-
operators with trace class commutators. These formulae together with the operator theoretic definition of К
10
homology discussed above, lead naturally to cyclic cohomology as the natural receptacle for the Chern char-
character (not the usual one but the Chern character in A* homology). Motivated by my work J.L. Loday and
D. Quillen developed the dual theory: cyclic homology and related it to the homology of the Lie algebra of
matrices. This result was obtained independently by Tsygan who, not having access to my work, discovered
cyclic homology from a completely different motivation, that of additive К theory.
We shall come back in chapter 4 to cyclic cohomology as the natural receptacle for the Chern character of
К homology classes and to the quantized differential calculus. In chapter 3 we develop both the algebraic
and analytic properties of cyclic cohomology with as motivation the construction of К theory invariants,
generalizing to the non commutative case the Chern Weil theory. It is worthwhile to consider the most
elementary example of that theory, namely the Gauss Bonnet theorem. Thus, let ? С R3 be a smooth closed
surface embedded in three space. Let us recall the notion of curvature. Through a point P of the surface, one
can draw the normal to the surface and, to reduce dimension by 1, cut the surface by a plane containing the
normal. One obtains in that way a curve which, at the point P, has a curvature: the inverse of the radius of
the circle, with center on the normal, which fits best with the curve at the point P. Clearly the curvature of
the above curve depends upon the plane containing the normal, and an old result of Euler asserts that there
are two extreme values K\ and /Cj for this curvature, reached for two perpendicular planes. Moreover given
any plane containing the normal, and making an angle 9 with the plane corresponding to Ki, the curvature
of its intersection with the surface is:
A", = A', cos3 9 + K2 sin2 9.
The Gauss Bonnet theorem then asserts that, provided S is oriented, the integral over ? of the product
K1K2 of the principal curvatures is equal to 2xB — 2y) where g is an integer, depending only upon the
topology of ? and called the genus. One remarkable feature of this result is that the above integral has an
extraordinary property of stability. One has indeed at one's disposal an infinite number of parameters to
modify the embedding of S in R3, say by making an arbitrary small bump on the surface. The theorem
nevertheless asserts that, in doing so, one will introduce equal amounts of positive and negative curvature
(here on the top of the bump and the sides of it respectively).
Fig.4. Curvature
One can of course give a rather straightforward proof of the above theorem by invoking the concept of degree
of a map in differential topology but we claim that, once considered from the algebraic point of view, the
above idea of stability has considerable potential as we shall now see. By the algebraic point of view I
mean that we let A be the algebra of all smooth functions on ?, i.e. A = C°°(?), and we consider on A the
following trilinear form: r(f°, f1, /2) = /E f°dfl Ad/2. In more geometric terms this trilinear form associates
11

to every map / : E —> R3 given by the three functions /",/', f- on E, the oriented volume bounded by
the image. This trilinear form possesses the following compatibility, reminiscent of the properties of a trace,
with the algebra structure of A:
We can now see the remarkable stability of the integral formula of the Gauss Bonnet theorem as a special
case of a general algebraic lemma, valid for ион commutative algebras. It simply asserts that if Л is an
algebra (over R or C) and т a trilinear form on A with properties 1) and 2), then for each idempotent
E = E2 € A the scalar т(Е, Е,Е) remains constant when E is deformed among the idempotents of A.
The proof of this statement is simple, the idea is that since the deformation of E is isospectral (all idempotents
¦ t
have spectrum С {0,1}) one can find Л' G A with E= [X,E] (where E is the time derivative of E) and
algebraic manipulations using 1) 2) then show that the time derivative of т{Е, Е, E) is zero.
It is quite important that this lemma does not make use of cammutatimly, even when we apply it in the
above case. Indeed, if we apply it to Л = C~(S) we get nothing of interest since as E is connected there
are no non trivial (i.e. different form 0, 1) ideinpotcnts in A. We may however use M?(A) = A ® Af3(C),
the algebra of 2 x 2 matrices with entries in A- We need first to extond the functional т to M2(A) and this
is done canouically as follows:
?(/°®/,/' eV./2©/r) = r(/°, /',/') Trace(,.'VV)
where }> e A, p' € Afa(C) and Trace is the ordinary (race on Л/2(С). One then checks easily that properties
1) and 2) still hold for the extended r. (Note however, at this point, that the original т did fulfill a stronger
property, namely 7-(/<'t0\/"• l),f<-)) = s(cr) 7-(/0,/',/5) for any permutation <r, but that due to the
cyclicity of the trace only the property I) survives after passing to т.) The next step is to find an interesting
idempotent E e M2(A). For this one uses the fundamental hut straightforward equality: An idempotent
of Af2(C7°°(S)) is exactly a smooth map from E to the Grassniannian of idempotents of Л'/2(С). Since the
latter Grassmannian is, using selfadjoint. idempotents to simplify, exactly the 2 sphere S2, we see that the
normal map of the embedding of E in R3 provides us with a specific idempotent E e M*{A) to which the
above stability lemma applies.
In fact the above lemma shows that, given a trilinear form т with properties 1) 2) on an algebra A, the
formula:
E -~ r(E,E,E)
defines an invariant of A* theory, i.e. an additive map оГ Л\|(.Д) to the complex numbers. This follows
because any finite protective module over an algebra .4 is the image of an idempotent E belonging to a
matrix algebra over A- The homotopy invariance provided by the lemma allows to eliminate the ambiguity
of the choice of E, in the stable isomorphism class of finite protective module: x € h'a(A).
The above simple algebraic manipulations extond easily to higher dimensions where conditions 1) 2) have
obvious analogues. The analogue of 1) in dimension и is.
where the — sign for n odd accounts for the ochlness of I lie cyclic permutation in that case.
The analogue of 2) is just the vanishing of Ьт where:
One checks that for n — 0 this vanishing characterizes the Ira en on A. An n + 1 linear form on an algebra A
satisfying 1) and 2) is called a cyclic cocycle (of dimension n). The above lemma and its proof easily extend
to the general case: an едал dimensional cyclic cocycle т givers an invariant of К theory by the formula:
E — r( E,E,...,E).
This immediately implies the construction of the usual Chern character of vector bundles for manifolds.
Indeed any homology class is represented by a closed de Rlinm current, i.e. by a continuous linear form С
on differential forms, which vanishes on coboundaries. One checks as above that the following formula then
defines a cyclic cocycle (of the same dimension as the current) on the algebra A of smooth functions on the
manifold:
r(f f")=(C,f°cif' л...лг/Г) v/yeA
Up to a normalization factor the above pairing of т with the Л' group of A gives the usual Chern character:
given a vector bundle E, the scalar (ch(E), G).
So far we have just reformulated algebraically the Chern Weil construction, dispensing completely of any
commutativity assumption. The remarkable fact, is that the highly non commutative group rings, i.e. the
convolution algebra СГ of a discrete group Г, possess very non trivial cyclic cocycles associated to the
ordinary group cocycles on Г. The group cohoinology //"(Г. С) is the cohomology of the classifying space
ВТ of Г, the (unique up to homotopy) quotient of л couiractihle topological space on which Г acts freely
and properly. It may be described purely algebraically in terms of group cocycles on Г: Given an integer
n, an n-group cocycle on Г (with complex coefficients) is a function of» variables jET which fulfills the
condition:
о
where for j = 0 one takes c( 1/3,1/3 i/,, + i) and for j = n + 1 one takes ( —1)"+1 c(i/i, ...,</„) as the
corresponding terms in the sum. Moreover, such cocycles can !>e normalized so as to vanish if any of the gt's
or their product: g, ... ;/„, is the unit element of Г.
Now the key observation is that such a normalized group cocycle defines uniquely a corresponding cyclic
cocycle of the same dimension и. on die group ring СГ. Since the latter is obtained by linearizing Г it is
enough to give the value of the cyclic cocycle on »+ I tuples (i/n,</i //„) € Г"+'. The rule is the following:
T(!hn<Ji .'/..) = r(i/i //„) if
= 1.
Observe that the conditions on the right are invariant under cyclic permutations, since those do not alter
the conjugacy class of the product r/0 .., 1/,,.
This formula was initially found in [Co] and later extended by M. Iviroubi and D. Burghelea who computed
the cyclic cohomology of group rings (cf. chapter :i seel ion 2).
We have used in a crucial manner the above cyclic cocycles on group rings, in collaboration with H. Moscovici
to prove the Novikov conjecture for the class of all hyperbolic groups in the sense of Gromov (cf. chapter 3
section 5). They play, in the 11011 commutative case, the role that de Rhain currents play in the commutative
case on the Pontrjagin dual of the discrete group Г. More precisely, when Г is abcliau, Lusztig, in his proof
of the Novikov conjecture, has shown how to obtain it from the Atiyah Singer index theorem for families of
elliptic operators, with parameter space X the Pontrjagin dual of Г. The idea is that each element X & X
being a character of the group Г determines a lint line bundle on any manifold Л/ with fundamental group
Г. Twisting the signature operator of the compact (oriented) manifold M by such flat bundles, one obtains
a family (Dx)x?x of elliptic operators, whose Atiyah Singer index (an element of Л"(Л')) is a homotopy
13

invariant of M. Next, the Atiyah Singer index theorem Tor families, as formulated in the cohomology of X
rather than К theory, gives the homotopy invariance of Novikov's higher signatures. Note that it is crucial
here to use the Chern character: сЛ : Л'(Л') —> Я*(Л') to express the index formula in cohomology. In
fact, even when X is a single point, the A* theory formulation of the Atiyah Singer index theorem only
becomes easy to use after translation, using the Chern character, in cohomological terms. The role of cyclic
cohomology in the context of the Novikov conjecture is exactly the same. The space X, Pontrjagin dual of
Г no longer exists when Г is non commutative, but the commutative C*-algebra C{X) is replaced by the
group C*-algebra С*(Г) which makes perfectly good sense. This C""-algebra contains as a dense subalgebra
of fairly smooth elements, the group ring СГ. The index of the above family of elliptic operators still makes
perfectly good sense and is the element of the A* group of С"(Г) given by the Miscenko-Kasparov signature
of the covering space of M. In fact it can even be obtained as an element of the К group of the group ring
ЯГ of Г over the ground ring R of infinite matrices of rapid decay. Moreover, when paired with the above
cyclic cocycle on СГ (extended to ЛГ) associated to a given group cocycle, the index of the signature family
gives exactly the Novikov higher signature associated to tlie group cohomology class of the cocycle. This
follows from a general index theorem due to II. Moscovici and myself (section 4 of chapter 3) for elliptic
operators on covering spaces. With such a result it would appear at first sight that one has proved the
Novikov conjecture in full generality. The difficulty is thai the homotopy invariance of the signature index
is only known at the level of the A" group of the C'"-algebra (Г'"(Г) but not for the A' group of the ring ЯГ.
One way to overcome it is to construct an intermediate algebra playing the role of the algebra of smooth
functions on X, which on the one hand is large enough to have the same К theory as С*(Г) and on the other
small enough so that the cyclic cocycle still makes sense. This is achieved, by the analogue of the Harish
Chandra Schwartz space, for hyperbolic groups (section 5 of chapter 3).
We shall give in section 6 of chapter 3 another striking application of cyclic coliomology and A' theory
invariants to the Godbillon Vey class, in the line of earlier work of S. Hurder. Using cyclic cohomology as
a key tool, we shall show that the Godbillon Vey class (the simplest instance of Gelfand Fuchs cohomology
class) of a codimension one foliation, yields a pairing with complex values between the К group of the leaf
space (defined as above using the C"-algebra of the foliation) and the flow of weights W(M) of the von
Neumann algebra of the foliation. Together with the. construction of the analytic assembly map of chapter
2 this result immediately implies that if the Godbillon Vey class of the foliation does not vanish then the
flow of weights W(M) admits a finite invariant probability measure. In other words the (virtual) modular
spectrum is a (virtual) subgroup of finite covolume in R+ and one is in particular in the type III situation,
the previous result of S. Ilurder. This shows that in the non commutative framework there is a very intricate
relation between differential geometry and measure theory, where the non vanishing of differential geometric
quantities implies the type 111 behaviour at the measure theoretic level. In fact as we shall see one can,
thanks to cyclic cohomology, give the following formula for the Godbillon Vey class as a 2-dimensional closed
current GV on the leaf space (i.e. in our algebraic terms as a 2-dimensional cyclic cocycle on the non
commutative algebra of the foliation)
GV = i,, [X].
It comes from the interplay between the transverse fundamental class [Л'] of the leaf space X and the
canonical time evolution <x, of the vou Neumann algebra of the foliation. The transverse fundamental class
[X] of the leaf space -Y is given by a one dimensional cyclic rocyclc which corresponds to integration of
transverse 1-forms. The time evolution at of the von Neumann algebra of the foliation is the algebraic
counterpart of its non commntativity and is one of the great surprises of non commutative measure theory
(cf. chapter I). Now these two datas, [Л] and <x, are in general slightly incompatible in that [X] is not in
general static, i.e. invariant under <rt. This is not too bad however, since one has:
It follows then, as in ordinary differential geometry, that if one contracts the closed current:
14
by the derivation Я generating the one parameter group <rt, then one obtains a 2-iimensional closed current.
The fundamental equation is then:
cv = ;„ [.v].
It gives a simple example of a general fact: that the leaf «pace of a foliation can have higher cohomological
dimension than the naive one: the codimension of the foliation.
Using a lot more analysis all this is extended t.o higher Gelfand Fuchs cohomology and many geometric
applications are given. They all show that provided one uses the tools of non commutative geometry one can
indeed think of a foliated manifold as a bundle of leaves over the non commutative leaf space. For instance
we show that if the A genus of a manifold V does not vanish, then V does not admit a foliation with leaves
of strictly positive scalar curvature, a result, which is easy for fibrations by a Kubiui type argument and a
well known result of A. Lirhncrowicz.
D. The quantized calculus (Chapter 4)
The basic new idea of non commutative differential geometry is a new calculus which replaces the usual
differential and integral calculus.
This new calculus can be succintly described by the following dictionary. We. fix a pair (W,F) where И is
an infinite dimensional separable Hillieit span- and F is a selfadjoint operator of square 1 in H. Giving F
is the same as giving the decomposition of H as I he direct sum of the two orthogonal closed subspaces:
Assuming, as we shall, that botli subspaces are infinitely dimensional, we. see that all such pairs («, F) are
unitarily equivalent. The dictionary is then the following:
CLASSICAL
Complex variable
Real variable
Infinitesimal
QUANTUM
Operator in W
Selfndjoinl operator in W
Compact operator in W
Infinitesimal ('oiiipnrl operator in H whose characteristic
of order u values /i,, satisfy /i,, = O{r,~") n —. <x>
Differential of real
or complex variable
Integral of infinitesimal
of order I
= [/¦••/]= Ff-fF
Dixniier trace
Let us comment in some detail eacli entry of the dictionary.
The range of a complex variable corresponds to the s/wchiuu SpG') of an operator. The holomorphic
lunct.onal calculus for operators in lfilben space gives meaning to f(T) for any holomorphic function /
denned on Sp(T) and only holomorphic functions act in i hat 8 ,-ality. This reflects the need for holomorphy

in the theory of complex variables. For real variables the situation is quite different. Indeed when the operator
T is selfadjoint, f(T) now makes sense for any borel function / on the line.
The role of infinitesimal variables is played by the compact, operators T in H. First К = {T 6 C(H) ; T compact}
is a two sided ideal in the algebra C{Ti) of bounded operators in H, and it is the largest non trivial ideal. An
operator Г in И is compact iff for any e > 0 the size of T is smaller than ? except for a finite dimensional
subspace of И. More precisely one lets for n 6 (SI:
/tn(T) = Inf{||T - R\\ ; R operator of rank < n}
where the rank of an operator is the dimension of its range. Then T compact о /in(T) —> 0 when n —* oo.
Moreover the Ц„(Т) are the eigenvalues, ordered in decreasing size, of the absolute value |7*| = (TTI'2 of
T. The rate of decay of the Ц„(Т) as n —¦ oo is a precise measure of (he size of the infinitesimal T.
In particular for each positive real a the condition:
(i.e. there exists С < oo such that ftn{T) < С n~" V;i > 1) clefiura the infinitesimals of order a. They
form a two sided ideal as is easily checked using the above formula Tor /i,,(T). Moreover if T, is of order ab
Г2 of order <*2 then T\T? is of order Qi + «•_>.
The differential df of a real or complex variable, usually given by tin' differential geometric expression:
In general the above sequence does not converge so that Tr^, a priori depends on a limiting procedure ш.
As we shall see however, in all the applications one can prove the independence of Тгш(Г) on ш. Such
operators T will be called measurable. For instance when T is a pseudodifferential operator on a manifold
it is measurable and its Dixmier trace coincides with the Manin-Wodzicki-Guillemin residue computed by
a local formula. In general the term residue (T) for the common value of Тгш(Г), Т measurable, would be
appropriate since for T > 0 it coincides with the residue at s = 1 of the Dirichlet series ?(s) = Тгасе(Г'),
s 6 С, Re(s) > 1.
We have now completed our description of the framework of the quantized calculus. To use it for a given
non commutative space X we need a representation of the algebra A of functions on X in the Hilbert space
H. The compatibility of this representation with the operator F is simply that all operators / in И coming
from Л have infinitesimal differential:
[F,f]eic
Such a representation is called a Fredholm module, and these are the basic cycles for the A'-homology ot A
when A is a C*-algebra.
is replaced in the new calculus by the commutator:
The passage from the classical formula to the above operator theoretic one is analogous to the quantization
of the Poisson brackets {/, g) of classical mechanics as coinmul ators: [/, </]. This is at the origin of the name
"quantized calculus". The Leibnitz rule d(frj) = (df)ij + f dij still holds.
The equality F2 = 1 is used to show that, tin; differentials df have vanishing anticommutator with F.
The next key ingredient of our calculus is the analogue of integration, it. is given by the Dixmier trace. The
Dixmier trace is a general tool designed to read in a classical manner a data of quantum mechanical nature.
It is given as a positive linear form Тг„ on the ideal of infinitesimals of order 1 and is a trace:
ТгшEТ) = Tr^TS) VT of order 1, S hounded.
In the classical differential calculus it is a great fact that one can neglect, all infinitesimals of order > 1.
Similarly, the Dixmier trace does neglect (i.e. vanishes on) any infinitesimal of order > 1, i.e.
(where the little о means, as usual, that nfi,t —¦ 0 as n —• со').
This vanishing allows considerable simplifications to occur, similar to those of the symbolic calculus, for
expressions to which the Dixmier trace is applied.
The value of Тгш(Т) is given for T > 0 by a suitable limit of the bounded sequence:
It is then extended by linearity to all compact operators of order 1.
10
To see how the new calculus works and allows operations not duable in distribution theory we shall start by
a simple example. There is a unique way to quantize in the above sense the calculus of functions of one real
variable (i.e. for A" — R) in a translation and scale invariant manner. It is given by the representation of
functions as multiplication operators in L-(R) while F is the Ililbert transform. Similarly for X = S1 one lets
L°°(Sl) act in Z2(S') by multiplication, while F is again the Hilbert transform, given by the multiplication
by the sign of n in the Fourier basis (е„)„ег of Z2(S'), with е„(9) = exp(m0) V0 S Sl.
The first virtue of the new calculus is that df continues to make sense, as an operator in L2El) for an
arbitrary measurable / e t°°E'). This of course would ,ilso hold if we define df using distribution theory
but the essential difference is the following. A distribution is defined as an element of the topological dual of
the locally convex vector space of smooth functions, here C^{Sl). Thus only the latter linear structure on
C°°(S') is used, not the algebra structure of C"(Sl). It is consequently not surprising that distributions are
incompatible with pointwise product or absolute value. Thus more precisely while, with / non diflerentiable,
df makes sense as a distribution, we cannot make any sense of \df\ or powers \df\p as distributions on S .
Let us give a concrete example where one would like to use such an expression for non differentiable /. Let с
be a complex number and let J be the Julia set given hy the complex dynamical system z —* z2 + с = <p(z).
More specifically J is here the boundary of the set В = {: ? C; sup |у>"(г)| < oo}. For small values of
С as the one chosen in fig. 5, the Julia set J is a Jordan curve and В is the bounded component of its
complement. Now the Riemann mapping theorem provides us with a conformal equivalence Z of the unit
disk, D = {z ? C, \z\ < 1} with the inside of B, and by a result of Caratheodory, the conformal mapping Z
extends continuously on the boundary S' of D to a lioineoniorphism, which we still denote by 2, from S1
to 3. By a known result of D. Sullivan, the Mausdorff dimension p of the Julia set is strictly bigger than 1,
1 < p < 2 and is close to 2 for instance^ in the example of fig.5. This shows that the function Z is nowhere
of bounded variation on S' and forbids a distribution interpretation of the naive expression:
f{Z)\dz\" v/ e C(J)

that would be the natural candidate for the llausdorff measure on 3'.
Fig.5. Julia set
We shall show that the above expression makes sense in the quantized calculus and that it does give the
Hausdorff measure on the Julia set J. The first essential fact is that as dZ = [F,Z] is now an operator
in Hilbert space one can, irrespective of the regularity of Z, talk about \dZ\, it is the absolute value \T\ =
(T'TI1'* of the operator T = [F,Z]. This gives meaning to any function h(\dZ\) where Л is a bounded
measurable function on R+ and in particular to \iIZ\r. The next essential step is to give meaning to the
integral of f(Z)\dZ\p. The latter expression is an operator in L2(Sl) and we shall use a result of hard
analysis due to V.V. Peller, together with the homogeneity properties of the Julia set to show that the
operator /B)|dZ|'> belongs to the domain of definition of the Dixmier trace TX,, i.e. is an infinitesimal of
order 1. Moreover, if one works modulo infinitesimals of order > 1 the rules of the usual differential calculus
such as:
= \4>'(Z)\F
are valid and show that the measure:
has the right conformal weight and is a lion zero multiple of the HausdorfT measure. The corresponding
constant governs the asymptotic expansion for the distance, in the sup norm on Sl, between the function Z
and restrictions to Sl of rational functions with at most n poles outside the unit disk.
This example of quantized calculus will be explained in section 3 of chapter 4. The first two sections of this
chapter deal with the respective properties of the usual trace (section 1) and the Dixmier trace Ttu (section
2)-
In section 1 we shall use the ordinary trace to evaluate differential expressions such as /° df1 ... df" where
f' G A and (%, F) is a Fredholm module over A.
The dimension of such modules is measured by the size of the differentials df — [F, f] for / e A, i.e. by the
growth of the characteristic values of these operators. More specifically a Fredholm module ("H, F) over an
18
algebra A (commutative or not) is called p-summable iff the operators df,f?A all belong to the Schatten
ideal С of operators (i.e. T 6 С iff ?/1„(Т)'' < oo). We shall show in section 1 that the Chern character
of a p-summable Fredholm module over an algebra A can be defined by a trace formula (in the line of the
earlier works of Helton-Howe and Сагеу-Pincus) and gives a cyclic cocycle of dimension n for any given
integer n larger than p— 1 and of the same parity as the Fredholm module. (For a module to be even
one requires that the Hilbert space is Z/2 graded, with F auticommutiug with the grading, as required by
general К homology definitions.) Moreover the various cyclic cocycles thus obtained are all images of a
single one by the periodicity operator S of cyclic cohomology. The cyclic cohomology class thus obtained
has a remarkable property of integrality, displaying the quantum nature of our calculus. Thus, when pairing
the cocycle Ch»{Ti, F), the Chern character of a Fredholm module over an algebra A, with the К group of
A one only gets integers:
(Ck.(n,F),K(A)) CZ.
This follows from a simple formula computing the value of the pairing as the index of a Fredholm operator
(section 1).
In section 2 we introduce the Dixmier trace and give я general formula, in terms of the Dixmier trace, for
the Hochschild class of the character Gh,G{,F) defined above.
The relevance of this class is that, though not the whole cohoinological information contained in the char-
character, it is exactly the obstruction to write the latter as the image, by the periodicity operator S of cyclic
cohomology, of a lower dimensional cyclic cocycle. (As follows from general results of chapter 3 section 1.)
The formula obtained for the Hochschild cocycle uses an unbounded selfadjoint operator D whose sign is
equal to F and which controls uniformly the size of the ila = [f, a], a 6 A by the conditions:
a) [D, a] bounded Vri e A , /J) |O|~" of order 1.
The formula obtained gives a Hochschild n-cocycle as the Dixmier trace applied to the product a°[?), a1]...
[Да"] \D\~" where [D,a>] is the commutator of D with the element а' оГЛ. The proof that this Hochschild
cocycle is cohomologous, in Hochschild cohoinology, to the character of G{, D) is very involved because it
relates the original cyclic cocycle, computed in terms of the ordinary trace, to an expression involving the
logarithmic or Dixmier trace. It implies in particular that if the obstruction to lower the dimension does not
vanish the Dixmier trace of D~" cannot vanish and the eigenvalues of D~l cannot be o(jb~'^n).
At the end of section 2 we show, using results of D. Voiculescu, that the existence of such a D controls the
size of abelian subalgebras of A, i.e. that the number of independent commuting quantities in the algebra
A is bounded above by n. This is another justification of the depth of the relation between the degree of
summability and dimension.
While section 3 is devoted to the quantized calculus in one real variable we show in section 4 that on a
conformal manifold the calculus can be caiionically quantized. This is related to the work of Donaldson and
Sullivan on quasiconformal 4-manifolds. We show that tlip Polyakov action of 2 dimensional conformal field
theory is given, in terms of the components „Y" of a map -V : D — Rrf of a Riemann surface to d-dimensional
space, by the formula:
I(X) = Tru.(»7,,v rf.Y" rf.Y") , <1X" = [F,X"}.
The remarkable fact is that the right hand side continues to make sense when S is now a 4-dimensional
conformal manifold. This is due to the discovery by M. Wodzicki of the residue of pseudodifferential operators,
a unique tracial extension of the Dixmier trace. We compute this new 4-dimensional action at the end of
section 4 and relate it to the Paneitz Laplacian. In section 5 we move to highly non commutative examples
coming from group rings and return to the origin of the quantized calculus as a substitute of differential
topology in the non commutative context. We exploit the integrality of the Chern character Ck,("H,F).
We show the power of this method by giving an elegant proof, in the spirit of differential topology, of the
Pimsner Voiculescu theorem solving the Kadison conjecture for the C""-algebra of free groups. As another
19

remarkable application of this integrality result we describe in details in section 6 (of chapter 4) the work of J.
Bejlissard on the quantum Hall effect. Experimental results, of V. Klitzing Pepper and Dorda, in 1980, have
shown the existence of plateaus for the transverse conductivity as a function of the natural parameters of the
experiment. This unexpected fact gave precise meaning to the numerical value of the conductivity on these
plateaus and yielded precision measurements of the fine structure constant independently of quantum field
theory. The numbers obtained have the same unexpected stability property as the integral occuting above in
the Gauss Bonnet theorem. Bellissard constructed a natural cyclic 2-cocycle on the lion commutative algebra
of "functions on the Brillouin zone" and expressed the Hall conductivity as the pairing between this cyclic
2-cocyle and an idempotent of the algebra, the spectral projection of (he Ilamiltonian. This accounts for the
stability part. He then went on and showed that his 2-cocycle is in fact the Chern character of a Fredholm
module, from which the remarkable integrality of the conductivity in units e'/h does follows. In doing so
he defined the analogne of the Brillouin zone as a non commutative space and the framework to apply the
above quantized calculus for this example. All this is explained in great detail in section 6. We also show
there how the ordinary pseudodifTerential calculus adapts to the uon commutative torus, the simplest and
probably the first example of a smooth non commutative space. We then apply it to prove an index theorem
for finite difference - differential equations on the real line.
The rest of this chapter 4 is devoted to the infinite dimensional case, i.e. to the construction of the Chern
character for Fredholm modules which are not finitely sinniiiablc. This is motivated by the following two
classes of infinite dimensional non commutative spaces: the dual space of non amenable discrete groups and
the phase space in quantum field theory (cf. section 9 of chapter i). The first step in order to adapt the above
tools to the infinite dimensional case is to develop entire cyclic cohomology which has the same relation with
ordinary cyclic cohomology as entire functions have with polynomials. From the very beginning of cyclic
cohomology, the possibility of defining it by means of cocycles with finite support in the F, B) fundamental
bicomplex did suggest the existence of infinite dimensional cocycles, not reducible to finite dimensional ones,
by considering cocycles with arbitrary support in the (Л, В) bicomplex. However a key algebraic result, the
vanishing of the E\ term of the first spectral sequence of the bicomplex (cf. chapter 3 section 1) shows that
the cohomology of cochains with arbitrary supports is always trivial. It is thus necessary to impose a non
trivial limitation to the growth of the components (^rJ) °f a corhain with infinite support, in order to obtain
a non trivial cohomology. This growth condition eluded me for a long time but turned out to be dictated by
the pairing with Л" theory. In section 7 of chapter '1 we shall adapt it to arbitrary locally convex algebras,
which shows in particular that entire cyclic cohomology applies to any algebra over С (endowing it with the
fine locally convex topology). We give in that section the equivalence between three points of view on entire
cyclic cocycles which can be viewed as (normalized) cocycles in the {btB) bicomplex, characters of infinite
dimensional cycles, or traces on the Cimlz algebra of A. i.e. the free product of A by the group with 2
elements. The growth condition of entire cocycles then means that they extend to a suitable completion of
the algebra QA (resp. of the Cuntz algebra), whose quasi nilpotency is analyzed. We end this section with
the computation of the entire cyclic cohomology of the algebra of Laurent polynomials and its relation with
a remarkable deformation of the Cuntz algebra in that case (also noticed independently by Cuntz Quillen).
In section 8 of chapter 4 we extend the previous finite dimensional construction of the Chern character in
К homology to the infinite dimensional case, For <i Fredholm module (Ti,F) over an algebra A the oo
dimensional summability condition is now that the commutators da = [F,a] for a G A all belong to the two
sided ideal of compact operators T such that /i,,(T), 'he »-"' characteristic value of T, is of the order of
(logn)*1'2 when Ti goes to oo.
We first show that given such a Fredholm module over A one can find a .selfadjoint unbounded operator D
whose sign is equal to F (i.e. D — F\D\ is the polar decomposition of D) and which satisfies the following
two conditions:
1) The commutator [D,a] is bounded for any a E A
2) Trace (e-D") < oo.
Such 0-summable modules (H, D) over an algebra will turn out, as we shall see in chapter VI, to be an excel-
excellent point of departure for the metric aspect in uon commutative geometry, i.e. the analogue of Riemannian
20
geometry in our framework. This fact is already visible in the proof of the existence of the operator D given
F- The heuristic formula for D~~ is indeed the following:
where the х* are generators of the algebra A, the дцг are the entries of a positive matrix, and the operator
dx for x G A is the quantum differential [F, x].
Such modules (И, D) will be called (9-summable) A' cycles for short. We showed how to construct the Chern
character of such К homology classes as an entire cyclic cocycle. Our formula for this character was quite
complicated and Jaffe Lesniewski and Osterwalder, motivated by supersymvnetric Quantum field theory then
obtained a much simpler formula for the same cohomology class.
All this is explained in detail in section 8 where the role of the quasinilpotent algebra of convolution of
distributions with support in R+ is put into evidence.
In section 9 we describe the two basic examples of 0-sumniable A' cycles not reducible to finitely summable
ones. The first example combines the construction of Miscenko and Kasparov of Fredholm representations of
discrete subgroups of semisimple Lie groups with the twisted tie Rliam operator of Witten in Morse theory.
We leave open the problem of completing the computation of the character of this К cycle for discrete
subgroups of higher rank Lie groups. The second example is the supersymmetric Wess Zumino model of
quantum field theory. We prove there the new unexpected result that, even for the free theory, the К
theory of the algebra A{0) of observables localised in an open subset О of space time is very non trivial, of
infinite rank as an abelian group, and pairs non trivially with the A* homology class given by the supercharge
operator Q of the theory.
We have already explained that chapter V is a detailed account of the theory of von Neumann algebras. In
its last section we explain our joint work with J.-B. Dost, motivated by earlier work of B. Julia, on a natural
system of Quantum statistical mechanics related to the arithmetic of Z and the Riemann zeta function. We
show that our system, formulated as a C" dynamical system, undergoes a phase transition with spontaneous
symmetry breaking with symmetry group the Galois group of the field of roots of unity.
E. The metric aspect of uon commutative; geometry
In the last chapter of this book we shall develop operator theoretic ideas about the metric aspect of geometry
and then apply these ideas to a fundamental example: space time.
Given a (not necessarily commutative) * algebra A corresponding to the functions on a "space X", our basic
data in order to develop geometry on Л* is simply а A' cycle Gi, D) over A in the above sense (cf. D). A
first example is provided by ordinary Riemannian geometry. Given a Riemannian manifold (/C-oriented for
convenience) we let its algebra A of functions act by multiplication in the Hilbert space И of i2 spinors
on X, and we let D be the Dirac operator in H. One thus obtains a A* cycle over A which represents the
fundamental class of X in A' homology. We shall give (section 1 of chapter 6) four simple formulae showing
how to recover, from the A' cycle Gi,D) the following classical geometric notions on X:
1) The geodesic distance d on X
2) The integration against the volume form det(gfi[/)ll- с/л:1 Л .. .Adxn
3) The affine space of gauge potentials
4) The Yang Mills action functional.
The essential fact is that all these notions will continue to make sense in our general framework, and in
particular will apply non trivially even to the case of finite spaces Л*.
The operator theoretic formula 1) for the geodesic distance is in essence the dual of the usual formula in
terms of the length of paths 7 in X. Recall that given two points p and </ in a Riemannian manifold X, their
geodesic distance d(p, q) is given as the uifimuni of the lengths of paths 7 from j) to q. The latter length
being computed by means of an integral along 7 of the square root of ijttl/ dx11 dxv. Our formula, for the
21

same quantity d(p,q) is different in nature in that it is a supremum (instead of an infimum) and it invokes
as a variable л Junction f on Л' instead of a path 7 in Л'. It is the following:
The norm involved in the right hand side is the norm ofoperators in Ililbert space. Note also that the points
p, q of X are used to convert the element / of A inlo a scalar, namely they are used, in conformity with
Gelfand's theorem, as characters of the algebra A.
There are two important features of this formula. The first is that since it does not invoke paths but
functions it continues to be meaningful and gives a finite 11011 zero result when the space X is no longer
arcwise connected. Indeed one can take a space ,Y with two points: афЬ, the algebra A of functions on X is
then simply A = СфС, the direct sum of 2-copies of С and a natural A' cycle is given by the diagonal action
of A in P(X) with operator D = \ 't • One then checks that, with our formula the distance between a
and 6 is given by I = /t~' •
An equally important feature of our formula is that, since it uses functions instead of paths it is directly
compatible with the formalism of quantum mechanics in which particles are described by wave functions
rather than by any classical path.
The next step is to recover the tools of Riemannian geometry which go beyond the mere "metric space"
attribute of such spaces such as 3) and 4). First, the volume form and the corresponding integral are
recovered by the Dixmier trace, already mentioned in the discussion of chapter 4. In fact the case already
covered in chapter 4 of the HausdorH" measure on Julia sets in terms of the Dixmier trace was far more
difficult. In section 1 we shall develop the formalism of gauge theory (commutative or not of course) and the
Yang Mills action functional from our operator theoretic data. The main mathematical result of section 2 is
the analogue of the basic inequality between the second Cliern number of a hermitian vector bundle and the
minimum of the Yang Mills functional on compatible connections on ihis bundle. This is proved in section
2 using the main theorem of chapter IV section 2. In section 3 we determine the Yang Mills connections on
the lion commutative torus, a joint result of M. RicflVl and myself, after defining the metric structure of that
simple non commutative space in great detail. As it turns out, even though the space we start with is non
commutative, the moduli space of Yang Mills connections is nicely behaved and finite dimensional.
The end of the chapter (and the book) is occupied by the analysis, with the above new geometric tool, of
the structure of space time. The idea can he simply stated as follows: it was originally through the Maxwell
equations that, by the well known steps due to Lorentz, Michelson Morley, Poincare and Einstein, the model
of Minkowski's space time was elaborated. In more modern terms the Maxwell Dirac Lagrangian accounts
perfectly for all phenomenal involving only the electromagnetic interaction. However a century passed since
then and essential discoveries, both experimental and theoretical have shown that in order to account for
weak and strong forces a number of modifications in the above Lagrangian were necessary. (See the small
chronological table below for the early development of weak forces.)
Chronological table
1896 Discovery of radioactivity by II. Becquerel
1898 M. Curie shows that radioactivity is an intrinsically atomic property
1901 Rutherford shows the inhomogeneity оГ iiranic rays and isolates the /3 rays (i.e. the weak interaction
part)
1902 Kaufman shows that, the /J rays are formed of electrons
1907-14 Chadwick shows that the spectrum of /} rays is continuous
1930 W. Pauli introduces the neutrino to account for the above continuous spectrum
1932 W. Heisenberg introduces the concept of isospiu (in the other context of strong forces)
1934 E. Fermi gives a plienoiiienological theory of weak iiit-eraci-ions
1935 Yukawa introduces the idea of heavy bosons mediating short, range forces.
22
Among the next steps which led to the Glashow Weiuberg Stilain Lagrangian of the standard model let us
mention the following:
The discovery of the vector-axial form of the weak currents, the intermediate boson hypothesis which replaces
the 4-fermion interaction of tlie Fermi theory by a renormalizable interaction. This imposes a considerable
mass for the intermediate boson 111 order to get a possible unification of the weak and electromagnetic forces
using the ideas of non abelian Gauge theory of Yang and Mills. There was a major difficulty at this point
since anon abelian pure Yang Mills field is necessarily massless. This difficulty was resolved by the theoretical
discovery of the Higgs mechanism, allowing for the generation, by spontaneous symmetry breaking, of non
trivial masses for fermions and intermediate bosons. The Higgs field thus introduced appears in three of the
five natural terms of the Lagrangian and completely spoils their geometric significance.
Now our idea is quite-simple. We have at our disposal a more flexible notion of geometric space in which
the continuum (i.e. say manifolds) and the discrete (cf. the 2 point space example above) are treated on the
same footing, and in which the Maxwell Dirac Lagrangian does make sense: its bosonic part is the already
treated Yang Mills action and the fermionic part is easy to get since we are given the Dirac operator to start
with. Thus we can keep track of the above modifications оГ the Lagrangian as modifications of the geometry
of space time. In other words, we look for a geometry in our sense, i.e. a triple (A,ft,D) as above, such
that the associated Maxwell Dirac action functional produces the standard model of electroweak and strong
interactions with all its refinements dictated by experimental results.
The result that we obtain is canonically derived from the standard model considered as a phenomenological
model. What we find is ageometric space which is neither a continuum nor a discrete space but a mixture of
both. This space is the product of the ordinary continuum by a discrete space with only two points, which
for reasons which will become clear later, we shall call L and R. The geometry, in our sense, of the finite
space {L, R) is given by a finite dimensional Ililbert space representation H and a selfadjoint operator D in
Ti. These have direct physical meaning since the operator D is given by the Yukawa coupling matrix which
encodes the masses of the elementary fcrmious as well as their mixing, i.e. the Kobayashi Maskawa mixing
parameters. Computing the distance between the two points L and Я thus gives a number of the order of
10~16cm, i.e. the inverse mass оГ the heaviest fermion. The naive picture that emerges is that of a double
space time, i.e. the product of ordinary space time by a very tiny discrete two point space. By construction
purely left handed particles such as neutrinos, live on the left handed copy Xt, while electrons involve both
Xi, and Xr in X = Xl U Xr. Note that each point p of A'i is extremely close to a corresponding point p1 of
Xr and when computing the differential of a function / on Л' = Л'? U XR we shall find that it invokes not
only the ordinary differential of / on Xt or on Л'д but also the finite difference f(p)-f(pf) of the values of /
on corresponding points of A'i and Xr. It is this finite difference, occuring as anew "transverse" component
of the gauge potential which yields the famous Higgs fields which appears in the three non geometric terms
of the standard model Lagrangian (cf. chapter VI) ? = ?« + ?/ + Cv + Cy + Cv¦
Note that there is no symmetry whatsoever between the points L and R and in fact the natural vector bundle
used over that space has fiber C2 over L and С over R. All this works very well for the electroweak sector
of the standard model but in order to account for the (already essentially geometric) strong structure, much
more work was necessary. It lias been done in collaboration with .]. Lott and is described in the last two
sections 4 and 5. It hinges on the fundamental problem of defining correctly what is a "manifold" in the set
up of non commutative geometry. Our proposed answer, which is essentially dictated by the 51/B) X 51/C)
structure of up down quark iso-doublets, is given by Poincare duality discussed in section 4. It fits remarkably
well with the work of D. Sullivan on the fundamental role of Poincare duality in A'-homology for ordinary
manifolds (cf. section 4).
In section 5 we discuss the full standard model, and account also for the hypercliarge assignment of particles
from a general unimodularity condition. One should understand this work as pure phenomenology, at the
classical level, interpreting the detailed structure of the best pheuomenological model, as the fine structure
of space time rather than as a long list оГ new particles. Applications of this point of view in quantum field
theory, such as a model building device are still ahead of us.
23

CHAPTER I
NON COMMUTATIVE SPACES AND MEASURE THEORY
My goal in this chapter is first of all to show the extent to which lloisouborg's discovery of matrix mechanics,
or quantum mechanics, was guided by the exprrimonl.nl results of speotroscopy. The resulting replacement
of the phase space by a 'noncommutat.ive space', more precisely the replacement of the algebra of functions
on the phase space by the algebra of matrices, is one of the uiosl important conceptual steps, on which it is
important to dwell. I will then attempt to place into evidence, tlmiiks to quantum statistical mechanics, the
interaction between theoretical physics and pure mathematics in the realm of operator algebras, and to show
how this interaction was at the origin of the classification of factors summarized below. Next I will show
how the theory of operator algebras replaces ordinary measure theory lor the 'non commutative spaces' that
one encounters in mathematics such as the space of leaves of a foliation. This will first of all give a detailed
illustration of the classification of factors by geometric examplcs.lt will also allow us to extend the Atiyah
Singer index theorem to the non compact leaves of a measured foliation, using the full force of the Murray
and von Neumann theory of continuous dimensions.
The content of this chapter is organized as follows:
1. Heisenberg and the non commutative algebra of physical quantities associated to a microscopic system.
2. Statistical state of a microscopic system and quantum statistical mechanics.
3. Modular theory and the classification of factors.
4. Geometric examples of von Neumann algebras: Measure theory of non commutative spaces.
5. The index theorem for measured foliations.
Appendix A. Transverse measures and averaging sequences.
Appendix B. Abstract transverse measure theory.
Appendix C. Non commutative spaces and set theory.
I.I. Heisenberg and the noncoiuiniitative algebra of physical quantities associated with a
microscopic system
The classification of the simple chemical elements into Mendeleev's periodic table is without doubt the
most striking result in chemistry in the 19th century. The theoretical explanation of this classification,
by Schrodinger's equation and Pauli's exclusion principle, is an equivalent success of physics in the 20th
centf-*' ~"J — - —-i- -<" > ¦ ~
of v
the uitisj ictcu \ji ustinawis. vviui ooiir, it is tne discretization of angular momentum. For de Broglie
and Schrodinger, it is the wave nature of matter. These diverse points of view are all corollaries of that of
Heisenberg: <Ae noncommulative algebra of physical quantities. My first goal will be to show how close the
latter point of view is to experimental reality. Towards the end of the 19th century, numerous experiments
permitted determining with precision the lines of the emission spectrum of the atoms that make up the
elements. One considers a Geissler tube filled with a gas such as hydrogen. The light emitted by the tube is
analyzed with a spectrometer, the simplest being a prism, arid one obtains a certain number of lines, indexed
by their wavelength. The configuration thus obtained is the most direct source of information on the atomic
structure and constitutes, as it were, the signature of the element under consideration. It depends only on
the element considered, and characterizes it. It is thus essential to find the regularities that appear in these
configurations or atomic spectra. It is hydrogen that, in conformity with Mendeleev's table, has the simplest
spectrum (Fig.l).
Fig.l. The visible part of the spectrum of the hydrogen atom. ©CNRS. Paris Observatory photo. L. Pamia
The numerical expression of the regularity of the lines #o, Hp, #7,... was obtained by Balmer in 1885 in
the form
я. = f*. л, = {§*,
,,
where the value of the length L is approximately 3645.6 X 10~8 cm. In other words, the wavelengths of the
lines in Figure 1 are of the form
25

A =
- L , where » is an integer equal to 3, 4, 5 or 6.
„2-4'
Around 1890, Rydberg showed that for complex atoms the lines of the spectrum can be classified into series,
each of them being of the form
— = —- - with n and m integers, ra fixed.
А m* п*
Here R = 4/i is Rydberg's constant. From this experimental discovery one deduces that, on the one
hand, the frequency i/ = с/А is a more natural parameter than the wavelength A for indexing the lines of
the spectrum and, on the other hand, the spectrum is a set of differences of frequencies, that is, there exists
a set I of frequencies such that the spectrum is the set of differences f;; = i/,- - Vj of arbitrary pairs
of elements of /. This property shows that one can combine two frequencies i/1;- and vjt to obtain a
third i/it = Vij + Vjt. This important corollary is the Ritz-Rydberg combination principle: the spectrum is
naturally endowed with a partially defined law of composition; the sum of certain frequencies of the spectrum
is again a frequency of the spectrum (Fig. 2).
Fig.2. The vertical arrows represent tlie transitions. They are indexed by a pair of indices (i, j) .
Now, these experimental results could not be explained in the framework of the theoretical physics of the
19th century, based on Newton's mechanics and Maxwell's electromagnetism. If one applies the classical
conception of mechanics to the microscopic level, then an atom is described mathematically by the phase
space and the Hamilton]an, and its interaction with radiation is described by Maxwell's theory. As we shall
see, this theory predicts that the set of frequencies forms a subgroup Г of R. Let us begin with the phase
space. In the model of classical mechanics, to determine the later trajectory of a particle it is necessary to
know both its initial position and its velocity. The initial data thus forms a set with six parameters, which are
the three coordinates of position and the three coordinates of velocity, or rather of the momentum p = mv .
If one is interested in a number n of particles, it is necessary to know the position and momentum of each of
them. One is thus dealing with a set with 6» parameters, called the phase space of the mechanical system
under consideration. Starting with a function on this space which is called the Hamiltonian and which
measures energy, classical mechanics prescribes the differential equations that determine the trajectory from
26
the initial data. The natural structure of the phase space is that of a syinplectic manifold X whose points
are the 'states' of the system. The Hamiltouian Я is a function on Л" that intervenes to specify the
evolution of every observable physical quantity, that is, of every function / on Л', by the equation
where { } denotes the Poisson bracket and / = ^-/ .
In the good cases, as for example for the planetary model of the hydrogen atom, the dynamical system
obtained is totally integrable. This means that there are sufficiently many 'constants of motion' so that, on
specifying them, the system is reduced to an almost periodic motion. The description of such a system is
done very simply. For., on the one hand, the algebra of observable quantities is the commutative algebra of
almost periodic series
'/@ = ][] «»¦ ¦¦¦»» «4>Bi"(n,i/H-,
where the n,- are integers, the i/; are positive real numbers called the fundamental frequencies, and
{n,i/) = ^п,-!/,- • On the other hand, the evolution in time is given by translation of the variable t .
The interaction between a classical atom and the electromagnetic field is described by Maxwell's theory.
Such an atom emits an electromagnetic wave whose radiative part is calculated by superposing the plane
waves Wn , „=(„],...,„),) with frequencies (?»,!/) = ]T n,i/, , and whose amplitude and polarization are
calculated simply from the fundamental observable (hat is the clipolo moment. The dipole moment Q has
three components Q,, Qy and Q., each of which is an observable quantity.
and which give the intensity of the emitted radiation of frequency (n,v) by the equation
1 = IT = з?B;г<"'1/))''A''—I" + l«».»l3 + l''--.»l3b
where с denotes the speed of light.
It follows in particular that the set of frequencies of the emit-tcnl radiations is an additive subgroup Г С R
of the real numbers. Thus, to each frequency emitted there corresponds all of its integral multiples or
harmonics.
In fact, spectroscopy and its numerous experimental results show that this last theoretical result is contra-
contradicted by experience. The set of frequencies emitted by an atom does not form a group, and it is false that
the sum of two frequencies of the spectrum is again one. What experience dictates is the Ritz-Rydberg
combination principle, which permits indexing the spectral lines by the set Д of all pairs (i,j) of elements
of a set / of indices. The frequencies Vij and v^t only combine when j = k to yield i/ц — Uij + Vjt. The
theory of Bohr, by artificially discretiziug the angular momentum of the electron, succeeds in predicting
the frequencies of the radiations emitted by the hydrogen atom, but it is unable to predict their intensity
and polarization. It is by a fundamental calling into question of classical mechanics that Heisenberg arrived
at this goal and went well beyond his predecessors. This calling into question of classical mechanics is ap-
approximately as follows: in the classical model, the algrbra of observable physical quantities can be directly
read out of the дтопр Г of emitted frequencies; it is the convolution algebra of this group of frequencies.
Since Г is a commutative group, the convolution algebra is coiiuiniiative. Now, in reality one is not dealing
with a group of frequencies but rather, due to the Ritz-Rydbi'rg combination principle, with a groupoid
Д = {(»,j) : »,j 6/) having the composition rule {i,j) ¦ (j, t) = (MO • The convolution algebra still has
meaning when one passes from a group to a groupoid, and the convolution algebra of the groupoid Д is
none other than the aigebnt of matrices. For, the convolution product may be written
27

which is identical with the product rule-for matrices.
On replacing the commutative convolution algebra of (lie group 1' by the nonconunutative convolution
algebra of the groupoid Д dictated by experimental results, Hnisrubcrg replaced classical mechanics, in
which the observable quantities commute pairwise, by malvix mechanics, in which observable quantities as
important as position and momentum no longer commute.' In Heisenberg's matrix mechanics, an observable
physical quantity is given by its coefficients 4(jj) indexed by elements («, j) 6 A of the groupoid Д . The
time evolution of an observable is given by the homoniorphism {i,j) e Д — щ e R of Д into R which
associates with each spectral line its frequency. One has
»(ij)(O = »(ij> expB*ii/,v<) • A)
This formula is the analogue of the classical formula
9n,...n,(() = 4n,...nt expBiri{n, vH .
To obtain the analogue of Hamilton's law of evolution
37» = <"¦*}.
one defines a particular physical quantity II that plays the role of the classical energy and is given by its
coefficients #(,¦ j), with
H(i,j) =0 if i ф j, tf, j.j, = /)!/,-, where u, - i/j = i/h (V i,j e I),
and where h is Planck's constant- that convert-s frequencies ini,o energies. Oiu) sees that H is defined
uniquely, up to the addition of a multiple of the identity in.itrix. Moreover, the above formula A) is
equivalent to
B)
d 2*i
""P*? = ——(//// — qll) .
Ш h
This equation is similar to the one of Hamilton that uses the Poisson brackets. It is in fact simpler, since
it only uses the product of the observables, and more precisely the commutator [A, D] — AB — В A which
plays the role that the Poisson bracket plays in llaniiltouian morhaiwcs. liy analogy with classical mechanics,
one requires the observables q of position and ]> of momentum' to satisfy [p, (/] = —ift, where ft = Л/2т .
The simple algebraic form of the classical energy as a function of j) and q then gives the equation of
Schrodinger for determining the set {i/,-, i ? 1} , or spectrum of // .
The quantum system thus described is much simpler and more rigid than its classical analogue. One thus
obtains a nonnegligible counterpart to abandoning the coinmutativity of classical mechanics. Though less
intuitive; quantum mechanics is more directly accessible by virtue of its simplicity and its contact with
spectroscopy.
In fact, the results of the theory of «-products show that a ttgmplcclic structure on a manifold such as the
phase space is none other than the indication of the existence ofa deformation with one parameter ( ft here)
of the algebra of functions into a noiiconimutative algebra. I refer the reader to the literature ([Arn-C-M-
P], [Bay-F-F-L-Si], [Bay-F-F-L-So], [DeW-L], [Dr], [Li], [Hi.,], [Vey]) for a description of the results of this
theory.
1 Heisenberg's rules of algebraic calculation were imposed on him by the experimental results of spec-
spectroscopy. However, Heisenberg did not understand right away that the algebra he was working with was
already known to mathematicians and was called the algebra of matrices. It was Jordan and Born who per-
perceived this. In fact, Jordan had remarked that the conditions which, in Heisenberg's formalism, correspond
to the Bohr-Sommerfeld quantification rules, signified that the diagonal elements of the matrix [p,q] were
equal to — ift.
28
1.2. Statistical state of a macroscopic system and quantum statistical mechanics
A cubic centimeter of water contains a considerable number, (on the order of Af = 3 x 1022 ), of molecules
of water agitated by an incessant movement. The detailed description of the motion of each molecule is not
necessary, any more than is the precise knowledge of the microscopic state of the system, for determining the
results of macroscopic observations. In classical statistical mechanics, a microscopic state of the system is
represented by a point of the phase space, which is of dimension QN for N point molecules. A statistical
state is described not by a point of the phase space but by a measure ft on that space, a measure that
associates with each observable / its mean value:
'fd,,.
For a system that is maintained at fixed temperature by means of a thermostat, the measure ft is called
the Gibbs' canonical ensemble; it is given by a formula that invokes the Hamiltonian H of the system and
the Liouville measure that arises from the symplectic structure of the phase space. One sets
¦ Liouville measure ,
C)
where /J = 1/kT, T being the absolute temperature and к the DolUniann constant, whose value is
approximately 1.38 X 10~2s joules per degree Kelvin, and where Z is a normalization factor.
The thermodynamic quantities, such as the entropy or the free energy, are calculated as functions of /3 and
a small number of macroscopic parameters introduced in the formula that gives the Hamiltonian Я. For
a finite system, the free energy is an analytic function of these parameters. For an infinite system, some
discontinuities appear that correspond to the pim.se transition phenomenon. The rigorous proof, starting
with the mathematical formula that specifies // , of the absence or existence of these discontinuities is a
difficult branch of mathematical analysis ([Kneo]).
However, as we have seen, the microscopic description of matter cannot be carried out without quantum
mechanics. Let us consider, to fix the ideas, a .solid having an atom at each vertex ofa crystal lattice Z3 . The
algebra of observable physical quantities associated with each atom x = (x (, xn, ?3) is a matrix algebra Qx ,
and if we assume for simplicity that these atoms are of the same nature and can only occupy a finite number
n of quantum states, then Qj. = Л/,,(С) for every x. Now let Л be a finite subset of the lattice. The
algebra Q\ of observable physical quantities for the system formed by the atoms contained in Л is given
by the tensor product QA = ®гел Q*
The Hamiltonian Ял of this finite system is a self-adjoint matrix that is typically of the form
where the first term corresponds to the absence of interactions between distinct atoms and where A is a
coupling constant that governs the intensity of the interaction. A statistical state of the finite system Л is
given by a linear form tp that associates with each observable A ? Q\ its mean value tp(A) and which
has the same positivity and normalization properties as a probability measure /i, namely,
a) Positivity: ip(A'A) > 0 (УЛ e QA );
b) Normalization: tp(l) = 1.
If the system is maintained at fixed temperature T, the equilibrium state is given by the quantum analogue
of the above formula C)-.
ip(A) =
(V.-1
D)
where the unique trace on the algebra Q,\ replaces the Uouville measure.
29

As in classical statistical mechanics, the interesting phenomena appear when one passes to the thermody-
namic limit, that is, when Л — Z3 . A state of the infinite system being given by the family (ipA) of its
restrictions to the finite systems indexed by Л , one obtains in this way all of the families such that
a) for every A, tp\ is a state on QA ;
b) if Ai С Аг then the restriction of ip\, to Q\, is equal to tpAl .
In general, the family tp\ defined above by means of ехр(-/?Ял) does not satisfy the condition ,b) and it
is necessary to better understand the concept of state of an infinite system. This is where C*-algebras make
their appearance. In fact, if one takes the inductive limit Q of the finite-dimensional C""-algebras Q\ , one
obtains a C""-algebra that has the following property:
An arbitrary slate tp on Q is given by a family (ip\) satisfying the conditions a) and b).
Thus, the families (tp\) satisfying a) and b), that is the states of the infinite system, are in natural byective
correspondence with the states of the C""-algebra Q. Moreover, the family (Яд) uniquely determines a
one-parameter group (at) of automorphisms of the C*-algebra Q by the equation
This one-parameter group gives the time evolution of the observables of the infinite system that are given
by the elements A of Q , and is calculated by passing to the limit starting from Heisenberg's formula. For
a finite system, maintained at temperature T, the formula gives the equilibrium state in a unique manner
as a function of H\ , but in the thennodynamic limit one cannot have a simple correspondence between
the Hamiltonian of the system, or, if one prefers, the group of time evolution, and the equilibrium state of
the system. Indeed, during phase transitions, distinct states can coexist, which precludes uniqueness of the
equilibrium state as a function of the group (o,). It is impossible to give a simple formula that would define
in a unique manner the equilibrium state as a function of the one-parameter group (»i) . In compensation,
there does exist a relation between a state ip on Q and the one-parameter group a, that does not always
uniquely specify tp from the knowledge of a, . but which is the analogue of the formula D). This relation
is the Kubo-Martin-Schwmger condition ([Ku], [Mart-S]) as formulated by Haag, Hugenholtz and Winnink
[HHW]:
Given T, a state tp on Q and the one-parameter group a, of automorphisms of Q satisfy the KMS-
condition if and only if for every pair Л, В of elements of Q there exists a function F(z) holomorphic in
the strip (.-eC Im .- e [0, hff] j such that (Fig. 3)
F(l) = <p(Aa,(B)), F(l + \f43) = 4>{a,(B)A) (V< € R) .
Here t is a parameter of time, as is hji = h/ЬТ which, for T = 1 A', has value approximately 10"ns.
Fig.3. The function F is holomorphic in the unshaded strip and connects ip(Aat(B)) and tp(at(B)A).
This condition allows to formulate mathematically, in quantum statistical mechanics, the problem of the
coexistence of distinct phases at given temperature T, that is, the problem of the uniqueness of tp, given
(at) and /3. We shall give in chapter V section 10 an explicit example of phase transition with spontaneous
symmetry breaking coming from the statistical theory of prime numbers.
This same condition has played an essential role in the modular theory of operator algebras. It has thus
become an indisputable point of interaction between theoretical physics and pure mathematics.
30
1.3. Modular theory and the classification of factors
Between 1957 and 1967, a Japanese mathematician, Minoni Tomita, motivated in particular by the harmonic
analysis of nonunimodular locally compact groups, proved a theorem of considerable importance for the
theory of von Neumann algebras. His original manuscript was very hard to decipher and his results would
have remained unknown without the lecture notes of M. Takesaki [T2], who also contributed greatly to
the theory. Before giving the technical definition of a von Neumann algebra, it must be explained that the
theory of commutative von Neumann algebras is equivalent to Lchesguc's measure theory and to the spectral
theorem for self-adjoint operators. The noncommutaUve theory was elaborated at the outset by Murray and
von Neumann, quantum mechanics being one of their motivations. The theory of noncommutative von
Neumann algebras only achieved its maturity with the modular theory; it now constitutes an indispensable
tool in the analysis of noncommutative spaces.
A von Neumann algebra is an involutive snbalgebra Л/ of the algebra of operators on a Ililbert space И
that has the property of being the commutant of its commutant: (M1)' = A/ .
This property is equivalent to saying that M is an involutive algebra of operators that is closed for the
operation of weak limit. To see intuitively what the equality (Л/')' = M means, it suffices to say that it
characterizes the algebras of operators on Ililbert space that are invariant under a group of unitary operators:
The commutant of any subgroup of the unitary group of the llilbert space is a von Neumann algebra and they
are all of that form (given M take as a subgroup, the unitary group of XI'). In the general noncommutative
case the classical notion of probability measure is replaced by the notion of state. A typical state on the
algebra M is given by the linear form V[A) = (A(,() , where { is a vector of length 1 in the Ililbert
space. Tomita's theory, which has as an ancestor the notion of quasi-Hilbert algebra ([Di2]), consists in
analyzing, given a von Neumann algebra A/ on the llilhert space П and a vector { e 4 such that-Щ
and M'Z are dense in H , the following unbounded operator S:
Sxi = x'{ (Vx G Л/).
This is an operator with dense domain in 4 that is conjugate-linear; the results of the theory are as follows:
1) S is closable and equal to its inverse.
2) The phase ] = S\S\~l of S satisfies JM J = A/'.
3) The modulus squared A = \S\- = S'S of S satisfies Л"Л/Д-" = Л/ for every leR.
Thus, to every state ip on A/ one associates a one-parameter group a? of automorphisms of M af (x) =
Д«хД-« Ух G M V( e R, the group of modular automorphisms of y. It is precisely at this point
that the interaction between theoretical physics and pure mathematics takes place. Indeed, M. Takesaki and
M. Winnink showed simultaneously [T2][\V] that the connection between the state y> and the one-parameter
group of of Tomita's theorem is exactly the KMS condition for Л/3 = 1 .
These results as well as the work of R. Powers [Pow], 11. Araki and E. J. Woods [Ar-W] on factors that are
infinite tensor products, proved to be of considerable importance in setting into motion the classification of
factors.
The point of departure of my work on the classification of fnrtors was the discovery of the relation between
the Araki-Woods invariants and Tomita's theory. For this it had to be shown that the evolution group
associated with a state by that theory harbored properties of the algebra Л-/ independent of the particular
choice of the state tp
A von Neumann algebra is far from having just one stale ^, which lias as consequence that only the
properties of <rf that do not depend on the choice of <? have real significance for M . The crucial result
that allowed me to get the classification of factors going is the following analogue of the Radon-Nikodym
theorem [Co3]:
For every pair ср,ф of states on M, there exists a canonical \-cocycle I — m , u(,+i, = «t.^Г,(u<,)
V(,,B e R with values in the unitary group of M , sncli that of l>) = u,<rf(x)u't Vx e M, V( e R.

Morover, v^djuOi-o coincides
1) in the commutative case, with the logarithm of the liacloii-Nikodym derivative
2) in the case of statistical mechanics, with the difference of the Hamiltonians corresponding to two equi-
equilibrium states, or the relative Ilamiltonian of II. Araki [Ar-j].
It follows that, given a von Neumann algebra M , there exists a canonical homomorphism 6 of R into the
group Out M = Aut M/Int M (the quotient of the automorphism group by the normal subgroup of inner
automorphisms), given by the class of rrf independently of the choice of tp. Thus, KeriS = T(M) is an
invariant of M , as is SpS = S(M) =f| Sp Av .
Thus von Neumann algebras are dynamical objects. Such an algebra possesses a group of automorphism
classes parametrized by R . This group, which is completely canonical, is a manifestation of the noncom-
mutativity of the algebra M . It lias no counterpart in the commutative case and attests to the originality
of non commutative measure theory with respect to the usual theory.
Twenty years after Tomita's theorem, and after considerable work (cf. Chapter 5), we now have at out
disposal a complete classification of all the hyperfinite von Neumann algebras. Ratlier than give a definition
of this class, let us simply note that
1) If G is a connected Lie group and n 6 Rep 6' is a unitary representation of G, then its commutant
ir(G)' is hyperfinite.
2) If Г is an amenable discrete group and т? Hep Г, thru я{Г)' is liyperfinite.
3) If a C*-algebra A is an inductive limit of finiUxlimensioiiiil algebras und if 7 € Rep .4 , then ir(A)"
is hyperfinite.
Moreover, the classification of hyperfinite von Neumann algebras reduces to that of the hyperfinite factors
on writing M = J Mtd[t(t), where each Aft is a factor, that is. has center equal to С . Finally, the list of
hyperfinite factors is as follows:
In М = М„(С).
Ico M = ?Gt), the algebra of all operators on an infinite-dimensional llilbert space.
Hi R — СИЩЕ), the Clifford algebra of an infinite-dimensional Euclidean space E .
"со До,1 = Я®1со-
III» Rx = the Powers factors (A e ]0, 1[).
IIIi Roz-R\,®Rx, (VA,,A3, A,/A2 ? Q), the Araki-Woods factor.
IIIo Rw , the Krieger factor associated with an ergodic flow \V .
After my own work, case 111] was the only one that remained Ю be elucidated. U. llaagerup has since shown
that all the hyperfinite factors of type 111, are isoniorpliic. All these results are explained in great detail in
chapter V.
32

Fig.4. The lady in blue by Gainsborough
33
Fig.4.

1.4. Geometric examples of von Neumann algebras: Measure theory of nnucouimutative spaces
My aim in this section is to show by means of examples that the theory of von Neumann algebras replaces
ordinary measure theory when one has to deal with non commutative .spaces. It will allow us to analyse
such spaces even though they appear as singular when considered from the classical point of view, i.e. when
investigated using measurable real valued functions.
We shall first briefly review the classical Lebesgue measure theory and explain in particular its intimate
relation with commutative von Neumann algebras. We then give the general construction of the von Neumann
algebra of the noncommutative spaces Л" which arise naturally in differential geometry, namely the spaces of
leaves of foliations. Our first use of this construction will he to illustrate the classification expounded above
by numerous geometric examples.
a) Classical Lebesgue measure theory
H. Lebesgue was the first to succeed in defining the integral fa }{x)<lx of a (hounded) function of a real
variable x without imposing any serious restrictions on /. At a technical level, for the definition to make
sense it is necessary to require that the function / he measurable. However, this measnrability condition is so
little restrictive that one has to use the uncountable axiom of choice to prove the existence of nonmeasurable
functions. In fact, a very instructive debate took place at t he time A005) between Borcl, Baire and Lebesgue
on the one hand and Iladamard (and Zcrmelo) on the other, as to the "existence" of a well ordering on the
real line (see Lebesgue's letter in Appendix C). A result of the logician Solovay shows that (modulo the
existence of strongly inaccessible cardinals) a nonmcasuiable function cannot be constructed using only the
axiom of conditional choice.
This result on measurability still holds if the interval [«,/'] is replaced by a standard Borel space X. It
shows in particular that among the classical structures on a set Л* obtained by specifying a suitable class
of functions / : X —» R such as continuous or smooth functions, the measure space structure, obtained by
specifying the measurable functions, is the c0ar.4e.st possible.
In particular, a measure space -V is not modified by a transformation T such as the one indicated in Figure
4, which consists in making it explode like n jigsaw puzzle whose pieces are scattered. This transformation
has the effect of destroying all shape; nevertheless, a particular subset Л of .V does not change in measure
even though it may be scattered into several pieces.
This wealth of possible transformations of Л" is bound up with the existence, up to isomorphism, of only
one interesting measure space: an interval equiped with Lebesgue measure. Even spaces as complex in
appearance as a functional space of distributions equipped with a functional measure are in fact isomorphic
to it. In particular, the notion of dimension has no intrinsic meaning in measure theory. The key tools of
this theory are positivity and the completeness of the Hilbert. space L-(X,/i) of square integrable functions
/, J\f\2dn < 00 on Л'. A crucial result, is the Radon-Nikodym theorem, by which the derivative df-i/di/ of
one measure with respect to another may be defined as a function on .V.
The topology of compact metrizable spaces Л' has a remarkable compatibility with measure theory. In fact,
the finite Borel measures on Л' correspond exactly i.o the continuous linear forms on the Banach space
C{X) of continuous functions on Л', equipped with the norm j| / ||= suPreA" |/(.r)|. The positive measures
correspond to the positive linear forms, i.e., the linear forms кр such that <p (//) > 0 for all / € C{X).
To go into matters more deeply, it is necessary to understand how this theory arises naturally from the
spectral analysis of self-adjoint operators in Hilbcrt space, and thus becomes a special case (commutative)
of the theory of von Neumann algebras (Chapter 5), which is itself the natural extension of linear algebra to
infinite dimensions.
Thus, let Ti be a (separable) Hilbert space and T a bounded selfadjoint operator in H, i.e. (T?,r)) =
(?, Trj) V?, ц?Н. Then if p(u) is a polynomial of the real variable u,;;(«) = rio«" H 1- a,,, one can define
the operator p{T) as
34
T"'".
It is rather amazing that this definition of p(T) extends by continuity to till bounded Borel functions / of a
real variable u. This extension is uniquely given by the condition:
if the sequence /„ converges simply to /, i.e. /„(") — /(») for all u 6 R .
There exists then a unique measure class /i on R, (carried by the compact, interval / = [— j( T ||, |j T ||]) such
that:
J(T) = О О
/ l/k'/< = 0.
Moreover the algebra M of operators in "H of the form /('/*) for some bounded Borel function / is a von
Neumann algebra, and is the von Neumann algebra generated by T. In other words if S is a bounded
operator in Ti which has the same symmetries as T (i.e.. which commutes with all unitary operators U,
U'U - UU' = 1 in П which fix T : UTV = T), then there exists a bounded Borel function / with
S = f(T). This commutative von Neumann algebra is naiurnlly isoiuorphic to f^f/./j), die algebra of
classes modulo equality almost everywhere, of bounded measurable functions on the interval /.
/3) Foliations
We shall now describe a large class of noncoiuiiiiitativc spaces arising from differential geometry, see why
the Lebesgue theory is not able to analyse such spaces and replace iL by the theory of non commutative von
Neumann algebras.
The spaces -Y considered are spaces of manifolds which are solutions of a differential equation, i.e. spaces of
leaves of a foliation.
Let V be a smooth manifold and TV its tangent bundle, so tluit for each x € V, TVV is the tangent space
of V at x. A smooth subbuudle F of TV is called ivtegrahle iff one of the following equivalent conditions is
satisfied:
a) Every x 6 V is contained in a subnianifold IV of V such that
TV(\V)=FV Vj/elV.
b) Every x 6 V is in the domain U С V of a submersion p : V — R' A/ = codim F) with Fy = Ker(p»)s Vy 6
U.
c) C°°(V, F) ~ [X 6 C~(V, IV), -V, 6 F1: Vx ? 1-"} is a Lie subalgebra of the Lie algebra of vector fields on
V.
d) The ideal J{F) of smooth differential forms which vanish on F is stable by differentiation: dj С J.
Any 1-dimensional subbuudle F of TV is integrable, but for dim /^ > 2 the condition is non trivial, for
instance if P —> В is a principal //-bundle (with compact, structure group //) the bundle of horizontal
vectors for a given connection is integrable iff this connection is flat.
A foliation of V is given by an integrable .subbundle F of TV. The leaves of the foliation (V, F) are
the maximal connected submauifolds L of V with T,(L) — Ft Vx g L, and the partition of V in leaves
V = ULa, a € A is characterized geometrically by its "local triviality": every point r? ^ has a neighborhood

U and a system of local coordinates(*) (x')j=i,.,^imv so that the partition of U in connected components
of leaves(**) corresponds to the partition of R?"" v = RdlmF x №°<"™р т the parallel affine subspaces
RdimF x pi cf fig 5
dimensional,
= C, and essentially useless.
Fig.5. Foliation
The manifolds L which are the leaves of the foliation are defined in a fairly implicit manner, and the simplest
examples show that:
1) Even though the ambient manifold V is compact, the leaves L can fail to be compact.
2) The space X of leaves can fail to be Hausclorff for Lhe quotient topology.
For instance let V be the two dimensional torus V = R2/Z2, with the Kronecker foliation associated to a
real number 9, i.e. given by the differential equation:
dy = Odx.
Then if 9 is an irrational number, the leaves L are diffeoniorphic to R, while the quotient topology on the
space X of leaves is the same as the quotient Lopology of S1 — R/Z divided by the partition into orbits of the
rotation OE51 н? а+ в, and is thus the coarse topology. Thus there are no open sets in X except 0 and X.
Similarly the ergodicity of the rotation Rg on Sl shows that if we endow Лг with the quotient measure class
of th^Jebesgue measure class on V we get the following "pathological" behaviour: any measurable function
/ : X —> R is almost everywhere equal to a constant. This implies that classical measure theory does not
distinguish between Л' anil a one point space, and in particular the Lp spaces of analysis, W^X) are one
(*) Such charts are called foliation charls.
(**) Called plaques, they are the leaves of the restriction of the foliation to the open set U.
Fig.6. Ivronecker foliation
7) The von Neumann algobra of a foliation
Let [V, F) be a foliated manifold. We shall now construct a von Neumann algebra W(V,F) canonically
associated to (V,F) and depending only on the Lebesgue measure class on the space X of leaves of the
foliation. The classical point of view: L^iX) will only give the center Z(W) of W.
The basic idea of the construction is that while in general we cannot find 011 X interesting scalar valued
functions there are always, as we shall see, plenty of operator valued functions on X. In other words we just
replace c-numbers by (/-numbers. To be more precise let us denote, for each leaf I of (V, F), by L2(() the
canonical HilberL space of square integrable half densities ([Gu-S]) on (. Then we define:
ates to
ch leaf
Definition l.(*) A random operator q = (qt)t€X is a measurable map t —> qt which r
t ? X a bounded operator qt m the IIMbcrt space L~(?).
To define the measurabiliiy of random operators we note the following simple facts:
a) Let A : V —> X be the canonical projection which to each x 6 V assigns the unique leaf t — A(x) passing
through 1. Then the bundle (L"(A(j')))j-€l- of HilberL spaces over V is measurable. More specifically consider
the borel subset of V x V : G ~ {(x7y) ? l; x ^ ; у ? A(x)}. Then, modulo the irrelevant choice of a j
density \dy\ll- along the leaves, the sections (^)гег of the bundle (L2(X{x)))xiV are just scalar functions
on G and measurability has its ordinary meaning.
b) A map ( —' qt as in definition 1 defines by composition with A an endomorphism (?л(г))«е1' of the bundle
(L2(A(x)))xgi/. We shall then say thai, q = (^)^eA' 's measurable iff the corresponding endomorphism
(9A(x))x€K 's measurable in the usual sense, i.e. iff for any pair of measurable sections {?x)X?v, ('Hx)xGV °f
(L2(A(x)))r?i/ the following function on V is Lebesgue measurable:
— (qX(I
6 С
(*) We assume here to simplify the discussion that the set of leaves with non trivial holonomy is Lebesgue
negligible. In general (cf. chapter 2) one jusi. replaces с by 7, its holonomy covering.
37

There are many equivalent ways of defining mcasurabilily of random operators. As we already noted (П
section a) any natural construction of families (iu)tiX will automatically satisfy the above moasuri»b.iUty
condition.
Let us give some examples of random operators.
1) Let / be a bounded borel function on V¦ Then for each leaf t ? X let ft be the multiplication operator:
(h «)(*) = /(*) «(x) v* e e, v« e L2(C).
This defines a random operator / = {fc)ttx ¦
2) Let X 6 C"(V,F) be a real vector field on V tangent to the leaves of the foliation. Let then фг = '¦
be the associated group of diffeomorphisnis of V. (Assume for instance that V is compact to ena
existence of x/>t)- By construction 4'i{x) 6 A(x) for any x 6 V so that it defines for each I a din"
of I. Let (Ut)it x be the corresponding family of unitaries (the map ( — L2(t) is functorial). It is а гэдДога,
opertor U = (Uijttx.
Let q = {qe)eex be a random operator. The operator norm \\(it\\ °f f/r in L~{f) defines a measurable fuacJjtQjfc
on V by composition with A : V — X which gives meaning to the norm:
(*) \\q\\ = Essential Supreniuni of ||</f|| , I' ? X.
We say that q is zero almost everywhere iff q o, A is so.
The von Neumann algebra W(V, F) of a foliation is obtained as follows:
Proposition 2. The classes of bounded random operators (г//.);.€л' modulo equality almost everywhere,
endowed with the following algebraic rules, form a von Neumann algebra ll'( V, F):
(P + '/)/. =1'L + 'IL W. e ,Y
(;»/k = pi.4l v/, e x
(P")L =(/'(.)• VLe.Y
The norm, uniquely defined by the involutive algebra structure, is given by (*). We have used here the
possibility of defining a von Neumann algebra without a specific reprnseuial ion in 1 lilbert space, cf. chapter
5. If one wishes to realise concretely \V{ V, F) as operators in a Ililbert space one can for instance let random
operators act by:
in the Hilbert space "H of square integrable sections (?s ).,gr of the bundle A" L1 of Ililbert spaces on V. One
can also use the restriction of \mL~ to a sufficiently large transverse suhmaiiifold T of V. The invariants of
von Neumann algebras are independent, of the choice of a specific representation in Ililbert space and depend
only upon the algebraic structure.
The von Neumann algebra bV(V,F) only depends upon the space Л* of leaves with its Lebesgue measure
class, one has:
Proposition 3. Two foliations {Vt,Fi) with the same kaf space- have isomorphic von Neumann algebras:
W(VuFi)~W(V2,Fi).
To be more precise one has to require in proposition 3 that dim !¦]i > 0 and the assumed isomorphism
Vi/F\ ~ V1/F2 to be given by a bijection ip : V\/F\ — V«/F? which preserves the Lebesgue measure class
and is borel in the sense that {(*,;/) 6 Ц X 1A ; t;'(Ai(.r)) = Av(;/)} is л horel subset of Vi X Vi, with
A( : Ц —> Ц/Fi the quotient map.
For any von Neumann algebra M its center:
Z(M) = {x 6 M ; *.!/ = ух V.y ? M)
is a commutative von Neumann subalgebra of Л/.
In the above context one has the easily proved:
Proposition 4. Let (V, F) be a foliated manifold then the construction 1) of random operators yields an
isomorphism:
L"a(X)~Z(W(\\F))
of the algebra of bounded measurable (classes of) functions on X with the center of W(V, F).
In particular W{V, F) is a factor iff the foliation is ergodic.
We shall now illustrate the type classification of von Neumann algebras by geometric examples given by
foliations. This will allow a non algebraically minded leader t.o foi-m a mental picture of these notions.
Type I von Neumann algebras
By definition a von Neumann algebra Л/ is of type I iff it is algebraically isoinorphic to the commutant of
a commutative von Neumann algebra. Thus type 1 is the stable form of the. hypothesis of commutativity.
We refer to chapter V for the easy classification of type I von Neumann algebras (with separable predual)
as direct integrals of matrix algebras Л/„(С), и е {1.2 oo].
A simple criterion for a von Neumann algebra Л/ to be of type I is that it contains an abelian projection e
with central support equal to 1. A projection e — e_" = cr 6 M is called abclian iff the reduced von Neumann
algebra:
Mt = {x e M ; jce = ex = x]
is commutative. The central support of e is equal to 1 iff one lia.4 ex ф 0 for any i 6 Z(M) , x ф 0.
Using this criterion one easily gets that the von Neumann algebra of a foliation (V',F) is of type I iff the leaf
space X is an ordinary measure space. More precisely:
Proposition 5. Let (V,F) be a foliated manifold. Then Ihc associated von Neumann algebm is of type I iff
one of the following equivalent conditions is satisfied.
a) The quotient map A : V —* Л" admits a (Lc.licsgue) nu as и cable section.
/?) The space of leaves X (with its quctticni structure) is. up to a null set. a standard borel space.
A measurable section a of A is a measurable map from V to V constant along the leaves and such that
In general an abelian projection e 6 W(V, F) is given by a measurable section С —• ?( e L~(C) of the bundle
(L2(t))tix such that ||?(|| € {0,1} Vf 6 -Y. The formula for the random operator e = (e;)<ex is then:
e< 4=(.4.Zt)tt V,;eL-(C) , Vf 6 ,Y.
The central support of e is equal to 1 iff ||?{j| = 1 a.e. The above type 1 property rarely holds for a foliation.
It does of course for fibrations or if all leaves are compact hut it also holds for some foliations whose leaves
38

are not compact such as the Reeb foliation (fig.7). In this last example the map a, which assigns to each
leaf С the point a(l) 6 I where the extrinsic curvature is maximal, yields a measurable section of A : V —* X.
Fig.7. Example of type I foliation
Type II von Neumann algebras
A finite trace т on an algebra is a linear Conn т such that 7-(iy) = т(ух) for any pair of elements of that
algebra. Let Ti be an infinite dimensional Ililbert space and Л/^С) = CCH) be the von Neumann algebra
of all bounded operators in H. It is the only factor of type I^. This factor does not possess any finite trace
but the usual trace of operators, extended by +00 for positive operators which are not in С'(Н), is a positive
semi-finite faithful normal trace on Л/оо(С). We refer to chapter V section 3 for the technical definitions.
By definition a von Neumann algebra M is semi-finite iff it has a positive semi-finite faithful normal trace.
Any type I von Neumann algebra is seini-fiuite. Any semi-finite von Neumann algebra M is uniquely
decomposed as the direct sum M = JWi ф Л/ц of a type I von Neumann algebra M\ and a semi-finite von
Neumann algebra M\\ with no non zero abelian projection. One says hat M\\ is of type 1Г.
Let (V, F) be a foliated manifold and M = H'(V',F) be the associated von Neumann algebra. We shall see
now that the positive faithful semi-finite normal traces т on At correspond exactly to the geometric notion
of invariant transverse densities on V which we first describe.
Given a smooth manifold Y a density p on Y is given by a section of the following canonical principal bundle
P on Y. At each point у 6 V the fiber Py is the space of non zero homogeneous maps p : Ad Ty —•• R+,
p{\v) = |A| p(v) VA e R, в e Ad T,, where T, = T,j(Y) is the tangent space of Г at у 6 У, and where d is
the dimension of Y.
By construction P is an R+ principal bundle over Y.
The choice of a measurable density determines a measure in the Lebesgue class and all measures in that
class are obtained in this manner. Let us now pass to the case of foliations (V, F). We want to define the
notion of density on the leaf space Л". Given a leaf ( 6 Л' the tangent space 7>(X) can be naively thought of
as the «/-dimensional real vector space, 17 = dim V - dim F obtained as follows. For each x ? V the quotient
40
Nx = Tt(V)/Fc is a good candidate for T((.V), it remains to identify canonically the vector spaces Nr, N3
for x, у 6 I.
Fig.8. Holonomy, i and у belong to the same leaf, U is a domain of foliation chart
containing x,y and x,y belong to the same plaque
Let U С V be a domain of foliation chart so that r. and у belong to the same plaque of U i.e. to the same
leaf p of the restriction оГ F to U (fig.8). The leaves of (U, F) are the fibres of a submersion x : U -> R'
whose tangent map at x,y gives the desired identification: Nx ~ Ny. This identification is independent of
any choice when the leaf I has no holonomy and is transitive. Since we assume that the set of leaves with
non trivial holonomy is Lebesgue negligible we thus get a well defined real measurable bundle T(X) over
A', with fiber T/(X) ~ Nx for any i?(. As above in the case of manifolds we let P be the associated
R+ principal bundle of densities. The fiber Pt оГ P at t e X is the space of non zero homogeneous maps
p ¦ л« Te — R+, p{\v) = |A| p(v) VA 6 R, Vn e Л* Т/. (Here q = dimT, is the codimension of F.)
We shall now see that the densities on X, i.e. the measurable sections p of P on X, correspond exactly to
the positive semi-finite faithful normal traces on the von Neumann algebra W{V% F):
Proposition 6. Let (V,F) be a foliated manifold, X its leaf space.
1) Let p be a measurable section of P on X, and T = (Tc)tiX be a positive random operator. Letlt = {Ua)a61
be a locally finite open covering of V by domains of foliation charts, (f.a)atl « smooth partition of unity
associated to this covering. For eack plaqve p ofUa let Trace(((.aT)/p) be the trace of the operator^12 Te (.1 2
in L2(p) С L2(e) where С is lite leaf of p. Then Hi e following number тр(Т) 6 [0,+oo] is independent of the
choices of Ua, ?a:
*>cn =.
Pip)-
2) The functional тр : M+ — [0, +oo] tints obtained is a positive faithful semi-finite normal trace on M and
all such traces on M are obtained in this may.
In particular we get a simple criterion for 1V'( V, F) to be semi-finite:
41

Corollary 7. W(V, F) is semi-finite iff the R'+ principal bundle P over X admits some measurable section.
For в fc Q the Kronecker foliation dy = Oilx of the 2 torus fulfills this condition; its von Neumann algebra
W(T2,Fe) is the (unique) hyperfinite factor type lie namely Л01 (cf. section 3). Similarly any flow which
is ergodic for an invariant measure in the Lebesgue measure class gives the same hyperfinite factor of type
IIqo. Any product (Vi x Vz,F\ x F?) of the above examples will also yield this factor йод.
Let us now describe a type II foliation giving rise to a 11011 hyperfuiite factor. Let Г = "i(S) be the
fundamental group of a compact Riemann surface 5 of genus > 1 ami a : Г — SO(N) be an orthogonal
faithful representation of Г. Let V be the total space of the SO(N) flat principal bundle over 5 associated to
ct. Then the horizontal foliation of V is a 2 dimensional foliation F whose associated von Neumann algebra
is a type lloo non hyperfinite factor.
Type III von Neumann algebras
Any von Neumann algebra M is canonically the direct sum Л/ = ЛЛфЛ/пФЛ/ш of von Neumann algebras of
respective types. A type Ш von Neumann algebra Л/ is .such that the reduced algebras Me, e = e2 = e*, e ф О
are never semi-finite. Of course this is a negative statement, hut the modular theory (cf. section 3) yields
a fundamental invariant, the flow of weights Н'(Л/), which is an action of the group R"+ by automorphisms
of a lebesgue measure space, as well as the canonical continuous decomposition of Л/ as the crossed product
of a type II von Neumann algebra by an action of R+. We refer to Chapter 5 Appendix A for the technical
definitions but we shall now spell out for the case of the von Neumann algebrii Л/ of a foliation (V, F) what
these invariants are. Let (V, F) be a foliated manifold. We fust, need to come back to the construction of
the R+ principal bundle P over the leaf space ,V and interpret the total space of P as the leaf space of a
new foliation (V',F'). Let N, Nx - TI(V)/FI, be the transverse bundle of the foliation (V, F). It is a real
vector bundle, of dimension q — codim F, over the manifold V. We let Q be the associated R+ principal
bundle of densities. Thus the fiber of Q at x ? V is the R + homogeneous space of non zero maps:
6 :
R+ , «(Ai>) = |A| *(
VA 6 R , Vt> 6 A*JVr.
By construction Qisa smooth principal bundle and we denote by \" its total space. The restriction of
Q to any leaf t of the foliation has a canonical /lal connection V. This follows from the above canonical
identification of the transverse spaces iV^, j?(. Being a local statement it does not use any holonomy
hypothesis. Using the flat connection V we define я foliation F' of V", dim F' = dimF as the horizontal
lifts of F. Thus Гог у 6 V = Q sitting over x 6 V, the space F? С Т„ {1") is the horizontal lift of F*. The
flatness of V insures the integrability of F'. One then checks Mint:
Proposition 8. 1) (V,F') is a foliaial manifold canoxiicttlly associated lo {V, F).
2) The group R^_ acts by automorplnsm.fi of the foliation {V1. F1).
3) With the above holonomy hypothesis I he leaf space of ( V"', /*'') is the total space of the R* principal bundle
P over the leaf space X of(V,F).
In 2) the action is of course that of the R+-bundle Q = I", we shall denote it, by (Ox)xeu' , "л 6 Aut(V, F').
With the above notations we now have:
Proposition 9. a) The von Neumann algebra W(V, /•") is aliiuii/s semi-finite.
b) W(V, F) и of type [If iff\Y(V, F1) is of type IF.
c) The flow of weights ofW(V, F) is r/incii btj the action 9 ofH"+ on the commutative uon Neumann algebra:
t~(f) = Z(\V(V',F')).
d) The continuous decomposition of W( V, F) is given ha the crossed jirodaci of W( V, F') by the action 9 of
r;
With this result we can now illustrate the classification of section :i l>y geometric examples.
42
All invariants discussed in section 3, S(M), T(A[)... are computed in terms of the flow of weights W(M),
thus for instance (for factors)
S(M) = {A € R; , 0a = id}
T(M) = Point spectum of 9 = {* 6 R , 3u ф 0 , 0x(u) = A"u VA e R+}.
In particular M is of type II^ iff W(M) is trivial.
Thus an ergodic foliation W(V, F) is of type III, iff (V, F') is ergodic.
A simple example of this situation is given by the Anosov foliation F of the unit sphere bundle V of a compact
Riemann surface S of genus g > 1 endowed with its Riemannian metric of constant curvature —1. Thus (cf.
for instance [Mi4]) the manifold V is the quotient V = G/Г of the semi simple Lie group С = PSLB,H) by
the discrete cocompact subgroup Г = 1^E) and the foliation F of V is given by the orbits of the action by
left multiplications on V — G/Г of the subgroup T С G of upper triangular matrices.
The von Neumann algebra M = W(V, F) of this foliation is the (unique) hyperfiuite factor of type II7, : Я,».
One can indeed check that the associated type II foliation (V", F') is ergodic. It has the same space of leaves
P as the horocycle flow given by the action on V = G/Г of the group of upper triangular matrices of the
fcrm[J J
Let us give an example of a foliation whose associated von Neumann algebra is the hyperfiuite factor Rx of
type III\, A 6 ]0, 1[. Let G = P5?B,R), Г С G as above and P he the subgroup of G of upper triangular
matrices ? , ; t € R, a ? R+. Let V be the manifold V = G/Г X T where T is the one dimensional
torus, T = R^./Az. The group P acts on G/Г by left translations and on T by multiplication by a. The
product action of P on V gives a two dimensional foliation (V, F) ami:
\V(V,F)~lh-
The same construction with T and the action of R^. on T replaced by an arbitrary ergodic smooth flow Y
yields a foliation (V, F) whose associated von Neumann algebra is the unique hyperfinite factor algebra of
type IIIa with Y as flow of weights, namely the Krieger factor Il\- (<:f. chapter 5). In fact the hyperfinite
factors of type II h, A e ]0,1] and a large class of hyperfinite factors of typo Ilia already arise from smooth
foliations of the 2 dimensional torus ([Kazu]).
We cannot end this section without mentioning the deep relation between the Godbillon Vey class and the
flow of weights which will be discussed in chapter 3 section fi.
¦13

1.5. The index theorem for measured foliations
In the previous section we used the geometric examples of noncoiumutative spaces coming from the spaces
of leaves of foliations to illustrate the classification of factors. In this section we shall show how the theory
of type II von Neumann algebras is used as an essential tool in measuring the continuous dimension of the
random Hilbert spaces of ?2-solutions of a leafwise elliptic differential equation. An extraordinary property
of factors of type IIt such as the hyperfiuite factor R is the complete classification of the equivalence classes
of projections e € R by a real number diiu(e) 6 [0, 1] that can take on any t'«/«e between 0 and 1. The
grassmannian of the projections e ? R thus no longer describes the lines, planes, etc. of ordinary geometry
but instead "spaces of dimension a ? [0,1]", in other words a continuous geometry. The force of this
discovery comes across very clearly as one reads the original texts of Murray and von Neumann ([Mur-Ni]).
As in the usual case, one can speak of the intersection of "subspaces"; the corresponding projection e Л / is
the largest projection majorized by e and /. One can likewise speak of the subspac.e generated by e and /;
the corresponding projection is denoted e V /. The fundamental equation is then
dim(eA/) + dim(e V/) = (lini(c) +<lim(/) Ve, /.
These properties make it possible to extend the notion of continuous dimension to the representations of
a II\ factor N in Hilbert space, so to speak to JV-niodiiles, so that those modules are classified exactly by
their dimension dimwCW), which can be any real number in [t),+oo] (cf. chapter 5, section 9). We first
explain in detail the notion of transverse measure for foliations and the Ruelle-Sullivan current associated
to such a measure. We insist on this notion because it is the geometric counterpart of the notion of trace
on a C-algebra and is at the heart of noncomiuntative measure theory, We then discuss the special case of
the leafwise de Rham complex and show how the continuous dimensions of Murray and von Neumann allow
to define the real-valued Betti numbers. We finally describe the index theorem which replaces, for the non
compact leaves L of a measured foliation (V, F), the Atiyah Singer index theorem for compact manifolds,
and is directly in the line of the index theorem for covering spaces due to Atiyah [At,<j] and Singer [Sino].
a) Transverse measures for foliations
To get acquainted with the notion of transverse measure for a folia I ion, we shall first describe it in the simplest
case: dimi** = 1, i.e. when the leaves are one dimensional. The foliation is then given by an arbitrary smooth
1-dirnensional subbuudle F of TV, the iutegrability condition being automatically satisfied. To simplify even
further, we assume that F is oriented, so that the complement of the zero section has two components F+
and F~ = —F+. Then using partitions of unity, one gets the existence of smooth sections of F+, and any
two such vector fields X, X' ? C°°(V, F+) are related by .V = <!>.V where ф ? C°°(V,R+). The leaves are
the orbits of the flow expi.Y. The flows //, = exp(.Y and ll[ = pxp/.V have the same orbits and differ by
a time change:
я.'(р) = "г(«,,.)Ы «eR,fe v.
The dependence in p of this time change T[t,p) makes it clear Mint л measure /i on V which is invariant
under the flow H (i.e. Iltft — /(, V/ 6 R), is not. in general mvarinut under IV (\.ake the simplest case
X = щ on 51). To be more precise let us fust translate I he iiivariance H,/j = /i by a condition involving
the vector field Л' rather than the (low H, = ехр/Л'. hVcall that a de RIihiii current С of dimension q on V
is acontinuous linear form on the complex l.opological vector space (."^(V, Л*?^) of smooth complex-valued
differential forms on V of degree </. In particular a measure /i on V defines a O-cliinensional current by the
equality (l*,u) = Jш dp, Vuj 6 C°°(V). All the usual operations, the Lie derivative d\- with respect to a
vector field, the boundary d, the contraction i\- with a vector field, are extended to currents by duality,
and the equality dx — di x + i xd remains of course true. Now the condition Ittft — /*, Vf ? R is equivalent
to dxli = 0, and since as ц is 0 dimeusioual, its boundary d/i is 0, il is equivalent to d(%xv) — 0. Tliis
condition is obviously not invariant if one changes ,Y into Л'1 = ФХ. ф 6 f?°°(W',H+). However if we replace
X by X' = фХ and ц by ft' = ф~1ц, the current ix'l>' is equal to i\/« and hence is closed, so that ц' is now
invariant for H't = exp((A"').
4-1
So while we do hot have a single measure fi on V invariant under all possible flows defining the foliation, we
can keep track of the invariant measures for each of these flows using the l-dimensio»al current ixn = C.
To reconstruct \i from the current С and the vector field Л", define
(f,f) = (C,w) , Vui eHl'.A1^) , u(.Y) = /.
Given a 1-dimensional current С on V and a vector field -Y 6 C~(V, F+) the above formula will define a
positive invariant measure for Ht = ехр*Л' iff С satisfies the following conditions:
1) С is closed, i.e. dC = 0
2) С is positive in the leaf direction, i.e. и is a smooth 1-forni whose restriction to leaves is positive then
(O)>0.
We could also replace condition 1) by any of the following:
1)' There exists a vector field Л' 6 C°°(V,F+) such that exp/.Y leaves С invariant.
1)" Same as 1)', but for all Л" 6 C™( V, F+).
In fact if Csatisfies 2) then (C,ui) = 0 for any и whose restriction to F is I), thus ixC = 0, VA' 6 C°(V,F+),
and dxC = dixC + ixdC = ixttC is zero iff dC = 0.
From the above discussion we get two equivalent points of view on what the notion of an invariaut measure
should be for the one dimensional foliation F:
a) An equivalence class of pairs (.Y,/i), where Л' 6 C"°(V',/•'+), /i is an exp(.Y invariant measure on V, and
(X,fi) ~ (X',ii<) when Л" = фХ,/1 = V for ф e C'~(KR;).
b) A 1-dimensioual current C, positive in the leaf direction (cf. 2)) and invariant under all (or equivalently
some) flows ехр<Л', Л' е C°°( V, F+).
Before we proceed to describe a) and b) for arbitrary foliations wo relate them to a third point of view c)
that of holonomy invariant transverse measure. A submanifold .V of V is called a transversal if at each
P 6 N, Tf(V) splits as the direct sum of the subspaces T,,(jV) and Fv. Thus the dimension of N is equal
to the codimension of F. Let then p ? N, aud let U, j> ? U, be the domain of a foliation chart. One can
choose U small enough so that the plaques of U correspond bijectively to points of N П U, each plaque of U
meeting N in one and ouly one point.
Starting from a pair (A',I*) as in a), one defines on each transversal N a positive measure as follows: the
conditional measures of ц (restricted to U) on the plaques, are, since fi is invariaut by Hi = exptX,
proportional to the obvious Lebesgue measures determined by ,Y, so the formula Л(В) =lim i ц(Вг),
Be = U Ht(B) makes sense for any Borel subset П of /V
If one replaces (A',yu) by an equivalent pair (,Y',/i') it is obvious that A does not change, since ц' = ф~11г
while A" = фХ. By construction Л is invariant under any of the (lows Ht = expdA' i.e. K(HtB) = A(B),
V< 6 R, and any Borel subset В of a transversal. In fact much more is true:
Lemma 1. Let Nt, N2 be two Iransuersals, Bt a Borel subset of Л',-, i = 1,2 and ф : Bi —> S2, я Borel
bijection such that for each x ? B\, i'(x) is on the leaf of x, Ihcn we have Л( flt) = A(S2).
To prove this, note that if p, e ЛГ, and H,(pi) = p-> ? Л'5 for some / 6 R, then there exists a smooth function
ф defined in a neighborhood of p\ and such that <^(pi) = (, lf^P)U>) e .V2. Thus there exists a sequence фп
of smooth functions defined on open sets of .'Vi and such thai
{(p,<) € /V! x R , H,(jj) e Л'.,) = U Graph ф„.
45

Let then (Р„) be a Borel partition of Д, such that for p 6 Pn one has ф{р) = tf^,(p)(p). It is enough to
show that Л(У>(Р„)) = Л(Р„) for all n. But since on Р„, ф coincides with p ^ Я*„(Р)(р) the result follows
from the invariance of Л under all flows exptY, Y 6 C°°(V, F+).
Thus the transverse measure K(B) depends in a certain sense only on the intersection number of the leaves
L, of the foliation, with the Borel set B. For instance if the current С is carried by a single closed leaf L
the transverse measure Л(В) only depends on the number of points of intersection of L and В and hence is
proportional to й м (В П L)*.
By a Borel transversal В to (V, F) we mean a Borel subset В of V such that В П Lis countable for any leaf
L. If there exists a Borel injection ф of В in a transversal /V with ф(х) e Leaf (x) Vx 6 B, define Л(В)
as К(ф(В)) (which by lemma 1, is independent of the choices of N and ф). Then extend Л to arbitrary
Borel transversals by <r-additivity, (remarking that any Borel transversal is a countable union of the previous
ones).
We thus obtain a transverse measure Л for (V,F) in the following sense:
Definition 2. A transverse measure Л for the foliation (V, F) is a a-additive map В —* Л(В) from Borel
transversals (i.e. Borel sets m V with V П L countable for any leaf L) to [0, +oo] such that
1) /"/У> : fl] -. fl2 is a Borel bijeciion and Ф(х) is on the leaf of r. for any x g Д,, then \(B\) — Л(Вг).
2) Л(А') < oo if A' is a compact subset of a smooth transversal.
We have seen that the points of view a) and b) are equivalent and how to pass from a) to c). Given a
transverse measure Л as in definition 2, we get for any distinguished open set lf(*), a measure fiu on the set
of plaques ж of U, such that for any transversal В С U one has:
MB)
¦I
Card (flfli)
Put (C(/iU>) = J (/,ui) d/i(/(')i where w is a differential form on (Г and J,w is its integral on the plaque ж
off/. Then on U П У the currents C'ti and Су agree so that one gets a current С on V which obviously
satisfies the conditions b), 1) and 2). One thus gets the equivalence between the three points of view a) b)
and c).
Let us now pass to the general notion of transverse measure for foliations. VVe first state how to modify a)
and b) for arbitrary foliations (dimF ф 1). To simplify we assume that, the bundle F is oriented. For a) we
considered in the case dimi** = 1, pairs (Xtft) up to the very simple equivalence relation saying that only
X ®c°°(V) /* matters. In the general case, since F is oriented we can talk of the positive part (Adlm FF)
of Ad"nFF, and, using partitions of unity, construct sections u of this bundle. These will play the role of
the vector field X. Given a smooth section t> 6 C°° (V, Adlmrf) , we have on each leaf L of the foliation
a corresponding volume element. With k = A\mF, it corresponds to the unique t-form ы on L such that
u){v) =¦ 1. In the case dim F = 1, the measure p had to be invariant under the flow Л[ = exp(fX). This occurs
iff in each domain of foliation chart U, the conditional measures of ft on the plaques ofU are proportional
to the measures determined by the volume element v. Thus we shall define the invariance of the pair (b,ju)
in general by this condition. So a) becomes classes of invariant pairs («>,/») where v ? C°° (V, Ad""FF) ,
while {v,n) ~ (»',ц') iff u' = фи and ц = <pft' for some ф ? C^K.R+J. The condition of invariance of ц is
of course local. If it is satisfied and У ? C^tK, F) is a vector field leaving the volunie element invariant,
i.e. dy(v) — 0, then У also leaves ;i invariant. Moreover since Lebesgue measure in R* is characterized
by its invariance under translation, one checks that if dyfi = 0 VV 6 C°°(V, F) with 3y(t>) = 0, then
(v,/*) is invariant. To see this one can choose local coordinates in V transforming v in the &-vector field
v — jfrA ¦ ¦ Л j|i- 6 C°°(V,AF). With tlie above trivialization of i> it is clear that the current ivn one gets
by contracting v with the 0-dimensional current. //. is a closed current, which is locally of the form:
(*) By this we mean that U is the domain of a foliation chart.
46
where ы is a fc-form with support in U, J^ ы is its integral on a plaque ж of U, and pv is the measure on the
set of plaques coming from the disintegration of /i restricted to U with respect to the conditional measures
associated to v.
Clearly С satisfies conditions 1) and 2) of b) and we may also check that dyC = 0, Vy e C°°(F). Thus
for b) in general we take the same object as in the case Is = 1, namely a closed current positive in the leaf
direction, the condition "closed" being equivalent to
dvC = 0 W€CIOC(V,F).
To recover p, given С and », one considers an arbitrary it-form ш on V such that w(») = 1 and puts
(/.^> = (C, /w) V/ € C°°(V). One checks in this way that, a) and b) are equivalent points of view. For
c) one takes exactly the same definition as for Is = 1. Given a transverse measure Л as in definition 2 one
constructs a current exactly as for k = 1. Conversely given a current С satisfying b), one gets for each
domain V of a foliation chart a measure /hi on the set of plaques of U, such that the restriction of С to U
is given by:
Thus one can easily define Л(В) for each Borel transversal fl: if В is contained in U then:
Л(Д) = / Card (Bni)#(i(i|.
One easily checks, as in the case of flows, that Л satisfies definition 2.
/3) The Ruelle-Sullivan cycle and the Eufcr number of a nu'asnrod foliation
By a measured foliation we mean a foliation (V, F) equipped with a transverse measure Л. We assume that
F is oriented and we let С be the current, denning Л in the point of view b). As С is closed, dC = 0, it
defines a cycle [C] 6 Ht(V,R), by looking at. its de Rham hoinology class. The distinction here between
cycles and cocycles is only a question of oriontability. If one assumes that F is transversally oriented then
the current becomes even and it defines a cohomology class (cf. [Rue-S]).
Let now e(F) € Hk(V,R) be the Enler class of the orienicd real vector bundle F on V (cf. [Mi4]). Using
the pairing which makes Hk(VM) the dual of the finite dimensional vector space #i(V,R), we get a scalar
X(F,Л) = (e(F),[C\) 6 R which we shall first interpret in two ways as tlie average Euler characteristic
of the leaves of the measured foliation. First recall that, for an oriented compact manifold M the Euler
characteristic %(M) is given by the well known theorem of 11. Po'mcare and II. Hopf:
pi Zero A"
where X is a smooth vector field on Л/ with only finitely many zeros, while w(.Y,p) is the local degree of X
around p. Given generically p is a non degenerate zero, i.e. in local coordinates, Л" = J2 o' j§7, the matrix
||j (p) is non degenerate and the local degrre is the sign of its determinant.
Also, choosing arbitrarily on M a Riemannian metric, one lias the generalized Gauss Bonnet theorem which
expresses the Euler characteristic as the integral over Л/ of a form il on Л/. equal to the Pfaffian of Bт) К
where К is the curvature form. These two interpretations of the Killer characteristic extend immediately to
the case of measured foliations (V, F, Л).
•17

First, if Л' 6 C°°( V, F) is a generic vector field tangent to the foliation, its set of zeros, T = {p € V, X(p) = 0},
defines asubinanifold of V which is not in general everywhere transverse to the foliation F but is so Л-alrnost
everywhere. Thus Л-almost everywhere the local degree ы(Л',р) = ±1 of the restriction of X to the leaf of p
is well denned, and using the transverse measure we can thus form J g Zero x u>(X, p) dk(p). This scalar is
again independent of the choice of X and equals x(^, Л) = (e(F), [C]>, as is easily seen from the geometric
definition of the Euler class by taking the odd cycle associated to the zeros of a generic section.
Fig.9. Vector field on the generic leaf of a foliation
Second, let || || be a Euclidean structure on the bundle F. Then each leaf L is equipped with a Euclidean
structure on its tangent bundle, i.e. with a Riemannian metric. So the curvature form of each leaf L allows us
to define a form UL, of maximal degree, on L, by taking as above the Pfaffian of Bт)-1 times the curvature
form. Now QL is only defined on L, but one can easily define its integral, using the transverse measure Л,
it is formally:
(In point of view a), tliis integral is ii(<j>) where ф is the function ф(р) = UV(XP), in b) it is (fi', C) where ft'
is any i-Гогт on V whose restriction to each leaf L is UL, and in c) it is the above integral computed using
a covering by domains G, оГ foliation charts and partition of unity ф(, the value of J (JL ф<0) dk{L) being
given by J (Jw ф{п) dh( л-) where t varies on t.he set of plaques of J/,-.)
To show that the above integral is equal to \-( F, A), choose a connection V for the bundle F on V compatible
with the metric, and using its curvature form A', take П' = P/(A'/2t). This gives a closed i-forrn on V,
and by [Mi4] p.311 the Euler class of F is represented by П' in tf*(V,R). Now the restriction of V to leaves
is not necessarily equal to the Riemannian connection. However both are compatible with the metric. It
48
restriction of П' to L. Since uij. is canonical, it is the restriction to L of а к — 1 form ы on V and hence:
¦> = ИЛ.[С]>.
So in fact both interpretations of ,\(F, Л) follow from the general theory of characteristic classes. One is
however missing the third interpretation of the Euler number \(Л/) of a compact manifold:
where the pi are the Betti numbers:
Pi = dim (//•'(Л/.Я)).
The first idea to define the pi in the foliation case is to consider the transverse measure Л as a way of defining
the density of discrete subsets of the generic leaf and thon take Pi as the density of holes of dimension i.
However the simplest examples show that one may very well have a foliation with all leaves diffeomorphic
to R2 while x(^, A) < 0, so that p\ > 0 cannot be defined in the above naive sense. Specifically, let p be
a faithful orthogonal representation of the fundamental group Г of a Hieinann surface S of genus 2. Then
the corresponding principal bundle V on S lias a natural foliation: V is the quotient of 5 x SO(n)(*) by
the action of Г, and the foliation with leaves Sx points is globally invariant under Г and hence drops down
as a foliation on V. The bundle F is the bundle of horizontal vectors for a flat connection on V, and each
fiber p~'{x} is a closed transversal which intersects each leaf in exactly one orbit of Г. So the Ilaar measure
of SO(n) defines a transverse measure Л. Since the transverse, measure of each fiber is one it is clear that
x(F, Л) = —2. Moreover, since the representation /> of Г in $O(n) is faithful, each leaf L of the foliation
is equal to the covering space 5 i.e. is confonnal to the unit disk in C. So each leaf is simply-connected
while "po — P\ + /?:>"< 0. This clearly shows that one cannot obtain 6i by counting in a naive way the
handles of this surface. However though the Poincare disk (i.e. the unit, disk of С as a complex curve) is
simply connected it has plenty of nou zero harmonic 1-forms. For instance if / is a bounded holomorphic
function in the disk D, then the form ы = f(z)dz is harmonic, and its L2 norm /ы Л *ш is finite. Thus the
space Hl(D, C), of square integrahle harmonic 1-forms, is infinite dimensional and p\ will be obtained by
evaluating its "density of dimension", following the original ide-л of Atiyali [ЛЦ], for covering spaces.
Given a compact foliated manifold we can, in many ways, choose a Euclidean metric || || on F. However since
V is compact, two such metrics || ||, || ||' always satisfy an inequality оГ the Гогт С || ||'< || || < С \\ ||'. So
for each leaf L the two Riemanniau structures defined by || || and || ||' will be well related, the identity map
from (L) to (L)' being a quasi-isometry. Letting then Ti be the llilbert space of square integrable 1-forms
on L with respect to || [j (resp. H! with respect to j| ||') the above quasi-isometry determines a bounded
invertible operator T from H to V.'. We let P (resp. P') be the orthogonal projection of ~H (resp. ~H') on the
subspace of harmonic 1-forms. Then P'T (resp. PT~l) is a bounded operator from Я'(?,С) to H'(L',C)
(resp. from ff'(L',C) to //'(L,C). These operators arc inverse of each other since for instance the form
PT~lP'Tu> — ы is harmonic on L, is in the closure of the range of the boundary operator, and hence is
0. (At this point, of course, one has to know precisely the domains of the unbounded operators used, the
compactness of the ambient manifold V shows that each leaf with its quasi isometric structure is complete, in
particular there is no boundary condition needed to define the Laplacian. its minimal and maximal domains
coincide as in [Ats]).
From the above discussion we get that the llilbert space //J(L,C) of square iutcgrable harmonic j-forms on
the leaf L is well defined up to a qiiasi-isonietry. Of course this fixes only its dimension, dimffJ(L,C) 6
{0,1,... ,+oo}. In the above example all leaves wen; equal to tlie Poincare disk D so that for each L:
A\mHl(L,C) = +oo. However in this example \(F, Л) = —2 was finite, which is not compatible with a
definition of/?! as the constant, value +00 of dim(#'(L,C). One can easily see that /7°(L,C) = 0 and
H2{L, C) = 0, thus we should have /J, = 2.
(*) S is the universal covering of S.
49

Now note that the quasi-isometry defined above from 1P(L,C) to IP((L'),C){j = 1) is canonical. This
means that, on the space of leaves V/F of (V\ F), the two "bundles" of Ililbert spaces are isomorphic as
"bundles" and not only fiberwise. The point that there is more information in the "bundle" than in the
individual fibers is well-known. However it is also well-known that in classical measure theory all bundles are
trivial. If (X,ij.) is a Lebesgue measure space and (HT)Tix, (H'Artx are two measurable fields of Hilbert
spaces (cf. [Di2]) with isomorphic fibers (i.e. dim HI = dim ll'x a.e.) then they are isomorphic as bundles.
Since in our example: dim H'(L, C) = +00 VL € V/F, one could think that this bundle of Hilbert spaces is
measurably trivial. In fact it admits no measurable cross-section of norm one. This follows as in section 4
Type I examples, using the ergodicity of the transverse measure Л.
Thus we see that the measurable bundle Я1 (L, С) is not trivial. It is however isomorphic to a much simpler
measurable bundle, which we now describe. Let В be a Borel transversal, then to each leaf L of the foliation
we associate the Hilbert space Hi = C2(LnB) with ort.lionormal basis (e^) canonically parametrized by the
discrete countable subset BClL of the leaf L. (To define the measurable structure of this bundle, note that its
pull-back to V assigns to each x € V the space B(LX Л Д) where Lx is the leaf through x, so given a section
(s(x))l?v of this pull-back, we shall say that it is measurable iff the function (x,y)€ V x flu (s(x),e3) is
measurable.)
Lemma 3. Let B,B' be Borel transversals. Then if Hit two bundles of [Filbert spaces (Hl)lsv/f, Hl =
12(ЫЛВ); (H'L)L(.viF H'L = (!(?П B') are measurably isomornkie, one has A(B) = A(B').
Proof. Let {Ul)l?V/f be a measurable family of unitaries from II[. to II'L. For each z 6 В let Xx be
the probability measure on B' given by A,({»/}) = \{Uc(X]cr,<:,j)\". By construction Xx is carried by the
intersection of the leaf of x with B'. From the existence of this nia|> x >— \r which is obviously measurable
one concludes that A(B) < A{B'). Q.E.D.
From this lemma we get an non-ambiguous definition of the dimension for measurable bundles of Ililbert
spaces of the form Hl = C(in B) by taking:
We can now state the main result of this section. We shall assume that the set of leaves of (V, F) with
non-trivial holonomy is A-negligible. This is not always true and we shall explain in remark 1 how the
statement has to be modified for the general case.
Theorem 4. a) For each j = 0,1, 2,..., dim F, there c::i.ih a llorrl transversal Bj such that the bun-
bundle (№(L,C))L6v/f of the square intcgrahle harmonic form» on L is measurably isomorphic to (<2(in
b) The scalar Cj = \{Bj) is finite, independent of the choice of Bj and of the choice of the Euclidean
structure on F.
c) One has ?(-!)'ft = \(F, Л).
Of course if F = TV so that there is only one leaf, a Bore! transversal В is a finite subset of V, A{B) is its
cardinality, so one recovers the usual relationship between Euler characteristic and Betti numbers. Let us
specialise now to 2-dimensional leaves, i.e. dim F — 2. Then we get /i0 - /ii + j3-, = 57 J Kdh where К is the
intrinsic Gaussian curvature of the leaves. Now 0a is the dimension of the measurable bundle Hr, — {square
integrable harmonic 0-forms on L]. Thus, as harmonic O-fornis are constant, there are two cases:
If L is not compact, one has Hi, = {0}.
If L is compact, one has Hl — C.
Using the * operation as an isomorphism of n°[L, C) with //-'(Л.С) one gets the same result for H2(L,C),
and hence:
50
Corollary 5. ff the set of compact leaves of{V,F) *s A-negligible, then the integral J КdA of the intrinsic
Gaussian curvature of the leaves is < 0.
Proof.
i- J
KJA = l)a - Л + A = -/?, < 0.
Q.E.D.
Remark 6. 1) The above theorem was proven under the hypothesis: "The set of leaves with non-trivial
holonomy is negligible". To state it in general, one has to replace wherever it appears, the generic leaf L by
its holonomy covering L. Thus for instance the measurable bundle I1>(L,C) is replaced by W(L,C), and
f2fLnS) is replaced by P(LnB) where LnB is a shorthand notation for the inverse image of L П В in the
covering space L. One has to be careful at one point, namely the holonomy group of L acts naturally on both
H'(L,C) and 12(LC\B), and the unitary equivalence Ul ¦ H'(L,C) t— B(BnL) is supposed to commute with
the action of^the holonomy group. Then with these precautions the above theorem holds in full generality.
Now unless i is compact one has H°(L,C) = 0. Thus unless L is compact with finite holonomy (which by
the Reeb stability theorem implies that nearby leaves are also compact) one has /?0 = 02 = 0. This of course
strengthens the above corollary; to get J KdA < 0 it is enough that the set of leaves isomorphic to S2 be
Л-negligible. Of course, one may have J h'dA > 0, as occurs for foliations with 52-leaves.
2) Using the analytical proof of the Morse inequalities for manifolds ([Wit]) one can prove their analogue for
foliations ([Co-F]). The main point here is a result of Igusa [Ig] which shows that given a foliation (V, F), one
can always find, a smooth function ф 6 C°°(V) such that the singularities associated to the critical points
of the restriction of ф to the leaves are at most of degree 3. (As above for vector fields X(r) € Fx Vx € V,
it is not possible in general to assume that the restriction of ф to the leaves will be a Morse function, since
this would yield a closed transversal to the foliation.)
3) If T С V is a closed transversal to a foliation F. Then one can perform surgery along the leaves at
the points of Tni. This will not affect the space of leaves or the transverse measure, but will in general
modify the real Betti numbers $. Using this operation for products of Kronccker foliations (T^,/1^.), one
gets examples where the real Betti numbers /};,/= 1 fc/2, к = dim F have given (irrational) preassigned
values.
4) The homotopy invariance of the real Betti numbers /3i has been proved by J. Ileitsch and C. Lazarov
[Hei-La].
7) The index theorem for measured foliations
The above formula J2(—lVft = xC^A), which relates the real Betti numbers of a measured foliation with
the Euler characteristic of F evaluated on the Rnelle-Sullivan cycle, is a special case of a general theorem
which extends to measured foliations the Atiyah Singer index theorems for compact manifolds [At-Sii] and for
covering spaces [Ats] [Sino]. As we have already seen above, a typical feature of the leaves of foliations is that
they fail to be compact even if the ambient manifold V is compact. However, the continuous dimensions of
Murray and von Neumann allow us to measure by a finite positive real number, dinu(KerO), the dimension
of the random Hilbert space (\\erDL)LiX °f L2 solutions of leafwise elliptic differential equation Dl? — 0.
This continuous dimension vanishes iff the solution space KerDt, vanishes for Л-almost all leaves.
More specifically, one starts with a pair of smooth vector bundles F.\, E? on V together with a differential
operator D on V from sections of E\ to sections of En such that:
1) D restricts to leaves, i.e. (D?)r only depends upon the restriction of ? to a neighborhood of x in the leaf
of x (i.e. D only uses partial differentiation in the leaf direction).
2) D is elliptic when restricted to any leaf, i.e. the principal symbol (td{i\?) € Поп^Ец, Е2,х) is invertible
for any?€ F
For each leaf I, let Dl be the restriction of D to L (replace L by the holonomy covering L if L has holonomy).
Then Dl is an ordinary elliptic operator on this manifold, and iLs Л2-кегме1: {(, 6 L~{L,E\),Dl{?) = 0} is
51

formed of smooth sections of Ei on L. As in [At5] one does not have problems of domains for the definition
of Dl as an unbounded operator in V since its minimal and maximal domains coincide. In this discussion
we fix once and for all a 1-density o/ on the leaves. This choice determines L-(L, Ei), i = 1,2 as well as the
formal adjoint D"L of ?>/., which coincides with its Hilbert space adjoint.
The principal symbol (Гц of D gives, as in [At-Sio], an element [o*d] of the /v-theory with compact supports
K'(F') of the total space F" of the real vector bundle F~ over V. Using the Thorn isomorphism in
cohomology as in [At-Si2] yields the Chern character(*) ch(crD) ? H"(V,Q) as an element of the rational
cohomology of the manifold V. We can now state the general result ([Cog]).
Theorem 7. Let (V, F) be a compact foliated manifold (*), D a longitudinal elliptic operator on V, X = V/F
the space of leaves,
a) There exists a Borel transversal В io F such that the bundle (Ker DL)Lex is measurably isomorpkic to
{t?{L П В))ьех, and the scalar \(B) is finite and independent of the choice of B.
b) dimA(Ker(?>)) - dimA(Ker(?>')) = ?(cl,<rDTd(Fc), [С]), г = (-1)*^, к - dim F.
In this formula [C\ is the Ruelle-Sullivan cycle, Td(Fc) is the Todd genus of the complexified bundle F.
Using the flatness of F in the leaf direction together with the orthogonality of С with the ideal of forms
vanishing on the leaves, one can replace Td(Fc) by the Todd genus of TqV.
Let us note that in b) the two sides of the formula are of a very different nature. The left hand side gives
global information about the leaves by measuring the dimension of the space of global L2 solutions. It
depends obviously on the transverse measure Л. The right hand side only depends upon the homology class
[C] € Hk(V, R) (a finite dimensional vector space) of Ihe Ruelle-Siillivaii cycle, 011 the subbundle F of TV
which defined the foliation, and on the symbol of D which is also a local data. In particular, to compute
the right hand side it is not necessary to integrate the bundle F. This makes sense since the conditions on a
current С to correspond to a transverse measure on (V, F) are meaningful without integrating the foliation
(C should be closed and positive on &-forrns w whose restriction to F is positive).
For a more thorough discussion of this theorem and several applications we refer to the book [Mo-S] of C.
Moore and C. Schochet. We shall just illustrate it by the following simple example which shows how the
usual Riemann Roch theorem extends to the non compact case.
Let (V, F) be a compact foliated manifold with 2 dimensional leaves. Let. 11s assume that F is oriented, Then
every Euclidean structure on F determines canonically a complex structure on the leaves of (V,F). This is
analogous to the canonical complex structure of an oriented Hieinann surface. Next, let us assume that the
foliation (V, F) has some leaf of subexponential growth (Appendix A) and let Lj be an associated averaging
sequence, assumed to be regular ([Mo-S]), and Л the associated transverse measure. Then the index theorem
for measured foliations, applied to the longitudinal 0 operator with coefficients in a complex line bundle E,
takes the following form ([Mo-S]):
dimA (Кегдя) — diniA (Кег(с^ь') ) =
Number of zeros of E in Lj Number of poles of E in L, 1
lim ., ,, , . =— Inn - - H— Average Euler Characteristic
j-oo Vol(ij) j-00 Vol(tj) 2
where the zeros and poles of the line bundle E are defined using a classifying map ([Mo-S]), but take the
usual concrete meaning when E is associated to a divisor. More explicitly, consider the case of a foliation
of a 3 manifold V by surfaces. Let {71, 7,;} be a collection of d embedded closed curves in V which are
transverse to F and {nb ..., nrf} non-zero relative integers. This data defines a. complex line bundle E onV
,1
whose restriction to each leaf is the complex line bundle with divisor ^ ",G,- П L). Then the above formula
i-\
becomes ([Mo-S]):
(*) We assume for simplicity that F is oriented.
52
dimA (КегЙЕ) - dimA (Ker (Be)') =
in perfect analogy with the usual Riemann Roch theorem, with the counting of poles and zeros replaced by
the counting of their densities.
Problem. Use the above "Riemann Roch" theorem in conjunction with the method of construction of
measures used in the proof of Szemiredi's theorem [Fu] to obtain results 011 non compact manifolds of
subexponential growth independent of any foliation with transverse measure. Important results in this
direction have been obtained by J. Roe (cf. [Ron]).

Appendix A. Transverse measures ant] averaging sequences
Let (V, F) be a compact foliated manifold. Let us fix ;i Euclidean structure || || on the bundle F, tangent to
the leaves, and endow the leaves with the corresponding Riemannian metric. Let L be a leaf and for x ? L
and r > 0 let B(xtr) be the ball of radius r and center x in the leaf L.
Definition 1. A leaf L has non-exponential growth t/lim infr—oo \ log(Vol(fl(z,r))) = 0.
This condition is independent of the choice of x 6 L and of the Euclidean structure.
For each leaf of non-exponential growth there is a sequence i-j — oo for which the compact subsets Lj =
B(x, rj) satisfy: ^o|;^ V ~* ^- Such a sequence of compact, snbinanifolds with boundary is called an averaging
sequence ([G-P]).
Proposition 2. ([G-P]) Given an averaging sequence Lj the following equality defines a transverse measure
С for(V,F)
(C,u) = lim
1
j-oo Vol(Lj)
for any differential form и on V of degree eqtial to the dimension of F.
This construction of transverse measures is rather general but it. is not true that all transverse measures can
be obtained this way, as one can see using, for instance, the 2 dimensional foliation discussed in section 5 /?).
It shows, however, that any foliation of a compact manifold with some leaf of non exponential growth does
admit a non trivial transverse measure.
Appendix B.' Abstract transverse iueastu'e theory
Let X be a standard Borel space. Then given a standard Borel space У and a Borel map У Д X with
countable fibers (p" {x} countable for any x 6 X), there exists fj a Borel bijection ф of У on the subgraph
{(i,n), 0 < n < F(x)} of the integer valued function F defined by F(x) = card(p~'{i}) on X. In particular
if (yi.Pi)i (У2,Рг) are as above, then the two integer valued-functions Fi(x) = Card(p,rl{i}) coincide iff
there exists a Borel bijection ф : Yt —• Y2 with p2 о ф = p,. Let X be a set, and Yt, У2 be standard Borel
spaces with maps p,- : Yi —> X with countable fibers and such that {(yi,!/j) 6 У, х У;,р,(у,) = Pj(yj)) is
Borel in У, x Yj, i,j — 1,2. Then we say that the "functions" from Л' to the integers defined by (yi.pi)
and (Уг,рг) are the same, iff there exists a Borel bijection ф : Y\ — K3 with рз о ф = pl. By the above, if:
Vi 6 X, Card (pj"!{z}) = Card (pj'fz}) and if the quotient Borel structure on X is standard, then the two
"functions" are the same. However, in general we obtain a more refined notion of integer valued function and
of measure space. Let (V, F) be a foliation with transverse measure Л, we shall now define, using Л, such
a generalized measure on the set X of leaves of (V, F). Each Borel transversal В is a standard Borel space
gifted with a projection р:«бВь (leaf of я) 6 Л' with countable fiber. Clearly for any two transversals
we check the compatibility condition that in Bt x fl2. the set {(i|,i3),ii on the leaf of 62} is Borel.
Definition 1. Let p be a map with countable fiber's from a sUmdanl Horel space Y to the space X of leaves.
We say thatp is Borel iff {(y,x),ye Y , x 6 Icafpdi)} is Доге/ in Y x V.
Equivalently, one could say that the pair (Y,p) is compatible with the pairs associated to Borel transversals.
Given a Borel pair (У,р) we shall define its transverse measure Л(У,р) by observing that if (B, q) is a Borel
transversal with q(B) = X, we can define on /i и У the eqni valence relation coming from the projection to
X and then find a Borel partition У =L) Yn and liorel maps i/1,, : Yn —* В with q о фп = p, фп injective.
oo
Thus Yl A(i/>n(y>0) is an unambiguous definition of A(V,/>).
l
We then obtain probably the most interesting example of the following abstract measure theory:
A measure space (Xt3) is a set .Y together with a collection В of pairs (У, p), Y a standard Borel space, p
a map with countable fibers, from Y to A' with the only axiom:
A pair (Ytp) belongs to Б iff it is compatible with all other pairs of В (i.e. iflf for any (Z, 7) in Б one has
{(y.z),p(y) = q(z)} Borel in Y x Z).
A measure Л on (А', Б) is a map which assigns a real number: A(Y,i>) 6 [0, +00] to any pair (У, p) in Б with
the following axioms:
cr-addiiivity ЛE^(Уп,Р»)) = Yl Л(У„,;)„) (here ?2(У„,р„) is the disjoint union У of the У„ with the obvious
projection p).
Invariance. If ф : У] —' У2 is a Borel bijection with pn о ф = pi then:
Of course the obtained measure theory contains as a .special case the usual measure theory on standard
Borel spaces. It is however much more suitable for spaces like the space of leaves of a foliation, since giving
a transverse measure for the foliation (V, F) is the same as giving a measure (in the above sense) on the
space of leaves, which satisfies the following finiteness condition: Л(/\\ p) < oo for any compact subset К of
a smooth transversal.
The role of the abstract theory of transverse measures ([Cot,]) thus obtained, is made clear by its functorial
property: if A is a Borel map of the leaf space of (V'i, Ft) to the leaf space of (IA, Fj) then Л.(Л) is a "measure"
on V2/F2 for any "measure" Л on Vi/Ft (V/F is the space of leaves of (V, F)).

Appendix C. Non-commutative spaces and set theory
We have seen in appendix В that it is possible to formulate what is a transverse measure for a foliation using
only the space X of its leaves provided we take in account in a crucial manner the following principle: one
only uses measurable maps between spaces. Thus, using tins principle we saw that the notion of an integer
valued function on X becomes automatically more refined and leads directly to transverse measures. In fact,
the space X, viewed as a set, has, if one applies the above principle an effective cardinality which is strictly
larger than the cardinality of the continuum. Indeed, it is easy to construct a Borel injection of [0,1] in X
but in the ergodic case it is impossible to inject X in [0,1] by a measurable map since such maps are almost
everywhere constant.
The impossibility of constructing effectively an injection of Л' into [0, 1] is equivalent to the impossibility of
distinguishing the elements of X from each other by means of a denumerable family of properties Pn each of
which defines a measurable subset of X. The non-commutative sets are thus characterized by the effective
indiscernability of their elements. In connection with the above "measurability" principle, let us reproduce
the following letter of H. Lebesgue to E. Borel. Taken from Ocuvrcs de Jacques Hadamard, © Publications
CNRS, Paris, 1968. Authorized reproduction.
"You ask me my opinion on the Note of Mr. Zermelo (Malk. Ann., v. 59), on the objections that you have
made to him (Math. Ann., v. 60) and on the letter of J. lladamard that you communicated to me; here it
is. Forgive me for being so lengthy, for I have tried to be clear.
First of all, I am in agreement with you in this: Mr. Zermelo has very ingeniously proved that one knows
how to solve Problem A,
A. Put a set M into well-ordered form,
whenever one knows how to solve Problem B:
B. Make correspond to each set M' fanned of elements of M a particular element m' o/M'.
Unfortunately, Problem В is not. easy to solve, it seems, except, for the sets one knows how to well-order;
consequently, one does not have a general solution of Problem A.
I strongly doubt that a general solution of this problem can he given, at least if one accepts, along with
Mr. Cantor, that to define a set M is to name a property P belonging to certain elements of a previously
defined set N and characterizing, by definition, the elements of M. In effect, with this definition, nothing is
known about the elements of M other than this: they possess all of the unit!own properties of the elements
of N and they are the only ones that have the viiituown property P. Nothing in this permits distinguishing
between two elements of M, still less classifying them as one would have to do in order to solve A.
This objection, made a priori to every attempt at solving Л, obviously falls if one specializes N or P; the
objection falls, for example, if N is the set of numbers. All that one can hope to do in general is to indicate
problems, such as B, whose solution would imply that of A and which are possible in certain special but
frequently encountered cases. Whence the interest, in my opinion, of Mr. Zcrmelo's reasoning.
I believe that Mr. Hadamard is more faithful than you to Mr. Zermelo's thinking, in interpreting that
author's Note as an attempt, not at the effective solution of A, but at proving the existence of a solution.
The question comes down to this, which is hardly new: can one prove the existence of a mathematical entity
without defining if.
This is obviously a matter of convention; however, I believe that one can build solidly only by accepting
that one can prove the existence of an entity only by defining it. From this point of view, close to that of
Kronecker and of Mr. Drach, there is no distinction to be made between A and the problem C:
C. Can every set be well-ordered?
I would have nothing more to say if the convention I indicated were universally accepted; however, I must
confess that one often uses, and that I myself have often used, the. word existence in other senses. For example,
when one interprets a well-known argument of Mr. Cantor by saying that there exists a nondenumerable
56
infinity of numbers, one nevertheless does not give the means of naming such an infinity. One merely shows,
as you have said before me, that whenever one has a denumeraMe infinity of numbers, one can define a
number not belonging to this infinity. (The word define always has the meaning of naming a characteristic
property of what is defined.) An existence of this nature may be used in an argument, and in the following
way: a property is true if its denial leads to admitting that one can arrange all of the numbers into a
denumerable sequence. I believe that it can only intervene in this way.
Mr. Zermelo makes use of the existence of a correspondence between the subsets of M and certain of their
elements. You see, even if the existence of such correspondences were not in doubt, because of the manner
in which this existence had been proved, it would not be evident that one had the right to use this existence
in the way that Mr. Zermelo does.
I now come to the argument that you state as follows: "It, is possible, in a particular set M', to choose
ad libitum the distinguished element m'; it being possible to make this choice for each of the sets M', it is
possible to make it for the set of these sets"; and from which the existence of the correspondences appears
to result.
First of all, M' being given, is it obvious that one can choose in'? This would be obvious if M' existed
in the nearly Kroneckerian sense I mentioned, since to say that M' exists would then be to affirm that
one knows how to name certain of its elements. But let us extend the meaning of the word exists. The
set Г of correspondences between the subsets M' and the distinguished elements ro' certainly exists for
Messrs. Hadamard and Zermelo; the latter even represents the number of its elements by a transfinite
product. Nevertheless, does one know how to choose an element of Г? Obviously not, since this would yield
a definite solution of В for M.
It is true that I use the word choose in the sense of naming and that it would perhaps suffice for Mr. Zermelo's
argument that to choose mean to think of. But it would still be necessary to observe that it is not indicated
which one is being thought of, and that it is nevertheless necessary to Mr. Zermelo's argument that one
think of a definite correspondence thai is always the same. Mr. Iladamard believes, it seems to me, that it is
not necessary to prove that one can determine an element (and one only); this is, in my opinion, the source
of the differences in appreciation.
To give you a better feeling for the difficulty that I see, 1 remind you that in my thesis I proved the
existence (in the non-Kroneckerian sense and perhaps difficult to make precise) of measurable sets that are
not measurable B, but it remains doubtful to me that anyone can ever name one. Under these circumstances,
would I have had the right to base an argument on this hypothesis: suppose chosen a measurable set thai is
not measurable B, when I doubted that anyone could ever наше one?
Thus, I already see a difficulty in this "in a definite M', I can choose a. definite m'", since there exist sets
(the set С for example, which one could regard as a set M' coming from a more general set) in which it is
perhaps impossible to choose an element. There is then the difficulty that you note relative to the infinity of
choices, the result of which is that, if one wants to regard the argument, of Mr. Zermelo as entirely general, it
must be admitted that one speaks of an infinity of choices, perhaps an infinity of very large power; moreover,
one gives neither the law of this infinity, nor the law of one of the choices; one does not know if it is possible
to name a law defining a set of choices having the power of the set of the M'; one does not know whether it
is possible, given an M', to name an m'.
In summary, when I examine closely the argument of Mr. Zermelo, as indeed with several general arguments
on sets, I find it insufficiently Kroneckerian to attribute a meaning to it (only as an existence theorem for C,
of course).
You refer to the argument: "To well-Older a set, it suffices to choose an element of it, then another, etc." It
is certain that this argument presents enormous difficulties, even greater than, at least in appearance, that
of Mr. Zermelo; and I am tempted to believe, along with Mr. Iladamard, that there is progress in having
replaced an infinity of successive and mutually dependent choices by an infinity, unordered, of independent
choices. Perhaps this is nothing more than an illusion and the apparent simplification comes only from the
fact that one must replace an ordered infinity of choices by an unordered infinity, but of much larger power.
So that the fact that one can reduce to a single difficulty, placed at the beginning of Mr. Zermelo's argument,
57

all of the difficulties of the simplistic argument that you cite, perhaps proves simply that this single difficulty
is very great. In any case, it does not seem to me to disappear because it involves an unordered set of
independent choices. For example, if I believe in the existence of functions y{x) such that, given any z,
у is never bound to x by an algebraic equation with integer coefficients, it is because I believe, along with
Mr. Hadamard, that it is possible to construct one; but, for me, it is not the immediate consequence of the
existence, given any x, of numbers у that are not bound to x by any equation with integer coefficients.1
I am fully in agreement with Mr. Hadamard when he declares that the difficulty in speaking of an infinity
of choices without giving the law for them is equally grave whether it is a matter of a denumerable infinity
or not. When one says, as in the argument that you criticize, "it being possible to make this choice for each
of the sets M', it is possible to make it for the set of these sets", one hasn't said anything if one does not
explain the terms employed. To make a choice could be the writing or the naming of the chosen element;
to make an infinity of choices cannot be the writing or naming of the chosen elements one by one; life is
too short. Thus one has to say what it means to do it. By this is meant, in general, giving the law that
defines the chosen elements, but this law is for me, as for Mr. Hadamard, equally indispensable whether it
is a matter of a denumerable infinity or not.
Perhaps, however, I am again in agreement with you on this point, because, if I do not establish theoretical
distinctions between the two infinities, from a practical point of view I make a great distinction between
them. When I hear someone speak of a law defining a transfuiite infinity of choices, I am very suspicious,
because I have never yet seen any such laws, whereas I do know of laws defining a denumerable infinity of
choices. But perhaps this is only a matter of habit and, on reflection, [ sometimes see difficulties equally
grave, in my opinion, in arguments in which only a denumerable infinity of choices occur, as in arguments
where there is a transfinity. For example, if I do not regard as established by the classical argument that
every set of power greater than the denumerable contains a set whose power is that of the set of transfinite
numbers of Mr. Cantor's class II, I attribute no greater value to the method by which one proves that a
non-finite set contains a denumerable set. Although [ strongly doubt that one ever names a set that is
neither finite nor infinite, the impossibility of such a set does not seem to me to be proved. But I have
already spoken to you of these questions."
' In correcting proofs, 1 add that in fact the Argument, whereby one usually legitimises the statement A
of Mr. Hadamard (p. 262), legitimizes at the same time the statement B. And, in my opinion, it is because
it legitimizes В that it legitimizes A. {Translator's note: The footnote is Lebesgue's. Here A and В refer to
the statements in Hadamard's letter to Borel, reproduced on pp. 335-337 of the previously cited Collected
Works of Hadamard. Lebesgue's letter is the third of Five letters on the theory of sets (between Hadamard,
Borel, Lebesgue and Baire), originally published together in the "Correspondence" section of the Bulletin of
the French Mathematical Society [Bull. Soc. Math. France 33 A905), 261-273].}
58
CHAPTER II
TOPOLOGY AND Л'-THEORY
In this chapter we shall extend topology beyond its classical set theoretic framework in order to understand
from the topological point of view the following spaces which are ill behaved from the set theoretic side:
1) The space X of Penrose tilings of the plane.
2) The dual space Г of a discrete group Г.
3) The orbit space of a group action on a manifold.
4) The leaf space of a foliation.
5) The dual space G of a Lie group G.
One could base this extension of topology on the notion of topos due to Grol.hendieck. Our aim however
is to establish contact with the powerful tools of functional analysis such as posilivity and Hilbert space
techniques, and with К theory. The first general principle at work is that to any of the above spaces X
there corresponds in a natural manner a C'-algebra which plays the role of the involutive algebra С(Л') of
continuous functions on X. To the non-classical nature of such spaces corresponds the vov commutativity of
the associated C-algebra. When it happens that the space X is well behaved classically then its associated
C-algebra is equivalent (in the sense of strong Morita equivalence, cf. Appendix A) to a commutative
C'-algebra: C(X).
The second general principle is that for the above spaces the A' theory of I.lie associated C-algebra "С(Л')"
is the natural receptacle for invariants of families of ordinary spaces (Ys)x^x parametrized by Л". Thus, for
instance, the signature Sign(i) of the generic leaf of a foliated compact manifold (V', F) is a specific element:
Sign(t) 6 /v'(.Y)
of the К theory group of the C-algebra of the foliation. Moreover this clement is an invariant of leafwise
homotopy equivalence. This type of result extends the result of Miscenko on the homotopy invariance of
the Г-equivariant signature of covering spaces ([Mis2]) which corresponds to example 2. There are several
reasons, reviewed in section 1, which make the К theory of C*-algebras both relevant and tractable. In fact,
computing the К groups of the C-algebra associated to the above spaces (of examples 1 to 5) amounts to
classify the stable isomorphism classes of virtual "vector bundles" over such spaces and as sucli, is a natural
prerequisite for using them as geometric spaces.
The third principle at work in all the above examples is the construction of a geometric group, noted K,(BX),
easily computable by standard techniques of algebraic topology, and of a map /i, the analytic assembly map,
from the geometric group to the К group K(X) of the C-algebra associnted to A":
/i : K.(BX) -^ K(X).
59

The general conjecture ([Bau-Ci]) that this map /i is an isomorphism is a guiding principle of great relevance.
The notations BX and K.(BX) will be explained later, but essentially BX stands for the komotopy type
of an ordinary space which fibers over X with contractible fibers. Thus for instance if (V,F) is a compact
foliated manifold with contractible leaves, then BX is homotopic to V, since V fibers over the leaf space X
with contractible fibers: the leaves of the foliation. Also, in example 2 if Г is torsion free then BX is the
usual classifying space ВГ of the discrete group Г. In this last example A', is A' homology and the map /i
is the assembly map of Miscenko and Kasparov.
The bivariant К К theory of Kasparov [Kasi] plays a crucial role in the construction of the map ft and in the
computation of К theory of C*-algebras. We shall use a variant of that theory, the E theory or deformation
theory which is both easier to develop (appendix B) and more appropriate for К theory maps since unlike
KK it is half exact in its two arguments. However we shall come back to К К later in Chapter 4 and use
explicitely the more precise geometric significance of the KK cycles. In this chapter we shall construct our
К theory maps from deformations of C* algebras (section 6). In particular we shall get the Atiyah Singer
index theorem as a corollary of the Thorn isomorphism theorem (section 5). The content of this chapter,
mainly based on examples, is the following:
1. C*-algebras and their К theory.
2. Elementary examples of quotient spaces.
3. The space of Penrose tilings.
4. The dual space of discrete groups.
5. The tangent groupoid of a manifold.
6. Wrong way functoriality as a deformation.
7. The orbit space of a group action.
8. The leaf space of a foliation.
9. The longitudinal index theorem.
10. The analytic assembly map and Lie groups.
Appendices.
A. C* modules and strong Morita equivalence of C" algebras.
B. Е theory and deformations.
C. Crossed products of C'-algebras.
D. Penrose tilings.
60
II.1. C*-algebras and their A'-theory
Given an ordinary compact (or locally compact) space У, Urysohn's lemma [Rn] shows that there exist
sufficiently many continuous functions / 6 C(Y) on У to determine uniquely the topology of Y. This is true
for the real-valued functions and a fortiori for the complex-valued ones. Thus, the topology of У determines
and is determined by the algebra A — C(Y) of continuous complex-valued functions on У, equipped with
the involution / -> /', where f"(y) = f(y) (Vy 6 У). (The analogous statement would be false if R or С
were replaced by a totally disconnected local field.)
Moreover, the class of involutive C-algebras A obtained in this way can be characterized very simply: they are
the commutative C*-algebras (with unit). General C*-algebras have a very simple axiomatic characterization
that expresses exactly what we expect from functions of class C°. The theory of C*-algebras begins in 1943
with a paper of Gel'fahd and Naimark [G-N]. (See also M.H. Stone [Sto].)
Definition 1. А С-algebra is a Banack algebra over С with a conjugate-linear involution x —» x* such that
(xy)" - y'x' and
for z,y ? A.
A C-algebra A is an involutive Banach algebra for the norm x
involutive algebra structure by the spectral radius of x*x:
x || uniquely determined from the
|| x ||2=spectral radius (x'x)
= sup{|A|,i*x — A not invertible}.
For an involutive algebra A to be a C'-algebra, it is necessary and sufficient that it. admit a ^representation
топ a Hilbert space, such that
1)ж(х) = 0 => x = 0;
is norm closed.
D)
The theorem of Gelfand, which is based on complex analysis and the structure of the maximal ideals of A,
shows that if A is a commutative C*-algebra then there exists a compact topological space Y = Sj>(A) such
that A =C(Y).
Let A be a commutative C*-algebra and suppose that A has a unit. The set Sp/1 of homomorphisms x of
A into С such that x(l) = ', equipped with the topology of pointwise convergence on A, is compact; the
compact space Sp^4 is called the spectrum of A.
Theorem 2. Let A be a commutative C'-algebra with unit and let X = Sp.l be its spectrum. The Gel'fand
transformation
x e A^the function x(x) - x{*) [X
is an isomorphism of A onto the C* -algebra O(X) of continuous complex functions on Л".
Thus, the contravariant functor С that associates to every compact space .V the C"-algebra C(X) effects
an equivalence between the category of compact, spaces with continuous mappings, and the opposite of the
category of commutative unital C*-algebras with unit-preserving homomorphisnis. To a continuous mapping
/:Х^У there corresponds the homomorphism C(f) : C(Y) —> C(X) that sends h 6 C(Y') to fto/ 6 C(X).
In particular, two commutative C*-algebras are isomorphic if and only if their spectra are homeoniorphic.
The basis for non commutative geometry is the possibility of adapting most of the classical tools, such as
Radon measures, A'-theory, cohomology, etc., necessary for the study of a compact, space У, to the case
where the C-algebra A = C(Y) is replaced by a non commutative C*-algcbra. In particular, the general
61

theory is not limited to the spaces X given in examples 1) to 5) above. It is however important to note that
to a general C*-algebra A there corresponds a space X namely the space of equivalence classes of irreducible
representations ж of A on Hilbert space. One can still prove the existence of sufficiently many irreducible
representations of A, and when A is simple, i.e. has no nontrivial two-sided ideals, the natural topology
of the space X has the same triviality property as that of the set A' of Penrose tilings encountered below
(section 3). Moreover exactly as one can find any finite portion of a given tiling in any other tiling, two
irreducible representations Xl and ж2 of A can, by means of a unitary transformation, be made as similar to
each other as one likes. This follows from [K] and [Уог].
The cohomology theory for compact spaces X which is the easiest to extend to the non commutative case is
the АГ-theory K(X).
The group K°{X) associated to a compact space X has a very simple description in terms of the locally
trivial, finite-dimensional complex vector bundles E over X. To such a bundle E there corresponds an
element [E\ of K°(X) that depends only on the stable isomorphism class of E, where Ei and ?2 at« s&id
to be stably isomorphic if there exists a vector bundle F such that the direct sum ?, ф F is isomorphic
to Ei © F. Moreover, the classes [E] generate the group A'°(X) and the relations [Ei ф Вз] = [E,] + [E2]
constitute a presentation of this group.
One obtains in this way a cohomology theory, the topological A'-theory (.lint is both simple to define and to
calculate, thanks to the Bott periodicity theorem (cf. below).
More precisely, the functor A" is a contravariant functor from the category of compact topological spaces and
continuous mappings in the category of Z/2 graded abelian groups: Л" = А ф Л1. The group К*(Х) is
obtained as the group ir0(G?oo) of connected components of the group G?oo = Un>i Gtn(/1), the increasing
union of the groups GLn(A) of invertible elements of the algebra Mn(A) — Mn(C(X)) of n x n matrices with
elements in the algebra A = C{X). The inclusion mapping of GLn into GL,,+ \ is given by g —» j? . It
is because GLn(A) is an open set in Mn(A) (a general property of Banach algebras A, commutative or not)
that the group A-1(X) = ir0(G?oo) is denumerable (as soon as A is separable as a Banach space) and can
be calculated by the methods of differential topology.
The fundamental tool of this theory is Bott's periodicity theorem, which remains valid for a not necessarily
commutative Banach algebra A [Woo]. It can be stated as the periodicity of period two ^(Gioo) —
irfr+2(G?oo) of the homotopy groups, but this periodicity has as most important corollary the existence of a
short exact sequence for the functor A", valid for every closed set У С A':
K°(X\Y) -. A'°(X) — Л'°(У)
T 1 (*)
Л''(У) <- A"(A') - Л''(А'\У)
One defines the group A"' for a locally compact space such as Z = А'\У, as the kernel of the restriction
IC(Z) -> A''({oo}), where Z = Z\J {oo} is the one point compactification of Z.
It follows from the Bott periodicity theorem that the group A'°(A') is the fundamental group ir\
), where A = С(Л')
(Gtoo()), ()
All of the features of the topology of compact spaces we have just described adapt remarkably well to
the noncommutative case where the algebra C(X) of continuous complex functions on Л' is replaced by a
not necessarily commutative C*-algebra A. As for A'-theory, the definitions of the group A'1, in the form
Л',(Л) = 'k+i+i(GLCK,(A)) where k 6 N is an arbitrary integer, and the Bott periodicity theorem are
unchanged. The exact sequence (*) remains valid for every closed two-sided ideal J of A in the form
(*)'
A'o(J) -» К „(A) -» Ko{B)
T I
A-,(B) - K,(A) - A',(J)
where В = A/J is the quotient C*-algebra of A by J. Also as above, if A is non miital one defines Hj(A) as
the kernel of the augmentation map: Kj(A) —*¦ A'j(C) where A is obtained from A by adjoining a unit, i.e.
62
A = {e + Al ; a 6 A , A 6 C} , e(a + Al) = A 6 С
Moreover, the description of A'0 in terms of vector bundles is carried out in terms of projective modules
of finite type over the algebra A. Indeed, by a theorem of Serre and Swan [Ser3] [Sw] the locally trivial
finite dimensional complex vector bundles E over a compact space ,Y correspond canonically to the finite
projective modules over A = C{X). To the vector bundle E one associates the С(Л') module ? = C(X,E)
of continuous sections of E. Conversely, if ? is a finite projective module over A = C(.Y), the fiber of the
associated vector bundle E at a point p 6 X is the tensor product
where A acts on С by the character x, x(f) = f(p) V/ 6 С(Л').
The direct sum of vector bundles corresponds to the direct sum of the associated modules. Isomorphism
and stable isomorphism thus keep a meaning in general and for a unital C*-algebra A the group Ko(A) is
the group of stable isomorphism classes of finite projective modules over A (or more pedantically, of formal
differences of such classes). If ? is a finite projective module over A there exists a projection (or itlempotent)
e = (', e { М„(А) and an isomorphism oi ? with the right A module e.4" = {(?;);= 1 n; ?« 6 A, e? = {}.
For non unital C*-algebras the group A'o is the reduced group of the C'*-algebra A obtained by adjoining a
unit.
One important feature of the A'-theory groups A'o, A'i of a C-algebra is that if .4 is separable as a Banach
space (which in the commutative case A = С(Л') is equivalent to Л' mctrisable), then these A' groups are
countable abelian groups. This feature is shared by the A'-theory of Banach algebras but not by /\*-f.heory of
general algebras. (One can for instance show that the group A'o of the convolution algebra C~(R) of smooth
compactly supported functions on R is an abelian group with the cardinality of (.lit; continuum cf. section
10.)
There is one crucial feature of the A'-theory of C*-algebras which differs notably from the case of general
Banach algebras. It was discovered in connection with the Novikov conjecture [Mis3]. Tlie point is that for
C"*-algebras one has a canonical isomorphism:
Л'о(Л) ~ Witt(^)
where the Witt group Witt(^4) classifies the quadratic forms Q over finite projective modules over A. More
precisely let us first recall the precise definition of the Witt, group of an involutive algebra A over C. Given
a finite projective (right) module ? over A a hermiiian form Q over ? is a sestjuilincar form ? x ? —*¦ A such
that:
1) Q(?a, t,b) = a'Q(t,4)b V?, П 6 ? , Va, Ь е A.
2)О(ч,«) = О«,чГ 4t,iее-
то a hermitian form Q on ? corresponds a linear map Q of ? into ?' = Попы (?,/)), given by
The hermitian form is called invertible when Q is invertible. There are obvious notions of isomorphism and
direct sum of hermitian forms. The Witt group Witt(^4) is the group generated by isomorphism classes [Q]
of invertible hermitian forms Q on arbitrary finite projective modules over A with the relations:
<*)[Qi®Qj] = [Qi] + [Q2]
0) [Q] + l-Q] = 0.
When A is a C*-algebra (with unit), there exists on any given finite projective module ? over A an invertible
hermitian form Q satisfying the following positivity condition:
63

(cf. chapter 5 for the notion of positive elements in aC* algebra). Moreover, given ? all the positive invertible
hermitian forms on ? are pairwise isomorphic ([Mis3]) thus yielding a well defined map <p : Ko(A) —» Witt(/1)
such that ip(?) is the class of (?,Q) with Q positive. This map is actually an isomorphism ([Mis3]) and this
property uses in an essential manner the fact that the involutive Banach algebra A is a C*-algebra, and in '
particular that:
T = T* , T e A =s> Spectrum(T) С R.
(In other words if T is a selfadjoint element of A and A 6 С is not real then T — A is invertible in A.) This
property, as well as the isomorphism Kq(A) ~ Witt(^4) fails for general involutive Banach algebras, but one
should expect that for any involutive Banach algebra В one has:
Witt(S) = Witt(C" B)
where C*B is the envelopping C*-algebra of B, i.e. the completion of В for tlio seiuinorm: || x ||c>= Sup{||
ж(х) || , ж unitary representation of В in a Hilbert space}.
This equality has been proved by J.-B. Bast for arbitrary commutative involutive Banach algebras [Boss].
64
II.2. Elementary examples of quotient spaces
We shall formulate the algebraic counterpart of the geometric operation of forming the quotient of a space
by an equivalence relation, and show how to handle non-IIausdorfT quotient» by using non commutative
C*-algebras.
Let Y = {a, 6} be a set consisting of two elements a and b, so that the algebra C(Y) of (complex valued)
functions on Y is the commutative algebra СфС of 2 x 2 diagonal matrices. There are two ways of declaring
that the two points a, 4 of У are identical, i.e. of quotienting У by the equivalence relation: a ~ 6.
1) The first is to consider the snbalgebra А С C(Y) of functions / on У which take the same value at a and
4: f(a) = f(b).
2) The second is to consider the larger algebra В D C(Y) of all 2 x 2 matrices:
\fha
The relation between the two algebras is the notion of strong Morita equivalence of C""-algebras, due to M.
Rieffel [Ri2] cf. Appendix A. This relation preserves many invariants of C*-algrbras, such as A'-theory and
the topology of the space of irreducible representations. Thus, in our example the pure states wa and ш, of
В which come directly from the points a, 4 of У by the formula:
yield equivalent irreducible representations of В = A'/afC), which corresponds to the identification a ~ b in
the spectrum of B. We shall now work out many examples of the above algebraic operation of quotient,
first (example a)) to give an alternate description of existing spaces, but mainly to give precise top о logical
meaning to non-existing quotient spaces for which the first way above yields a trivial result while the second
yields a non trivial non commutative C*-algebra.
a) Open covers of manifolds.
Let X be a compact manifold obtained by pasting together some open pieces Ut С Л" of Euclidean space
along their intersections {/, П Uj. To obtain Л' from the Euclidean open set. V = Ц Vj, disjoint union of the
Uj 's, one must identify the pairs {z, -') of elements of V where the pasting tak'es place. To be more precise,
there is an equivalence relation 1Z on V generated by such pairs, and for z% z' ? V one has z — г'G2.) iff
p(z) = p(z') where p : V —> X is the obvious surjection. Since Л' is compact, we shall assume that the above
covering is finite. To this description of X we associate the following C*-aJgcbra:
г, г1 6 V
Fig. 1. One identifies г with z' using the algebra of matrices
Proposition 1. Let 7J. 4e the graph: 7J. = {(г, ;') E V x V; z ~ z'} of the equivalence relation endowed with
its locally compact topology. The following algebraic operations (tint the vector space CoGv) of continuous
functions vanishing at infinity onii. into а С-algebra, C*Gv.) which is niroagly Morita equivalent to С(Л'):
65

(/•)(-,-')=/(-'.--)¦
Indeed, by construction C'(Ti) is identical with the C*-algebra of compact endomorphisms of the continuous
field of Hilbert spaces (НТ)Т?\- over Л' with 4X = €2(p*)) Vl ^ X. (cf. Appendix A).
The locally compact space 7J is the disjoint union of the open sets 7J.jj = {(г, г') 6 У. X t/jj р(г) = р(г')} —
(У; П Vj. Thus a function / 6 CoGJ.) can be viewed as a matrix (Д,) of functions where each fa belongs to
Co(Ui П Uj), while the algebraic rules are just the matrix rules. Since (Ui) is an open covering of X one can
easily construct an idempotent e 6 C""GJ.), e = e" = e2, e = (ey) which at each point i 6 X gives a matrix
(e;j(i)) of rank one, thus exhibiting the strong Morita equivalence:
С-(Я-) == С(Л').
The C-algebra C*GZ) is net commutative but is strongly Morita equivalent to a commutative C*-algebra.
The latter algebra is uniquely determined since strong Morita equivalence between two commutative C-
algebras is the same as isomorphism.
In trading С(Л') against C"*GJ.) we lose the commutativity, but we keep the same spectrum, i.e. the topo-
logical space of irreducible representations (canonically isoniorphic to Л"), the same topological invariants
such as A* -theory. Л*(С*G?.)) = Л'(Л'). But the main gain is that we do not use anywhere the topology of
the quotient space X in the construction of CGI). The crucial ingredients were the topology of V. and its
composition law: (г, ;')»(;',;") = (z,:").
As we shall see later, these ingredients still exist in situations where the quotient topology of X is the coarse
topology and hence is useless.
/?) The dual of the infinite dihedral group Г = lx/
Let us now describe another example in which the two above algebraic operations of quotient 1) and 2) yield
obviously different (non strongly Mori La equivalent) algebras. Thus, let Y = 1\ U h be the disjoint union of
two copies I\ and U of the interval [0, I] and on У let 7J. be the equivalence relation which identifies (s, i)
with (s, j) for any i,j 6 {1,2} provided s 6 ]0,1[. In other words the quotient space X = Y/V. is obtained
by gluing the two interiors of the intervals /[ and U but not the end points. If we apply the operation 1) we
get a subalgebra of C(l\ U /j) and since two continuous functions / 6 C[0,1] which agree on a dense (open)
set are equal, we see that this subalgebra is O([0,1]). In particular it is homotopic (Appendix A definition
10) to С and its A' theory group is A'o = 1.
Fig.2. The dual of the infinite dihedral group.
The points inside the intervals are identified, but not the endpoints
If we apply the operation 2) we get the C" subalgebra of the C"-algebra М2(С([0,1])) = Af2(C) ® C([0,1])
given by:
A = {(J:(())i6[o.i] 6 M2(C([0,l])) ; j:@) and x(\) are diagonal matrices} .
66
(We viewed a generic element x 6 Mo(C([0,1])) as a continuous map * i— r{l) 6 Д A>(C), t 6 [0,1]).
Obviously the space of irreducible representations of A is the space Л' = Y/Tl with its nou II ausdorfT topology.
The К theory of this C"-algebra A is much less trivial than that of C'[0,1]. Indeed from the exact sequence
of C*-algebras given by evaluation at the end points
where J = M2(C) ® Cb(]0,1[), one gets using the six terms exact sequence of A' groups (section 1) that
Ko(A) is isomorphic to Z3.
The above example shows clearly that the two operations 1) and 2) do not in general yield equivalent results,
and 2) is obviously much finer.
67

II.3. The space X of Peuvose tilings
Fig.3- Penrose tiling
I had the good fortune to attend a lecture by R. Penrose on quasiperiodic tilings of the plane (Fig.3).
Penrose has constructed such tilings using two simple tiles A and B. The tiling T depends on successive
choices made in the course of the construction. More precisely, to construct such a tiling one must choose a
sequence (zn)n=o i ?. of O's and l's that, satisfies the following coherence rule:
--„ = 1 => --„+, = 0. A)
We use the value of zn as indicated in the construction of T in Appendix D. However, it can happen that
two different sequences ; = (г,,) and -' = (;'n) lead to the same tiling; in fact,
T is identical to T О Зн such that. Zj = z'j (Vj > n). B)
Thus, the space Л' of tilings obtained is the quotient set K/Ti. of a compact space К by an equivalence
relation 7J. The compact space A", honieomorphic to the Cantor set, is the set of sequences г = (г„)пем of
O's and l's that satisfy the rule A); it. is by construction a closed subset of the product of an infinite number
of two-element sets. The equivalence relation 7v. is given by
68
г~г'»3п such that z, - z'j (Vj > n). C)
Thus X is the quotient space K/V.. It is exactly in this form that Penrose presented his set of tilings in
his exposition: If one tries to understand the space X as an ordinary space, one sees very quickly that the
classical tools do not work and do not distinguish A' from the space consisting of a single point. For example,
given two tilings T\ and Тг and any finite portion P of 7\, one can find exactly the same configuration P
occuring in Тг, thus no finite portion enables one to distinguish Ti from Tj ([Gru-S]). Thus any configuration
which does occur in some tiling T such as those of fig. 3 will occur (and infinitely many times) ill any other
tiling T. This geometric property translates into the triviality of the topology of Л'. The natural topology
on X has for closed sets F С X the closed sets of К that are saturated for 7J. But, every equivalence class
for "R is dense in К and it follows that the only closed sets F С X are F — 0 and F = X. Thus the topology
of X is trivial and does not distinguish X from a point; it is, of course, not Ilausdorff and it contains no
interesting information.
One possible attitude toward such an example would be to say that, up to fluctuations, there is only one
tiling T of the plane and to not be disturbed by the distinction between A' and a point. However, we shall
see that X is a very interesting 'noncommutative' space or 'quantum' space, and that one of its topological
invariants, the dimension group, is the subgroup of R generated by Z and the golden number '+/^. This
will show, in particular, why the density or frequency of appearance of a motif in the tiling must, exclusively
on account of the topology of the set of all tilings, be an element of the group Z -f f 1-t/^ J Z.
Let us now give the construction of the C-algebra A of the space X = Л"/ft of Penrose tilings (compare
with [Cu-K]). A general element a 6 A is given by a matrix (аг„.<) of complex numbers, indexed by the pairs
(г, г') 6 72- The product of two elements of A is given by the matrix product, (аб)-.г" — ?3^ я*,.г'&л',г".
To each element x of X there corresponds an equivalence class for 7J, which is a dennmerahle subset of A*.
One can therefore associate to x the Ililbert space 1\ having this denumerable set for an orthonormal basis.
Every element a of A defines an operator on l\ by the equation
For а 6 A the norm | a(x) \\ of the operator a(x) is finite and does not depend on x. 6 Л'; it is the C-
algebra norm. Of course, to get more technical, one must give a precise definition of the class of matrices
a that we are considering. Let us do it. The relation TZ С К х К is the increasing union of the relations
7Jn = {(г, z')\zj = z'j (Vj > n)}, and as such 7J. inherits a natural locally compact topology (not equal to
its topology of subset of К х A'). Then A is by definition the norm closure of СД7?.) for the above norm.
One can show that every element a 6 A of this norm closure comes from » matrix (oSi.i), (:,-') 6 7J..
We can summarize the above construction by saying that, while the space Л" cannot be described nontrivially
by means of functions with values in C, there exists a very rich class of operator-valued functions on X:
The algebraic structure of A is dictated by this point of view, because one has
(Xa + цЬ)(х) =Xa(x) + цЬ{х)
(ab)(x) =a(x)b(x)
for all a, b € A, A, ft ? С and x 6 X.
To exhibit clearly the richness of the C*-algebra A, I will now give an equivalent description of this C-
algebra, as an inductive limit of finite-dimensional algebras. The Cantor set A' is by construction the
projective limit of the finite sets Л*„, where Kn is the set of sequences of n + 1 elements (jj)j=o.i n of O's
and l's, satisfying the rule :j — 1 => zj+i = 0. There is an obvious projection A',,+ i — Л"„ that consists in
'forgetting' the final zn+t. On the finite set К„, consider the relation
69

TV = {(z,:1) € К„ х Л'„ ; г„ = 4}-
Every function a =: aZiZt on 7tn defines an element a of the C*-algebra A by the equations
(
= О
if (*
Moreover, the subalgebra of-A obtained from the functions on 72.n is easily calculated: it is the direct sum
An = Л^*Я(С) Ф Mk' (C) of two matrix algebras, where ibn (resp. lc'n) is the number of elements of Kn
that end with 0 (resp. with 1). Finally, the inclusion An —* -An + i is uniquely determined by the equalities
ibn+i = jfcn + k'n, fcn-ы = kn which permit embedding Mkn Ф Л/t; as block matrices into Мкщ+1 wid by the
homomorphism {a, a1) —* a into Л/f
The C*-algebra A is then the inductive limit of the finite-dimensional algebras A,,, and one can calculate its
invariants for the classification of Bratteli, Elliott, Effros, Slien and Handclinan ([Br], [Eli], [Eb], [E-H-S]) of
these particular C*-algebras. The invariant to be calculated, due to G. Elliott, is an ordered group, namely
the group Л'о(-А), described earlier, generated by the stable isomorphism classes of finite projective modules
over A. In an equivalent way, it is generated by the equivalence classes of projections, as in the work of
Murray and von Neumann on factors.
A projection e € ¦Mfc(.A) is an element of Mk{A)sncb that e2 = e and e = сщ. The projections e,/ are
equivalent if there exists и € Mjt(-A) such that u*u = с and uu* = f. In order to be able to add equivalence
classes of projections, one uses the matrices Mn(A), with n arbitrary; this permits assigning a meaning to
e0/
-[:
is obtained canouically from the
mmet.mation operation by which
for two projections e, / € Mt(A). The ordered group (K0(A),I\0(A)'
semigroup of equivalence classes of projections e 6 Mn(A) by the usual
one passes from the semigroup N of nonnegative integers to the ordered group (Z,Z+) of integers.
This ordered group is very easy to calculate for finite-dimensional algebras such as the algebras An, and for
the algebra A = (J A> encountered earlier. Since А„ is the direct sum of two matrix algebras, we have
Ko(An) = Z2 , Л'о(Д„)+ = Z+ ф Z+ С Z Ф Z.
The ordered group (Ko(A), Ka(A)+) is then the inductive limit of the ordered groups (Z Ф Z , Z+ ф Z+),
the inclusion of the n'th into the (n + l)'th being given by the matrix
[ii]
that corresponds to the inclusion An С Ап±\ described earlier.
Since this matrix defines a bijection (a,b) — (a + b,a) of Z2 onto Z2, the desired inductive limit is the group
Ko(A) — Z2. However, this bijection is not a bijection of Z+ ф Z+ onto Z+ ф Z + , and in the limit the
semigroup K0(A)+ becomes [<0(A)+ = {(a,6) 6 Z2; f '"'г ) о +4 > 0}, as one sees on diagonalizing the
70
matrix |J [NFig.4).
\
\
;:;;:;: ;Hj
\ • ••••••••
FiS/1.
= {(»,
in) 6 Z2
It follows that, modulo the choice t>f bcisis of ZJ i.e. modulo PSLB,T), the golden number appears as a
topological invariant of the space Л* through the C-algebra Л. In another formulation, one shows that this
C*-algebra has a unique trace r. Thus there exists a unique linear form г on A such that т(ху) = т(ух)
for all x,у 6 A and r(l) = 1. The values of г on the projections form the intersection [Z + 1-гУ° Z) ЛН+.
This trace r is positive, i.e.. satisfies
т(ч'а) > 0 (Vn € A)
and it may he calculated for a = «,.._,., directly a.s the integral of the diagonal entries of the matrix a:
т(а)
= / "[.¦.-¦ I
Jk
M-),
where the probability measure /r on /\" is uniiiucly doteimined by the condition т(аЬ) = т(Ьа) (Vo,6 6 A).
In classical measure theory, given a Radon measure p on л compact space Y, the Hubert space L2(Y,p) is
obtained as the completion of C(V} for t.ln1 sealnr product (f,g) = f f]fdp, and L°°{Y,p) is the weak closure
of the algebra C(Y) acting by multiplication on ?~(V', p). In the case that we are interested in, the algebra A
replaces C(Y), the positive trace г replaces the 11,-ulou measure p, and the weak closure of the action of A by
left multiplication on L-'(.l,r) (the completion of .1 for the scalar product (a, 6) = r(a*6)) is the hyperfinite
factor of type Hi of Murray and vou Neumann [Mui-N], which we shall denote by R.
Whereas the continuous dimension of Murray and von Neumann can take on all the positive real values
for the projections of R (or of tlie matrix rings .M,,(]{)), as dim(e) = т(е), for the projections that belong
to A this dimension can only take values in the subgroup Z + ( '"t/^J Z, which accounts for the role of
these numbers in measuring i.he densities of t.iles or of patterns of a given type in a generic Penrose tiling
in accordance with our interpretation in chapter I of the continuous dimensions as densities. To summarize,
we have shown in this example that the 'topology' of -V is far from being trivial, that it gives rise to the C-
algebra A, which is a simple C"-algebra, uniquely characterized as a C*-algebra (up to Morita equivalence)
by the properties
71

1) A is the inductive limit of finite-dimensional algebras; it is said to be approximately finite (or AF);
2) (Ko(A),Ko(A)+)= (z2, {(«,(.); (^) a + 6 >o}).
Finally, we note that the C*-algebra A is exactly the one that intervenes in the construction by Vaughan
ones [Jone2] of subfactors of index less than 4, for index equal to the golden number (Chapter 5, Section
0). This can be exploited to describe explicitly the factor in this geometric situation.
10)
72
II.4. Duals of discrete groups and the Novikov conjecture
We have seen in the above example of the space Л' of Penrose tilings how the pathologies that a lion IlausdorfT
space exhibits from the set theoretic point of view disappear when it is considered from the algebraic and non
commutative point of view. To a conservative mathematician this example might appear as rather special
and one could be tempted to stay away from such spaces by dealing exclusively with more central parts of
mathematics. Since every finitely presented discrete group Г appears as the fundamental group: Г = т,(Л/)
of a smooth compact 4-manifold, it would be hardly tenable to exclude discrete groups as well. As we shall
see shortly however, as soon as a discrete group Г fails to be of type 1 (a finitely generated discrete group
Г is of type I only if it contains an abelian normal subgroup Го С Г of finite index, which is of course very
rare), its dual space, i.e. the space X — Г of irreducible representations of Г, is of the same nature as the
space of Penrose tilings. Fortunately, as in the example of Penrose tilings all the unpleasant properties of
this space from the set theoretic point of view disappear when we treat it from the algebraic point of view.
When the group Г is abelian, its dual space Г = X is a compact space, the Pontryagin dual of Г, whose
topology is characterized by the commutative C*-algebra C(X) of continuous functions on Лг. This C*-
algebra has, thanks to the Fourier transform, an equivalent description as the norm closure С*(Г) of the
group ring СГ of Г in the regular representation of Г in С'(Г). More precisely every element n = (аг)г6р of
СГ is a function with finite support on Г, and it acts in the Ililbert space (~(T) as a convolution operator:
(«*«),= ? «„ („ vc ее2(г).
The C*-algebra С*(Г) is the norm closure of this algebra of operators. Let us now give a simple example
of a non type I discrete group and investigate its dual from a set theoretic point of view. Let Г be the
semidirect product of the abelian group 7? by the group Z acting on 1? by the powers of the automorphism
a 6 Aut(Z2), given by the two by two matrix а = .By construction Г is a finitely generated solvable
group. Let Y be the 2-torus which is the dual of the normal subgroup Z~ С Г. The theory of induced
representations [M4] shows immediately that the dual space Г of Г contains the space V'/Z of orbits of the
transformation a of Y, since each such orbit defines an irreducible representation of Г, and different orbits
yield inequivalent representations. The quotient space Y/Z. is of course not IlausdorfF and is of the same
nature as the space X of Penrose tilings, thus a fortiori the dual space Г of Г lb%s, if one tries to understand it
with the classical set theoretic tools, a pathological aspect. This aspect disappears if, as we did for the space
of Penrose tilings, we analyse Г by means of the associated non commutative algebra. Thus for instance
the measure theory of Г for the Plancherel measure class is described by the von Neumann algebra А(Г)",
weak closure of the left regular representation of Г in ?2(Г) (cf. Chapter 5). This remains true for arbitrary
discrete groups. In our example А(Г)" is the hyperfinite factor R of type II\. For an arbitrary discrete group
one has always a finite trace r: the Plancherel measure, on the von Neumann algebra А(Г)", and moreover
the following equivalence holds:
Г is amenable <=> А(Г)" is hyperfinite
(cf. Chapter 5).
The topology of Г is of course described by the convolution C"-algobra of Г but it requires some elaboration
when the group Г is not amenable. The point is that in this case there is a natural closed subset ГР of the
dual Г, the support of the Plancherel measure, called the reduced dual of Г, so that one has two natural
convolution C*-algebras of Г, namely:
cr(T) = "C(fy , с;(г) = "C(fr)".
By definition С*(Г) is the completion of the group ring СГ for the norm; || a ||max= Sup{|| тг(«) ||, к unitary
representation of Г}. It is a C*-algebra whose representations are exactly tlie unitary representations of Г.
73

The reduced dual Гг of Г is the space of irreducible representations of Г which are weakly contained in the
regular representation ([Dia]). It is the space of irreducible representations of the C-algebra С?(Г) which
is the norm closure of СГ in ^(Г).
To the inclusion Гг С Г corresponds a surjection of C-algebras:
The К theory of the C*-algebra of a discrete group Г plays a crucial role in the work of Miscenko and
Kasparov on the conjecture of Novikov on homotopy invariance of the higher signatures for non simply
anifolds. Let us first recall the statement of this conjecture. Let Г be a discrete group and
fi h Nik jt iff the
Kasparov on th ojecture of Novikov on homotopy invariance o
connected manifolds. Let us first recall the statement of this conjecture. Let Г be a dr gp
(H) l thn the pair (Г,х) satisfies the Novikov conjecture iff the
) f t itd nifolds A/ ifi
nected manifolds. Let us first recall the statement
i € H*(BT,R) = H'(T,H) a group cocycle, then the pair (Г,х) satisfies the Novikov conject
following number is a homotopy invariant for maps (Л/, Ф) of compact oriented manifolds A/ to the classifying
space ВТ:
Signr(M,*) =
In other words what is asserted is that if (А/', Ф') is another pair of a compact oriented manifold M' and a
map Ф', Ф' : M' -* ВТ, which is homotopic to the pair (M, Ф), then:
SignI(JM,«) = Sign,(A/',»').
As a corollary it follows that one gets a homotopy invariant for compact oriented manifold M wilh_fun-
damental group Г; the map Ф is then automatic, it is the classifying map of the universal cover M of
M.
The cohomology class L(M) is the Hirzebruch L genus which enters in the Hirzebnich signature theorem.
The latter asserts that for a 4k manifold the signature of the intersection form on H2t is given by a universal
polynomial X-t(pi, ¦ ¦ • ,pt) in the rational Pontryagin classes of the manifold:
Sign(M) = (Z,t(Pl,...,Pt),[Af]>.
The Hirzebruch formula only involves the top dimensional component of the L genus but the above higher
signature formula uses all its components (cf. [M14]):
(P;j ^
One says that a discrete group Г satisfies the Novikov conjecture if the conjecture is satisfied by any group
cocycle x 6 H*(T, R). The natural generalization of the signature Sign(M) of a Ah oriented compact manifold
to the non simply connected situation yields (cf. [Mis2]) an element Signp( Д7) where Г =: тгх(М), in the Wall
group /„n(Q[r]). This follows from the theory of algebraic Poincare complexes of Miscenko [Mis2]. Here
Q[F] is the group ring of Г with rational coefficients, and the Wall group La, of a ring with involution is
equal to the Witt group which classifies symmetric bilinear forms modulo the hyperbolic ones. The crucial
reason for the role of the К theory of the С algebra of the group Г is the isomorphism
Witt(.4) ~ Ko(A)
valid for any C*-algebra (but false for involutive Banach algebras in general), which allows to extract from
the homotopy invariant Signr(M) 6 Z,4n(Q[r]), a homotopy invariant belonging lo a A" theory group, namely
the image pSignr(M) € Л'0(С*(Г)).
Simple examples, such as the case of commutative Г (cf [Lus]) show that in such cases the relevant information
is not lost in retaining only the К theory signature, and that the latter, being in the t.opological A" theory
74
of the Pontryagin dual Г of Г which is a torus, is much easier to use than Iho L theory signature. This point
is an essential motivation to study the A" theory of the C"-algebras of discrete groups.
In L theory there is a natural spectrum, the L theory spectrum and a map, called (.he assembly map
a : h,(T,L) —> ?«(Q(r)) whose range contains all the elements Signr(.W) and whose rational injectivity
implies the Novikov conjecture. Kasparov and Miscenko have constructed an analytic assembly map from
the К homology K.(BT) = Л.(ВГ.Ви) (where BU is the К theory spectrum) of the classifying space ВГ
of Г to the К theory of the С -algebra of Г:
А'.(ВГ) — Л-.(С*(Г))
which makes the following diagram commute:
(*)
h.(BT,L) — L.(C(T))
Л'.(ВГ) -^ Л-.(С*(Г))
where the horizontal arrows are the assembly maps, the left vertical arrow comes from the natural map
from the L theory spectrum to the BU spectrum and the right vertical arrow comes from the equality for
C*-algebras of L theory with A' theory.
The simplest description of the map /i : А'.(ВГ) —> А'»(С*(Г)) is based on the existence for every compact
subset К of В Г of a canonical "Miscenko line bundle":
(к 6 Ло(С(А-)©С(Г))
which is described as a finite projective С module over C(A') © С*(Г) by the following proposition:
Proposition 1. Let К be a compact space, <p : A' —* ВТ a continuous map and К —- A' the T principal
bundle over K, pull back of ET — ВГ. Lei ? be the completion ofCc(K~) for the norm: || (. \\=\\ ({,?) Ц1'2,
иЛеге for cJ.tj 6 CC(K) the element (f,77) ofC(K) & C"(T) is defined by:
Then ? is a right C* module, finite and projective, over C(A')® С*(Г), for the riyhl action uniquely defined
by the equality:
V? € CC(K), f € C(K), g&T, x e\.
The proof is straightforward. The local triviality of the Г principal bundle .V over ,V yields in fact, an explicit
idempotent e 6 М„(С(К) ® С*(Г)) whose associated right module is ?.
The construction of 1ц is naturally compatible with the inductive system of compact subsets of ВТ. By
definition the К homology of ВТ is the inductive limit:
75

h.(Sr,BU) = !г.(ВГЛВи) = jim ir.(AABU)
where A* runs through finite subcomplexes of ВГ. To construct ft one uses the following formula which holds
both in the Kasparov KK theory and in E theory:
(**)
= (z® l)otK € ?(С,С-(Г)) = К (С (Г))
Vz € h,(K,BU) = ?(C(A'),C).
One has z® 1 € E(C(K) ® С*(Г),С*(Г)) and ** € ?(C,C(A') ® С*(Г)) so that /i(j) € ?(С,С*(Г)) =
Л"(С*(Г)).
One then has the following index theorem of Kasparov and Miscenko ([Kas?] [Kas;,] [M1S3] [Mis-S])
Theorem 2. The analytic assembly map, /j : А'.(ВГ) —> К.(С"(Г)) defined by (**) makes the diagram (*)
commute.
In particular, let M be a compact oriented manifold of dimension n and <p : Л/ —- ВТ a continuous map.
Then there is a corresponding element г g К.(ВГ) whose image /<(z) € А\,(С""(П) is the Г equivariant
signature of M, i.e. the image p(z') = Signr(A/) of the Miscenko L theory class. One has z = <p*(<r(M))
where <r(M) 6 A'.(M) is the A' homology class of the signature operator on M. What matters here is that
the Chern character of <r(M), ch.(<r(M)) e H.(A/,Q), is given by:
ch.(<r(M)) = L(M)n[M].
Thus the Novikov conjecture for a discrete group Г is equivalent to the homotopy invariance of the image
(p*(<r(M)) in the rational A" homology of ВГ. Since the Miscenko I theory class :' e Х,„(СГ) is homotopy
invariant, one gets as a corollary of theorem 2,
Corollary 3. (Loc.Cit) The Novikov conjecture for Г is implied by /he mlional injectivity of the analytic
assembly map:
li : Л'.(ВГ) - Л-,(С-(Г)).
This, so called, strong Novikov conjecture has been proved by Miscenko for fundamental groups of nega-
negatively curved compact Riemannian manifolds ([Misi]) and by Kasparov for discrete subgroups of Lie groups
([Kass]).
Thanks to the work of Pimsner and Voiculescu [Pi-V3], of Cuntz [Cui] for free groups, and of Kasparov
[Kas4], Fox and Haskell [F-H] and Kasparov Julg [Ju-K] for discrete subgroups of SO(n, 1) and SU(n, 1),
one knows for many torsion free discrete groups Г that the analytic assembly шар /j is an isomorphism.
However there is no discrete group Г, infinite with property T of Kazhdan, for which Л'(С7 (Г)) or Л"(С(Г))
has actually been computed. One difficulty is that due to property T the natural homomorphism:
с-(г) л с;(г)
fails to be а К theory isomorphism. Indeed the idempotent corresponding by property T to the trivial
representation of Г defines a non zero К theory class [e] € А'о(С*(Г)) whose image ir(e) is zero. Thus
there are indeed two A" theories to compute. The map \x does not contain [e] in its range and thus cannot
yield an isomorphism with Х'(С"*(Г)). The possibility that л"о/л is an isomorphism is still open, but by the
counterexample of G. Skandalis ([Sk4J) it cannot be proved by a KK theory or E theory equivalence. When
the group Г has torsion the above map ц is too crude to be expected to yield the A' theory of С*(Г). Indeed
even with Г finite, say Г = Z/2Z, one has K0(C"(T)) = A'o(C ф С) = Z" while K0(BT)q = Q. We shall see
later how to modify Kt(BV) and /j when Г has torsion.
76
II.5. The tangent gronpoid of a manifold
Let M be a compact smooth manifold. ТЫ1 pseudodifferoiit.ial calculus on M allows one to show that an
elliptic pseudodifferential operator D on .17 is a Frudholin operator and hence has an index, Ind(?>) 6 Z.
The Fredholm theory then shows that this index only depends upon the A'-theory class of the symbol of D.
One thus obtains ([At-Sii]) an additive map, the analytic index map:
Ind,, : К„{Т'М) — Z
from the К theory of the locally compact space T'M to Z.
In this section we shall show how the same шар 1ж1„ arises naturally from a geometric construction, that of
the tangent groupoid of the manifold .1/. This groupoid encodes the deformation of T'M to a single point,
using the equivalence relation on Д/ x [0, 1] which identifies any pairs (.i,f),(y,f) provided e > 0.
;l implicitely the notion of groupoid. All our algebra
In the above section, as well аи in Chapter I, we ht
structures could be written in the following form:
where the 7's vary in a groupoid G, i.e. iu a small category with inverses, or more explicitely:
Definition 1- .4 (jnni/toid consists of n sfl G. a suhsd G' С О, 1wo maps r,s : G —* G^°' and a law of
composition:
: С"-1 = {(-и.-,-,) 6 С, x G : .ч(-ц) = r(To)} — G
such that:
A) *Gi»72) = «G2) , r(Tl.-,,) = '-Gi) ¦ V@i ,12) € C1-1
B) s(x) = ,-(x) = x vjrer;""
C) TosG) = 7 , г(т>7 = ¦) VT € G
D) G1°7з)°7з = 71°G2«7з)
E) Each 7 Ляд a twv-xulcd turrrxr ~,~l. irtlti "и = i'[j) , 7~l7 =
= sG2)
Kig.5. Groupoid

Here are a few important examples of groupoids.
Equivalence relations.
Given an equivalence relation Tl С X x X on a set X, one gets a groupoid in the following obvious way:
G = V., G<°> = diagonal oflxXcH, r(x, y) = i, s(z, у) = у for any 7 = (x, y) € ft С X x JV and:
(i,3/)o(y,2) = A,2) , (х.з/Г1 = (y,x).
Groups.
Given a group Г one takes G = Г, G'0' = {e}, and the law of composition is the group law.
Group action. (Section 7).
Given an action Xxr^Xofa group Г on aset X, a(x,g) = xg, x(g,gi) = (xgi)gi V* € X, gt € Г; one
takes G = X X Г, G<°> = X x {e}, and:
r(x, g) = x , s(x, g) = ij V(i, j) 6 A" X Г
(*.Ji)(y.ff2) = (I.Sift) if ifll = y
In all the examples we met so far, the groupoid G has a natural locally compact, topology and the fiber
G' = r~1{x}, x € G<°) of the map r, are discrete. This is what allows to define very simply the convolutio
algebra by:
(a*6)G)=
We refer to [Rent] [ВГ02] for the general case of locally compact groupoids. Our next example of the tangent
groupoid of a manifold will be easier to handle than the general case; though no longer discrete, it will be
smooth in the following sense:
Definition 2. ([Pra]) A smooth groupoid G is a groupoid together with a differentiable structure on G and
G^ such that the maps r,s are submersions, the object inclusion map: G^0' -~ G is smooth, as is the
composition map G'2' —» G.
The general notion is due to Ehresmann and the specific definition here to Pradines [Pra]. He proved that
in a smooth groupoid G, the maps s : G" —> G'0' are subimmersions, where Gr = {7 € G;i'G) = 1}.
The notion of ^ density on a smooth manifold allows to define in a canonical manner the convolution algebra
of a smooth groupoid G. More specifically, given G, we let fW2 be the line bundle over G whose fiber ftj.'
at 7 6 G, r(y) = 1, s(f) = у is the linear space of maps:
p : /\
such that p(Xv) = W2p(v) VA 6 С Here Gy = {7 € G;s(f) - x) and к = diml^C*) = dimT,(Gy) is
the dimension of the fibers of the submersions r : G —> G(°) and s : G —> G^°).
Then we endow the linear space Cf(G, П1/2) of smooth compactly supported sections of Q1'2 with the
convolution product:
(a*(.)G)=/ аG,ЖТ2) Va,& e Cf (С.П1'2)
78
where the integral on the right hand side makes sense since it is the integral of a 1-density, namely
аG)Н7Г'т)> on "le man'f°'d G*, x = rG).
As two easy examples of this construction one can take:
a) The groupoid G = M x M where M is a compact manifold, r, s are the two projections G —> M = G(o' =
{(x,i);x € M] and (x,y)o(y, 2)= (x, z) Vz,i/,z € M.
The convolution algebra is then the algebra of smoothing kernels on the manifold M.
/3) A Lie group G is in a trivial way a groupoid with G'0' = {e}. One then gets the convolution algebra
Cf (G,n1/2) of smooth 1-densities on G.
Coming back to the general case, one has:
Proposition 3. Let G be a smooth groupoid, and let Cf (G, fl1^2) be the convolution algebra of smooth
compactly supported | densities, with involution *, f'(j) = /G"')-
Then for each x € G'0' the following defines an involutive representation ir» of Cf(G, fW2) in the Hilbert
space L2(GX):
Ы/Ю G) = J Н-пШ-ПЧ) V7 € Gx , f € X.2(GJ.
Tie completion ofC^(G,U1'2) for the norm: || / ||= Sup || т,(/) || is а С"-algebra, denoted C"r(G).
We refer to [Coc,][Reni] for the proof.
Let us now pass to an interesting example of this general notion, namely we construct the tangent groupoid
of a manifold M. Let us first describe G at the groupoid level; we shall then describe its smooth structure.
We let G = (M x Mx ]0,1]) U (TM) where TM is the total space of the tangent bundle df'Af.
We let G<°> С G be M x [0, 1] with inclusion given by:
(x,e) — (x, 1, г) € M x A/x ]0,1] for 1 € М,г > О.
(x, 0) — 1 € M С TM as the 0 section, for г = 0.
The source and range maps are given respectively by:
fr(i,j,f) = (i,?) for x€M,e>0
r(x, X) = (i,0) for x € M, Л' € T,(A/).
The composition is given by:
{
s(x,X) = (
for y€M,€>0
for xe A/,X eT
(x,y,г)о(у,г,г) = (x,z,?) for г > 0 ; х,у,г е Af
(x, X)»(x, У) = A, A' + Y) for 1 € A/ ; Л", V € Tr( Af).
Putting this in other words, the groupoid G is the union (a union of groupoids is again a groupoid) of the
product G\ of the groupoid M x M of example a) by ]0,1] (a set is a groupoid where all the elements
belong to G^) and of the groupoid G2 = TM which is a union of groups: the tangent spaces TX(M).
This decomposition G = G, U G2 of G as a disjoint union is true set theoretically but not at the manifold
level. Indeed we shall now endow G with the manifold structure that it inherits from its identification with
the space obtained by blowing up the diagonal Д = M С M x M in the cartesian square M X M. More
explicitely the topology of G is such that Gy is an open subset of G and a sequence (xn,2/n,?») of elements
79

of Gt = M x Mx ]0,l] with ?n -> 0, converges to a tangent vector (x, X); X € TX(M) iff the following
holds:
Х„ -* X , Уп — X , ; > Л.
Fig.6. The tangent groupoid of Af
The last equality makes sense in any local chart around x independently of any choice. One obtains in this
way a manifold with boundary, and a local chart around a boundary point (г,Л") € TM is provided, for
instance, by a choice of Riemannian metric on M and the following map of an open set of TM x [0, 1] to G:
¦ (x,X,e)= A,ехрх(-гА'),г) € M x Mx ]0,l] , for s > 0
¦ (х,Л,0) = (х,Х)€ТМ.
Proposition 4. With the above structure G is a smooth groupoid.
We shall call it the tangent groupoid of the manifold M and denote it by Сл,. The structure of the C*-
algebra of this groupoid Gm is given by the following immediate translation of the inclusion of Gi = TM
as a closed subgroupoid of Gм, with complement Gx.
Proposition 5. 1) To the decomposition Gm = G\ U Go of Gm as a union of an open and a closed
subgroupoid corresponds the exact sequence of C* -algebras:
0 —
C'(G) — C'(G2) -0.
2) The C'-algebra C""(Gi) is isomorphic to Co(]0,1])®? where /C is the elementary C"-algebra (all compact
operators in Hilbert space).
3) The С-algebra С(Сг) is isomorphic to Co(T*M), the isomorphism being given by the Fourier transform:
C'(TtM) ~ Co(T^M), for each x € M.
It follows from 2) that the C*-algebra C*(Gi) is contractible, it admits a poiutwise norm continuous family
вх of endomorphisms, A e [0,1], such that 90 = id and 0i = 0. (This is easy to check for Co(]0, 1]).) In
particular, from the long exact sequence in /v-theory we thus get isomorphisms:
<r. : Ki(C"(G)) ~ Ki(C"(G2)) = КНТ'М).
In the right hand side K'(T* M) is the /C-theory with compact support of the total space of the cotangent
bundle. We now have the following geometric reformulation of the analytic index map Ind0 of Atiyah-Singer.
Lemma 6. Let p : C'(G) — К = C'(M x M) be the transpose of the inclusion M x M ->• G; (x,y) -*
(x,j/, 1) Vx,у 6 Л/. Then the Atiyah-Singer analytic index is given by:
80
Indo =
1 : K°(T'M) — Z = K0{K).
The proof is straightforward. The map a : C'(G) — C'(G2) ~ C0(T'M) is the symbol map of the
pseudodiflerential calculus for asymptotic pseudodiflerential operators ([Gu-S]).
We shall end this section by giving a proof of the index theorem, closely related to the proof of Atiyah
and Singer ([At-Si i]) but which can be adapted to many other situations. Lemma б above shows that the
analytic index Inda has a simple interpretation in terms of the tangent groupoid Gm = G. If the smooth
groupoids G, Gi, G2 involved in this interpretation were equivalent (in the sense of the equivalence of small
categories) to ordinary spaces Xj (viewed as groupoids in a trivial way, i.e. Xj = A'j°l), then we would
already get a geometric interpretation of Inda, i.e. an index formula. Now the groupoid G\ = M x Mx ]0,1]
is equivalent to the space ]0,1] since MxMis equivalent to a single point. Thus the problem comes from
Gi which involves the groups TrM and is not equivalent to a space. Given any smooth groupoid G and a
homomorphism h from G to the additive group R^ one can form the following smooth groupoid Gh'.
= G x R"
l0) = G<01 x R'v
гG,Л) = (r(y),X), s(j,X) = (sG),X + h(f)) V7 € G, X € R", G,, -V,)oG2,A'3) = G1,73, Xi) for any
composable pair.
Heuristically if G corresponds to a space X, then the homomorphism h fixes an RN principal bundle over X
and G/, corresponds to the total space of this principal bundle. At the level of the associated C*-algebras
one has the following:
Proposition 7. Let G be a smooth groupoid, h : G —• R^ a homomorphism.
1) For each characterx € Ям of the group RN the following formula defines an automorphism ax ofC"(G):
K(/))G) = X(hG))fh) V/ € C?(G,Q).
2) The crossed product C*(G)XaRw ofC'(G) by the above action a of RN = (RN)A is the С-algebra
C'(Gh).
Thus we see in particular that if N is even, the Thorn isomorphism for C*-algebras (appendix C) gives us a
natural isomorphism:
Ko(C'(G))~Ko(C-(Gh)).
In the case where G corresponds to a space Л', the above isomorphism is of course the usual Bott periodicity
isomorphisms. We shall now see that for a suitable choice of homoniorphism G -Л RN where G = Gm is the
tangent groupoid of M, the smooth groupoids Gh, Glh, G2.I1 will be equivalent to spaces, thus yielding a
geometric computation of Inda and the index theorem.
Let M — RN be an immersion of M in a Euclidean space R". Then to j corresponds the following
homomorphism Л of the tangent groupoid G of M in the group R":
Л(х,у,г) = Л*';?'?' ?>o
h(x,X) =j.(X) УЛ'еГ^ЛП.
One checks immediately that 7G1072) = Д71) +7G2) whenever G1,72) € G*2'.
This homomorphism h defines a free and proper action of G by translations, on the contractible space RN.
This follows because j is an immersion, so that j, is injective. The smooth groupoid G/, is thus equivalent
to the (classifying) space BG which is the quotient of G'0' x RN by the equivalence relation:
81

I
(x, X) ~ (y, У) iff З7 € G ; rG) = x , sG) = у , Л' = У + ЛG).
Since the action is free and proper the quotient makes good sense. Similar statements hold for G\ and Gj.
A straightforward computation yields
BG= (]0,1] x.RN)\Ju(M)
where v(M) is the total space of the normal bundle of M in RN.
In this decomposition, BG = BGt U BG2, one identifies BG2, the quotient of GJ,0) xR" = MxR"by the
action of G2 = TM with the total space of v, ux = RN/ТХ(М). The isomorphism a of (]0, 1] x R'') with
the quotient BG\ of gJ°' хН"= (]0,1] x M) x R'4' by the action of G\ depends upon the choice of a base
point 10 € M and to simplify the formulae we take j'(io) = 0 € RN. One then has
a(e,X)=((xo,?),X) V<r >0 , A" € RN.
With these notations the locally compact topology of BG is obtained by gluing ]0,1] x R'' to v(M) by the
following rule:
iff Х„ — j(x)
and
(?„,*„) -> (х,У) for ?„ — 0 , i € Л-/,У € »,(Л-/)
*--'W — у in мм)-
Using the Euclidean structure of RN we can view ux(M) as the orthogonal subspace of j.Tx(M) С RN and
use the following local chart around (x,Y) € u(M):
?>(x,Y,f) = (?,j(x) + eY) € ]0,1] x RN for г > 0.
To the decomposition of G/, as a union of the open groupoid C|,i and the closed groiipoid G2/, corresponds
the decomposition BG = BG\ UBG2. As in proposition 5, BG\ is properly contractible and thus we get a
well-defined ЛГ-theory map:
ф : K°(BG2) ~ K°(BG) — K°(RN)
which corresponds to Inda = p,o(o,)~l under the Thom isomorphisms: A'o(C*(G()) ^ h'o(C"(Gh,i)) =
K°(BGi). Now from the definition of the topology of BG it follows that V is the natural excision map:
K°(v(M)) — K°(RN)
of the normal bundle of M viewed as an open set in R". Moreover the Thom isomorphism:
is the Bott periodicity, while the Thom isomorphism
K°(T'M) ~ Ka(C(Gi)) =: K0(C(G,,h)) ~ h°(BG,)
is the usual Thom isomorphism r : K°(T"M) ~ K°(v(M)). Thus we have obtained the following formula
hid, = /?o0or
which is the Atiyah Singer index theorem ([At-Sii]), the right hand side being the topological index Ilid,-
We used this proof to illustrate the general principle of first reformulating, as in lemma 6, the analytical
index problems in terms of smooth groupoids and their A'-theory (through the associated C"-algebras). And
then of making use of free and proper actions of groupoids on contractible spaces to replace the involved
groupoids by spaces, for which the computations become automatically geometric.
82
Ц.6. Wrong way functoriality in K-theory as a deformation
Let X, Y be manifolds and f : X —* У a Л'-oriented,smooth map. Then tlie.Gysin or wrong way functoriality
map: ' ' '
/! : K(X) - K(Y)
can be described ([Co-S]) as an element of the Kasparov group KK(X, Y). We shall give here the description
of p. as an element of E(X, У), i.e. as a deformation. The construction is a minor elaboration of the
construction of the tangent groupoid of section 5. The advantage of this construction is that, given a smooth
map: f '¦ X —> У, it yields a canonical element of:
E(T"X®f'TY,Y).
Thus it adapts to the equivariant and to the non-commutative cases.
a) The index groupoid of a linear map
Let E, F be finite dimensional real vector spaces and L : E ¦— F a linear map. Then the group E acts by
translations on the space F and we can thus form the groupoid F >dt E = liul(Z,) which is the semidirect
product of F by the action of E. More specifically, with G = Ind(Z,) we have: G = F x E, G(o> = F x {0},
and for (tj,0 € F x E, r(q,{) = 77, s(t],0 = r] + Щ), while:
By construction G = Ind(i) is a smooth groupoid and is equivalent to the product of the quotient space:
F/lm(L) by the group Ker L. Thus using Fourier transform the C-algebra of Ind L is strongly Morita
equivalent to Co((F/lm L) x (KerZ,)') which justifies the notation Ind L.
Proposition 1. a) Let L : E —> F be a linear map. Then the. family ludffZ,), e 6 [0, 1] gives a canonical
deformation
H & E(Co(F x E'),C'(lndL)).
b) Let L: E -* E', V : E' — E" be linear maps. Then the family \z, € " ,E' of linear maps from E $ E'
Lu L J
to E' ф ?" jines a canonical deformation of (Ind L) x (Ind //) (o lnd(Z/oZ.).
Statement a) follows because Ind@) = F Xo ? has Coff x ?') as associated C-algebra, while for г > 0,
F x,i E is canonically isomorphic to F Xi ff = Ind(Z,).
To get b) one uses the equivalence of Iud .f with the smooth groupoid \ni\(L'oL).
Lu L J
All of the above discussion goes over for families of vector spaces and linear maps, i.e. for vector bundles
E, F over a smooth base В and vector bundle maps L : E —> F. We shall still denote by hid L = U Iud /,,
the corresponding smooth groupoid. Proposition 1 then applies without any change, where in a), F x E*
denotes the total space of the vector bundle F x E'.
/3) Construction of p € E(T' M ®f'TN,N)
Let M, N be smooth manifolds and / : M —> N a smooth map. The tangent map /. of / is a vector bundle
map:
/. :TM->f'(TN)
and thus by a) it yields a smooth groupoid,
83

As a manifold Ind(/») is the total space of the vector bundle TM Ф f"(TN) over M, while its groupoid
structure is as a union of the groupoids Ind(/,,i), /.,* : TXM —> Tj^N. We shall construct a canonical
deformation of the groupoid Ind(/») to the product of the space N (viewed in a trivial way as a groupoid, with
N = JV<°)) by the trivial groupoid (equivalent to a point) M x M. Thus, let G\ = Ind /,, G2 = N x (M x
M) x ]0, 1] and let us endow the groupoid G = G\ U Gi with the relevant smooth structure. The topology
of G is uniquely specified by the closedness of G\ and the following convergence condition for sequences of
elements of G2: for ?„ > 0, ?„ —> 0 a sequence (<„, (х„, у„),г„) of elements of 6'? = N x (M x M) x ]0, 1]
converges to (x,r;,0 € Gb with x € M, f] € T,(l)(iV), f € Tt(A/), ifT one has:
х„ -» x , yn — x , tn — /(x)
One checks that this topology is compatible with the groupoid structure of G which becomes a smooth
groupoid with the following local diffeomorphism y>: G\ x [0,1] —* Gi around any point of G\ x {0}:
?>((*, V,O.<0 = (exp/(,)(?4),(*,exPi(-ff),f)) € G2
(where the exponential maps are relative to arbitrary Riemannian metrics on M and JV).
Thus C"(G) gives a canonical deformation of C*(Ind(/.)) to the C""-algehrn of N x (Л/ х М), and combining
it with the element 8 of proposition 1 a) and the strong Morita equivalence C(<'V x (Д/ x M)) = C(N)®tC ~
C(N), we get a canonical element:
f\&E(T'M®f(TN), N).
In order to convert /! to an element of E(M,N) we need а Л' orientation of / as an element of:
7) A-orientations of vector bundles and maps
Let M be a compact space and i*1 a real finite-dimensional vector bundle over M with p : F —> Л/ the
corresponding projection of the total space of the bundle to the base Л7. We assume for simplicity that the
fibers of F are even dimensional, the odd case being treated similarly (with /\l instead of /v"°). There is a
natural map a -» ? from K°(F) to E(C(M), C0(F)) which we now describe: first let p :: F — F x M be the
proper continuous map: p(?) = (?,?(?)) alu^ P* 6 Hom(Co(F) ® C(M), Cq(F)) (he corresponding morphism
of C*-algebras. Then for a 6 A'°(F) = E(C,C0(F)), let ? be given by:
? = p*o(<r®idM)
(one has<r®idM € E(C(M), Ca(F) ® C(JW)) and p* € E(Ca(F) ® C(M),C0(F)) so that ? is well defined).
Let In be the fiber dimension of F and SpincB;») be the Lie group SpinB«) Xz/iz ^A). Then a Spinc
structure on the bundle F is a reduction of its structure group GL?n(R) to SpincBn)-
Definition 2. The vector bundle F ts К oriented if H is endowed with a Spin' structure.
In particular this implies a reduction of the structure group of F to 50B»). i.e. an orientation and a
Euclidean structure on F. The latter is in general not invariant in the Г equivariant case, in which case
the definition 2 has been adapted by P. Baum (cf. [Ili-Si]), replacing the group Sl>incB») by the group
МЦп = A«2n(R) xI/2 GA), where M?2n(R) is the nontrivial two fold covering of GLta(R). The following
proposition then still holds in the equivariant case ([Hi-Si]).
84
Proposition 3.' a) A complex vector bundle has a canonical K-orientation given by the homomorphism
SU(n) - SPineBn).
b) The dual F' of а К-oriented bundle is K-oriented.
c) For any bundle F, F ф F* has a canonical K-orientation.
d) Let 0 —> F —* F' —* F" —* 0 be an exact sequence of vector bundles. If two of these bundles are K-oriented
the third inherits a K-orientation.
e) Let M —> JV* be a continuous map, F a K-oriented bundle over N. Then f*(F) is a K-oriented bundle
over M.
(cf. [Hi-Si]). The group H2(M,I) of complex line bundles on M acts transitively and freely on the set
of /i'-orientations of F." Any given Л'-orientation on the bundle F yields canonically ([At-Sii]) an element
a of K°(F) given by the Spinor bundle S± with its natural Z/2 grading, and the morphism of p"S+ to
p'S~ (over F) given by Clifford multiplication 7(f), { € F. The latter morphism is an isomorphism outside
the 0-section М С F and thus defines ([At-Sii]) an element of K°(F). Moreover the Thorn isomorphism
in A*-theory ([At-Sii]) then shows that the corresponding element ? 6 E(C(M),Co(F)) is invertible. (In
particular a is a generator of K(F) as a K(M) module.)
Definition 4. Let Af, N be smooth manifolds, f : M —> N a smooth map. Tkcv а К -orientation of f is a
K-orientation of the real vector bundle F = T"M в f"(TN).
We shall denote by <Tj, 07 the corresponding elements of K(F) and E(C(M),C0(F)).
As an immediate corollary of proposition 3 one gets:
Proposition 5. a) // the tangent bundles of M and N are I\-oriented then any f : M —* N inherits a
K-orientation.
b) The identity map Ыц : M —> M has a canonical K-orientation.
c) The composition of K-oriented maps is K-oriented.
Remark 6. The above discussion of ЛГ-orientations of vector bundles and the construction of? 6 E(Co(M), G>(F)
extends (cf. [Co-S]) to the case of non-compact manifolds. (The element a, which we shall not use, is however
no longer an element of K(F).)
S) Wrong way functoriality for Л'-orieuted maps
Let M and JV be smooth manifolds and let f : M —* JV be a smooth map. We have constructed above (/?) a
deformation /! € E(T" M® f(TN),N) canonically associated to /. Let us now assume that / is K-oriented
and let <tj € E(M,T'M® f"(TN)) be the corresponding invertible element. Ну a slight abuse of language
we shall still denote by /! the following element:
/! = p.oo-j € E(M,N).
We can then formulate as follows the results of section 2 of [Co-S]:
Theorem 7. a) The element /! 6 E(M,N) only depends upon the К-oriental homotopy class of f.
/3) One has (IdM)! = 1 € E(M,M).
7) For any composable smooth maps f\ : M\ —> Mi, /3 : Mi —> Л/3 one /in» with the corresponding K-
orientations:
(ho fi)l = fro fii ? E(MltM3).
This result can be deduced from [Co-S] and the natural functor from the A"Л'-category to the E category
(appendix B). However the direct proof is easier due to the canonical construction of section /3) and we invite
the reader to do it as an exercise.
85

As in [Co-S] one gets the following strengthening of Bott periodicity.
Corollary 8. Let M,N he (not necessarily compact) smooth manifolds and f : И — N a (not ncccssanly
proper) homotopy equivalence. Then /! is invertible.
In particular f\ yields natural isomorphisms for arbitrary C""-algebras A and Д, between the groups:
E(A,
=s E(A, В & С0(Л/))
E(A®Ca{M),B)~ E(A®C0[N),D).
The Bott periodicity is the special case of M = R2" and N =pt; it also applies to any (non-compact)
contractible manifold.
86
jj#7. The orbit space of a group action
In this section, as a preparation for the case of leaf spaces of foliations, \vk shall consider spaces of the same
nature as the space of Penrose tilings, namely the spaces of orbits for an action of a discrete group Г on
a manifold V. We assume that Г acts on V by diffeornorphisms and use the notation of a right action
(i, g) € V x Г — xg € V with
x(gifJ) = (xg,)g2 Vffi, g? € Г , Vz € V.
Such an action is called free if each g 6 Г, д ф 1 has no fixed point, and proper when the map (x, ij) —» (i, xg)
is a proper map from V x Г to V x V.
When the action of Г is free and proper the quotient space X = V/T is HausdorfT and is a manifold of
the same dimension as V. The next proposition shows that the space Л' is eqnivalently described, as a
topological space, by the C*-algebra crossed product Co(V) ХГ (Appendix C).
Proposition 1 ([Ri2D Ш Г act freely and properly on V and let X = V/T. Then the C'-algebra C0(X) is
strongly Morita equivalent to the crossed product С-algebra С0(У)
It is important to mention that in this case, and for any Г amenable or not, there is no distinction between
the reduced and unreduced crossed products. Also the equivalence О*-Ьткм1п1е ? is easy to describe; it is
given by the bundle of Hilbert spaces CHx)i^x over X, whose fiber at j: S Л' = V/T is the C2 space of the
orbit x eX. This yields the required C'o(K) x Г, Со(Л') C-bimodule.
When the action of Г is free but not proper, for instance when V is compact, and Г infinite, the quotient
space X = К/Г is no longer HausdorfT and its topology is of little use, so neither is the algebra Co(X) of
continuous functions on X. But then Proposition 1 no longer holds and the topology of the orbit space is
much better encoded by the crossed product C-algebra Ca(V) »Г. Our aim in this section is to construct
A"-theory classes in this C*-algebra from geometric data. For that purpose we shall not need the hypothesis
of freeness of the action of Г. In fact the case of Г acting on the space V reduced to a single point, already
treated in section 4, will also guide us. We shall assume that Г is torsion free and treat the general case
later in section 10.
The motivation for the construction of A'-theory classes, due to P. Ваши and myself ([Bau-Ci]) is the
following. While there are in general very few continuous niaps / : A' — 11' froni a non commutative space
such as V/T to an ordinary manifold, there are always plenty of smooth maps from ordinary manifolds W
to such a space X = V/T. Thus since we dispose by section 6 of the wrong way func.toriality map /! in
AT-theory, we should expect to construct elements of Л'-theory of the form:
/!(*) € A'(C0(V) ХГ) V-- € K{Co(W))
for any smooth Л'-oriented map / : W —* V/T from an ordinary manifold to our space X = V/T.
Given a smooth map p : W —» V, one obtains by composition with p : V — V/T = Л' a "smooth" map to
X but clearly there are other natural "smooth" maps to Л' which do not factor through V; indeed if we are
given an open cover (Wi)ier of W and smooth maps p; ¦ \V{ —> V such that on auy non-empty intersection
Wi П Wj one has pop,- = po pj then po p : W —> V/T still makes sense. Since we do not want to assume
that the action of Г on V is free, we strengthen the equality po pt = po pj on H'v П Wj by specifying a Cech
1-cocycle:
Pij
i П Wj - Г
such that:
Pj(x) Pij(x) = Pi(x) Va: € Wi П Wj.
To the Cech 1-eocycle (Wi,pij) with values in Г there corresponds a Г-principal bundle

w -Lw
and local sections Si : Wi — W such that
4 (*) PH (*) = »i(*) Vi € Wi П Wj .
There exists then a unique Г-equivariant smooth map from W to V such that:
P(si(x)) = Pi(x) view,-.
To summarize, we obtain the following notion:
Definition 2. Let Г be a discrete group acting by diffeomorphisms on a manifold V. Then a smooth
map W —> V/T from a manifold W to the orbit space, is given by a T-prtncipal bundle W -2* W and a
Г-equivariant smooth map p : W —> V.
The reason for the existence of sufficiently many such maps is that homotopy classes of smooth maps
W i—*- V/T correspond exactly to homotopy classes of continuous maps:
W — V x r ?T = Vr
where ET —> ВТ is the universal Г-principal bundle over the classifying space ДГ of Г.
First, given W -i. W and p : W —> V as in definition 2, we get, a Г-eiiuivariiinl. continuous map:
where ip is a classifying map W —> ET for the bundle W, The map (p,<?) then yields a continuous map
ф : W —> V Xp ET. Second, given a continuous map Y> : W -~ V xp ?T, one can pul! back to W the
Г-principal bundle over К хр ЕГ given by
V x ET -^V xrET.
One then gets a Г-principal bundle W ^->W over IV and a continuous r-eqiiivarianl. map v : W —> V x ?T.
The composition of ф with the projection pry : Kx ЕГ —> К gives us a Г-eqiiivananl continuous map W Д K.
Since V Xp ?T is a locally finite CIV complex there is a natural notion of smooth map W i— К х р ЕГ and
every continuous map from W to К хг ЕГ can be smoothed, thus yielding smooth maps W ^- V/T in the
sense of Definition 2. Thus:
Proposition 3. Lei T be a discrete group acting by tlifjeomor]ihhvis on the manifold V, and let W be
a manifold. Then homotopy classes of continuous maps W i— Vp = V x\- ET correspond liijectinel) to
homoiopy classes of smooth maps W —* V/T in the sense of Definition 2.
Any Г-equivariant bundle F on V is still Г-equivariant on V x ET and hence drops down to a bundle on Kr.
This applies in particular to the tangent bundle TV of V, yielding a bundle г on Vp. At a formal level the
geometric group K, r(Vp) is defined as follows:
Definition 4. K. r(Vp) is the K-homology of the pair(Br,Sr) of the unit halt, unit sphere bundle of r over
Vr.
Since Vp = V x p ЕГ is not in general a finite simplicial complex we have to be precise regarding the definition
of К -homology for arbitrary simplicial complexes A'. We take А'.(Л') = bin Л'.(У) where V runs through
compact subsets of X. In other words we choose A' homology with compact supports in the sense of [Sp]
Axiom lip. 203.
Let Hl(Vv, Q) be the ordinary singular homology of the pair (Bt,St) over V'p, with coefficients in Q, Since
it is also a theory with compact supports, the Chern character
is a rational isomorphism.
Since we are interested mainly in the case of orientation preserving dilfeomorphisms, let us assume that V
is oriented and that Г preserves this orientation. Then the bundle r over V'r = V Xp ?T is still oriented,
and letting U be the orientation class of г on Vp, we can use the Thom isomorphism ([Mi-S] Theorem 10, p.
259):
Ф ¦ Hr,+n(Vr,Q) - H,(Kp,Q) (n = dim г = dim V)
(where ф(х) =р,((/Пг) Vz € Я,+„((ВТ, Sr, Q) and where p is the projection from Вт to the base Vr).
Thus <^<>ch is a rational isomorphism:
To construct the analytic assembly map ft : Krl(Vr) —> A',(Co( V) >i Г) requires a better understanding of
the Л'-homology of an arbitrary pair (here the pair Br,ST over Vp). This follows from:
Proposition 5. a) Let M be a Spin" manifold with boundary, and assume that M is compact. Then one
has a Poincare duality isomorphism:
K'(M)<* K.(M,dM).
b) Let (X, A) be a topological pair of simplicial complexes and let x 6 Л".(Л", Л). Then then exists a compact
(Spinc) manifold with boundary (M,dM), a continuous map f : (М,дМ) — (Л',.4) and an element у of
K.(M,dM) with f.(y) - x.
The Chern character is then uniquely characterized by the properties:
l)/.ch(») = ch(/.(»)).
2) If M is a compact Spin0 manifold with boundary, and z 6 [\.(M,dM) is the image of у € A'*(M) under
Poincare duality, one has:
ch(*) = (ch(y) • Td(M)) П (Л/, дЩ.
(Here Td(M) is the characteristic class associated to the Spin' structure of .\f as in [Bau-D]).
Now let (N,F,g) be a triple where N is a compact manifold without, boundary, F € I\'(N), and g is
a continuous map from N to Kr = V xp ET, which is A'-oriented, i.e., such that the bundle TN ®д"т
is gifted with a Spin0 structure. To such a triple corresponds an element of K.(Br,Sr) as follows. Let
B,S be the unit ball, unit sphere bundle of g*r on N. Then В is a Spinc manifold with boundary so
that the Poincare duality isomorphism assigns a class у € h\(B,S) to the pull-back of F to B. Then put
l(N,F,g)] = g.(y) 6 K,(Bt,St). For convenience any triple (N,F,ij) as above will be called a geometric
cycle.
Proposition 6. a) Any element of A'.,T(Vp) = h'.(Br,Sr) is of the form [[N,F, <f)] for some geometric
cycle (N,F,g).
b) Let (N,F,g) be a geometric cycle , N' he a compact manifold and f ; N' —> ;V a continuous map which is
K-orienUd: i.e. TN' ®f'TN is gtfled with a Spin" structure. Then, for any F' 6 K'{N') with /!(F') = F
one has:

l(N',F',gof)] = [(N,F,g)] in K,(Bt,St).
Here /! : K'(N') — K'(N) is the push-forward map in A' theory (cf. [Bau-D]).
Proof, a) By Proposition 5 b) there exists a compact Spin" manifold with boundary (M,dM), an element
у of K,(M,dM), and a continuous map / : (M,dM) —• (Bt,St) with i = /.(!))¦ By transversality one
may assume that the inverse image in M of the 0 section of r is a submanifold JV of M with normal bundle
v the restriction of f(j) to JV. Since the boundary of M maps to Sr, the manifold JV is closed without
boundary. Let g be the restriction of / to JV. Then g is a continuous map from N to Vr, and the bundle
TN © 0*7- = TN © i/ is gifted with a Spin' structure. Let В be the unit ball bundle В i JV of the bundle
v = g*r. Then since p* : A'*(JV) —» K'(B) is an isomorphism the answer follows from Proposition 5 a),
b) Let (B,S), (B',S') be the unit ball, unit sphere bundle of д"т and f"g"т over N and JV', and / :
(B',S') -» (B,5) the natural extension of /. One has f~\(p'(F')) = P"(F)- Tlllls tlie conclusion follows
since /' is Poincare dual to /. : K.(B', S') -» K.(B, S).
If we translate the Chern character ф о ch in terms of geometric cycles we get:
Proposition 7. Let (N,F,g) be a geometric cycle. The
<pocb[(N,F,g)] = g.(chF Td(TN®g'r)n[N}) 6 W.(V'r,Q)-
Note that we assumed that г was oriented, hence N is oriented since the ЫшсПе TN iBj'r is Spinc and
hence oriented.
So far we have just described the general elements of the group A'W>T( VY-) aIU' computed l.hcir Olieru character.
We shall now construct the analytic assembly map:
I* : K.,T(Vr) ^ K.(Co(V)xT).
Given a geometric cycle (N,F,g) let (by Proposition 3) JV be the corresponding Г-principal bundle over JV
and 7j : N —> V the corresponding Г-equivariant A'-oriented map. By Proposition 3 we can assume that g is
smooth. Then by section 6 we get a well-defined element g\ 6 E(C'o(JV),C'o(V')), and since Ihe construction
of <?! is natural, the corresponding deformation passes to the crossed products by Г, hence yielding:
g\r e E(Co(N)xT , Co(V)xT).
As the action of Г on N is free and proper, the crossed product Ca(N) X Г is strongly Moriia equivalent to
Co(JV/F) = Co(JV) so that we can view <?!r as an element of:
E(C0(N),C0(V)xT).
We then define /j(N, F, g) as the image gl-r(F) of the A'-theory class F 6 A'.(Cn(JV)) under the element g\r.
Theorem 8. There exists an additive map ft o/A'*>T(Vr) to A\,(Co( V) X Г) such that for any geometric cycle
(N,F,g) as above /j(N,F,g) = g'.r(F), where F 6 A',(Co(JV)) «5 viewed as <m element of A'(Co(JV ХГ))
through the Moriia equivalence, and J'r as an element of E(Co(N) x Г, Co( V) X Г).
We shall now apply the theorem to give examples of crossed products A = (.'(У)>чГ, with V a compact
manifold, where the A"-theory class of the unit 1 of A is trivial. We first need to find a geometric cycle
(JV, F, g) whose image ^(JV, F, g) is the unit of A.
Lemma 9. Let F 6 K»(Co(V)), and consider ihe geometric cycle (V, F, p) where, jj : V —, V/Y is the quotient
map with its tautological К-orientation. Then ii(V,F,p) is equal to j.(F) where j : C»(V) —» A — Co(V) »Г
is the canonical homomorphism of Co(V) into the crossed product by Г.
90
The proof is straightforward ([CO]3]).
With this we are ready to prove:
Theorem 10. Let V be a compact oriented manifold on which Г acts by orientation preserving diffcomor-
phisms. Assume that in the induced fibration over ВТ with fiber V : V x p Er — ВГ, Ihe fundamental class
[V] of the fiber becomes 0 m Hn(V xr?T,Q). Then the unit of the С-algebra Л = С(К)>!Г is a torsion
element of Л'о(Л).
Proof. By Theorem 8 we know that the map ц : K'(V, Г) —» K.(C(V) X Г) is well-defined. Thus it
is enough to find x ? A';(Vr) with /л(х) — \a and ch(ar) = 0 (since the Chern character is a rational
isomorphism (see above)). Now Lemma 9 shows that /i(x) = 1A where x is the A'-cycle (V, lv,g) where
lv stands for the trivial line bundle on V, g is the map from V to Vr given by g(s) = (s x *о)/Г for some
t0 6 ?T, and where the Л'-orientation of the bundle TV ®TV comes from its natural complex structure.
By Proposition 7 the Chern character of this A'-cycle is equal to g.(Td(Tc(V)) D[V]). Let r be the bundle
on Vr associated to TV. Since the restriction of г to any fiber coincides with TV, we see that Ihe Chern
character of x is given by Td(rc)g»[V], and hence is equal to 0.
As a nice example where this theorem applies let us mention:
Corollary 11. Let Г С PSL B,R) be a torsion free cocompact discrete яиЬдпир. Lei Г act on V = Pi(R)
in the obvious waj/. Then in the С-algebra A = C(V)xT the unit \A ts a torsion element of K0(A).
Proof. Let H — {z 6 C,Imz > 0} be the Poincare space, and Л/ = H/T be the quotient Riemann surface.
Let us identify H to ?T since it is a contractible space on which Г acts freely and properly. Then as in
[Mi-S] p.313 we can identify the induced bundle Vr = Pi(R)xr ?T = Pi(R) Xp II over BY = M to the unit
sphere bundle of M, denoted i]. It follows that the Euler class e(rj) of ц is equal to 2 — '2g times the generator
of H2(M, Z). Applying this with the Gysin sequence of the tangent, vector bundle TM; (cf. [Mi-S] p.143).
shows that ir"([M]") is a torsion element of Н2(т],2), where ж : r/ —• M is the projection and [M]* is the
generator of H2(M, Z). But >j is an oriented manifold and the honiology class of the fiber, [Pi (R)], is Poincare
dual to ir'([M]), and hence is also a torsion element so that ch(j;) = 0.
91

II.8. The leaf space of a foliation
a) Construction of C'(V,F)
Let (V, F) be a foliated manifold of codimension q. Given any ifV and a small enough open set W С V
containing x, the restriction of the foliation F to W has, as its leaf space, an open set of R', which we shall
call for short a transverse neighborhood of x. In other words this open set W/F is the set of plaques around
x. Now given a leaf L of (V, F) and two points x, у G L of this leaf, any simple path 7 from x to у on the leaf
L determines uniquely a germ /1G) of diffeomorphism from a transverse neighborhood of x to a transverse
neighborhood of y. One can obtain /1G) for instance by restricting the foliation F to a neighborhood N of 7
in V sufficiently small to be a transverse neighborhood of both x and у as well as of any 7(/). The germ of
diffeomorphism h(j) thus obtained only depends upon the homotopy class of 7 in the fundamental groupoid
of the leaf L, and is called the holonomy of the path 7. The holonomy groupoiil of a leaf L is the quotient of
its fundamental groupoid by the equivalence relation which identifies two paths 7, 7' from x to у (both in L)
iff ftG) = h(Y). The holonomy covering L of a leaf is the covering of L associated to the normal subgroup of
its fundamental group Ж\(Ь) given by paths with trivial holonomy. The holonomy groupoid of the foliation
is the union G of the holonomy groupoids of its leaves. Given an element. 7 of G, we denote by x = s(j) the
origin of the path 7, and by у = r(j) its end point, and v and 5 are called the range and source maps.
An element 7 of G is thus given by two points x = s(j), у = r(y) of V together with an equivalence class of
smooth paths: j(t), t G [0,1]; 7@) = x, 7A) = y, tangent to the bundle F (i.e. vvith 7 (t) 6 F-,(t), Vt 6 R)
up to the following equivalence: 71 and 72 are equivalent iff the holonomy of the path 73 7J at the point
x is the identity. The graph G has an obvious composition law. For 7,7' 6 G, the composition 707' makes
sense if 5G) = r(Y). The groupoid G is by construction a (not necessarily Ilausdor(f) manifold of dimension
dimG = dimVr + dimF ([Win]).
If the leaf L which contains both x and у has no holonomy (this is generic in the topological sense of dense
G«'s) then the class in G of the path j(t) only depends on the pair (y,x). In general, if one fixes x — .4G),
the map from Gx = G,5G) = x} to the leaf L through x, given by 7 6 Gr >— у = r(f) is the holonomy
covering of L.
Both maps r and 5 from the manifold G to V are smooth submersions and G is a smooth groupoid in the
sense of definition 5.1. The map (r,s) to V x V is an immersion whose image in V x V is the (often singular)
subset {(y, iNfxy;j and x are on the same leaf}. The topology of G is not in general the same as that
of a subset of V x V, and the map G —• К x V is not a proper map. By consl.rncl.ion, the C*-algebra of the
foliation is the C*-algebra C'(G) of the smooth gronpoid G constructed according 10 proposition 5.3, but
we shall describe it in detail. We shall assume for notational convenience that, the manifold G is llausdorff,
but as this fails to be the case in very interesting examples we shall refer to [C011)] to remove this hypothesis.
The basic elements of C'(V, F) are smooth half-densities with compact support, on G, / 6 G'f (G, f21/2),
where Q7 for 7 G G is the one dimensional complex vector space Qj- " О Qy " where s(y) = r, '"G) = y,
and fii'2 is the one-dimensional complex vector space of maps from the exterior power Ak Fx, k = dim F, to
С such that:
p(Xv) = \X\1/2p{v) Vu 6 AkFx , VA 6 R.
Of course the bundle (Q^2)X?V is trivial, and we could choose once and for all a trivialization v making
elements of C^(G, fi1'2) into functions, but we want to have canonical algebraic operations.
1/2), the convolution product / * g is given by the equality:
For /, <?
(f*9)h)=
This makes sense because for fixed 7 : x —<¦ y, fixing vx G Л*/1*, vy G hk'Fy, the product /G1 )il(i^
a 1-density on G" = {71 G G;i'Gi) = y}, which is smooth with compact support (it vanishes if -jy
92
) defines
support
/), and hence can be integrated over G" to give a scalar, (/ * g)(j), evaluated on ur, vy. One has to check
that f *g is still smooth with compact support, which is trivial here, and remains true in the non-IIausdorff
case.
The * operation is given by /"G) = /<7~'). '-e- if 7 : * — У and п„ G KkFc, vy G SkFy, then /"G)
evaluated on o,, vy is equal to /G"') evaluated on vy, 11,. We thus get. a «-algebra С?°(О,?2''2). For
each leaf L of (V?F) one has a natural representation jr^ of this *-algebra on the i2-space of the holonomy
covering I of L. Fixing a base point x G L, one identifies Z with Gx = {7; s(j) = x] and defines:
where ? is a square integrable half-density on Gx. Given 7 : x —> у one has a natural isometry of L2(G,:)
onto L2(Gy) which transforms the representation irx to iry. Applying proposition 5.3 we thus get:
Definition 1. C'(V,F) is the C'-algebra completion ofC™(G,uil2) with the norm \\ f ||=Snp|| »,(/) ||.
If the leaf L has trivial holonomy, then the representation ir^, x G L, is irreducible. In general its conimutant
is generated by the action of the (discrete) holonomy group G* in L2(Gr). If the foliation conies from a
submersion p : V —> B, then its graph G is {(я,у) G V x ViH1) = р(и)}, which is a subinanifold ofKx1
and C*(V, .F) is identical with the algebra of the continuous field of liilhori spaces L'1 (p~' {x})t€B. Thus
(unless dimF = 0) it is isomorphic to the tensor product of Са(П) with the elementary C-algebra of
compact operators. It is always strongly Morita equivalent to Co(B). If l.hr foliation comes from an action
of a Lie group H in such a way that the graph is identical with V x //, then C"[V,F) is identical with the
reduced crossed product of Co(V) by H (cf. Appendix C). Moreover the construction of C*(V, F) is local in
the following sense:
Lemma 2. IfV С V is an open set and F' is the restriction of F to V", then the tjmph G' of(V',F') is an
open set in the graph G of(V, F), and the inclusion Cf (C, fi1/2) С Cf (G, Q'/2) extends to an isometric *
homomorphism ofC"(V',F') into C*(V,F).
This lemma, which is still valid in the non llausdorff case ([Co,0]), allows one to reflect, algebraically the
local triviality of the foliation. Thus one can cover the manifold V by open sets IK, such that. F restricted
to Wi has a Hausdorff space of leaves: S, = Wi/F, and hence such that the G""-algobras C'[Wt,F) are
strongly Morita equivalent to the commutative C*-algebras Co(Ri). These subalgehras C~(\Vi,F) generate
C'(V,F), but of course they fit together in a very complicated way which is related to the global properties
of the foliation.
/?) Closed transversals and idempotents of C*(V,F)
Let (V,F) be a foliated manifold, and N С V a compact submanifold with diniA' = codimF, everywhere
transverse to the foliation Then the restriction F' of F to a small enough tubular neighborhood V of N
defines on V a fibration with compact base N. In other words, there exists a fibrat.ion V -?-» N with
fibers R*, к = dim-F, such that for any x 6 V, one has J^, = Ker(/i.),. The C-algebra C(V'.F') of
the restriction of F to V' is thus strongly Morita equivalent to G{N), and in fact isomorphic to G(N) ® 1С
where К is the C*-algebra of compact operators. In particular it contains an idmnpotent e — e2 = e",
e = In ® / 6 C(N) ® K, where / is a minimal projection / G ?¦ Using the inclusion C'(V,F') С C*(V, F)
given by the above lemma 2, we thus get an idempotent of C'(V,F), which we shall now describe more
concretely. The transversality of N with the foliation ensures the existence of a neighborhood V of G'0' in
G such that:
7 G VoV , 5G) G N , r(y) G /V => 7 G G@1,
93

where G(o) = {(x,x);x G V] is the set of units of G. Let then ? 6 Cf {r-l(N),s"(W12)) be a smooth
section on the submanifohl r~i(N) С G of the bundle s*(fi'/2) of half-densities, such that:
a) Support (CV
0) Jrb)=
)=y
6 JV.
The equality «G) = ? .(•,<>_.<„ {(У7'1){(У) defines an idempotent e G Gf(G,«1/2) С С*(У,^). Indeed
for each x G V the operator зг^е) in L~{GX) is the orthogonal projection on the closed subspace of L?(Gx')
spanned by a set of orthogonal vectors labelled by the countable set / = r~*(N) П Gr, namely (v-t)-relt
where:
')-rG') = «GG')-1) V7'GGI.
A more canonical way to define the (equivalence class of the) above idempotent e G C*{V, F) is to show that
the C*-niodule eA, A = C"(V,F), is the same as the completion of Cf (r~l(N),s'(uxl2)) with the right
action of Cf(G, fi1/2) given by convolution:
(
)G)=/
and with the Cf(G, n''2)-valued inner product given by:
The construction of this C-modnle ^';v over C'*(V, F) makes sense whether N is compact or not, but in the
former case the identity is a compact ciuloiuorplusin of €n and ?/v = eC(V, F) for a suitable idempotent
e € C*{V,F) (the identity being of rank one).
Fig,7. Transversal
If N is compact and meets every leaf of [\\ F), then ?/v gives a strong Morita equivalence between C*(V, F)
and the C*-a.lgebra of compact endomorphisms of ?д- which, if we assume that N is compact, is unitai. This
C*-algebra is the convolution C*-algebra of l.he reduced groupoid Gn-
G.v = {"/ €C?;rG) € N,s(j) e N) .
The groupoid G/v is a manifold of dimension q — codimi^, and the convolution product in C^
by:
94
is given
/Gi)!7G2)
The G*-algebra norm on C^°(Gn ) is given by the supremum of the norm || *x(f) || where for each x 6 N =
G$\ ** is the representation of C?°(Gn) in f2(Gyv,x) given by:
(тД/ЮG)= E /Gi)fG=) VT 6 GN , 5G) = x.
We shall denote this C-algcbra by C^(V,F), and compute it in a, simple example.
Let thus (V, F) be the Kronecker foliation dy = OtU of the 2-torus V = T2 = R2/Z2 with natural coordinates
(x,y) & R2' Here в 6 ]0,1[ is an irrational number.
The graph G of this foliation is the manifold G = T2 x R with range and source maps: G —• T2 given by:
'•((J-, .'/),<)= (J.- +
and with composition given by ((j;, 1/),/)((*',,1/),/') = ((*',!/'),/ + '¦') for any pair of composable elements.
Fig.8. Tin' kroucckcr foliation with a closed transversal T
Every closed geodesic of the Hat torus T" yields a rampart transversal. More precisely, for every pair (p, q)
of relatively prime integers we let /V,, , be the siibmainfokl of T" given by:
The graph G reduced by JV = /V,,,,, i.e. GN = {7 G G\ r(-j) G N,s(j) G N], is then the manifold GN = T x Z
with range and source maps givon by:
where 0' G R/Z is determined uniquely by any pair ())','/') of integers such that pq' — j/q = 1, 9' = F~e~\ ¦
The G*-algebra G^( V, F) is the crossed produci. of C(T) by the rotation of angle в', i.e. it is the irrational
rotation G*-algebra ([H.ii]) An1 ¦ generatetl by two unitaries U, V such that:
VU =expBTi'0')?;Vr.
9-r)

Since N = JVj,,, meets every leaf of the foliation, it follows that C"(V,F) is strongly Morita equivalent to
Agi for any relatively prime pair (p, q). In particular for p = 0, q = 1 we get As and by transitivity of strong
Morita equivalence we see that if в, в' are on the same orbit of the action of PSL{2, Z) then Д» is strongly^
Morita equivalent to Agi, an important result due to M. Rieflel [Rii]. The finite projective C*-module
? — ?-fA over As which achieves the strong Morita equivalence with Agy (i.e. At' ^ End^,(?)) is obtained
directly from the above geometric picture. One first determines the manifold:
E?.i = {? 6 G ; r(j) G ЛГр,, , sG) 6 JVo.i} •
Let us assume that p > 0 to avoid the trivial case p = 0. One finds then that. EP<1 — {(@,y),2) ; i ? H,
py = t(q — p$) modulo 1}. It is thus the disjoint union of p copies of the manifold R, since the value of
у 6 R/Z is not uniquely determined by the equality:
py = i(q — рв) modulo 1.
Working out the Дц-valued inner product on С^(ЕР,Я) one finds the following description of the C"-module
?p,q-
One lets ? = i - 9 and lets Wu W2 be unitary operators in С = /v such that rt'f = VKf = 1, WxWi =
exp Bxi'i) W2W1. Then V\ and V2 are the operators on C"(R) given by:
(Vif )(*) = ?(* - <0 , (V2f )E) = expBTis)f («) V.f G R.
Thus the action of As on ?,,,, is determined using the identification C^( ?,,.,) = C™(R)QK by the equalities:
?U = (V, ® IVi)f , ?V = (V2 ® VK2)? V? G С~(/?Р.„)
where U, V are the above generators of Ae.
The До-valued inner product which allows one to complete Cf(E,,4) and gel. ?r я is given by the value of
the components (f, п)т,„ of
m,,i i/mV":
We shall come back to these modules in later chapters.
Of course foliations can fail to have such a closed transversal N, and we shall show in an example that even
C"(V F) can fail to have any non zero idempotent. We let Г be a discrete cocompact subgroup of 5LB,R).
Then V = 5LB,R)/F has a natural flow Ht, the horocycle flow, defined by the action by left translations of
the subgroup:
( Гл ni 1
of SLB,R).
We let F be the foliation of V in orbits of the horocycle How. First, the How is minimal, so, [F-S][IIi-S2]
C*(V, F) is a simple C*-algebra. Then, letting /i be the measure on V associated to the Haar measure of
SLB,R), we can associate to p (which is Ht invariant for all ( G Щ a transverse measure Л for (V, F) and
hence a trace г on C'(V,F). By simplicity of C'(V,F) this trace г is faithful. Thus for any idempotent
eeC'(V,F) one has:
0 < т(е) < оо if e^O.
Now let Gj, s G R, be the geodesic flow on V, defined by the action by left translation of the subgroup
< _j ; s6R>. For every s, G, is an automorphism of (V,F), since the equality G,HiG~x =
0 1 Г1 01 \c-° 0"
96
Ht-,.t (which follows from:
). shows that for у = G.(x),
jr = (G,),FX. bet 9, be the corresponding automorphism of C"(V, F). For every i G C"(V, F) the map
s ,_> 9,(x) from R to C"(V,F) is norm continuous, which shows that if e is a self adjoint idempotent, then
в,{е) is equivalent to e for all s G R and hence т(в,(е)) = т(е), Vs G R-
But though ii is invariant obviously by the geodesic flow, the transverse measure Л is not invariant by G,;
indeed the equality CHtGj1 = Яе-"( shows that GS(A) = е'Л for all s G R. Thus rod, = е'т, and
so т(е) = 0 for any self adjoint idempotent e. So C'(V, F) does not have any non zero idempotent though
it is simple (cf. [Bl2] for the first example of such a C*-algebra). We shall describe in section 9 an other
example with a unital C*-algebra.
T) The analytic assembly map ^ : K,T(BG) — K(C*(V, F))
In this section we shall show how the construction of Л'-theory classes for the orbit space of a group action,
discussed in section 7, adapts to the leaf spaces of foliations. We shall thus get a very general construction
of elements of K(C'(V, F)) from geometric cycles, i.e. A'-oriented maps of compact manifolds to the leaf
space. Our first task will thus be to define carefully what we mean by a smooth map:
/ : W — VIF
where W is a manifold.
Any smooth map Wi V gives by composition with the canonical projection ]i : V —• V/F a smooth map
/ = pop from W to V/F. But as in the case of orbit spaces (section 7) a general smooth map / : W —» V/F
does not factorize through V and is given by a Cecil cocycle (Oi,jij) on \V with values in the graph G of
(V, F). More precisely (П(),€; is an open cover of W and -щ : tliflflj — G is a smooth map such tliat:
7ij(*)»7ji U) = lit (¦») Vx 6 Qi П Qj П nt..
2,- —• V patch together as maps p<>7jj from Qi to V/F.
Thus the smooth maps
Given such a Cech cocycle (?),-,7,-j) one constructs as follows a principal right G bundle over W which
captures all the relevant information on the map /. Let Gj be the niajiifold obtained by glueing together
the open sets:
Qi = {(x,1)eQi xG; тф) = гG)}
by the maps (x,y) —' (x,jji(z)°j). Then let r/, sj be the smooth maps given by:
r; : Gj — W , г;(х-,т) = * V(j.-,7) G &
S/ :G;- V, M*,7) = sG) V(x,7)eiV
The groupoid G acts on the right on Gj; for a G Gj and 7' G CV such that s/(re) = rG#) the composition
007' G G/ is given by:
Given ct|,ft2 6 C?/ such that r/(ai) = »7(a2) there exists a unique 7' G Ci' such that or«2 = О-1О7'. Thus Gj
is a principal right G-bundle over \V in an obvious sense.
Definition 3. a) A smooth map f : \V —* V/F is given by its graph which us a principal light G-bundte Gj
over W.
/?) The map f is called a submersion iff the map sj is a submei'sion: Gj — V¦
As in the case of orbit spaces, the reason for the existence of sufficiently inniiy smooth maps / : W —* V/F
is that homotopy classes of such maps correspond exactly to homotopy clnwsrs of continuous maps:
97

W -* BG
where BG is the classifying space of the topological groupoid G. The space BG is only defined up to
homotopy, as the quotient of a free and proper action of G on contractible spaces. More precisely a right
action of G on topological spaces is given by a topological space У, a continuous map sy : У —> G^ and a
continuous map Ух,СДу where У x ,G = {(y, 7) 6 У x G ; sy(u) = >'(т)} such that A/071 @72 = 1/0G,72)
for any у G У, 7i 6 G, 72 6 G with r^j) = sy(y), 5G1) = rG2).
In other words the map which to each t G G^ assigns the topological space Y'j = Sy1^} and to each 7 G G,
7 : t —' t'y assigns the homeomorphism у —* 3/07 from Yf to У(, is a contravariant functor from the small
category G to the category of topological spaces. We shall say that such an action of G on У is free iff for
any у ?Y the map 7 G G'Y^V^ —> 1/07 G У is injective and proper iff the map ((/,7) —* (у, У07) is proper- A
free and proper action of G on У is the same thing as a G-principal bundle on the quotient space: Z = Y/G,
quotient of У by the equivalence relation:
yi ~ У2 iff З7 6 G , У107 = y2.
We shall say that the action of G on У has contractible fibers iff the fibers of ,vy : У —¦ G^ are contractible.
Then exactly as for groups (and with the usual paracompactuess conditions) the classifying space BG is the
quotient EG/G of a G-principal buudle У = EG with contractible fibers. It is unique up to homotopy, and
it classifies, when G is the graph of a foliation (V, F), the homotopy classes of smooth maps to V/F. In fact
we have already seen in sections 5 and 7 two other examples of classifying spaces for smooth groupoids. In
section 5 we computed BG where G is the tangent groupoid of a manifold M by using the action of G on
У = G<°) x R" coming from an immersion 3 : M —. R". In section 7 the homotopy quotient Vxr ET is
the classifying space of the semidirect product G = V ХГ, which is a smooth groupoid.
For the case of foliations one can, assuming that the holonomy groups GJ arc torsion free, construct ([Co,0])
EG as the space of measures with finite support on the leaves of (V, F). What matters is that BG is an
ordinary space which maps to V/F with contractible fibers. More precisely any foliation (V, F') with the
same leaf space as (V, F) determines a G-priucipal bundle, and if llie holonomy coverings of the leaves of
(V, F') are contractible then the total space V is the classifying space BG.
We shall now, under the above assumption that G* is torsion free for all x G V, construct, the geometric
group K. T(BG) and the analytic assembly map: ([Bau-C,] [Co-S])
K,.T(SG) — K(C'(V,F)).
First the transverse bundle rr = TX(V)/FX of the foliation is a G-ettuivariant. bundle, the action of G on т
being given by the differential of the holonomy: /1G). : тх -^ ту "ij : x — у. 'Го г corresponds an induced
real vector bundle, which we abusively still denote by r, on BG = EG/G. Thus the group K.T(BG) is
well-defined, exactly as in definition 7.4, as the twisted Л'-homology, with compact supports, of the space
BG. (The twisting by т is defined as in 7.4 from the pair (Bt,St) of the unit ball, unit sphere bundle of
r.) We moreover have the exact analogue of proposition 7.6: By a geometric cycle wo. mean a triple (W,y,g)
where W is a compact manifold, у G K'(W) a Л'-theory class and ij a smooth map: W —• V/F (definition
3) which is К oriented by a choice of a Spin" structure on the real vector bundle TW Ф ij't.
Proposition 4. a) Any element of K*iT(BG) is represented by a geometric cycle [W,ti,g).
b) Let [W, y, g) be a geometric cycle, f : W —* W a K-orientcd smooth map. Then for any x G A'*(W),
the following geometric cycles represent the same element of A', T(BG):
(W',x,gaf)~(W,f\(x),,l).
For any smooth K-oriented map W -Д V/F one can factorize /1 as 7 о / where / : W —» IK, is К -oriented
and smooth, while g : W\ —• V/F is a submersion in the sense of definition ,'J. 0) ([Co,0]). Thus using
98
orm the
oupoid
proposition 4.b We see that in order to define the analytic assembly map /i : K.T(BG) — K(C'(V,F)) we
just need to define ^(W, y, g) = g\(y) in the case where g : W — V/F is a smooth К-oriented submersion.
Using the pull back Fw by g of the foliation F, which is a foliation of W because g is a submersion (cf.
[Co-S]) one can then easily reduce to the construction of the wrong way fonrtoriality map:
p! K'(F')~K(C'(V,F))
where F" means the total space of the integrable bundle F" 011 V. (We refer to [Co-S] for the reduction to
this case which invokes the construction of a homomorphism of C'(W, Fw) to the 6"-algebra of compact
endomorphisms of a C*-module over C'(V, F), thus yielding an element s3 G E(C'(W, fw),C"(V, F)).)
We shall construct p! G E(Co(F"), C*(V, F)) as a deformation of the convolution C"-algebra of the vector
bundle F to that of the -smooth groupoid G of the foliation. In order to do so, we just need to defo
groupoid F (i.e. F^ = V С F as the 0-section and F is the union of the /ravpe FX, x ? V) to the gr
G. In fact we just need the relevant topology on the union:
G' = FU(Gx ]0,1])
where both terms are groupoids, ]0, 1] being, as in section 5, viewed as space. Here F is a closed subset of
G', and a sequence Gn,?n), 7n 6 G, г„ > 0 of elements of Gx ]0, 1] converges to (л, Л') G Fx С F iff ?„ —» 0,
х„ = s(jn) —» x, у„ = r(jn) —• x, V--1' —> X and length G,,) — 0. Hero the condition length G) < s
defines a basis of neighborhoods of the diagonal G'0' in G. In fact, using a Euclidean structure on F, we
can give the following local diffeomorphism: F x [0,1] -^ G' near ((x,A'),0);
v((!/,y),?) = ((y,expy(-?y)),E)GGx]0, 1] for <• > 0
where the latter pan- of points у and expv(fV) G Leaf of у define the end points ol'thc; path y(t) = expy(etY),
7GG.
As in section 5 one checks that with the above structure G' Is a smooth groupoui, and this suffices, using
the associated G*-algebra, to get the required deformation:
p\€E(Ca(F-),C(V,F))
(where we identified C*(F) with Co(F') using Fourier transform). Thus given a smooth /\"-oriented sub-
submersion g : W — V/F we get an element. g\ G E(C(W), C( V, F)) as the composition of eg о (?>iv)! G
E(Co(Fw),C'(V,F)) with the Thorn isomorphism 0 G E(C(W),C0(Ffv)) given by the Л' -orientation of g.
We can then state the main result of [Co-S] as:
Theorem 5. a) Let W be a compact manifold and let g : \V —• V/F be a smooth К oriented map. Tien
the composition f\ о j\ = g\ G E(C(W), C*(V, F)) is independent of the factorization of g = / о j through a
K-oriented submersion / : W — V/F.
b) The element g\ only depends upon the K-oriented komolofg class of ij and one has (у о li)\ — g\ о ft! for
any K-oriented smooth map h : X —* W.
The construction of the analytic assembly map ^ follows immediately from this theorem. One has:
Corollary 6. Let x G K.T(BG) and (W,y,g) be a geometric cycle repirsctilinq x. The element g\(y) G
K(C*(V, F)) only depends upon x, and 11 is an additive man:
Let us note also that this construction of Л'-Uieory classes of C'{V, F) contains, of course, the construction
of the classes of idempotents eT G C*(V,F) associated to closed transversals T of (V, F). More precisely the
composition of the inclusion T С V with the projection p : V — VjF yields a smooth map: / : T —• V/F
99

which is by construction etale and hence A'-oriented. One then has a geometric cycle (T, It,/) where It is
the class in Ka(T) of the trivial one-dimensional vector bundle. One checks c?asily that /i(T, It, /) = [er].
The construction of g\ for a smooth map g : W —> VjF has been extended by M. Ililsum and G. Skandalis
([Hi-Si]) to smooth maps of leaf spaces, after solving the very difficult case of the submersion V/F —> pt.
We shall come back to this point in chapter 3.
Remarks 7. a) We have only defined the geometric group as /CiT(#G) when the holonomy groups G%,
x G V, are torsion free. As in the case of discrete groups, the general case requires more care and will be
treated in section 10 (cf. also [Bau-C-H]).
b) Exactly as in section 7 the Chern character Ch, is a rational isomorphism of Л'„ T(BG) with H+(BG,Q),
(if we assume to simplify that r is oriented i.e. that the foliation is transversally oriented). Thus, as in
section?, any element z G K(C'(V,F)) of the form/i(z), where x G K.,T(BG) and Cli.(x) = 0, is a torsion
element.
100
II.9. The longitudinal index theorem for foliations
Let (V, F) be a compact foliated manifold. Let E\, &, be smooth complex vector bundles over V, and let
D : С°°(У, Ei) —• С°°(У, Ез) be a differential operator on V, from sections of ?, to sections of Ег. Let us
make the following hypothesis'.
1) D restricts to leaves, i.e. (D?)x only depends upon the restriction of ? to a neighborhood of i in the leaf
ofx.
2) D is ettiptic when restricted to leaves, (i.e. the principal symbol <т/э(?, j]) G llom(?i x, E-* T) is invertible
for any J) 6 F', т)ф О.)
In any domain of a foliation chart U = T x P the operator D appears as a family, indexed by t 6 T, of
elliptic operators on the plaques Pt. One can then use the local construction of a parametrix for families
of elliptic operators and patch the resulting operators by using a partition of unity in V subordinate to a
covering (Uj) by domains of foliation charts. What one obtains is an inverse Q for D modulo the algebra
Cf(G,Q}l2) = J of the foliation (section 8). To be more precise let us fix for convenience a smooth non
vanishing 1-density along the leaves and drop the ti's from the notation. Then to D corresponds a distribution
section D 6 C^°°(G,s*(E;) ® r'(Ei)) with support on G'(cn. To the quasi-inverse Q of D corresponds a
distribution section Q 6 C~'x(G,s'(E^) ® r'(E\)). The quasi-inverse property is then l.lie following:
QD- lBl G C
DQ-\e7 6
where 1e, corresponds to the identity operator C°°(V, Ei) —• C™(V, Ei).
Now the algebra J = C?°(G) is a two-sided ideal in the larger algebra A, under convolution, of distributions
T G Cr°°(G) which are multipliers of Q°(G), i.e. satisfy T * f G C~(G). mul f *T € C™(G), for any
/ G Cf(G). Thus, neglecting the bundles Ej for a while, the existence of Q means that D yields an invertible
element of A/J. It is however not always possible to find a representative 1У G A of the same class, i.e.
D' G D+J, which is invertible in A. The obstruction for doing so is an element of the /v'-t.heory group /Co(J),
as follows from elementary algebraic A'-theory ([Мц]). We shall however recall in cle.Lail the construction of
this obstruction Ind(O) G A'o(J), since this allows us to take the bundles ?"< into account and will also be
useful in chapter 3.
a) Construction of Ind(Z)) G A'o(J)
Let J be a non-unital algebra over C, and define l\o(J) as the kernel of i.he map:
Л'оG) — A'o(C) = Z
where J is the algebra obtained by adjoining a unit to J, that is, J = {(a. A); a G J, A G C}, and s(a, A) = Л
V(a,A) G J-
(For a unital algebra A'q is the group associated to stable isomorphism clnsses of finite pi-ojective modules
and direct sum.)
Let then A be a unital algebra (over C) containing J as a two-sided ideal, and let j : A —• A/J = Л be the
quotient map. Recall that finite projective modules push-forward under inorphinis of algebras.
Definition 1. Given J С A as above, a quasi-isomorpliism is </iven by <i ti'iple (?ь?:),Л) where ?\,?i яге
finite projective modules over A and h is an isomorphism:
h ¦ j.S, - j.?2.
Any element D of A which is invertible modulo J determines the qiiasi-isomorphisin (A,A,j(D)). A quasi-
isomorphism is called degenerate when h comes from an isomorphism T : и — ?V There is an obvious
101

notion of direct sum of quasi-isomorphisms, and a simple but crucial lemma ([Mi,]) shows that the direct
sum (?,,?2,/i) ф (?2,?i, h~l) is always degenerate.
More explicitely, let D 6 Horru(?i,?2) and Q 6 Hoitu(?2,?i) be such that j(D) = h and j(Q) = ft.
Then the matrix
T_\D + (l- DQ)D DQ-1
- [ \~QD Q
defines an isomorphism of ?, © ?2 with ?2 © ?i such that j(T) = / -1 I
It follows that quasi-isomorphisms modulo degenerate ones form a group which, as we shall see now, is
canonically isomorphic to /v'o(J) independently of the choice of A.
Let us first consider the special case A = J. Then the exact sequence: 0 — J —> J -^ С — 0 has a natural
section r : С — J, ? о г = id. Thus for any finite projective module ? over J the triple (?, (г о е).?, id) is a
quasi-isomorphism, which we denote p(?).
Let then A be arbitrary, and let a be the homoniorphism a : J — A. <\(a, A) = я + Al Va 6 J, A 6 C.
Proposition 2. Given J с A as above, the map a. о p is an isomorphism from h'0(J) lo the group of
classes of quasi-isomorphisms modulo degenerate ones.
The proof follows from the computation ([Mil]) of the Л'-theory of the fiborod product algebra:
{(<M,«2) 6 A x A,j(<n) = Л«з)}-
Given a quasi-isomorphism {?i,?i,h) we shall let Ind(/i) G h'o(J) be the associated element of Ka(J)
(prop.2). For instance if D is an element of A which is invertible modulo J then !nd(D) is the element of
'1 О]
Kq(J) given by [e] — [eg] where the idempotents e, во G Л/о(./) аго
above,
r= [O + (l - DQ)D DQ- 1
Thus, with So = 1 - QD, Si = 1 - DQ one gets:
0 0
, e = Te0T~' with, as
S0Q
Si
One has Ind(/i2o/ii) =
+ Ind(/i2) for any pair (?i,?2,/ii), (?¦_>, ?3. Л¦_>) of quasi-isomorphisms.
Let (V, .F) be a compact foliated manifold and let D be, as above, a longitudinal elliptic operator from the
bundle Ei to Ei- Then the inclusion С°°(К) С .4 of multiplication operators on V as longitudinal differential
operators (of order 0) allows to induce the vector bundles Ei to finite projective modules ? over A. The
above existence of an inverse for D modulo CC°°(G) is then precisely encoded in:
Proposition 3. The triple (?\,?i,D) defines a quasi isomorphism oner the algebras J — C™(G) С А.
We shall then let Ind(Z)) 6 fi'o(Cc°°(G)) be the index associated to (?u?-j, D) by proposition 2.
/?) Significance of the C*-algebra Index
By the construction of С(V, F) as a completion of the algebra С™(С) one has a natural homomorphism:
102
and we shall denote by lnda(D) the image h, lnd(D). In general we do not expect, the map Л,:
ЫС?{С))-+ Ka(C(V,F))
to be an isomorphism, and thus we lose information in replacing lnd(D) by Fnda(D) = /i,(Ind(D)). It is
however only for the latter index that one has vanishing or homotopy invariance results of the following type:
Proposition 4. ([С01З]) Lei (V, F) be a compact foliated manifold. Assume thai the real vector bundle F is
endowed with a Spin structure and a Euclidean structure whose leaf wise scalar curvature is strictly positive.
Let D be the leafwise Dirac operator. Then D is a longitudinal elliptic operator and 1т1а(О) = О.
In other words, with F of even dimension and oriented by its Spin structure, one lets S* be the bundle of
Spinors and D : C'X(V,S+) — C°°(V,S~) be the partial differential operator on V which restricts to leaves
as the leafwise Dirac operator.
The proof of the vanishing of \nda(D) is the same as the proof of .]. Rosenberg for covering spaces (cf.
[Gro-L]); using the Lichnerowicz formula for D"D and DO' one shows that, these two operators are bounded
from below by a strictly positive scalar. This shows the vanishing of Ind(I(D) 6 l^o(C""(V, F)) because
operator inequalities imply spectral properties in C*-algehras. It. is however not sufficient for proving the
vanishing of Ind(Z)) 6 A'o(C^°(G)). As an other example let us consider the leafwise homotopy invariance
of the longitudinal signature, i.e. of Inda(.D) where D is the longitudinal signature operator. This question
is the exact analogue of the homotopy invariance of the Г-invariant signature for covering spaces which was
proved by Miscenko and Kasparov. One gets ([Bau-Co] [IIi-S:i]):
Proposition 5. Let (V, F) be a compact foliated manifold with F even dimensional and oriented. Let D be
the leafwise signature operator. Then }nda(D) (z f\'o(C*[V, F)) is preserved under leafwise oriented homotopy
equivalences.
7) The longitudinal index theorem
Let V -^ В be a fibration where V and В are smooth compact manifolds, л\\Л lot F be the vertical foliation,
so that the leaves of (V, F) are the fibers of the fibration and the base В is the space V/F of leaves of the
foliation. Then a longitudinal elliptic operator is the same thing as a family (Oy)y^/j of elliptic operators
011 the fibers in the sense of [At-Si^]. Moreover the O"-algel>ra of I.he foliation [V, F) i.s strongly Morita
equivalent to C(B), and one has a canonical isomorphism:
K{C-(V,F))~K(C(B))= K[II).
Under this isomorphism our analytic index, Ind,,(D) G /\'(C*( V, P')) is the same as the. Atiyah-Singer index
for the family D = (Dy)yiB, Inda(D) S K(B) (cf. [AS4]). In this situation the Atiyah-Singer index theorem
for families (Loc.Cit.) gives a topological formula for lnd,,(Z)) as an equality:
Ind,,(D) = Ind,(J})
where the iopological index Ind((?>) only involves the Л'-theory class ar> G A'(F') of the principal symbol of
D and uses in its construction an auxiliary embedding of V in the Euclidean space R". (cf. Loc.Cit).
We shall now explain the index theorem for foliations ([Co-S]) which extends the above result to the case of
arbitrary foliations of compact manifolds and immediately implies the index theorem for measured foliations
of chapter I.
As in the Atiyah-Singer theorem we shall use an auxiliary embedding of V in RA' in order to define the
topological Index, Indi(Z)), and the theorem will be the equality JikI, = Ind,. This equality holds in
K(C"(V,F)) and thus we need an easy way to land in this group. Now given a foliation (V',F') of a
not necessarily compact manifold, and a not necessarily compact snhnuiuifold jV of V which is everywhere
transverse to F', then lemma 8.2 provides us with an easy map from A'(.'V) = I\(Cn(N)) to K(C(V, F')).
Indeed for a suitable open neighborhood V" of N in V' one has:
103

Ca(N) ~ C(V",F') С
(where the first equivalence is a strong Morita equivalence). Of course tlie resulting map: K{N) —¦
K(C'(V', F')) coincides with the map e! where e : N —¦ V'/F' is the obvious etale map, but we do not need
this equality (except as a notation) to define e!. The main point in the construction of the topological index
Ind( is that an imbedding i : V —• RN allows one to consider the normal bundle и — i*(F)± of i*(F) in RN
as a manifold N, transversal to the foliation of V' = V x R" by the integrable bundle F' = F x {0} С TV'.
First note that the bundle v over V has a total space of dimension d — dim V + N — dimF, which is the
same as the codimension of F' in V'. Next consider the map <p from the total space и to V' = V x R^ given
by
and check that on a small enough neighborhood N of the 0 section in j/ the map ip : N —¦ V' is transverse
to J7". (It is enough to check this transversality on the 0 section V С v whore it is obvious, and only uses
the injectivity of i. onfC TV.)
Theorem 6. [Co-S] Lei (V, F) be a compact foliated manifold, О a longitudinal elliptic differential operator.
Let i : V -> RN be an embedding, let и = (i.(F))x be the normal bundle ofi.F in RN, and let N ~ v be
the corresponding transversal to the foliation ofVx. RN = V bij F' = F x {0}. Then the analytical Index,
Inda(-D) is equal f5'lndt(a(D)), where &(D) G K(F') ts the K-thtory class of the principal symbol of D
while Ind((CT(-D)) is the image of a(D) G K(F") ~ A'(JV) (through the. Thom isomorphism) by the map:
K(N) 5 K(C'(V',F')) ~ K(C~(V,F))
(through the Bott periodicity isomorphism since C'(V',F') = SNC'(\',F) = C'( V, F) ® C0{RN).)
The proof easily follows from theorem 8.3 and the following analogue; of Icinma 5.0:
Lemma 7. With the notation of theorem 6, let x — (F, <x(?>), j»;r) 6c the i/eomctnc cycle given by the
total space of the bundle F, the K-theory class a(D) 6 f\~~(F) of the symbol of D. and the K-orienied map
F _» V ->¦ VIF. Then
\nda(D) =
K(C'(V. F)).
Note that a direct proof of theorem 6 is possible using the Tliom isomorphism as in section 5.
Of course the above theorem does not require the existence of a transverse measure for the foliation, and it
implies the index theorem for measured foliations of chapter I. A transverse meaAiire Л on (V, F) determines
a positive semi-continuous semi-finite trace, traceA, on C*(V, F) ami houre an additive map:
traceA : K(C'( V, F)) — R.
The reason why traceA о Ind( is much easier to compute than Ц-асед о liuln is that the former computation
localizes on the restriction of the foliation F' to a neighborhood V" of N in \" on which the space of leaves
V"IF' is the usual space N, so that all difficulty disappears.
Let us apply these results in the simplest example of a one dimensional oriented foliation (with no stable
compact leaves) and compute:
dimA K(C"(V,F)) cfl.
For such foliations the graph G is V x R and the classifying space DC is thus lioniotopic to V. Thus up to
a shift of parity the geometric group Л'„ T(BG) is the /\-t.heory group /\'* + t(l'). The projection
p-.V—V/F
is naturally Л'-oriented and the following maps from I\'(V) to A'(C"(V', F)) coincide:
1) p\ (cf. theorem 8.3) which, up to the Thoin isomorphism, I\'(F) ~ /\* + I(K) is equal by lemma 7 to
Ind,, = Indi.
2) The Thom isomorphism (appendix C):
ф: K'(V) = K(C(V))^ A'(C(V)>iR) = K(C'(V, F))
where we used an arbitrary flow defining the foliation. In particular by appendix С, ф is an isomorphism
and so are the map p! and the index Inda : K''{F) -^ K(C'(V, F)).
Taking for instance the Horocycle foliation, i.e. the foliation of V = SL['2, R)/F where Г is a discrete cocom-
pact subgroup of SLB,R), by the left action of the subgroup of lower triauguler matrices of the form ,
t 6 R, one sees that the analytical index gives a (degree 1) isomorphism of /\"( V) onto K(C'( V, F)), while
for the only transverse measure Л for (V,F) (the Horocycle flow being strongly ergodic), the composition of
the above analytical index with diniA : K"(C"(V, F)) i-> R is equal to 0, so in particular diniA is identically
0. This example shows that even when the foliation (V,F) does have a non-trivial transverse measure,
the ^-theoretic formulation of the index theorem (theorem 6) gives much more information than the index
theorem for the measured foliation (F, Л). As a corollary of the above we get:
Corollary 8. Let (V, F) be one dimensional as above, and let X he a transverse measure for (V, F). Then
the image dimA( K{C'( V, F))) is equal to {{ch(E),[C]);[E] G K'[V)}.
Here ch is the usual Chern character, mapping K"(V) to II"(V,Q), and [C] is the Rnelle-Sullivan cycle of
chapter I.
Corollary 9. Let V be a compact smooth manifold, and let <p be a minimal diJJcomoiyliLsm of V¦ Assume
that the first cohomology group H*(V, Z) is eqval to 0. Then the crossed product A — C(V) Xy, Z, ts a stmple
umial C*-algebra without any non-trivial idempotents.
As a very nice example where this corollary applies one can take the difFoomorphi.sm <p given by left translation
by 6 SLB, R) of the manifold V = SLB, R)/f, where the group Г, discrete and cocompact is chosen
in such a way that V is ahomology 3-sphere. For instance one can take in the Poincare disk a regular triangle
T with its three angles equal to ir/4 and as Г the group formed by product» of an even number of hyperbolic
reflections along the sides of T.
In order to exploit the results on vanishing or homolopy invariaiice of the analytical index Inda we shall
construct in chapter 3 higher dimensional generalizations of the above maps:
dimA . K(C'(V,F)) — R
associated to higher dimensional "Currents" on the space of loaves V/F whose differential geometry, i.e. the
transverse geometry of the foliation, will then be fully used.
104
105

11.10. The analytic assembly map and Lie groups
In this section we shall construct for general smooth groupoids G a group l\'"of)(G) which is computable from
the standard tools of equivariant A*-theory, к"ц, where H is a compact, group, and a map:
f ¦ 'Cp(G) - K(C'(G))
which will extend to the general case the constructions of the analytic assembly map of sections 4, 5, 7, 8
above.
The new ingredient needed to reach the general case is to understand the case when G has torsion, i.e. when
the groups G* = {76 G,rG) = 5G) = x} contain elements of finite order. We have carefully avoided this
case in the examples of sections 4, 7, 8 by requiring in sections 4 and 7 that the discrete group Г be torsion
free and in section 8 that the holonomy groups be torsion free.
After formulating the analytic assembly map ft in general, we shall then deal in detail with the case of Lie
groups.
a) Geometric cycles for smooth groupoids
In the examples of sections 4, 5, 7, 8 we constructed an additive map /1,
where BG is the classifying space of the topological groupoid G and Л'„ т is /\-homology, twisted by a
suitable real vector bundle on BG. As we said above, this is only appropriate in the case when G is torsion
free, and obviously does not surject onto K(C'(G)) when G is say 11 linite group. Elements of K.tT(BG)
were described as geometric cycles (cf. proposition 7.6 and proposition S.'l)
W -L BG
or equivalently, since BG classifies G-principal bundles, as free and proper 6'-spaces.
But since G was assumed to be torsion free, the hypothesis of freeuess did follow automatically from the
properness of the action. We shall now give the general definition of a geometric cycle and then explain how
it specializes to the previous one.
Definition 1. Let G be a smooth groupoid. Then a G-manifolil « given by:
a) A manifold P with a submersion a : P —» G'0'.
b) A right action of G on P : P xa G — P, (p,j) — jnf such that (;'о~ 1 )»7'J = /»G17з) vGi>72) 6 C?<2),
where P xa G - {(p, 7) 6 P * G ; <*(p) = rG)}.
Such a G-manifold provides us with a contravariant functor from the small category G to the category of
smooth manifolds, because since a is a submersion its fibers: гу~'{г}, j; a (jin\ are manifolds PXy while
p —> P07 is for 7 6 G, 7 : x —»]/, a diffeomorphism from Py to Pr.
We shall say that a G-manifold P is proper when the following map is proper:
PxaG^ Px P , (p,7) —(p,;«7)-
Let us define a groupoid structure on the manifold P xa G, by lotting (P xo 6')'°' = P with range, source
maps and composition given by.
= P .
= P°7
(P.7)"(P'.7') = (P.7-7')-
106
We thus get a smooth groupoid, which if the G manifold P is proper satisfies the following definition:
Definition 2. A smooth gronpoid G is proper iff the тир G —> C'(u) x G@1,- 7 — (>'G).sG)) is a proper
map.
The computation of /"L'(C*(G)) for smooth proper groupoids is perfectly duable by the usual tools of equiv-
equivariant /С-theory, Km of spaces, where H is a compact group. Indeed, if we look at the space of objects G^
of G, modulo isomorphism, i.e. at the quotient of G* by the equivalence relation TZ ~ (;-, s)(G), we get a
locally compact space X = G^°'/li- because of the properness hypothesis. Next, given x ? G*°\ its automor-
automorphism group: G% = {7 6 G;r(j) — 5G) = x] is compact, so that we are essentially dealing with a family
(#г)гех of compact groups indexed by z 6 X. For a compact group // the Л'-theory of the C*-algebra
C"(H) is naturaly isomorphic to the representation ring R(ll) which is the free abelian group generated by
classes of irreducible representations of H. The general case of K(C'(G)) for G a smooth proper groupoid,
though more complicated, is part of standard geometry.
We shall now complete the definition of geometric cycles for .smooth groupoids. Given a G-manifold P we
get another G-manifold Tg{P) by replacing Px, x ? G'o), by its tangent bundle TPr. Thus the total space
P' = Ta(P) is the space of tangent vectors to P which belong to the kernel of the map a, tangent to a.
Note that Tq(P) is a proper G-manifold as soon as P is a proper C>'-nixnifold.
Definition 3, Let G be a smooth groupoid. Then a geometric cycle for (! is ywcv by a proper G-manifold
P and an element у 6 A',(C*(TG
Note that TqP is a proper G-manifold, so that as we explained above the /\-theory of C(TgP>^G) is
computable by standard geometric tools; thus the qunlificativt; "geoinH fie" in definition 3 is appropriate.
Note also that the properness of the action of G on P yields the existence of a (.'-invariant Euclidean metric
on the real vector bundle Tg{P) over P so that there is no real distinction between TV,- and Tq. Let now Pi
and P2 be two proper G-manifolds and let /:
/ •• P, - P-i
be a smooth G-eqnivariant map. This means, with obvious notation, that. ui_(f(f>\)) = <ч(pi) Vpi 6 Pi, and
that:
/(Pi«7) = /(Pi)»7 vPi
, 7 S G , a(;-,) = r(T).
It implies that / defines a morphism from the functor j: —- Pi.,. to (.he functor j: — /\j.- and because of the
properness of both Pj's we can put together the deformations (Tfr)\:
defined in section 6. Note that the tangent maps Tfx are fi'-eqiiivarmntly /\'-oriented. This thus yields:
(Т/)! в E(C"(TaPi yoG) , C"(TOI>., xf.'))
and hence a corresponding map of Л'-theories:
- K(C'(Tt!P-,XiG)).
Proposition 4. Let G be a smooth groupoid. The fallowing relation 1.1 an equivalence relation among
geometric cycles: (Pi,.Vi) ~ {Pi.Vi) iff there exult a proper G-manifold P and G-eqiiivnriant smooth maps
/j : Pj — P such that
107

In order to show the transitivity of the above relation, one shows that given the following diagram of proper
G-manifolds and G-equivariant smooth maps:
P"
h'\
one can find a proper G-manifold P'" and G-equivariant smooth maps / : P — P'", f : P' —> P" such that
foh is G-equivariantly homotopic to f'°h'.
Under the above equivalence relation any finite number of geometric cycles (Pj, i/j) are equivalent, to geometric
cycles (P, Zj) with the same P, using say P = UPj and the obvious maps /,• : Pj — P. Moreover the group
law on equivalence classes given by:
is independent of the choice of P and coincides with l-he disjoint sum. We. shall denote by A"*op(G) the
additive group of equivalence classes of geometric cycles and call it. the group of lopological G-'mdices.
Proposition 5. In the examples of sections 4, 5, 7, 8 one has
Proof, let us treat the case (section 7) of a torsion free discrete group Г acting on a manifold V. Thus
G = V X Г is the semidirect product of V by Г.
Let (P, y) be a geometric cycle for the smooth gronpoid G. Thou the action of G on P yields an action of Г
on P, which is proper by hypothesis and is thus free since Г has no torsion. The map n : P —• V = G1-0' is
a smooth submersion and is Г-equivariant since:
zg = z*{a(z),g) V: G P , у е Г
so that a(zg) = s(a(z),g) = a(z)g.
We thus get according to definition 7.2 a smooth map from the manifold W = Р/Г to К/Г, and a corre-
corresponding map
a' ; \V -* Vxr?T.
Next, since the action of Г on P is free and proper, the crossed product (?*-alg<;hra Ca{TaP) ХГ is strongly
Morita equivalent, by proposition 7.1, to Co(TGP/T) = Co(E), where E is Hie total space of a real vector
bundle E over W = P/T such that:
E ®а"(т) = TW.
One has у G K(Ca(TGP) XT) = K{Ca(E)) = K'(E) = K.,a:(T)(W) by Poinc.are duality on W.
Thus a'(y) G K.lT(V Xr ЯГ) is well-defined. One thus obtains a natural шар j : /\,*op(G) -. K.,T(BG). It
passes to equivalence classes as in proposition 7.6. To check that this map j is surjective one uses proposition
7.6 a), and remarks that in that statement we may assume Unit the map ~j : /V —> V is a submersion. (Indeed
one can factor g as pr2ou where ргз : N X V — V is the second projection while h : N — N x V is given by
l(z,v) = (j,J(i>)) V; G -V , ii g I'.)
The injectivity of j follows from the definition of the equivalence of geometric cycles.
108
This proves the proposition for the examples of sections '1 and 7. The case of foliations follows from [Hi-Si].
The case of the tangent groupoid of a manifold follows from the following proposition, which allows one to
compute Ktof>(G) in many cases:
Proposition 6. Let G be a smooth groupoid and let Co be Ihe. category of proper G-manifolds and homoiopy
classes of smooth G-equivariant maps. Then if P is a final object for Co one has
This is easy to check.
This proposition will apply in particular to compute /\""O|>(G) when G is a Lie group (cf. /3) below).
We can now construct the analytic assembly map in general. Lot. (P,;/) be a geometric cycle for the smooth
groupoid G. Then for each x g G*0' let Ps — «~'{j:} be the corresponding manifold, and Gpx be its
tangent groupoid. Let G' be the smooth groupoid obtained by putting together the G'Pl, x G G'0', and
G" = G' X G be the semidirect product of G' by the natural action of G (which follows from the naturality
of the construction of the tangent groupoid). Then C"[G") gives vis a natural deformation of C*(TqP>jG)
to a G'-algebra which is strongly Morita equivalent to C"{G). We shall let [i[P,y) G [\,(C'(G)) be the
image of у G K,(C'(TaP' xG)) by this deformation.
Theorem 7. The above construction yields an additive map ft : /\t*o>(C-V) — Л'(С"(С)) which extends to the
general case ihe previously constructed analytic assembly тара of section /,, 5, 7, S.
We shall now investigate in more details the case of Lie. groups, where torsion plays an important role.
/3) Lie groups and deformations
Let G be a connected Lie group and let us compute the geometric group K'"op{G) and the analytic assembly
map /i in this special case.
Proposition 8. Let H be a maximal compact subgroup of G and V = T,(II\G) the tangent space at the
origin e to the homogeneous space H\G. Then one has a canonical isomoiphism:
~ K?op(G).
Here Kfj is equivariant A'-theory, and the vector space V is an 11~ъ\глсг for the isot.ropy representation of
H.
Proof. First by a result of A. Borel ([Bor]) the category i'u of proper G'-iiinnifolds and homot.opy classes of
smooth G-equivariant maps admits the homogeneous space P = //\G a-s a final object. Thus (proposition
6) one has:
I\op(G)= K,(C(TrxG)).
Next the smooth groupoid TPxG is equivalent to V X //, so that C'(TP xG) is strongly Morita equiva-
equivalent to G"(VX#) = Go(V)xtf. Now by [Ли] the K-tlmory of Cn(V) X // is the same as the Я-equivariant
Я-theory of V : KH(V).
Note that the latter group is easy to compute. If the dimension of V is even and the isotropy representation
of Я in V lifts to the covering Spin(V') of SO(V), one has a natural isomorphism r : T!.(FI) —» /i'h(V), the
Thorn isomorphism ([At-Sii]), where 7i(#) is the representation ring of //. We shall come back later to the
general case (proposition 14).
Let us now compute the analytic assembly map ft. We let. Go be the Lie group V X //, semi-direct product of
the vector group V by the compact, group H. As pointed out quite early in the theoretical physics literature
109

([Mui][Mu2] [Mi] [W-I] [Se5]), there is a close resemblance between the representation theories of G and of
Go which is tied up with a natural deformation from Go to G which we now describe. To that aim we just
need to describe the relevant topology on the groupoid G\ = Go U (Gx ]0, I]). Here Go is a closed subset,
and a sequence (д„,?„), gn 6 G, ?„ —» 0 converges to an element .'/ = (?,?) 6 Go = V X Я of Go iff the
following holds:
- ? e
= к
We thus obtain a deformation S 6 f?(C*(G0),C*(G)), and since C'(G0) = C"( V ХЯ) = C0(V)x/7, we
obtain a natural map from KH(V) = K.(C0(V)xH) to K(C'(C)).
Proposition 9. The above map is the same as the analytic assembly map ц : A"t'op(G) — A'(C(G)).
Proof. By definition (cf. theorem 7), the map [i is obtained from the semi-direct product of the tangent
groupoid Gp by the Lie group G. Let thus G2 = Gp XG. Then Px[0,1] is the space G2 of units of G2, and
G2 is equivalent to the smooth groupoid G2,z = {y 6 G3 ; Ht) 6 7 , sG) g 7} where 7 = e x [0,1] С GB0).
One then checks that G2,2 is the smooth groupoid G\ of the deformation from Go to G dicussed above.
Thus the deformation from Go to G gives, in all cases where /j is shown to be an isomorphism (cf. theorem
20 below), an isomorphism of the A'-theories of the group C*-algebras C^(G), which gives a precise content
to the observations of [Mi].
7) The G-equivariant index of elliptic operators on lioiiiogciwous spaces of Lie groups
Let as above G be a connected Lie group and Я a compact subgroup, P = 1I\G the corresponding ho-
homogeneous space. Every G-equivariant (hermitiaii) complex vector bundle E on P is obtained from a
corresponding (unitary) complex representation e of the isotropy group // in the fiber Es. We shall still
denote by €e the natural extension of this representation to the convolution algebra C'~°°(//) of distributions
on Я. Since distributions push-forward, the latter algebra C~™( //) is in a natural manner a snbalgebra of
the convolution algebra C~oa(G) of distributions with compact, support on G. When E is finite dimensional
the module Ce over C~°°(H) is finite and projective, and using the inclusion C'^iH) С C~"°{G) we can
consider the corresponding induced module <?? which is finite and projective over C~°°(G). It is given by
the action by convolution of C~°°(G) in the space C'^iP, E) of distribution sections of E with compact
support on P.
Proposition 10. 1) The convolution algebra J = C™(G) is a two sided ideal in the algebra A = C~00(G).
2) Let D be a G-invariani elliptic differential operator on P from a G-cquivanant bundle E\ to a G-
equivariant bundle ?2. Then the triple (C~°°(P,Ei), D) defines a qansi-isomorjiliism for Hie patr J,A.
Proof. 1) is standard ([C0-M4]). 2) follows from the existence of a quasi inverse for D modulo C%°(G) as
proven in [Со-МЦ] proposition 1.3.
Definition 11. We lei Ind(D) 6 Ka(C%°(G)) be the index of Hie quasi-isomorphism of proposition 10 2)
(cf. section 9 a)).
We shall now show in a simple example that this index is not. in general invariant, under homotopy of elliptic
operators, unlike the C*-algebra index Indo(D) obtained from the natural map:
coming from the inclusion Cf(G) С C'(G). This will in particular show that, the map j is far from injective
in general.
We take G = R and specialize to scalar operators so that, the two bundles Ej over P = R are trivial.
110
Proposition 12. The map D —> Ind(D) ? /\'o(G™(R)) is an injection of the projective space of non-zero
polynomials D = P (^), in /vo(C~(R)).
Proof. Consider the exact sequence of algebras:
0 — C~(R) — Л) — К — 0
where .4o is the convolution algebra of distributions on R with compact support and singular support {0}.
Then the quotient К ~ .40/G~(R) is a. commutative field (cf. (]). Any elliptic differentia] operator D on
R defines an element of .40 and o-(D) 6 A', o-(D) ф 0. By construction (.lie index, ind(D) 6 A'o(GJ°(R)), is
the image of o-(D) by the connecting map: d : A'i(A') — A'o(C™(R)) of algebraic Л'-tlieory ([M]). Moreover
since -4o is a commutative ring, one has a natural map:
det : A'i(^to) — -^0
given by the determinant. It follows that for any a 6 A", a / 0, #(a) = 0 there exists an element /3 of Ag
such that o-@) — a. Thus to show that d is injective on K'/C it is enough to show that, given any a 6 K,
а ф 0 it is impossible to find 0 invertible in Ao with a(fl) = a. This follows from the Paley-Wiener theorem
which shows that scalar multiples A60, A E С of the Dirac ni;iss at. 0 are tin1 only invertible elements of Ao-
(If a ? -45, then its Fourier transform 5 must vanish somewhere in С or be constant.)
The G*-alge,bra index, namely inda{D) — j. Xud(D) where:
is homotopy invariant, and ranges over a couni.rihlc ahcli;in group. It thus depends only upon the A'-theory
class of the principal symbol <7r of D, and is given in general by the following analogue of lemma 5.6:
Proposition 13. Lei G be a connected Lie group, P = J]\CJ (i proper homogeneous space over G, and D a
G-invanant elliptic diffcreiiiial operator on P. Then lnd,,( 0) = /«(hid D) ? I\{C*(G)) is given by
where the principal symbol а о of D defines a K-Uuorij class:
[>d] 6 K,,(Te(P)) ~ K(C-(TPxG))-
We have used the strong Morita equivalence of C''(TPxG) with C(T,,P X II) and the isomorphism
K(C*{T,>P >]Я)) ~ Кн(ТеР). A convenient description of the ll-v\\\i\variant. K~theory of the vector space
TBP — V which fits with symbols of differential operators is the following: Tin- basic objects to start with
are smooth .ff-equivariant maps ct : S ~* Iso(/'7i, E^), where 5 = S[V) is the unit sphere in V with respect
to an //-invariant metric, and E\. E2 are finite-dimensional unitary //-inociules. Two such maps
o-o e(C°°(S, lso(a,,fo))" , a, 6 (С'~(.Ь\Ьо( Е,,1-\))"
are called isomorphic, and we shall write n0 ~ alt if there exist у 6 коц(Еи, fc'i), Ф ? lso;/(F0, F\), where
IsoH denotes the Я-equivariant. isomorphisms, such that
^(tolfl = ац(?,)<р . for any ( 6 ,S'.
Further, ao and a, are said to be homotopic if there is an о 6 (C™{S x I. \ao(E. F))", where / = [0,1] with
the trivial action of Я, such that a\S x {0} ~ un and a\$ x {1} ~ щ. The set of all homotopy classes of
such maps will be denoted C. Phis is an a.belia.n semigroup, under the obvious direct sum operation. Let Co
denote the subsemigroup of all classes which can be irprpsenl.rd by i> couMnnl map a, a(?) = <p, ? ? S, with
<p E Isoj/(?i, ?:>)¦ Then C/Cq is not only a semigroup but actually а дгопр, wliich is isomorphic to Kn(V).
Ill

The elementary properties of the it-theory of C*-algebras and the G-invariant pseudo-differential
([C0-M4]) show a priori that the index Inda(D) € K(C"(G)) only depends upon the Л'-theory class [
Кц{ТйР) of the principal symbol of D (see [C0-M4] for more details). The proof of proposition 13 (
follows from the asymptotic pseudo-differential calculus ([Wh]).
Let us assume that the isotropy group Я is connected. Then if the homogeneous space P = H\G is (
dimensional the Bott periodicity theorem implies that the eqiiivariant Л'-theory K^(V) is zero, so |
proposition 13 has no content in this case. Let us thus assume that P = H\G Is et»en dimensional a
relax the condition of connectedness of Я by requiring only that Я preserves the orientation of V = JV|
We shall then determine /C//(V) and show that all its elements are obtained as symbols \gq\ where 0 i
G-invariant order one elliptic differential operator on P.
Let ?r : Я —> SO(V) be the representation of Я in V, and p : SpinV — SO(V) be the Spin covering^!
general it does not lift to SptnV, and so we form the double covering of II given by:
Я = {(h, s) 6 Я x SpinV ; jt(A) = p(s)}.
Let u 6 SpinV, к2 = 1, be the generator of the kernel of/), so that viewed as the clement A,и) 6 Я ii
central in Я and yields the central extension:
0 — Z/2 — Я — // — 1.
In particular the representation ring К(Я) inherits from 11 a. 2/2 grading and:
where *R.(H)± is generated by the equivalence classes of irreducible representations t of // such that e(wp
±1. Clearly one has:
7С-(Я)+~7г(Я)
and 'R.(H)- is by construction an X(//)-module. <
Let ? € Х(//)_ . We shall now construct a Dirac type operator fyc on P. let .5Г± be the half-spin represent*
of Spin(V). Since и € H acts by —1 on both ? and S it follows thai ?* = ? Q S^ are unitary //-moduR§|
We let E^ be the associated G-equivariant hermit-ian vector bundles over P. Let
Vf :C0O(P,E±) — C°°{P,TcPG E±)
be G-invariant connections, compatible with the Ilerniitian structure of
operator with coefficients in e as being the composition:
1
We define the (±) Difl
ft : Cf (P.
where с is the bundle homomorphism induced by the Clifford multiplication. Clearly ft are G-invarM
first-order elliptic differential operators.
Proposition 14. The following map /гоп»К.(Я)_ to K/,(V), V = Tt(P), is «» isomorphism: e 6 И(Л
We refer to [C0-M4] for details. It thus follows that when the dimension of P is even for II С G a mf
compact subgroup (with H\G equivariantly oriented) the analytic assembly map /i : f\'tOp(G) —¦ K(C
is the same as the Dirac induction map:
e e GгЯ)_ — inda(ft) e Л'(С"«7)).
112
Dirac induction was introduced by Schmid and Parasaralliy in [Sch] [Par] in order to solve the problem
of geometric realization of the discrete-series representations of semi-simple Lie groups. They proved the
following result:
Theorem 15. ([Sch] [Par]) Let G be a semmmple Lie group with finite center, H a maximal compact
subgroup. For each discrete senes representation к of G there is a unique II-module ? such thai
?r = Kernel ft , Kernel <Tt = {0}.
The relation between their use of Dirac induction and the map:
сеЩН)- — 1|кЦ?+)е K(C(G))
is provided by the following lemma:
Lemma 16. Lei D be a G-invanant elliptic differential operator on a proper homogeneous space P over
G. Assume that the corresponding Ililbert space operator L~(P, Et) — L~(P, E-2) is surjective. Then KerD
viewed as a C?(G)-module is finite and projective and
[Ker D] = Imln(O) m /\"o(<~:F')).
Of course in general one doe,s not have vanishing theorems for tin- tw^ted Dirac operator's ft and these
operators have non-zero analytical index Inda(D) in situations wliere both Keri} and Ker D* are reduced
to {0}.
In order to prove, in the context of semi-simple Lie groups G, thai, (.he i--kernel of Dirac type operators was
not {0}, Atiyah and Schmid relied on the index theorem for covering spaces [A<5] [Sin?], using the existence
of discrete cocompact subgroups Г С G. This theorem was then extended in [Co-Mi] to cover the general
case of homogeneous spaces of unimodular Lie groups. We shall now explain this result in details ([C0-M4]).
We assume as above that P = H\G is even dimensional oriented. Let. D : C™(P, Ei) — Cf{P,E2) be a
G-invariant elliptic differential operator. There is no ambiguity in extending D as a Ililbert space operator.
One specifies an Я-invariant volume form w 6 AmV, V — Te(P}t m ~ dim V, which fixes the orientation of
P as well as a G-invariant 1-density on P. The Hilbert. spaces L1{P, Ej} of /^--sections of the G-equivariant
bundles Ej are then well-defined, and ([C0-M4] lemma 3.1) the closure (in the Ililbert space sense) of the
densely defined operator:
D :
El) — Gf ( P, Ei
is equal to the distribution extension of D whose domain is {? 6 L-(P,E\) ; D? 6 ?-(P, i?2) in the
distributional sense}. Next, since G is unimodular, a choice of llliaar measure dg on G specifies uniquely the
Plancherel trace, trc on the von Neumann algebra of the (left) regular representation A of G in L'(G). It is
a semifinite faithful normal trace such that for any / 6 LJ{G) with A(/) bounded one has:
trGDf)'Hf)) = J\fM\2 <i«-
The restriction of this trace to C*(G) is semifiuite and semicoiit.inuoiis. This trace tve; gives uniquely a
dimension, dimGGr) (with positive real values), to any representation т of G which is quasi-contained in the
regular representation of G. The existence of a quasi inverse for D immediately yields:
Lemma 17. Let D be a G-invananl elliptic differential operator on P, anil let KerD be its L2-kernel. Then:
diinc(Ker D) < 00.
113

The index theorem for homogeneous spaces computes
IndG(D) = dimG(Ker?>)-diniG(KerZr)
in terms of the К theory class of the principal symbol of D:
Wo]eKH{V) , V = Tt{P).
By proposition 14 we just need to treat the case of a Dirac-type operator (ft, where ? 6 V.(H).
The first ingredient we need is the Chern character eft -. ЩН) —> H'ig,, II,R). Here II"(g, #,R) denotes the
relative Lie algebra cohomology with trivial coefficients, i.e. the cohomology of the complex (C(g, ff,R),d),
where
C(g, tf,R) = {a e Л'д* ; ixa = 0 for Л" 6 I) , A<r{h)a = a for h 6 Я}
and d:C»(g, H,R) — С»+1(в, H,R) is given by
da(x-i,...,xq+t) =
9 + 1к^,+1
Let us fix for the moment an Ad( ff)-invariant splitting of g, g = 1)фт. This specifies a O-invariant connection
on the principal bundle H —» G —» Л?/, whose connection form is given l>y the projection в : g —» h parallel
to m, and whose curvature form is prescribed by
Now let ? be aunitary representation of H on E. We denote by the same letter its differential, e : h —<¦ gH(E),
and we then define 6t 6 Л2т' ® glc(E) by the formula
ее(Л',У) = -^- ?F(ЛМ')) . Л', У 6 т.
2*7Г
Further, let us consider the form ir exp0e ? Am" © C. Actually, because ? is unitary, ir expS? (Е Лт'. In
addition, since for h 6 Я with image h E II one has
= е(А)в,(А',У)г(Л)-1 , Л', У 6 m,
one can see that ir exp@t is Я-invariant. This shows that its pull-back loAj, via the projection I—в : g —» m,
e
defines a cochain in ^ C4(g, H,H), which we will continue to dcnot.c- by the same symbol. Standard
q
arguments in the Chern-Weil approach to characteristic classes imply first that. /,r cxpQs is closed, and next
that the cohomology class in H*(g,H,R) it defines, and which will he. denoted eke, does not depend on the
choice of the vld(#)-invariant splitting of g. Finally, it is clear that ch?\ = c/iet if sl and ?2 are equivalent
unitary representations, so that we can define unambigoiisly cli : 1\.(!I) — Я*(д, /7,R).
The last ingredient we need is the analogue of the ./{-polynomial of Ilirzebrnch. To define it, we start from the
(real) /f-module V (which can also be viewed as an //-module) and form, as above, Qv *E Л2т" ® д?с(У)-
Then we construct the element in Am"
exp(i
-exp(-|
pull it back to an element in Лд* and, after noting that it is in fact a. cocycle, we define -4(д, Н) 6 H'(g, H,R)
as being its cohomology class.
Remark now that dim tfm(g,#,R) = 1, since G and II are niiimodular; in fact, Я'"@, H,R) = Cro@,//,R)
~ AmV. ]{Q = -?Q(i) e Я"(д, H,R), we define the scalar U[V] by the relation Q<"" = (Q[V])u.
We can now define a natural map Indt : K'n{V) —* R uniquely, using proposition 14, by:
Ind,(<r(^+)) = (ch(e) A{g, Il))[V]
for any element ? of 7J(#)_.
The index theorem for homogeneous spaces is then ([C0-M4]):
Theorem 18. Let D be a G-invariani elliptic differential operator on the proper oriented homogeneous
space P = ff\G, and let [aD] 6 Кц{У), V = Te(P), be /Ac I\-theory class of Us principal symbol. Then
dimG(KerD) and dimG(KerD*) are finite, and:
dimG(KerD)-dimG(KerD*) = Ind,([(TD]).
We refer to [C0-M4] for the detailed proof of this theorem. It is important to note that for large classes
of unimodular Lie groups the L2-kernels KerD of G-invariant elliptic operators on proper homogeneous
spaces are automatically finite direct sums of irreducible discrete-series representations. Thus for instance
([C0-M4]):
Theorem 19. Let G be a connected semisimple Lie group with finite center, and Id P = /7\G be a proper
homogeneous space over G. Then for any G-mvariant (pseudo) differential elliptic operator D on P the
unitary representation of G on the space of L~-soluiions of the equation Du = 0 in a finite sum of irreducible
discrete-scries representations.
6) The Л'-theory K(C;(G)) for Lie groups
Let G be a connected Lie group, H a maximal compact subgroup, if P = II\G is even dimensional then
(propositions 13 and 14) the Dirac induction map:
?6ВД_ - 1нс1„(#) 6 K(Cm{G))
coincides with the analytic assembly map
When P is odd dimensional the relevant part of the analytic assembly шар:
ft : Л7ор(С) - K.(C(G))
concerns Ktop instead of A't0op. Replacing G by G x R ajid using Bolt periodicity, one gets a similar
interpretation of /i in operator-theoretic terms (cf. [Co-M.,]).
When G is a connected solvable Lie group then ([C0-M.1] [Kas]) the Thom isomorphism (appendix C) shows
that the analytic assembly map is an isomorphism:
/v'op(G) •?¦ K.(C{G)).
For semisimple Lie groups the results of [Sell] [Par] [At-Si] [At-Sc-_>] on the geometric realisation of all
discrete-series representations by Dirac induction, together with [Ka.s] a-iid [C0-M4] section 7.5, suggested
that ft should be an isomorphism:
и : K-top{G) - K.{C;
It is crucial here to have composed ft with the natural inorphisin:
C'{G) — C;(G)
114
115

of restriction to the reduced C-algebra, i.e. to the support of tiie Planoherel measure. The point being that
ц itself fails to be surjective (proposition 21) as soon as О satisfies Ka/.hdan's property T.
After several important partial results [Vi] [Kas.(] [Pen-Pi] the hijectivity of /i was proven for general con-
connected linear reductive groups by A. Wassermann [VVa,].
Theorem 20. ([Wai]) Lei G be a connected linear reductive group. Then the analytic assembly map is an
isomorphism:
ц : A7op(G) - K.(C'r(G)).
The proof ([Wai]) relies on fundamental results of Arthur and IlarLsh-Chandra on the Plancherel theorem
for the Schwartz space of G, which yield a complete description of the reduced C-algebra C*{G) as a finite
direct sum of C*-algebras associated to each generalized principal series. The latter C*-algebras, though not
strongly Morita equivalent to commutative ones, are of the same nature as the C'*-algebra of example 2 /?).
It is of course desirable to obtain a direct proof of theorem 20 not relying on the explicit knowledge of C*(G)
provided by the work of Harish-Chandra. Such a direct proof for the injectivity of ft follows from Kasparov's
proof of the Novikov conjecture for discrete subgroups of Lie groups. II- implies in particular:
Theorem 21. [Kass] Let G be a connected Lie group. Then the analytic assembly map fi : /Чор(^) —>
K.(C;(G)) is injective.
Let us now explain why [i fails to be surjective on A*.(C*(G)) as soon as G has property T of Kazhdan,
which is the case for any semi-simple Lie group of real rank > 2.
First one has the following characterization of property T:
Proposition 22. ([Ak-W]) Let G be a locally compact group. The following properties are equivalent:
1) G has property T of Kazhdan ([Kaz]).
2) There exists an orthogonal projection e ? C~(G), e ф 0, such that т(е) = О for any unitary representation
7Г of G disjoint from the trivial representation.
We refer to [V2] for the simple proof.
Let then G be a Lie group with property T and let [e] ? I\o{C'{G)) he the class of the projection given
by 22 2). Since тго(е) = 1 where тго is the trivial representation, the class [e] gives a non-torsion element of
K0(C*(G)) whose image in K(,(C*(G)) vanishes by 22 2). In particular [c] does not belong to the range of
/i, say by theorem 21.
The surjectivity of ц : A't*op(G) —> K(C" (G)) is tied up to the completeness of the description of the. discrete
series by Dirac induction, as well as to a strong form of the Si'llierg principle on vanishing of non-elliptic
orbital integrals of discrete series coefficients:
The point is that the coefficient:
where { is a unit vector in H,, and d, the formal degree of the square-integrable representation ?r, is an
idempotent: p2 = p = p*, in the convolution algebra C*(G) of G. This follows immediately from the
orthogonality relations ([D13]) for locally compact unimodular groups. The regularity of p depends on the
integrability of the representation ir, p ? Ll(G) if tf is integrable ([Dia]). Now the orbital integral defines a
trace r on the convolution algebra of G, and the value r(p) only depends upon the A'-theory class of p in that
llfi
algebra. By construction the Dirac induction, or more generally the indeK map for G-invariant differential
operators on P ~ H\G, has the following property of localisation near И:
Let W D Я be any open neighborhood of tf. Then the Л'-tlieory class lnd(D) ? KQ(Cf(G)) is represented
by an idempotent p ? Afi(C?°(G)~), with p = (j>ij)ij = i t and
Support p^ С W.
In particular it follows that т(р) = 0 where r is the trace associated to the orbital integral of a g ? G whose
conjugacy class does not meet W. We refer to [Ju-Vj] [B-B] [J11-V3] for the relation between Dirac induction,
cyclic cohomology and the Selberg principle which we originally suggested.
It is of course desirable to find direct proofs of the surjectivity in theorem 20. In that respect the ideas
developed by Mackey in [Mi] or in the theoretical physics literature on deformation theory should be relevant.
e) The general conjecture for smooth groupoids
The case of Lie groups discussed above F)), though much simpler than the general case of smooth groupoids
(which includes discrete groups, foliations etc. for which C*(G) fails, as a rule, to he of type I) gives a very
clear indication that the analytic assembly map:
should be an isomorphism in general. Besides the case of discrete groups (section 4), this general conjecture
is supported by numerous explicit computations for foliations ([C01O] [Nat] [Tor] [St] [Pen]). It has many
interesting consequences; the injectivity of fi has as consequences the homotopy invariance of higher signa-
signatures (section 4), the Gromov-Lawson-Rosenberg positive scalar curvature conjecture [Gro-L], the homotopy
invariance of relative ц invariants ([We]). The snrjectivity of /j has implications such as the absence of non
trivial idempotents, e2 = e, e ф 0 in the reduced C-algebra C'(V) of torsion free discrete groups.
In essence the general conjecture is a form of G-equivariant Bolt periodicity. Indeed when the category Cc of
preposition 6 has a final object P which is a coniracitlilc proper G-manifohl (i.e. each Рх is contractible for
x ? G^0') then the bijectivity of ц asserts that, C7-equivariant.lv, (he tangent groupoid deformation (section
5) T'PX ~ point is a Л'-theory isomorphism, as in Bott's theorem.
We shall refer the reader to [Bau-C-H] for a detailed discussion of the implications of the above conjecture,
and for its formulation for locally compact groups, in particular for j>-adic groups.
One of the interests of the general formulation is to put many particular results in a common framework.
Thus for instance the following three theorems:
1) The Atiyah [At5] Singer [Sinj] index theorem for covering spaces
2) The index theorem for measured foliations (Chapter I)
3) The index theorem for homogeneous spaces (theorem 18)
are all special cases of the same index theorem for G-invariant elliptic operators D on proper G manifolds
(definition 1), where G is a smooth groupoid with a transverse measure Л In the sense of [C09]. While it is
desirable to prove such results in full generality, mast of the work goes in explicit computations of relevant
examples which motivated our presentation in this chapter. In chapter Л we shall construct higher invariants
of Л'-theory based on cyclic cohomology, and use them in particular as a tool to control the rational injectivity
of/i.
117

Appendix A. C-modules and strong Morita equivalence
In this appendix we expound the notions, due to M. RielTel ([Ш5][Шэ]) of C-module over a C-algebra and
of strong Morita equivalence of C"*-algebras. Let В be a C-algebra, by a B-valued inner product on a right
B-module ? we mean a B-valued sesquilinear form ( , >, conjugate linear in the first variable, and such that:
a) (?,?) is a positive element of В for any { 6 ?.
0) Ц,ЧУ = (i,tt V
7){Lni>) {i,i)i> we , {,,6J
By a pre-C-module over В we mean a right B-module ? endowed with a B-valued inner product. The
following equality then defines a semi-norm on ?:
IK INI («.«> ll1/2 v«6?
(where || <?,?) || is the C-algebra norm of (?,?) 6 B).
Definition 1. А С-module ? over В is a рге-C-module f. for which || || м a complete, norm.
By completion any рге-C-module yields an associated C-modnle. Given л C"-module ? over B, an
endomorphismT of ? is by definition a continuous endoinoiphisin of the right B-mockile ? which admits an
adjoint T", that is an endomorphism of the right B-module ? such that:
One checks that T", is uniquely determined by T and that gifted with this involution the algebra Ende(?)
of endomorphisms of ? is a C-algebra. One has
(Ti,Tt) <|| т||2 (?,о v? e t:, те EmiB(?)
where || T || is the С -algebra norm of T.
Of particular importance are the compact endomorphisms obtained from the norm closure of endomorphisms
of finite rank:
Proposition 2. ([Ri5]) Let ? be а С-module over B.
a) For any J,i|6j the following equality defines an eudomoriihi.im |?)(>/| 6 Endfl(?):
(Ю ('/!)(«) = ?('/.«> V«e?.
b) The linear span of the above endomorphisms is a self-adjoint nro-stded ideal o/
The usual properties of the bra-ket notation of Dirac hold in this set up, so that for instance:
We let Endg(?) be the norm closure in EndB(?) of the above two-sided ideal (prop. 2b)). An element of
End^(?) is called a compact endomorphism of ?. There are obvious corresponding notions and notations:
HomB(?i,?2), Нотв(?1,6>) for pairs of C-modules over B.
To get familiar with these notions, let. us consider the special case when В is a commutative C-algebra, so
that В is the *-algebra Cq(X) of continuous functions vanishing at. oo on the locally compact space X. Then
a complex hermitian vector bundle E on X gives rise to a C'*-modulc: ? = Cg(X, E) is the Co(X) module
of continuous sections of E vanishing at oo and the C0(A')-vaIin:d inner product is given by:
(«, ri)(p) = (S(p),4(p))e, V?,ц 6 ? , ,> 6 A'.
118
It is not true however that every C'-module over C0(X) arises in this way; they correspond to bundles
of Hilbert spaces but the finite dimensionality of the fibers and the local triviality are no longer required.
Instead one requires that one has a continuous field of Hilbert spaces in the following sense (cf. [Di3]).
Definition 3. Lei X be a iopological space. A continuous field E of Banach spaces over X is a family
(E(t))tix of Banach spaces, with a set Г С tt,€XE{i) of sections such thai:
(i) Г is a complex linear subspace о/П1ед-E(t);
(ii) for every t 6 X, the set of x(t) for x 6 Г is dense in E(t);
(iii) for every x 6 Г the function i —>|| x(t) || is continuous;
(iv) let x e UtixE(i) be a section; if, for every t 6 A' and every e > 0, there exists an x' 6 Г such that
|| x(i) — x'(t) ||< ? throughout some neigborhood oft, then x 6 Г.
The elements of Г are called the continuous sections of E. When each E(i) is a Hilbert space (i.e. || ? ||=
((?.?)I/2) jt follows that for any ?,rj 6 Г the function i -» (?((), i?(<)) is continuous. Every continuous field
of Hilbert spaces E over X yields a C-module over Co(A'), namely the space C0(X, E) of continuous sections
of E which vanish at oo, i.e. such that || {(<) ||—> 0 when / — oo. Moreover, now every C"-module ? over
C0(X) arises from a continuous field E of Hilbert spaces (canonically associated to ?). We shall see later
(proposition 4) how, when X is compact, the finite-dimensional hermitian vector bundles are characterized
among general continuous fields of Hilbert spaces. The latter notion implies the following semicontinuity of
i — d\m(Et):
{t 6 A', dim(?() > n} is an open subset of A".
Given an open subset V of X and a continuous field E of Hilbert spaces on V there is a canonical extension
E of E to X, where the fibers of E on the complement of V are {()}. This operation is particularly convenient
for problems of excision.
Let X be a locally compact space and E a (finite-dimensional) hermitian complex vector bundle over A', with
? = Ca(X,E) the corresponding C-module. Then EndCuU)(f) is (.he algebra Cb{\, End(?)) of bounded
continuous sections of the bundle End E = E' Q E of eiuloinorphisms of E. Moreover Е|к1с„(Х)(?) ^ tne
subalgebra C0(X, End(i?)) of sections vanishing at oo.
Let us go back to the general case of non-commutative C-algebras and characterize finite projeciive modules
among general C-modules.
Proposition 4. Let В be a unital С -algebra.
1) Lei ? be a C'-module over В such that le 6 EikIb(?). Then the underlying right B-module is finite and
projective.
2) Let ?o be a finite projective module over B. Then there exist B-valued inner products (,) on ?0 for which
?o is a C-module over В (and one has \e ? End1^(?),/.
3) Let (,) and (,)' be two B-valued inner products on ?0 as in 2). Then there exists an inveriible endomor-
endomorphism T of ?o such that:
For the proof see [Ri5] [Mis3].
Recall that by definition afinite projective module ?o over an algebra В is a direct summand of a free module
?i = BN, N finite. For arbitrary C-modules one has the following stabilization theorem of Kasparov
([Kas3]):
Theorem 5. ([Kas2]) Let В be а С -algebra, and let ? lie a C"-module oner В with a countable subset S С ?
such that SB is total in ?. Let d" ® В be the C-module over D sum of countably many copies of B. Then:
119

? Ф (f2 ® 5) is isomorphic to f ® В.
We have used the notation t2 <g> В as a special case of the following general notion of tensor product of
С -modules:
Proposition 6. ([Ri5]) let B,C be C"-algebras, ?' (resp. ?") be а С-module over В fresp. C) and p a
*-homomorphism В — Endc(?")- Then the following equality yields a structure of pre-C-module over С
on the algebraic tensor product ? = ?' ®в ?" ¦
Щ е?',ще ?"¦
We shall still denote by ?' <g>B ?" the associated C-modnle over С Given T 6 EndB(?') the following
equality defines an endomorphism T <g> 1 6 Endc(?' ®в ?")'¦
(T® 1)(? ® i?) =
«', ./6 ?".
By a (S -C) С-bimodule we shall mean a C"-module ? over С togotlier with a * homomorphisni from В to
Endc(?)- In particular given a C-algebra S, we denote by I л the П - В С bimodule given by: ? = B, the
actions of В by left and right multiplications, and the B-valued inner product: (*i, *-j> = Ь\Ьг V6bJ2 6 B.
Definition 7. ?e( B,C be С-algebras. A strong Monla equivalence В ~ С is (jiven by a pair ?u?2 of
C"-bimodules such thai:
?i &c ?2 = lfl , ?2 On ?\ - lc
One can then show that the linear span in С of the inner products (?, 4): ?, 4 6 ?1, is a dense two-sided ideal
and that the left action p : В — Endc(?i) is an isomorphism of В wil.li Endc.(^). It follows thus that ?x,
the complex conjugate of the vector space ?1, which is in a natural way а С - B-himodule:
is also endowed with a B-valued inner product:
Endowed with this inner product, ?, is a C-fl C*-bimoclule. The himodule ?, is then a S -C-equivalence
bimodule in the sense of [Ri2] and one checks that the abov.- definition 7 is equivalent to the existence of a
В -С equivalence bimodule. (One can then take ?2 = ?1.)
Let К be the C*-algebra of compact operators on an infinite dimensional separable Hilbert space.
Theorem 8. ([Bro3] [B-G-R]) Let В,C be two separable C'-olgelmis (more generally two C'-algebras with
countable approximate units). Then В is strongly Monla equivalent to С iff В ® K. is isomorphic ioC®K,.
Of particular importance is the strong Morita equivalence of л given С -algebra В with its full hereditary
sub C-algebras С. Неге С С В is:
a) hereditary iff" 0 < с < b and 6 6 С implies с EC.
0) full iff" the two-sided ideal generated by С is dense in B.
Hereditary sub-C" -algebras С of В correspond bijectively to closed left ideals L of В by С = L П ?¦*, and
for instance any self-adjoint idempotent e 6 В gives rise to such a hereditary C" siibalgebra, the reduced
C*-algebra:
120
B, = {x e В ; ex = .re = x).
The reduced subalgebra Bc is full iff the two-sided ideal generated by e is dense in B.
Let us now illustrate the general notion of strong Morita equivalence by a simple example:
Example. Let E be a real finite-dimensional Euclidean space and Cliffc(i?) the complexified Clifford algebra
of E, i.e. the quotient of the complex tensor algebra of E by the relation:
where 7 is the natural Jinear map of E to its tensor algebra. When E is even-dimensional, the C"*-algebra
Cliff(E) Is! non canonically, isomorphic to a matrix algebra.
Let then X be a compact space and E a real even-dimensional oriented Euclidean vector bundle over X.
Then the algebra A = C(X, Cliffc(-E)) of continuous sections of the bundle of Clifford algebras Cliffct-Ej,),
x 6 X, is a C*-algebra which in general fails to be strongly Morita equivalent to a commutative C*-algebra.
Recall that a spinc structure on the vector bundle E is a reduction of its structure group SOBn) to its
covering group SpincBn). Any such structure gives rise, using the Spin representation of SpineBn) in C3*,
to an associated hermitian complex vector bundle S on X which is a module over the complexified Clifford
algebra Clific(E). One thus obtains in this way an A - C(X) C'*-binradule which gives a strong Morita
equivalence of A = С(Х,СШс(Е)) with a commutative C"-algebra.
Proposition 9. The above construction yields a one-to-one correspondence between Spinc structures on E
and strong Morita equivalences between A = C{X, Cliffc(/?)) and a commutative C'-algebra.
Let B,C be С-algebras and ?, an equivalence B — C-C-bimodule. One obtains a functor from the category
of unitary representations of С to that of В by:
H € RepC
©c U E RepS,
and using ?2 = ~E\ as the inverse of ?t one gets a natural equivalence between the two categories of repre-
representations.
It follows in particular that two strongly Morita equivalent C"-algcbras have the same space of classes of
irreducible representations. In particular if a C-algebra В is strongly Morita equivalent, to some commutative
C-algebra then the latter is unique and is the C*-algebra of continuous functions vanishing at 00 on the
space of irreducible representations of В.
Strong Morita equivalence preserves many other properties. Л11 equivalence В — C-C-bimodule determines
an isomorphism between the lattices of two-sided ideals of В and C, and hence a homeomorphism between
the primitive ideal spaces of В and С It does also give a canonical isomorphism of the /v'-theory groups
K,(B) ~ K*(C). One should however not conclude too hastily that nothing will change when we replace
C-algebras by strongly Morita equivalent, ones. We shall illustrate this by the following example:
Example ^3E').
By Gelfand's theorem the category of commutative C-algebras and *-liomoinorphisnis is dual to the category
of locally compact spaces and continuous proper maps. To the notion of homotopy between continuous
proper maps corresponds the following notion of homotopy of *-lioniomorpliisms (morphisms for short) of
C-algebras:
Definition 10. Let А,В be С -algebras and let po, P\ be two morphisms Pj : A — B. Then a homotopy
between po and pi is a morphism from A to B®C[0,1] whose composition with evaluation at j 6 {0,1} gives
Pj-

In topology one usually deals with pointed spaces (Л» and base point preserving continuous maps. When
dealing with compact spaces X, this is equivalent to locally compact, spaces: Л"\{*} and proper maps. In
particular the homotopy groups тгп(Л» are obtained from liomotopy classes of morphisms:
The group law on п„(Х,*) comes from the natural morphisin
obtained from the n = 1 case and the usual map from a circle to a bouquet of two circles. All this is of
course just a transposition of the usual notions of topology.
All these notions continue to make sense if we replace C0(R») by the strongly Morita equivalent C'-algebra
Mk(C) <g> C0(R") = Mt(C0(Rn)) of к x к matrices over Co(R"). We may in particular define new homotopy
groups 7Г„,4(ЛГ, *) for acompact pointed space (X, *) as the group (using 6) of homotopy classes of morphisms:
Let us show now by exhibiting a specific p, that 7r3,2(S') is very different from the trivial ^(S1). For this we
need to construct a non-trivial morphism of 0,E'» to A/^ColR1)), or equivalent^ anon trivial morphism
p of C(Sl) to M2(C(S3)) whose range over the base point * of S'1 is formed of scalar multiples of the '^п^У
matrix 1 ? M2(C). To that effect let us identify S3 with unit quaternions S3 - {(«, 0) ? C-, |o| + |/3| - I)
\ 0Л
rep
trix 1 ? M2(C) T y
resented as 2 x 2 matrices: \^~ 0Ле М-ЛС). This yields a canonical unitary element U ? M2(C(S3)),
with value at q = a + 0j given by U(q) = I *-g _ I 6 A/2(C).
Let us choose the base point * of S3 to be q = 1, so that [/(¦) = 1 is th« «1™^У matrix- Then the morPhism
p from CE') to M2(CE3)) uniquely determined by:
yields an element of тг3
We leave it to the reader to check his understanding of /\-thcory by showing that the class of p in тгз.з^1)
is not trivial. We shall later (chapter IV) come back to this point and show that this morphism is of degree
1 in cyclic cohomology.
At an intuitive level i.e. trying to think of the above morphisni p at. the set theoretic level as a generalized
map from S3 to Sl, one finds that to a point q = a + 0j, H" + |^|2 = 1 of 53 it associates the two solutions
z, z' ? S1 = {z ? С, |г| = 1} of the equation:
= o
or equiva
lently: г2 - (a - a)z + 1 = 0.
122
Appendix B. S-theory and deformations of algebras
A functor F from the category of C*-algebras and *-homoniorphisms to the category of abelian groups is
called half-exact iff for any short exact sequence of C*-algebras:
0 — J — .4 — В — О
the corresponding sequence of abelian groups is exact at F(A). The functor A' which assigns to A its A'-
theory group K(A) is half-exact. G. Skandalis has shown ([Sk4]) that the bivariant functor Л'Л' of Kasparov,
which plays a crucial role in the construction of A'-theory maps, in general fails to be half-exact. It follows
in particular that in general the connecting map of Л'-theory:
Ki(D) t Ki+l(J)
associated to an exact sequence of C-algebras, does not necessarily come from a bivariant element (belonging
to KKl(B, J)). The natural receptacle for such bivariant elements is the extension theory ([Kasg]) but the
intersection product as constructed in [Kasg] is limited to Л'Л". In this appendix we shall expound the main
features of the bivariant i?-theory which solves the above dilficulties. Its abstract existence as a bivariant
semi-exact theory extending К К theory was first proved by N. Iligson [Hig]. In our joint work with N.
Higson [Co-H], it was shown that using the theory of deformations and the notion of asymptotic morphisms,
originating in [Co-Mi], one can define the intersection product of extensions. The resulting theory has a
number of technical advantages, in particular the complete proofs of the main properties of the intersection
product are quite short, and will be given in this appendix.
Deformations of C*-algebras and asymptotic morphisius
Let А, В be two C*-algebras. By a deformation from A to /? we mean a continuous field (A(i),T) of C-
algebrasover [0,1] (cf. [Di3] def. 10.3.1) whose fiber at 0 is Л@) = Л, and whose restriction to the half open
interval ]0,1] is the constant field with fiber A(i) = B, I > 0.
We shall now associate to such a deformation an asymptotic morphism from A to П, as follows. From the
definition ([Dij]) of a continuous field of C-algebras it follows that for any a ? A = A@) there exists a
continuous section a 6 Г of the above field such that o@) = a. Let us choose such an a = aa for each a ? A
and set:
Vt{a) = °>a (-Л e В V« 6 [l,oo[.
Using the continuity of || a(i) || as a function of I 6 [0, 1], for any continuous section a 6 Г one checks that
the family ipt of maps from A to В fulfills the following conditions:
(I) For any a € A, the map i —> Vi(«) is norm continuous.
(II) For any a, b ? A, A 6 C, the following norm limits vanish:
lim (ift(a) + Ay<F) - 4>t(a + M>)) = 0
^ (<pt(ab) - ip,(a) ip,{b)) = 0
We have thus associated to the given deformation an asymptotic morphism in the following sense:
Definition 1. Lei А, В be С-algebras. An asymptotic morphism from A to В is given by a family (^i)F[i,oo[
of maps from A io В fulfilling conditions (I) and (II).
Roughly speaking these conditions mean that given a finite suhset F of A and an ?¦ > 0, there exists a t0
such that for t > to, <pt behaves on Fas a true homoiuorphisni. up to the precision z.
123

The notion of deformation of algebras already plays a critical role in algebraic geometry ([Fu-M]), quanti-
quantization ([Li] [Bay-F-F-L-S2) and the construction of quantum groups ([Dr] [Fa-R-T]). We have given in this
chapter many concrete examples of deformations of quantum spaces (sections 5, 6, 7, 8, 10). To get some
feeling about asymptotic morphisms versus ordinary morphisins one can consider the following very simple
example.
Fig.9. A compact space with Ti(.Y) = {1}
We let X be the compact space shown in fig. 1. One can easily show that тг,(Л") = 0, i.e. there are no
interesting continuous maps / : Sl —» Л', or equival<-htly no interesting morphisms of C(X) to C(S').
However for each e > 0, the ^-neighborhood of Л",
Л'« ={--?C . dist(.-,.Y)<f)
obviously has iri(X.) = Z. With a little work one can get a family f, of degree one continuous maps
Л ¦ S' -»Xt, с > 0 such that ft(x) is for fixed x- 6 S1 a continuous function of ? > 0. Of course for ? -. 0,
these maps fc do not converge to a continuous map of 5' to X. One obtains a corresponding asymptotic
morphism ip, : C(X) -* C(S') by letting:
ipt(a) = no fUl Vrt 6 С(Л') , t € [l,oo[
where for each a 6 С(Л'), a is a continuous extension of a to C. This easy example shows that the notion of
asymptotic morphism is relevant even in the topology of ordinary compact spaces. Thus, in extending the
generalized homology theory associated to a spectrum S:
ffn(-V,S) = ir,1(A-A!:)
beyond spaces X which are simplicial complexes, so as to include arbitrary compact (metrisable) X, [Ka-
Kam-S], one should replace the continuous maps 5" — V involved in the definition of т„(У) by asymptotic
maps (i.e. the transpose of asymptotic raorphisms).
Coming back to the general discussion, we shall say that two asymptotic morphisins {ipt), ip't) from A to В
are equivalent iff:
lim ?>',(«) - Vt(«) = 0 Va 6 A.
t —CO
In other words, if we let B&, be the quotient C"-algebra:
124
(where C\, means bounded continuous functions), then the equivalence classes of asymptotic morphisms from
A to В correspond exactly to the morphisms: ip : A — B^ by the formula:
•P(a), = ?><(<») Va 6 A , t e [l,oo[.
Since ip is a morphism of C" algebras it is always a contraction. We say that ip is injective when ip is injective,
in which case it is an isometry.
Giving a deformation from A to В is equivalent, to giving the associated class of asymptotic morphism
ip, : A —» В constructed above. The convergence || ipt(a) ||—*|| о || Va 6 A shows that the classes thus
obtained are exactly the injective ones. Conversely any asymptotic morphism ip from Л to В is equivalent
to the composition фор where p : A -* A/J is the projection of A onto its quotient by a closed two-sided
ideal, while ф is a deformation from A/J to B.
We can now define homotopy between asymptotic morphisms.
Definition 2. Two asymptotic morphism (tp[) : A — B,i = 0,1 are komoiopic iff there exists on asymptotic
morphism (ipt) from A to B[0,1] = C([0, lj) ® В ivhose evaluation at i = 0, 1 gives (ip\).
It is an equivalence relation between asymptotic morphisms. Given two C-algebras A and B, we shall let
[[A, B]] denote the set of homotopy classes of asymptotic morphisms from A to B. A change of parameter
r(*) -»oo, i.e. the replacement of (ipt) by (?>r(()) = (</',) does not affect the horaotopy class of ip.
Composition of asymptotic morphisms
In this section we shall describe the composition of asymptotic morphisms, i.e. the analogue in ?-theory of
the main tool of the Л'/v-theory, which is the intersection product. The idea for defining the composition
фо<р of asymptotic morphisms: (ip,) : A —• B, (i't) : В —» С is entile simple. In general the plain composition
i/ii о ip, does not satisfy the conditions of definition 1 because of lack of uniformity of the hypothesis on фг;
but this can easily be compensated by a change of parameter, i.e. the replacement of 0i by i/v(() where r(t)
is determined by a diagonal process. The resulting composition is then well defined at the level of homotopy
classes of asymptotic morphisms. Let us now give the details.
Let (?>t)i6[ioo[, ipi '. A —* В,Ъе an asymptotic morphism and let Л' С A he a subset of A. We shall say that
(tpt) is uniform on К iff
a) (i,a) —» ?>((a) is a continuous map from [l,oo[ хЛ' to В
/3) For any e > 0, ЭТ < oo such that for any ( > T one lias:
|| (p,(o) + A(p,(a')-?>i(o+ Aa') ||< e , Va,a' 6 К , VA, |A| < 1
|| ip,(a)ipt(a') - ?>,(aa') ||< e Va, a' 6 A'
||(p((a)* -ip,(a-)\\<e , Va 6 К
II (ft(a) ll<ll a || +e , Vaetf.
The Bartle-Graves selection theorem applied to the quotient map
shows that each asymptotic morphism (ipt) is equivalent to an asymptotic morphism (ip't) which is uniform
on every (norm) compact subset К of A.
Let А С A be a dense involutive subalgebra of .4, and let us assume that A is a countable union A = UKn
of compact subsets Kn of A. One can choose the. A',,'s so that Kn + 1\„ С Л',,-и, К„1\„ С Л'п+i and
АЙ"„ С Кп VA 6 С, |А| < 1. It follows then that if К is a compact convex subset of A and А' с A then one
has К С Кп for some n.
125

Any asymptotic morphism (^i) from A to В which is uniform on compact convex subsets of A defines a
homomorphism from A to В„о and hence an equivalence class of asymptotic moiphisms from A to B. The
composition is given by the following lemma:
Lemma 3. Let (ipt) : A —* В, (Ф1) ¦ В —» С be asymptotic morpliisms of С-algebras. Lei A be a dense
a-compact involuiive subalgebra of A. Assume that (ip,) is uniform on every compact convex subset of A,
while A/11) is uniform on every compact subset ofB. Then There exists a continuous increasing function
r : [l,oo[ —» [l,oo[ such that for any increasing continuous function sit) > r(t), the composition 9, = 1/1,AH^,
is on asymptotic morphism from A to C, uniform on compact convex subsets.
The proof is quite simple; with A = UК„ as above, let („ 6 [l,oo[ be such that ipt satisfies conditions /?)
above on Kn for e = ?, ( > tn. Let then K'n С В be the compact subset:
and let г„ 6 [l,oo[ be such that ф, satisfies /?) он К'„ for ? = ? and * > r,,. Then one checks that any
continuous increasing function г : [l,oo[—» [l,oo[such that r(tn) > rn does the job.
Let (ip,) : A—> В, (ф,) : В -» С be asymptotic morphisms of C-algebras which are uniform on compact
subsets, and let А С A be a dense c-compact involutive. subalgebr.i of A. For 5 : [l,oo[ —»[1, oo[ as in lemma
3, we let 0( : A —» С be the extension to A of the composition (/',(,, oip, = 0t. It is an asymptotic morphism
(well-defined up to equivalence) from A to С
Proposition 4. 1) T/»e homoiopy class [5] 6 [[-4,6"]] is iudeptiideiit of the choices of A and of s(t), and
only depends upon the homoiopy classes [ifi] 6 [[Л,В]] and [Ф] 6 [[B,C]].
2) The composition [Ф]°[<р] of homoiopy classes is associative.
To prove the first statement note that the involutive subalgebra of /1 generated by two ст-compact subalgebras
is still G-compact, whence the independence In the choice of A. To prove the second statement, use the
involutive subalgebra В С В of В generated by the ip,{A) when- (y>i) is uniform on compact convex subsets
of A. Since В is still c-compact the conclusion follows.
One can define the external tensor product tp&> ф of asymptotic inorpliisms {(pt) : A —* С and (ф,) : В —» D
as an asymptotic morphism, unique up to equivalence:
(ф®ф): A 0max II — С О„,ах D
using the maximal tensor product of C-algebras ([T,]). This follows from the following useful lemma whose
proof is immediate using the C-algebra Coo-
Lemma 5. Let A,B,C be C-algebras and lei {tpt) : A -» C, @i) : В — С be asymptotic morphisms such
thai for any a 6 А, Ь е В the commutator [pi(a),i/',F)] converges to 0 in norm when t —> 00. Then there
exists an asymptotic morphism (9t) : A ®mllx В —С, unique up to equivalence, such that:
9,(a ® b) - ip,{a)
— 0 Vn 6 A , 6 6 B.
The external tensor product ip ® ф of asymptotic morphisms passes to hoinotopy classes.
Asymptotic morpliisms and exact sequences of (.""-algebras
Let 0 —v J -» A i В —> 0 be an exact sequence of separable C-algebras. By [Vo2] there exists, in the ideal
J, a quasi-central continuous approximate unit u, 6 ./, 0 < u, < 1 t e [l,oo[. This means that the following
holds:
a) || хщ - x Ц-» 0, || щх - x ||-> 0 when t -» 00, Vs 6 ./
b) II [«t, v] II-* ° when • -* °°, Vy 6 Л
120
cj ( _» m is norm continuous.
Let us use the shorthand notation, for any C-algebra A:
SA = C0(]0, 1[) © A
which is coherent with the topology notation for the suspension of a space.
Lemma 6. Let 0 —» J —» A —» В —» 0 бе ал esaci ге^яенсс o/ separable С-algebras.
1) /Ъг any cond'nuous quasi-central approximate unit (ut), 0 < 11, < 1, an</ any section b e В -* b' e A of
p, the following equality defines o?i asymptotic morphism (ip,) : SB — 7
= /(«•)*' V/ € Co(]O, 1[) , i 6 B.
2) Tie homotopy class [ip] = ?p o/ (yi) only depends upon the morphism p : A—> B.
The first statement is a direct application of lemma 5. The second follows from the convexity of the set of
quasi-central approximate units.
This lemma 6 is a key result which allows one to associate to any extension of C-algebras a class of asymptotic
morphisms, while to get a Л'Л'-class some further hypothesis, such as the existence of a completely positive
lifting of p, is required.
It is obvious that an asymptotic morphism of C*-algebras yields a corresponding map of Л'-theories, since
in a C-algebra the equations e = e' = e- are stable: if such an equation is fulfilled by x 6 A, up to e in
norm, then x is close to a solution (for 5 small enough). Thus if (9,) : A — В is an asymptotic morphism
and e 6 Proj(A) is a projection (e = e" = e2), then ^,{c) is close, for ( large, to a projection / 6 Proj(B)
whose K-theory class is well defined. We urge the reader to check directly at this point that the A'-theory
map thus associated to cp:
(?,). :K(SB) — Л'(У)
does coincide with the connecting map of the six-term exact sequence of Л'-groups associated to an exact
sequence of C-algebras.
To end this section let us note that the. above construction of ?p is coherent with the construction of the
asymptotic morphism (?>,) : A —> В associated to a deformation of C-algebras.
Indeed, given such a deformation one gets an exact sequence:
0 — SB — С ~ A — 0
where С is the C-algebra of the restriction of the continuous fickl (A(t), Г) to the half open interval [0,1[
and where it is the evaluation at 0. One then checks the following equality:
c, = l®f in [[5.4, SB]].
The cone of a map and half exactness
Before giving the precise definition of the B-thoory we shall, in this section, give, the main ingredient of the
proof of its half exactness. The point is that we shall then get a byproduct of this proof even for ordinary
compact spaces.
Let 0—>,/—>A—< В —• 0 be an exact sequence of C-algebras. The cone Ct, of the morphism p : A —> В is
by definition the C-algebra fibered product of CB = Сц(]0,1]) 0 В and A using tlie following morpliisms
toB:
127

p-.CB-^B , p(i) = 4(l) Vb = (*(s)).?io,n6CB
p: Л — В.
In other words an element x of Cp is a pair (F(s)).,6jo,ii, a) w'tn *0) = PW ^
One has a natural morphism i : J —> Cp given by:
and its composition with the evaluation map: ei» : Cp -~ A,
is the morphism j:
ev о i =: j : J — A.
The main point in proving half exactness is to invert tin- morpliism i. That, this is likely to be possible comes
from the contractibility of the third term (the cono CB of li) in the exact sequence:
0 — J -I Cr — CB — 0.
The inverse of i will be given by the asymptotic morphism ?q associated to the following exact sequence of
C-algebras:
0 — SJ — С A — Cp — 0
where С A = Co(]O,1])®Л is the coueof A and where a is given by: <t((«(s)),6j0i,|) = (W<4s)))>e|o,i|i o(l)) €
Cp. It is clear then that the Kernel of a is C0(]0,1[) © ,7 = SJ. Let (lemma 6) e, 6 [[5Cp,5Jj] be the
corresponding asymptotic morphism.
Lemma 7. Let Sj = id®j : SJ — SA. Then Iht composiliou Sj о г„ : SCV — SA is homoiopic to
Sev = id®e» where ev : CP — A «s Ihe evaluation morphism: t"(F,),ej0 ij, и) — a E A.
For the proof let («i)(e[i oo[ (resp. (/ii)I6[ioo[) be a quasi-central Approximate unit for the ideal J of Л (resp.
for C0(]0,l[)CC([0,l]))'. Thus 0 </>,(s)'< 1 Vs 6 [0,1], 'n(O) = />,(l) = 0 and k,(s) — 1 when ( — oo, for
any s g]0,1[. By construction the asymptotic, morphism e, : SCP — SJ is given by:
1Й(/ ® *) = /(/n © «iJ V/ 6 Co(]0,1[) , x E Cp
where x 6 СЛ is such that <r(x) = i. Here I = (J, ).<6]o.il and /(/>, О «()i 6 5J is given by the function:
s 6 ]0,l[ -/(/Hi»)»,)*", 6 7.
The composition Sjo?, is given by the same formula, but now f{li,{s)ut)x, is viewed as an element of A.
It is thus clear that it is homotopic among asymptotic niorpliisim to the following asymptotic morphism
(Л)|б[|,оо[, from SCP to SA
tfi(/®*) = (/D«)J,),€|o.i[ V/ 6 Gi(]0,1[) , x-€Cr.
As above the class of @i) is independent of the choices i — 2, and one checks directly that (i/>i) is homotopic
to the morphism p:
An easy
corollary of this lemma and of the technique of the Pnppe e.\act sequence ([Cu-S]) is the following:
1-28
Proposition 8. Lei 0 —> J —* A —* В —> 0 be an exact sequence of sepamble C* -algebras and D а С -algebra.
1) Let h € [[A,D]] be such that ho j is homotoptc to 0. Ткем there exists к е [[S2B,S2D]] such that S2h is
homotopic to к о Sheas'1 A,S2D}}.
2) Leih 6 [[D, A]] be such thaipoh is homotopic to 0. Then then exists к 6 [[SD, SJ]] such that Sh = Sjok.
Let us briefly sketch the proof of 1). With the notation of lemma Tone has Sev = Sjoea; thus the hypothesis:
h о j ~ 0 implies that Sh о Sev = (Sh о Sj) о е„ = S(h о j) о еа ~ 0. Thus, provided one applies S once,
one may assume that h о ev ~ 0. But such a homotopy yields by restriction to SB С Ср an asymptotic
morphism, к, from SB to SD with к о Sp~ Sh, thus the conclusion.
Similarly for 2) a homotopy po h ~ 0 determines precisely an element ti of [[D, C,,]] such that h = e» о jfcj.
Then by lemma 7 one gets Sh — Sev о Ski = Sj о (?„ о Sk\), so that fc = ?„ о S&i gives the desired
factorisation.
The proposition implies the half exactness of the i?-theory to be defined below, but it also applies to the
topology of ordinary compact spaces. Let us first recall a few definitions from topology. We work in the
category of pointed topological spaces and use the standard notations: Л' V Y = (Л* x *) U (* x Y) is called
the wedge of X and У; X Л Y is the space obtained from X X У by smashing the subspace X V У to a
point and is called the reduced join or smash product of Л' and V. In particular Л'Л Si = SX is called the
suspension of X.
Definition 9. [Ada] A spectrum S is a sequence of spaces S,, and maps crn : 5!С„ — S,,+i siici that each
?„ is a CW-compkx and ihe maps <т„ an embaldings of C\V-completes.
The generalized homology theory h, associated to a spectrum D is defined by the equality:
(*)
/»n(A',S)= Jim
This definition works well for spaces A' such as CIV'-coniplcxes, and Kahn, Kaminker and Schochet [Ka-
Kam-S] have shown, using duality, how to extend such a theory to the category of pointed (metrisable)
compact spaces so that the following axioms of Steenrod generalised homology are fulfilled:
Definition 10. A Steenrod homology theory Л. on the category of pointed (metrisable) compact spaces is
a sequence of covariant, homotopy invariant functors Un from this category to that of abelian groups which
fulfills the following axioms for all n and A':
Exactness. If Л is a closed subset of A' then the sequence
ЫЛ)-МЛ')-/|„(.\7.4)
is exact.
Suspension. There is a natural equivalence />„(А") — Л„+1EА").
Strong Wedge. Suppose Xj is a compact pointed metrisable space, j = 1,2,
Then the natural map
А„ | lim( A, V • ¦ • V At) ) — П /l»(-VJ)
is an isomorphism.
We shall now show that equality (*) gives a Steenrod homology theory provided one modifies the usual
definition of homotopy groups тгп(Л') of a pointed topological space A to take care of spaces such as the
compact of fig.1. The new homotopy groups: 7Г„(А") coincide with the usual ones for simplicial complexes
but do not agree with the usual ones for arbitrary compact spaces.
129

Let first К be a compact space with a base point * € К. The usual homotopy group л-„(Л') involves
homotopy classes of (base point preserving) maps of the ii-sphere to Л', or equivalently of morphisms:
С0(Л'УО — Co(R").
It is straightforward to see that the same construction can be done if we use homotopy classes of asymptotic
morphisms, i.e.
In particular the group structure of ir,,(/v) comes from the natural morpliism:
Co(R)eC0(R)iCo(R)
corresponding to the usual map: S1 -^ S1 V Sl.
When К is a finite simplicial complex, one can embed it in Euclidean space as a deformation retract of an
open neighborhood, and one checks in this way that the now definition of ;r,, agrees with the old one for
such spaces. However for the compact space of fig.l one has:
*i(A') = {0} , t,(/v) = Z.
For arbitrary pointed topological spaces (Л,*) we extend the above definition as follows:
ж„(Л', *) = ]т1т„(А\ *)
where К varies among compact subsets of Л* containing *.
Then proposition 8 yields easily:
Theorem 11. Let ? be a spectrum. The following equality defines a Steenrod limnology theory h,(-, S) on
the category of pointed meinsable compact spaces, which agrees will) the homologij theory defined by (*) on
simplicial complexes:
Let us now return to non commutative C*-algebras.
E- theory
Let K. be the elementary C*-algebra of all compact operators in a separable oo-dimensional Ililbert space.
Since such a Hilbert space H is isomorphic to "H в Н, one has a natural isomorphism:
p : M-,(K.) =: X
where M2(K) = K, ® M2(C) is the C"-algebra of 2 x 2 matrices over A.". We let E be the category whose
objects are separable C*-algebras while the set E(A, B) of morphisms from ,4 to В is:
E{A,B) = {{S.A О AT, SB О К]]
i.e. the set of homotopy classes of asymptotic morphisms from SA Q К- = Л 0 Co(R) <S> AC to SB ® K.
The composition: E(A,B) x E(B,C) —» E(A,C) is given by the composition of homotopy classes of asymp-
asymptotic morphisms.
Using the above isomorphism p : M2(?) — ? one defines the sum tp + ф of elements tp, ф of E(A, B) as the
asymptotic morphism: SA ® К —» SB ® Л/2(А-") = M-j(SB О A.') given by:
130
with obvious notations.
One checks that with this operation E becomes an additive category. The opposite - [tp] of a given element
of E(A, B) is obtained using the reflexion j--sas an automorphism of Co(R). Let C"-Alg be the category
of separable С-algebras and *-homomorphisms. Let then j : C'-Alg -* E be the functor which associates
to tp¦: A —> В the asymptotic morphism <pt = <p® id from SA © AC to SB ® K.
Theorem 12. [Co-H] (I) The bifuncior E(A, B) from the category C'-Alg to the category of abelian groups
is half exact in each of its arguments.
(II) Any functor F from the category C'-Alg to thai of abelian groups which is unchanged by A -> A <g> K,
homotopy invariant, and half-exact, factorises through ihe category E.
The proof of (I) follows from proposition 8.
See [Co-H] for the proof of (II). As the functor j verifies the hypothesis of the next corollary, the latter gives
a characterization of the category E.
Corollary 13. Let F : С -Alg —» Z be a functor to an addiliue category Z which is unchanged by A —» A®K.,
homotopy invariant, and half exact, as a bifuncior 1o abelian groups. Then F uniquely factorises through the
category E.
The E-theory is thus a concrete realisation of the category whose existence was proven by N. Higson in [Hig].
To get other corollaries of theorem 12 one specialises to the functors E(A, ¦) and E(-, B) the following general
properties of functors F from C-Alg to abelian groups, due to G. Kasparov [Kas9] and J. Cuntz [Cu3].
Lemma 14. ([Kasg]) Lei F be a covariavt, homotopy invariant, half exact functor from the category С-Alg
to abelian groups. Then to any exact sequence: 0 — J — A — В -~ 0 of separable C-algebras corresponds
a long exact sequence of abelian groups:
^- F.n(J) - F_,,(A) - F_,,(tf) ¦- F(J) - F{A) - F(B)
where F-a(A) = F(S"A).
There is a similar dual statement for contravariant functors.
Of course this lemma applies in particular to the functor E(-,D) and E(D,-) for a fixed C*-algebra D
and yields corresponding long exact sequences involving the functors E(Sn-,D), E(D,S"-). But in fact the
situation is every much simplified by the built in stability of E under the replacement of a C*-algebra by
its tensor product with K. Indeed one has natural isomorphisms in the E category between Co(R2) and
C. Moreover the suspension map 5 : E(A, B) — E(SA,SB) is an Isomorphism of abelian groups for any
C'-algebras A, B. Thus E{StA, S'B) = E(A, B) for k + (¦ even and is equal to E{A, SB) ~ E(SA,B) for
к + I odd. We shall denote the latter group by El(A, B). All these are variants of Bott periodicity, about
which the cleanest statement is given by the following result of J. Cuntz, whose direct proof is simple ([Cu3]).
Lemma 15. ([Сиз]) Lei F be a functor from the category C'-Alg to ihe category of abelian groups which is
homotopy invariant, half exact and unchanged by A — A® A.'. Then there is a natural equivalence between
F() and the double suspension F(S2).
The proof uses the Toeplitz C*-algebra, i.e. the C-algebra т generated by an isometry U, U'U = 1,
UU' ф 1. All such C*-algebras are canonically isomorphic, and one has an exact sequence:
131

0ycrC()
where C(S^) is obtained as a quotient of r by adding the relation UU' = 1. Taking the kernel of the
evaluation C(Sl) -* С at some point * € S1 yields an exact sequence
О -^ К -> то - C0(R) - 0.
The proof of J Cuntz consols in showing that F(r0) = {0}. In fact the corresponding asymptotic morphism
J Co(R') IT УС is the Heisenberg deformation, and the proof shows that e, yields an Я-theory .somorphism:
Finally we note that for any C'-algebra A one has a natural map from K\A) to E(Ca(R), AJ>K) which to
a unitary U € (А® ?)~, e(U) = 1 associates the corresponding ordinary morphism from Co(K) to A® A,.
One then easily checks
Proposition 16. The above map is a canonical isomorphism
E\C,A) ~ Ki{A) , ^(С.Л) ~ KoM).
In particular any Я-theory class у € Я(Л. B) yields by composition a corresponding A'-theory map:
x e к (A) = E(C,A) - ;/<>* € E(C, 0) =
13-2
Appendix C.-Crossed products of C*-algebras and the Thorn isomoi'i>liisui
Let Л be a C*-algebra, G a locally compact group and a : G —• Aut.(/1) a continuous action of G on A.
Thus for each g € G, ag € Aut(A) is a * automorphism of A and for each x € A the map g t—<• а,(х) is norm
continuous.
Definition 1. A covariant representation it о/(Л,а) is given by unitary representations жл, ir<j of A and
G on a Hilbert space 4 such thai: жа(д)жА(х)жа(д)-х = тд(а,<1)) Vg €G, x € A.
Of course the unitary representation of G is assumed to be strongly continuous ([Pe2]). The above definition
would continue to make sense if G were just a topological group but in the locally compact case, there exists
a natural C*-algebra, the crossed product В = A >0o G, whose unitary representations correspond exactly
to the covariant representations of (A, a). Indeed, let dg be a left Haar measure on G and let us endow the
linear space Ce(G, A) of continuous compactly-supported maps from G to A with the following involutive
algebra structure:
(/i *h)(g) = У/i(ffi) ar,l(/2(<7r1ff)№i V/j € Cc(G,A),g € G
V/
€ G
where 6 \G —» R!j_ is the modular function of the not. necessarily unimodular group G, i.e. the homomorphism
from G to R^. defined by the equality:
These algebraic operations on CC(G,A) are uniquely prescribed in order to get an involutive representation
ж of CC(G,A) from a covariant representation irof(A.u) by the following formula:
*(/) = J *л(П.ч))*а(у)<1у V/ eCc(G,A).
It is not difficult to check then ([Рез]) that the completion В = A »a G of CC(G, A) for the following norm
is a C*-algebra whose unitary representations correspond exactly to (lie covariant representations of (A, a).
| / j|= Sup {|| jt(/) ||, tt a covariant. representation of (Л,О')} .
Proposition 2. The map ж — т is a natural equivalence between the category of covariant representations
of (A, a) and the category of representations of the С -algebra crossed product A ~xa G.
In order to define the reduced crossed product, A ~Xa,r G, which differs from the above only when G fails to
be amenable (in fact when the action of G fails to be amenable), let us consider the (right) C*-module over
A given by:
? = L-(G,d<j)OA.
We can view ? as the completion, for the norm || ? || = || (?,?) Ц1'2 of CC{G, A) with Л-valued inner product
given by:
¦/<
(e.4> = / ZW'lW'l € Л , V?. ц € C,{G,A
while the right action of Л is given by.
((a)(9) = ?(»)a V{ ? Cr_(G, A), a e A, ij € G.
133

We then define the following left regular representation of (.4, a) as endomorphisins of ?:
Vf € Cc(G,A),a?A,g € G
One check that these formulae define elements of End^(f), and one thus gets a natural representation:
A : A»aG -~ EiuU(?J.
The image of A is a C'-algebra, called the reduced crossed product of Л by a and noted:
AxiOirG.
In the case when A = С the reduced crossed product C'Xr G is of course the reduced C'-algebra of G, C?(G),
i.e. the C*-algebra generated in the left regular representation of G by the left action of the convolution
algebra C,(G) or equivalently by L4G,dg). Similarly CxiG is the C"-algebra C"(G) of G. The nuance
between these two C*-algebras exists only in the non-amenable case. For instance when G is a semi-simple
Lie group the unitary representations of G which come from representations of C'(G) form the support of
the Plancherel measure and are called tempered representations.
Proposition 3. ([Pe2]) When G is amenable the representation A of A»aG in ЪлйА(?) is injeciive.
We shall now specialise to the case when G is abelian and describe the Takesaki-Takai duality theorem for
crossed products [To]. Since abelian groups are amenable the distinction between the two crossed products
disappears. Let G be the Pontrjagin dual of G and let {;/,</') for ij € G, g' € G be the canonical pairing with
values in T = {z € C; \z\ = 1}.
Proposition 4. ([To]) Let G he an abelian locally compact group, a an action of G on a C*-algebra A. The
following equality defines a canonical action a of the Ponlrjagiv dnalG ofG on the crossed product А~х„ G:
(S»'(/))C) = {Я,Я')ПЯ) V/ € СЛС, А) С AX>aG ; <j € G, g' e G.
Indeed one easily checks that the norm (*) is preserved by the ая', which are obviously automorphisms of
the involutive algebra CC(G, A). Also one can implement the automorphisms Ss/ in the representation A of
A ~Xa G as endomorphisms of ? = ?2(G, Aц) 0 A- One defines a representation p of G as automorphisms of
? by the equality.
{р(д'Шд) = {я,а'Ши) v« e cc(G, A) ¦ ., e G,</ e 5.
One then checks that the pair (A,/>) is а с ova riant representation of (Д >за G\ a) as endormorphisms of the
C*-module ?. We thus get a homomorphism A : ((А >за С) »~ G — End,i(?) and one easily checks that the
range of A consists of compact endomorphisms (cf. Appendix A). The main content of the Takesaki-Takai
duality theorem is that A is actually an isomorphism:
A : (
a ~
where к is the elementary C'-algebra of compact operators in L-(G,dy). To state this duality with the
required precision we need the following definition (compare with Chapter 5).
Definition 5. Let A be a C" -algebra, G a locally compact group and a,fx' two actions ofG on A. Then Oc
and a1 are outer equivalent iff there exists a map g — 11, from G to the group U of unitary automorphisms
of the C*-module A over A such that:
134
1) g
2) u
3) a
? is norm continuous for any ? ё А.
unan(u3i) v»bff2 € G.
«»<*,(o)u; VoeA,s€G.
is given by the following formula:
= «,/(.») V/ € CC(G, A).
Theorem 6. ([To]) Let G be an abelian locally compact group, and lei a be an action ofG on a C'-algebra
Л. There is a canonical isomorphism A of the double civ.ised pmduci {A'XaG)'>y^G with А® К which
transforms the double dual action a ofG into an action outer equivalent to the action a® 1 ofG on
Note that this theorem could not even be formulated had we decided to only consider commutative C-
algebras. Indeed even for A = С it expresses the fact that the C*-algebra C*(G) ~xG is isomorphic to the
non-commutative elementary C-algebra of compact operators. We shall see in particular how it immediately
implies a far reaching analogue of the Bolt, periodicity theorem [Coi9] (motivated by [Pi-Vi]). We specialise
to the case G = R, the additive group of real numbers, and our first task is to construct a natural map:
фа : Ki(A) -* K;+l(AxaR)
for any C-algebra A with one parameter automorphism group a. When the action a of R on A is trivial,
a, = id Vs € R, the crossed product A >3a R is canonically isomorphic to SA = A 0 Co(R) and the map фа
is the natural suspension isomorphism:
To define фа one reduces the general situation to the particular case of the trivial action using the following
lemma whose geometric meaning is the absence of curvature in the one dimensional situation.
Lemma 7. ([Coi9]) Lei A be а С -algebra, a a one parameter group of automorphisms of A, e = e* = e2
a self-adjoint projection, e € A, of A. There exists an equivalent projection f ~ e, f € A and an outer
equivalent action a1 ofH on A such thai a\(f) = /Vi€R.
One first replaces e by an equivalent projection / such that the map t —> a,(/) ? A is of class C°° (cf.
Chapter IV), and then one replaces the derivation t> — (;j7"i),_0 which generates a by the new derivation
6' = 6 + ad(h) where ad(h)x = hx - xh Ух € A, and /i ='.
From Lemma 7 and the canonical isomorphism /1
construction of
a R ^ A ~xai R for outer equivalent actions one gets the
Replacing A by SA one gets similarly
Ki(A) —
135

Theorem 8. ([Cox9]) Let A be a C* -algebra, and let a be a one parameter group of automorphisms of
A. Then фа : Ki(A) —» Ki+i(A ~XQ R) is an isomorphism of abelian groups for i = 0, 1. The composition
ф~офа is the canonical isomorphism of Ki{A) tuith Ki(AQK) where the double crossed product is identified
with A<&K by Theorem 6.
The proof is simple since it is clearly enough to prove the second statement, and to prove it for i = 0. One
then uses Lemma 7 to reduce to the case A = С where it is easy to check ([Coi9]).
The above theorem immediately extends to arbitrary simply connected solvable Lie groups H in place of R.
136
Appendix D. Penrose tilings
In this section, I review for the reader's convenience the classical results ([Gru-S]) on R.Robinson and
R.Penrose's quasiperiodic tilings of the plane.
Fig. 10. Elementary tiles [Gru-S]
One first considers two types GA anil Рл of triangular tiles (represented in Fig.10). The vertices are colored
white or black, and the edge between two vertices of the same color is oriented. The tile Ga has two black
vertices and one white, whereas the tile Pa has two whites and a black.
A tiling of type A of the plane is defined to be a triangulation of the plane by triangles isometric to Ga or
Pa, in such a way that the colors of common vertices, and the orientations of common edges, are the same.
The reader is referred to [Gru-S] for (he proof of the existence of such tilings of the plane (see Fig.3 in Section
3 of this chapter). We shall say that, two tilings T and V are identical if they can be obtained from each
other by an isometry of the Euclidean plane.
Let X, be the set of all type A tilings (up to isometry) of the plane. The essential result we shall use is the
possibility of parametrizing A' by the set. A' of infinite sequences (а„)„е1ч. "n € {0, 1} of zeros and ones,
satisfying
а„ = 1 => ап+1 = 0
137

in such a way that every tiling T of type A arises from a sequence, T = Г(а), and two sequences о and 6
yield the same tiling if and only if there exists an index N such that a,, — bn for all n > N. As this is an
important point, I shall describe the correspondence a — T(a) explicitly ([Gru-S], p.568).
Fig.ll.
If T is a tiling of type A and if we delete from the triangulation T all of the short edges that join two vertices
of different colors and separate a Gj-tile from a Рд-tile (Fig.11), we obtain a new triangulation T\ of the
plane whose triangles are isometric to one of the two triangles Go, Pq of Figure 10. We obtain in this way
a tiling of the plane of type B, that is, a triangulation of the plane by triangles isometric to Gb or Pe,
such that the colors of common vertices, and the orientations of common edges, are the same. If, in the
triangulation Ti, we delete all the edges that join two vertices of the same color and separate a Ge-tile from
a Рв-tile, we obtain a new triangulation Ti whose triangles are isometric to one of the triangles GTA', Рта'
of Figure 10. Iterating the procedure, we obtain in this way a sequence Tn of triangulations of the plane. To
each of them there correspond triangles Gn and Р„.
Given a tiling T of type A and a triangle a of the triangulation T (Fig. 11), we associate with the latter the
sequence (а„)„е1ч, where an is 0 or 1 according as the triangle of Т„ that contains a is large (i.e., isometric
to Gn) or small. We denote by i(T, a) the sequence obtained in this way.
We then denote by К the set of sequences (а„)„€м, а„ ? {0, 1}, such that а„ = 1 => а„+1 = 0. It is dear
that every sequence of the form i(T, a) belongs to k\ because the triangle Pn is not used for constructing
138
pn+1 = Gn- Conversely ([Gru-S], p. 568), one shows that every element a of К is of the form i(T, a) for a
suitable tiling of type A and a suitable a.
Two triangles a,/3 of a tiling T occur, for n sufficiently large, in the same triangle of Tn. It follows that the
sequences a = i(T, a) and b = i(T, /3) satisfy the condition
«m = *m (V"i > n).
(*)
Conversely ([Gru-S], p.568), one shows that if a = i(T,a) and b € A' satisfy (*), then there exists a triangle
/3 of T such that b = i(T,P).
One obtains in this way a bijection between the set Л' of Penrose tilings and the quotient k'/V. of the set К
by the equivalence relation Tt denned by the relation (*) ([Gru-S], p.568).
139

CHAPTER III
CYCLIC COHOMOLOGY AND DIFFERENTIAL GEOMETRY
In this chapter we shall extend de Rham homology beyond its usual commutative framework in order to
obtain numerical invariants of Л" theory classes on non commutative spaces.
In the commutative case, (A = C(X) for A' a compact space), we have to our disposal in К theory a tool of
capital importance, the Chern character
which relates the К theory of X to the cohomology of A
When X is a smooth manifold the Chern character may be calculated very explicitly by means of the
differential calculus of forms, currents, connections and curvature ([Mi4]). More precisely we have first of all
the isomorphism:
K0(A) ~ Ka(A)
where A = C°°(X) is the algebra of smooth functions on X, a dense subalgebra of the C* algebra C(X) = A
Then, given a smooth vector bundle E over X (or equivalently the finite projective module ? = С (A, t)
over A = C°°(X), of smooth sections of E) the Chern character of E:
Ch(E)eH'(X,R)
is represented by a closed differential form:
Ch(?) = trace (exp(V2/2iri))
for any connection V on the vector bundle E.
Any closed de Rham current С on the manifold X determines a map <pc from K'(X) to С by the equality:
where the pairing between currents and differential forms is the usual one.
One thus obtains in this way numerical invariants of К theory classes whose knowledge for arbitrary closed
currents С is equivalent to that of Ch(i?).
In this chapter we shall adapt the above classical construction to the non commutative case. This requires
defining the analogue of the de Rham homology, which was achieved, in [Co,6] under the name of cyclic
140
cohomology, for arbitrary non commutative algebras. This first step is purely algebraic: One starts with an
algebra A over C, which plays the role that C°°(X) was playing in the commutative case, and one develops
the analogue of the de Rham homology of A', the pairing with the algebraic К groups Ko(A) and K\(A)
and tools, such as connections and curvature, to perform the computations.
The resulting theory is a contravariant functor HC from non commutative algebras to graded modules over
the polynomial ring C(<r) with one generator a of degree 2. Being contravariant for algebras means that the
above functor is covariant for the corresponding "space" whence its homological nature. In the definition of
this functor the finite cyclic groups play a crucial role and this is why HC"(A) is called the cyclic cohomology
of the algebra A.
We shall expound the .theory below in a way suitable for the applications we have in mind, but we refer the
reader to several books [Kasti] [Кагз] [Lo] for more information.
The second step, crucial in order to apply the above construction to the К theory of С algebras of non
commutative spaces, involves analysis.
Thus the non commutative algebra A is now a dense subalgebra of а C* algebra A and the problem is,
given a cyclic cocycle С on A as above, satisfying a suitable condition relative to A, to extend the К theory
invariant:
•Pc ¦
- С
to a map from Ka(A) to C.
In the simplest situation the algebra А С A is stable under holomorphic functional calculus (cf. Appendix
C, this means essentially that if a € A is invertible in A its inverse is in A). The inclusion А С A is then an
isomorphism in К theory and the above problem of extension of tpc is trivially solved.
This method still applies when A is not stable under holomorphic functional calculus but when the cocycle
С extends to the closure A~ of A under this calculus. This will apply to the cyclic cocycles associated to
bounded group cocycles on the Gromov's word hyperbolic groups.
However a new step is required to treat the transverse fundamental class for the leaf space A' = V/F of
a transversally oriented foliation or for the orbit space A' = W/T where Г is a discrete group acting by
orientation preserving diffeomorphisms of a manifold W.
The main difficulty to be overcome is that in general the transverse bundle of a foliation admits no holonomy
invariant metric. Similarly an action by diffeomorphisms on a manifold does not preserve any Riemannian
metric (if so it would then factorize through the Lie group of isometries).
It follows then that the cyclic cocycle which is the transverse fundamental class is very hard to estimate with
respect to the C" algebra.
However we have already met a similar situation in chapter 1, in the measure theory discussion, where
the type III situation was precisely characterized by the absence of holonomy invariant volume element on
the transverse bundle. We saw moreover that one could reduce type III to type II in replacing the non
commutative space X by the total space Y of an R+ principal bundle over A', whose fiber Yl over each point
L ё X is the space of transverse volume elements at this point.
We adapted this method to overcome the above difficulty, through the following steps:
or) To the problem of finding a holonomy invariant metric on the transverse bundle r, (dim r = q), corresponds
a GL{q,H) principal bundle over the non commutative space A'. We let Y be the total space of the bundle
over X associated to the action of GL(q, R) on GL(q,R)/O(q, R). Then the transverse structure group of У
reduces to a group of triangular matrices with orthogonal diagonal blocks.
/3) Using the above reduction of the transverse structure group of Y we estimate the cyclic cocycle associated
to the transverse fundamental class of Y.
141

7) We use the non positive curvature property of the symmetric space H = GL(q, R)/
O(q, R) fiber of the projection:
p : Y — X
and the Kasparov bivariant theory, to construct a map K{C(X)) —<¦ K(C(Y)) of К theories, whose
composition with the map <pc:
K(C(Y))-*C
given by the fundamental class of У will yield that of Л'.
In the context of foliations we shall thus construct invariants of the К theory:
K(C'(V,F)) S- С
making the following diagram commutative:
K..ABG) —> K(C'(V,F))
H,(BG, R) -^-
2. Examples of computation of HC"(A); manifolds, group rings and crossed products.
3. Pairing of cyclic cohomology with К theory.
4. The higher index theorem for covering spaces.
5. The Novikov conjecture for hyperbolic groups.
6. The Godbillon Vey invariant.
7. The transverse fundamental class for foliations and geometric corollaries.
Appendix A. The cyclic category Л.
Appendix B. Locally convex algebras.
Appendix C. Stability under holomorphic functional calculus.
where p is the analytic assembly map of chapter 2, and w € H'(BG,R) is any cohomology class on the
classifying space BG of the holonoiny groupoid G of (V, F) belonging to the subring 7J С H"(BG,R)
generated by the Pontrjagin classes of the transverse bundle, the Chern classes of any holonomy equivariant
vector bundle on V and the pull back of the Gel'fand Fuchs characteristic classes у € H'(WO,).
As applications of the above construction we shall get geometric corollaries which no longer involve C*
algebras in their formulation and are:
1) The Novikov conjecture is true for the Gromov's word hyperbolic groups ([Co-Mi]).
2) The Gelfand Fuchs cocycles on Diff(M) satisfy the Novikov conjecture ([С01З] [Co-G-M]).
3) Let M be a compact oriented manifold and assume that the rational number A(M) is non-zero (since
M is not assumed to be a spin manifold A(M) need not be an integer). Let then F be an iniegrable spin
subbundle of TM. There exists no metric on F for which the scalar curvature (of the leaves) is strictly
positive f> e > 0) on M.
Moreover we shall see, in section 6 of this chapter, how the Godbillon Vey class appears naturally in our set
up from the lack of invariance of the transverse fundamental class [К/^ of a codimension 1 foliation, under
the modular automorphism group a, of the von Neumann algebra of V/F (chapter 1). In particular we shall
obtain the following corollary showing, in the non commutative case, the depth of the interplay between
characteristic classes and measure theory:
4) Let M be the von Neumann algebra of the foliation (K, F) of codimension one. Then if the Godbillon Vey
class GV €: HZ(V,H) is non zero, the flow of weights W(M) admits a finite invariant measure.
This implies in particular the remarkable result of S. Hurder [Hu-Ki] that under the hypothesis of the
theorem the von Neumann algebra M is not semifiniie and is of type III if the foliation is ergodic.
The content of this chapter is organized as follows:
1. Cyclic cohomology.
142
143

III.l. Cyclic cohomology
Given any (possibly non commutative) algebra A over C, HC'(A) is the cohomology of the complex (CJ, ft)
where CJ is the space of (n + l)-linear linear functionals ip on A such that
<p(a\ ..., o", a0) = (-1)" p(a°,..., a")
and where Ь is the Hochschild coboundary map given by
Va< € .4
(-1)"+1 p(a"+1 о0,..., а").
j=o
We shall develop the main properties of this cohomology theory for algebras in the following sections;
a) Characters of cycles and cup product in ffC*
In order to clarify the geometric significance of the basic operations such as В and the cup product <р#-ф of
cyclic cohomology we shall base our discussion on the following notion:
Definition 1. a) A cycle of dimension n is a triple (Q,d f) where Q = Ф Q? is a graded algebra over C, d
is a graded derivation of degree 1 such that d2 = 0, and J : il" —> С is a closed graded trace on il.
b) Let A be an algebra over C. Then a cycle over Л is given by a cycle (U,d,J) and a homomorphism
р:А-~п°.
As we shall see below a cycle of dimension n over A is essentially determined by its character, the (n+ l)-linear
function r,
т(а°,...,ап) = J p(a°) rffpfa1)) d(p(a2)) ¦ ¦ d(p(a")) Va> € A
and the functionals thus obtained are exactly the elements of Ker b П C".
Given two cycles п, п' of dimension n, their sum п ф О' is defined as the direct sum of the two differential
graded algebras, with J(ui,ui') = Jui + Jui'.
Given cycles п, п' of dimensions n and n', their tensor product 0" — ?2® ff is the cycle of dimension n + n'
which as a differential graded algebra is the tensor product of (O,d) by {Of ,d*), and where
j(w ®w') = I ы Iu' Vw € О , ш' € п'.
Examples 2. a) Let V be a smooth compact manifold, and let С be a closed de Rham current of dimension
q (< dim V) on V. Let п', i € {0,...,?} be the space C°°(V, Л'Т* V) of smooth differential forms of degree
i. With the usual product structure and differentiation п = ф п' is a differential algebra, on which the
i=0
equality /w = (C,w), for ш € пя, defines a closed graded trace.
b) Let V be a smooth oriented manifold and (x, g) € V x Г —> xg € V an action, by orientation preserving
diffeomorphisms, of the discrete group Г on V. Let A"(V) be the graded differential algebra Q°(K,A"TJ) of
smooth differential forms with compact support on V. The group Г acts by automorphisms of this differential
graded algebra: for any g € Г, let фд be the associated diffeomorphism фя(х) = xg Vx € V, then
= ,/>> Vw € A'(V).
It follows that the algebraic crossed product:
144
?Г =Л*(К)Х1Г
is also a graded differential algebra. Let us describe it in more details. As a linear space пр is the space
CJ°(Vxir, AT*) of smooth forms with compact support on the (disconnected) manifold V x Г. Alge-
Algebraically it is convenient to write such a form as a finite sum:
where the C/j's are symbols, the algebraic rules are then:
/Ewjfj = /v"« (e the unit of Г).
The invariance under diffeomorphisms (preserving the orientation) of the integral of top dimensional forms
then shows that the triple (u,d,J) defines a cycle of dimension n = dim V over the algebra C?°(K) xiT.
c) Let Г be a discrete group, A = СГ the group ring (over C) of Г. Let п'(Т) be the graded differential
algebra of finite linear combinations of symbols:
9adgidgi--dgn j,- € Г
with product and differential given by.
(godgi ¦ ¦ ¦ dgn)(gn+1dgn+2 ¦¦ ¦ dgm) =^ (-1)" > godgx ¦ ¦ ¦ d(gjgj + i )-dgn dgn+l ¦ ¦ ¦ dgm+(- l)agog,dg2 •¦¦dgn
l
d(gadgt ¦ • dgn) = dgodgy ¦ ¦ ¦ dgn.
Then any normalized group cocycle с ? Zk(V,C) determines a fc-dimensional cycle through the following
closed graded trace on ?2*(Г):
/ 9°dgl dgn =0 unless n - к and »0»'¦••»"= 1
Jsodg1---dgi^c{g\...,gt) if ga ¦ ¦ ¦ gk = I.
We recall that group cohomology Я*(Г,С) is by definition the cohomology of the classifying space ВТ, or
equivalently of the complex (C*, b) where:
С is the space of all functions у : Tp+i -* С such that f{gga,.. ,ggp) - j(g° gp) Vg,g> € Г
(*7)(ffo, • ¦ • ,9r+i) = J2(- 1)' 7C0, ¦ ¦ •,»,-!,3.-+1, ¦ ¦ • ,3P+i)-
i = 0
The group cocycle associated to 7 ? C*, 67 = 0, is given by
c(»i,- -,»i) = 7(l.»i,»iSf2>- -,9i92 -Як)-
The normalization required above is the following:
145

с = 0 if any 9i: = 1 or if gt ¦¦ ¦ gie = 1.
Any group cocycle can be normalized without changing its cohomology class because the above complex can
be replaced without altering its cohomology by the subcomplex of skew symmetric cochains:
7(ff»@), • • • > 9*{p)) = Sign (it) 7C0,..., ft,) Vji € Г , o- € Sp+i.
In this last example c) the differential algebra п"(Г) is independent of the choice of the cocycle c. The
construction of С2*(Г) from the group ring А = СГ is a special case of the universal differential algebra
п"(А) associated to an algebra A. ([Arv3] [Кагз]), which we briefly recall.
Proposition 3. Lei Л be a (not necessarily unital) algebra over C.
1) Let C2'(-4) be the linear space _4®c-4 (where А = -4ФС1 is the algebra obtained by adjoining a unit to A).
Then the following equalities define a structure о/Л-bitnodule on Cll(A) and a derivation d : A —> Ul(A):
(for any a,b,x,y ? .4, A € C)
x((a+
= (xa +\x)® by - (xab + \xb)
4=1®а?Й1(Д) Vog A.
2) Let ? be an A-bimochtle and 6 : A -* ? a derivation then there exists a bimodule morphism p : С2'(.4) —* ?
such that 8 — po d.
Thus (Q'(-4),d) is the universal derivation of A in an ,4-bimodule.
The universal graded differential algebra il(A) is then obtained by letting п"(А) = Cl'(.4) 8u ul(A) ¦ ¦ ¦ ®д
ul(A) be the n-fold tensor product of the bimodule п'(А), while the differential d : A — ft1 (.4) extends
uniquely to a square zero graded derivation of that tensor algebra. Note that one has a natural isomorphism
of linear spaces:
j : А® А®" -~пп(А)
with j((a° + A 1)© a1 ® •••© a") = a°dal ¦¦¦da" + Ma1 da1 -da" V»> € A , A € C.
(Note that the cohomology of the complex (Q(A),d) is 0 in all dimensions, including 0 since W(A) = A.)
We can now characterize cyclic cocycles as the characters of cycles over an algebra A.
Proposition 4. Let т be an (n + i)-linear functional on A. Then the following conditions are equivalent:
1) TAere exists an n-dimensional cycle (?2,d, /) and a. homomorghism p : A —> Cl" such that
r(aa,. . . , a") = J p(aa) d(p(a1)) ¦ ¦ ¦ d(p(a")) Va°,... , a" € A.
2) There exists a closed graded trace т of dimension n on 0(A) such thai
т(а°,...,а") = ?(a°dal -da") Va°, ...,o" € A.
3) One has т(а1,... ,a",a°) = (-1)" r(a°,. ¦ •, a") and
(-\Y r(a°,...,aV+1,...,a"+1) + (-l)"+1 r(a"+1 a0, ..., a") = 0
for any a", .. ., a" + i ? A.
146
Proof. The universality of п'(А) shows that 1) and 2) are equivalent. Let us show that 3) => 2). Given
any (» + l)-linear functional <p on A, define ф as a linear functional on пп(А) by
?oj((a° + A°l)®o1®---®a") =<p(aa,a\...,a").
By construction one has $(du) = 0 for all w € п"~у(А). Now, with т satisfying 3) let us show that f is a
graded trace. We have to show that
?((a°dai ¦ ¦¦dat)(at+ldak+2 ¦ -da"*1))
Using the definition of the product in il(A) the first term gives
к
r(a°
and the second one gives
n-k
Y^(- i)*f—*)+»-*-.» r(a*+i .. _ ai+i+Ja*+i+j + i ^ ai)
The cyclic permutation A, X(() = к + 1 + (, has a signature equal to (—!)"(*+') so that, as тх = ?(А)т by
hypothesis, the second term gives
Hence the equality follows from the second hypothesis on r.
Let us show that 1) => 3). We can assume that A = 0°. One has
г(а°,а',...,«")= j(a°dal)(da2--dan) = (-l)n-1 f(da2 ¦ ¦ dan)(a°da1)
= (-1)" f(da2-danda°)a' =(-1)" rfc1,. .., a", a0).
To prove the second property we shall only use the equality
/ aw = / ui for w ё ?2" , a € .4.
From the equality rf(a6) = (daN + adb it follows that
n
(do1 ¦¦•do")o"+1 =^(-l)"--> do1 ...d(aV+l)...do"+l + (-1)" oldo2..don+1,
thus the second property follows from
f a"+l (a°da1--da")= /(a0do1 ¦ ¦ da")a"+l.
Let us now recall the definition of the Hochschild cohomology groups H"(A,M) of A with coefficients in a
bimodule M ([Car-E]). Let A' = A ® A0 be the tensor product of .4 by the opposite algebra. Then any
bimodule M over A becomes a left A' module and by definition: H"(A,M) = ExtJi.(.4,.M), where A is
viewed as a bimodule over A via aF)c = abc, Va,i, с€Л As in [Car-E], one can reformulate the definition
of H"(A,M) using the standard resolution of the bimodule A. One forms the complex (C"(A, M), 6), where
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a) C"(A, M) is the space of n-linear maps from A to M\
b) for T € C"(A,M), ЬТ is given by
' a1 a' an+1) + (- 1)"+1 T{a\ ... ,a")a"+\
Definition 5. The Hockschild cohomology of A with coefficients in Ai is the cohomology Hn(A,M) of the
complex (C"(A,M),b).
The space A' of all linear functional on A is a bimodule over A by the equality (aipb)(c) — tp(bca), for
a,b,c ё A. We consider any T € C"(A, A') as an (n + l)-linear functional r on A by the equality
tV.o1....,!!") = T(al a")(aa) Va' € A
To the boundary 6T corresponds the (n + 2)-linear functional Ьт:
+ J2(-iy т(а°,...,а1а<+\...,а"+*) + (-1)"+1 г(ап+1Я° a").
•=1
Thus, with this notation, the condition 3) of proposition 4 becomes
a) ту = ?(j)t for any cyclic permutation у of {o, i,... , n};
b) Ьт = 0.
Now, though the Hochschild coboundary 6 does not commute with cyclic permutations, it maps cochains
satisfying a) to cochains satisfying a). More precisfey, let A be the linear map of Cn(A,A') to C(A,A")
defined by
7€Г
where Г is the group of cyclic permutations of {o, l,...»}. Obviously the range of A is the subspace C"(A)
of Cn(A,A') of cochains which satisfy a). One has
Lemma 6. Ь о А = А о Ь1 where b' : Cn(A,A') — C"+I{A,A') ts defined by the equality
j=0
Proof. One has
.vhere 0 < » < n, 0 < t < n + 1. Also
148
For j € {0,... n} one has
1 I*-1).
i=j+2
Also,
In all these terms, the x''s remain in cyclic order, with only two consecutive x^'s replaced by their product.
There are (n + l)(n + 2) such terms, which all appear in both bA<p and Aiftp. Thus we just have to check
the signs in front of Ttj (к ф j + 1) where TkJ = <р(хк,... ,x^x^+\ ... .x4). For Ab1 we get (_l)'+(»+»*
where i = j -k (modn + 2) and 0 < i < n. For bA we get (-iy+ni if / > к and (-l)»+n(*-0 if у < к.
When j > к one has i = j — к thus the two signs agree. When j < к one has i = n + 2 — к + j. Then as
n + 2 - к + j + (n + 1)* = j + n(k - 1) modulo 2
the two signs still agree.
Corollary 7. (C"(A),b) is a snbcomplex of the Hochschild complex.
We let HC"(A) be the n-th cohomology group of the complex (C",b) and call it the cyclic cohomology of
the algebra A. For n = 0, HC°(A) = Z^(A) is exactly the linear space of traces on A.
For A = С one has HC" = 0 for n odd but НС" = С for any even n. This example shows that the
subcomplex C? is not a retraction of the complex C", which for A = С has a trivial cohomology for all
n >0.
To each homomorphism /?: Л —> В corresponds a morphism of complexes: p* : C"(B) —* C"(A) defined by
(p»(a° а")=Г(р(а°) p(o"))
and hence an induced map p* : HC"(B) — #Cn(.4).
The map p" only depends upon the class of p modulo inner automorphisms as follows from:
Proposition 8. let и be an invertible element of A and в, 9(x) = uxu Vx ? A the corresponding inner
automorphism. Then $' : HC"(A) —» HC'(A) ts the identity.
Proof. Let a € A and let S be the corresponding inner derivation of A given by S(x) = ax — xa. Given
f € Z"{A) let us check- that ф,
i=0
a0
is a coboundary, i.e. that i/1 € B"(A). Let i/'ofa0,.. .,a"-') = y(a°,..., a"^) with a as above. Let us
compute Mi/'o = АЬ'фа. One has:
°,...,a",a)- (-1)" <p(a" a"-\ana)
+ (-1)" f>(aaa a"-1, a").
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Since btp = 0 by hypothesis, only the last two terms remain and one gets АЬ'ф0 — (—1)" ф. Thus ф =
(-1)" АЬ'фо = *((-!)" Лфо) € ЩЛ).
Now let u be an iuvertible element of A, let tp € Z\(A) and define #(x) = ttxu'1 for a; € A. To prove that
tp and tp о $ are in the same cohomology class, one can replace Л by M2{A), и by v =
tpi where, for a1 € Л and ft1 € M?(C)t
<fi2(aQ®bQ,a1®b\,...,an®bn) = tp(a°,... ,an) Trace F° ¦¦¦6n).
u 0
О и
and tp by
Now v = vlV2 with »i =
L ~j "
the result follows from the above discussion.
_?! 0
One has ut = expa,, a^ = «> v^, thus
We shall now characterize the coboundaries as the cyclic cocycles which extend to cyclic cocycles on arbitrary
algebras containing A. In fact extendibility to a certain tensor product С ®c Л algebra will suffice.
Following Karoubi [Kar4] [Karg], let С be the algebra of infinite matrices («ij)i,jeN with ay € C, such that
a) the set of complex numbers {a^} is finite,
/3) the number of non zero ay's per line or column is bounded.
Then for any algebra A the algebra С A = Cg>c Л is algebraically contractible in that it verifies the hypothesis
of the following lemma, and hence has trivial cyclic cohomology.
Lemma 9. let A be a unital algebra. Assume that there exists a homomorphism p : A —•¦ A and an inveriible
element X of M2(A) suck that X | ° ° x~l = | ° ° 1 for a ? A. Then HC"(A) = 0 for all n.
Proof. Let tp g Z"(A) and Kpi be the cocycle on M?(A) defined in the proof of proposition 8. For a € A,
let a(a) = . . and /}(a) = ... By hypothesis a and 0 are homomorphisms of .4 in Mi(A)
and, by proposition 8, tp^ooc and tp^a ft are in the same cohomology class. From the definition of tp2 one has
tpiiaia"), ..., a(a")) = v(a\ ... ,an) + tp(p(aa),... ,p(an)\
9?@(a\ • • •, /9(a")) = <p(p(aa), ..., p(a")).
Definition 10. We shall say that a cycle vanishes when the algebra fi° satisfies the condition of lemma 9.
Given an n-dimensional cycle (Q, d, J) and a homomorphism p : A —* Q°, recall that its character is given
by:
Proposition 11. Let т be an {n+ l)-linear functional on A; then
а) т 6 Z\(A) if and only if т is the. character of a cycle;
/9) г G B"(A) if and only if т is the character of a vanishing cycle.
Proof, a) is just a restatement of proposition 4.
/9) For (Q,d, J) a vanishing cycle, one has НС"(п°) = О, thus the character is a coboundary. Conversely if
г 6 B%(A), т - Ьф for some ф 6 CJ~'(-4), one can extend ф to CA = С ® A in an n-linear functional ф
such that
150
фA®а°,...,1®а"-1)=ф(а°,...,а"-1) for all a' € Л
and such that t/>A = e(A)^ for any cyclic permutation A of {0,..., n — 1}. Let p : A —> C4 be the obvious
homomorphism p(a) = 1 ® a. Then r' = it/1 is an n-cocycle on CA and r = p'r1 so that by the implication
3) =» 2) of proposition 4, r is the character of a cycle Q with Q° = C4.
Let us now pass to the definition of the cup product
HC(A) g> HC(B) — HC(A ® B).
In general one does not have п(А 8> В) = ЩА) ® п(В) (where the right hand side is the graded tensor
product of differential graded algebras) but, from the universal property of Q(A ® B), we get a natural
homomorphism т : Q(A ® В) -> ЩА) ® fi@).
Thus, for arbitrary cochains tp € С^ЛЛ") and ф € С"(В, В*), one can define the cup product <р#ф by the
equality
Theorem 12. 1) The cup product *р,ф к* 1р#ф defines a homomorphism
HC"(A) g> HCm(B) — tfC"+m(.4 ® 0).
2) The character of the tensor product of two cycles is the cup product of their characters.
Proof. First, let tp € Z"(A), Ф € Z"'(B); then tp (and similarly ф) is a closed graded trace on fi(.4), thus
!p ® ф is a closed graded trace on 1}(Л) ® Q(B) and v?#i/> € Z"+m(A ® 0) by proposition 4.
Next, given cycles Q, Q' and homomorphisms p : A -* li, /)' : В —' п', one has a commutative triangle
П(Л) ® ЩВ)
п(А ® B)
Thus 2) follows.
It remains to show that if tp ? B"(-4) tllen <Р*Ф is a coboundary:
I?®?
This follows from proposition 11 and the trivial fact that the tensor product of any cycle with a vanishing
cycle is vanishing.
Corollary 13. 1) HC"(C) is a polynomial ring with one generator <r of degree 2.
2) Each HC'(A) is a module over the ring HC"(C).
Proof. 1) It is obvious that #C"(C) = 0 for n odd and HC"(C) = С for n even. Let e be the unit of C;
then any tp € Z?(C) is characterized by tp(e, ...,e). Let us compute v?#V> where tp ? Zjm(C), ф € Z\m (C).
Since e is an idempotent one has in Q(C) the equalities
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de = ede + (de)e , e(de)e = 0 , e(deJ = (deJe.
Similar identities hold for e g> e and ж(е ® e) € Q(C) igi Q(C) and one has
ir((e ® e)d(e ® e)d(e ® e)) = edede ® e + e ® edede.
Thus one gets (<р#ф)(е, ...,е)= ^™!)'- <р(е, ...,е) ф(е, ...,е).
2) Let tp € Z\(A). Let us choose as a generator of ЯС2(С) the 2 cocycle <r(l, 1,1) = 1. Let us then check
that <r#tp = tp#cr and at the same time write an explicit formula for the corresponding map S • HC"(A) —>
HC"+2(A). У '
With the notations of 1) one has
(<p#<?)(a°,..., a"+2) = (?® ?)((a° ® e^a1 ® e) ¦ ¦ ¦ d(a"+2 ® e))
= ?(aVa2 da3 ¦ ¦ ¦ rfa"+2) + tp(aoda1(a2a3)da'1 ¦ ¦ ¦ da"+2) + ¦¦¦
+ tp(a°dal ¦ ¦ ¦ da'-^aW*1 )dai+2 ¦.. dan+2) + ¦¦¦
+ tp(a°dal ¦¦ da"(an+yan+2)).
The computation of <r#tp gives the same result.
For tp € Z%(A), let S<p = <r#tp = tp#* € Z^+2(A). By theorem 12 we know that SB%(A) С В"+2(Л) but
we do not have a definition of S as a morphism of cochain complexes. We shall now explicitly construct such
a morphism.
Recall that 1р#ф is already defined at the cochain level by (y>#i/>)A = (?® ф)° ж.
Lemma 14. For any cochain p € С?(Л) /e< Syj € CJ+2(^) ie deyined by S<p = ^ Л(<г#»?),• Йеп
a) ST3 ^(^«v) = <r*P for V € 2J(.4), so S ««ends Йе previously defined map.
/or у. €
Proof, a) If tp € ZJ(^) then (<r#tp)x - г(А)<г#у? for any cyclic permutation A of {0, 1 n + 2}.
b) We shall leave to the reader the tedious check in the special case ф = <т of the equality (ЬА1р)#ф = ЬАAр#ф)
for tp € Cm(A,A"). It is based on the following explicit formula for A(tp#a). For any subset with two
elements s = {i,j}, i < j, of {0,1 n + 2} = z/(»+3) one defines
or(s) = v?(a
in the special case j = n + 2 one takes
. .a'
<*(«) = ?>(a"+2a°,..., а'а'+\ ..., a»>) if i < n + 1,
a(s) = ?.(on+1o"+3o°,... ,a") " if . = n + 1.
l+?(n/J)
Then one gets yl(<r#v?) = J2 (-1)'+1 (n + 3 - 2г)ф{ where, for n even, one has
i = l
i'i = a({0, i}) + a({ 1,» + 1}) + ¦ ¦ ¦ + a({n + 2,» - 1}),
and for n odd
фi = a({0, (}) + у a({n + 2 - j,n + 2}) - a({n + 2 — i + 1,0}) a{n +2,:" - 1}.
We shall end this section with the following proposition. One can show in general that, if tp € Zn(A,A') and
V> € Zm(B,B') are Hoclischild cocycles, then y>#i/> is still a Hoclischild cocycle y>#i/> € Z"+m(A®B,A" ®B*)
152
and that the corresponding product of cohomology classes is related to the product V of [Car-E], p.216, by
[1р#Ф\ = ("П1Г/ Ы v M- Since " € Z2(C,C) is a Hochschild coboundary one has:
Proposition 15. For any cocycle tp ? Z"(A), Stp is a Hochschild coboundary: Stp — Ъф where
ф(а°,...,ап+1) = ^2(-1У ^(da1 -da'-1) a>(da'+l --da")).
Proof. One checks that the coboundary of the j-th term in the sum defining ф gives
ip(a°(dal ¦ -da'-1) a'a'+1(da'+2 ¦ ¦ ¦ da"+2)).
/J) Cobordims of cycles and the operator В
By a cAat'n of dimension n + 1 we shall mean a triple (u,du,f) where Q and dU are differential graded
algebras of dimensions n 4- 1 and n with a given surjective morphism r : Q —»¦ dQ of degree 0, and where
Г : Qn + i —* С is a graded trace such that
du - 0, Vu € 12" such that r(ui) = 0.
By the boundary of such a chain we mean the cycle (dQ, /') where for ш' € (dQ)" one takes f'w' = fdw for
any ш G fin with г(ш) = ш'. One easily checks, using the surjectivity of r, that f is a graded trace on 9J2
which is closed by construction.
Definition 16. let A be an algebra, and let A -^ Q, A ^» li' be iwo cyc/es over A. We shall say that these
cycles are cobordant (over A) if there exists a chain Q" with boundary Q ф Q' (where Q' is obtained from Q'
by changing the sign of J) and a homomorphism p" : A —> Q" such that г о p" — (p, p').
Using a fibered product of algebras one checks that the relation of cobordism is transitive. It is obviously
symmetric. Let us check that any cycle over A is cobordant to itself. Let Q° =r С°°([0,1]), Q1 be the space
of C°° 1-forms on [0,1], and d be the usual differential. Set dQ = СфС and take / to be the usual integral.
Then taking for r the restriction of functions to the boundary, one gets a chain of dimension 1 with boundary
(СфС, tp), tp(a, b) = a — b. Tensoring a given cycle over A by the above chain gives the desired cobordism.
(Equivalently one could replace smooth functions in the above chain by polynomials.)
Thus cobordism is an equivalence relation. The main result of this section is a precise description of its
meaning for the characters of the cycles. We shall assume throughout that the algebra A is unital.
Lemma 17. Let n, r2 be the characters of two cobordant cycles over A. Then there exists a ffochschild
cocycle tp e Z"+1(A,A") such that r, - rj ^ B0V, where
(Batp)(a0,. ..,an) = t
",.. •, <Л 1).
Proof. With the notation of definition 16, let
tp(aa,..., an+1) = fp"(a°) dp"(al) ¦ ¦ ¦ dp"(an+1) , Va1' € A.
Let
ы = p"(a°) dp"(a1) ¦ ¦ ¦ dp"(an) € ($)")".
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Then by hypothesis one has
(П - i)(o°,aI,...,on)= I dw.
Since p"(a°) = p"(\) p"(a°) one has
du = (dp"(/)) />>°) <*/>"(«') ¦ • ¦ dp"(a") + p"(l) <*/>>")' ¦ • <
Using the tracial property of J one gets
= (-1)" <p(a°,al an,l)+<p(l,a° a").
Using again the tracial property of J one checks that ip is a Hochschild cocycle.
Lemma 18. Let Ti.ro € -?"(.4) an*/ ass«me that r\ — r2 ~ Bof for some ip ? Z"
cycles over A with characters t\, Ti are cobordani.
/¦
(A, A"). Then any two
Proof. Let A —* 0 be a cycle over A with character r. Let us first show that it is cobordant with
{Q(A), r). In the above cobordism of Q with itself, with restriction maps r0, t"i, we can consider the subalgebra
{u>;ri(w) € il'}, where Q' is the graded differential subalgebra of il generated by p(A). This defines a
cobordism of Q with il'. Now the homomorphism p : Q(A) —* il' is surjective, and satisfies p" f — ?. Thus
one can modify the restriction map in the canonical cobordism of (Q(*4), r) with itself to get a cobordism of
(ЩА),т) with Of.
let us show that (il(A), ?i) and (il(A), ?¦>) are cobordaiit. Let ft be the linear functional on Q"+1 (A) denned
by
1) ni^da1 ¦ ¦ da"*1) = р(а° a"+1),
2) ^a1--Ja"+1) = (BoV)(a}-,...,an+l).
Let us check that /i is a graded trace on il(A). We already know by the Hochschild cocycle property of <p
that
H(a(bu)) = р((Ьш)а) , Va, 6 € Л , a) € Qn+1.
Let us check that /*(aw) = /*(u;a) for ш = da} ¦• -dan+i. The right side gives
)-..da"+1 rfa
1 a"+1,a) + (-1)"+'
(alda2 ¦ ¦ da)
1 ,a2, . .. ,an+1 ,
(-1)"
Now one checks that for an arbitrary cochian tp G C"+1(A,A*) one has
where Л is the cyclic permutation А(г) = j — 1. Here ip is a cocycle,
that /t(wa) = y(a, a1,..., a'1+1) = /*(аы).
It remains to check that for any a ? A and ы € fi™ one has
> = 0 and b'Вц/р — f = (—1)" ?"A so
= (-1)" /jurfa).
154
For w € rf^"~' tnis f°"°ws from the fact that B0V € C" (recall that Boy> = n — r2). For w = a°dal ¦¦¦ da" it
is a consequence of the cocycle property of Br>ip. Indeed one has ЬВо<р = 0, hence 6'Sov(a°, a1,..., a", a) =
(—I)" Bov(aa°'al> ¦ • • 'a") an<^ s'nce Ъ'Ва1? = f — (—l)n+1 у>л we get
(—I)
<p(a°, ...,a",a)- (-l)n+1 vj(a,a° a") = (-1)"+1 (B0<p)(aa°, a1 a"),
To end the proof of lemma 18 one modifies the natural cobordiam between (С1(А),т\) and itself, given by
the tensor product of Q(A) by the algebra of differential forms on [0, 1], by adding to the integral the term
uori, where n is the restriction map to {1} С [0, 1].
Putting together lemmas. 17 and 18 we see that two cocycles n, r2 € Z"(A) correspond to cobordant cycles
if and only if П - r2 belongs to the subspace Z"(-4) П B0(Zn+1(A,A')).
We shall now work out a better description of this subspace. Since At = (n+ l)r for any r € C%(A), where
A is the operator of cyclic antisymmetrisation, the above subspace is clearly contained B(Z"+1(A,A')),
where В = ABo : C"+1 — C.
Lemma 19. a) One has bB = — Bb.
b) One has Z$(A)n B0(Z"+1(A,A')) = BZ"+1(A,A').
Proof, a) For any cochain <p € C"+1(A,A"), one has
where A is the cyclic permutation А(г) = i — 1. Applying A to both sides gives ABobtp + Ab'Ba<p = 0. Thus
the answer follows from lemma 6.
b) By a) one has BZn+1(A,A') С Z%(A). Let us show that
BZ"+\A,A') С B0Zn+1(A,A').
Let/?€ BZn+1(A,A'), so that 0 = Bip, <p € Z"+l(A,A").
We shall construct in a canonical way a cochain ф € C"(A,A") such that ^y ,9 = Bo(<p — Ьф). Let
9 = Boy> — jji-j 0. By hypothesis Л0 = 0. Thus there exists a canonical ф such that i/> — e(A)V* = 0, where
A is the generator of the group of cyclic permutations of {0,1,..., n}, A(i) = a — 1. We just have to check
the equality
ВаЬф = 0.
Using the equality В0Ьф + b'ВоФ = i> — €{\)фх, we just have to show that b'Воф = 0. One has
ВоФ(а\ . ..,a"~l) = 0A, a0 a") - (-1)" ф(а°,..., a",1)
= (-I)" (V - €(\)фХ)(а° a",1) = (-I)" 0(a°, ... ,a"~\ 1)
= (-I)" (*>(l,a°, .. .,a"~\ 1) - (-1)"+1 ?.(a°,...,«""', 1, 1))
+ l^T (-1)" /3("°, ¦¦¦.«"" '.I)-
The contribution of the first two terms to Ь'Воф(аа,... ,an) is
j=0
(-1)" <p(a°,..
(-1)" (b^l.a0,...,^,!)-^1,...,^,!))
(ivj(a°,...,a",l,l)-(-l)" ip(a° a", 1)) = 0
155

since btp z= 0-
The contribution of the second term is proportional to
ra-l
-l)j 0(aa,..., a?a> + 1,..., a", 1) = b0(aa a", 1) = 0.
Corollary 20. 1) The image of В : C"+l -~ С is exactly CJ.
2) BJ(-4) С B0Zn+i(A,A').
Proof. 1) =*¦ 2) since, assuming 1), any bip, <p € CJ+I is of the form ЬВф = —ВЬф and hence belongs to
BZn+l(A,A") so that the conclusion follows from b). To prove 1) let <p 6 CJ. Choose a linear functional
?>o on A with y>o(l) = 1> and then let
ф(а0,... ,<,"+•) = v>o(a°Ma1,...,an+I)
+ (-l)n?>((ao-?.o(a°)l),aI a") W(a"«).
One has ^(l,a°, ...,a") = ip(aa,... ,a") and
ф(аа,... ,a", 1) = *>o(a°) <р(а1,. .. ,a", 1) + (-1)" *KaS... ,a")
+ (-1)"+1 Va(aa) <p(l,a\ ... ,a") = (-1)" ^(a0 a").
Thus BoV> = 2y> and y> € lm B.
We are now ready to state the main result of this section. By lemma 19 a) one has a well-defined map В
from the Hochschild cohomology group #"+1(A-4') to HC"(A).
Theorem 21. Two cycles over A are cobordant if and only if their characters rj, т^ € HC"(A) differ by an
element of the image of B, where
B: Hn+\A,A')
It is clear that the direct sum of two cycles over A is still a cycle over A and that cobordism classes of
cycles over A form a group M*(A). The tensor product of cycles gives a natural map: M*(A) x M*(B) —*
M" (A ® B). Since M~ (C) is equal to H^(C) = C[<r] as a ring, each of the groups M" (A) is a C[<r] module
and in particular a vector space. By theorem 21 this vector space is #C*(.4)/ImB.
The same group M"(A) has a closely related interpretation in terms of graded traces on the differential
algebra Q(A) of proposition 1. Recall that, by proposition, the map г ы г is an isomorphism of Z"(A) with
the space of closed graded traces of degree n on Q(A)-
Theorem 22. The map г i—» г jiiies an isomorphism of HC" (A)/ Ira В with the quotient of the space of
closed graded traces of degree n on Q(A) by those of the form a* /i, /i a graded trace on Q(A) (of degree n + 1).
Proof. We have to show that, given т € Z"(A), one has г = d'/i for some graded trace /i if and only if
г G ImB Э B". Assume first that г = rf'/i. Then as in lemma 17, one gets г = B0<p where <p € Z"+1(A,A')
is the Hochschild cocycle
<p(aa,al an+I) = /i(aoda1--da"+I) , W € A-
Thus г = ^ ylBov? € lm B.
Conversely, if г 6 ImB, then by lemma 19 b) one has г = Baip for some ip € Z"+i(A,A'). Denning the
linear functional /i on пп+1(А) as in lemma 18 we get a graded trace such that
156
H(da°dal ¦¦¦da") = r(a°,...,a") Va1' € Л
Thus M"(A) is the homology of the complex of graded traces on C1(A) with the differential <P. This theory is
dual to the theory obtained as the cohomology of the quotient of the complex (Cl(A), d) by the aubcomplex
of commutators. The latter appears independently in the work of M. Karoubi [Kari] as a natural range for
the higher Chern character defined on all the Quillen algebraic К theory groups Ki(A). Thus theorem 22
(and the analogous dual statement) allows:
1) to apply Karoubi's results [Kari] to extend the pairing of section 3 to all Kt(A);
2) to apply the results of section 7) (below) to compute the cohomology of the complex (fi(.4)/[ , ],d).
7) The exact couple relating HC"(A) to Hochschild cohomology
By construction the complex (C%(A),b) is a subcomplex of the Hochschild complex
(C"(A,A"),b), i.e. the identity map / is a morphism of complexes and gives an exact sequence:
0 -<¦ CJ -L C" — C/C1 — 0.
To this exact sequence corresponds a long exact sequence of cohomology groups.
We shall prove in this section that the cohomology of the complex C/C\ is H"(C/C\) = H"~l(C\).
Thus the long exact sequence of the above triple will take the form
О ^ HC°(A) - H"(A,A') -~ HC~l(A) - HC\A) ± Hl(A,A')
-* HC°(A) — HC\A) -i- • • ¦
H"(A) -i- Hn(A,A') - HCn~\A) - HC"+l(A) -i- Hn+l(A,A")~* ¦¦¦.
On the other hand we have already constructed morphisms of cochain complexes S and В which have
precisely the right degrees:
5 : HC"~l(A) — HC"+l(A),
В : Hn(A,A") -* HC"~l(A).
We shall prove that these are exactly the maps involved in the above long exact sequence, which now takes
the form
HC"(A) — H"(A,A') Z. HC"~\A) -S. HC+l(A) -^ • ••.
Finally to the pair b, В corresponds a double complex as follows: C1' = Cn'm(A,A') (i.e. C"'m is 0 above
the main diagonal) where the first differential d, : C"'m — C"*+1'm is given by the Hochschild coboundary 6
and the second differential d2 : С1- —> C"'m+1 is given by the operator B.
By lemma 19 one has the graded commutation of ditd2. Also one checks that B2 = 0 so that d\ — 0. By
construction the cohomology of this double complex depends only upon the parity of n and we shall prove
that the sum of the even and odd groups is canonically isomorphic with
157

It С {Л) Q C=ff"(A)
//C-1C)
(where #C*(C) acts on С by evaluation at. <r = 1).
/
/
m .
/
1
1
r
X
\
c2
/
\
\
c4
b
/
c3
\'
c6
/
в.
с5
в\
\
ca
/
b
c7
\
\
c10
л
с9
\
\
n
Fig.l. The {b, B) bicomplex
The second filtration of this double- complex [ F* = Y. <?"'"' 1 yields the same filtration of H"(A) as the
V '»>« /
filtration by dimensions of cycles. The associated spectral sequence is convergent and coincides with the
spectral sequence coining from the above exact, couple. All these results are based on the next two lemmas.
Lemma 23. Let ф e C{A,A") be such thai 1,ф G Q+1(-4). Then Ьф € Z"~'(-4) and БВф = п(п+\)Ъф in
HC(A).
Proof. One has Вф € C"~l by construction, and ЬВф = -ВЬф = 0 since Ьф € C^+1- Thus Вф € Z"'1.
In the same way Ьф G Z" + 1.
Let tp ~ Ьф, by proposition 10 one has Sip = ЬФ' where
ф'(а° a") = ?j- ty-1 ^(«"(rfa1 ¦ --d
j = l
It remains to show that.
Ф' -f(A) !/¦'* = n(n + 1)(*" ~e(X) ф"х)
where A(i) = i — 1 for )' € {0, I,... ,n + 1} and v'1" — </' € B". Let us first check that
a1' (dai+l ¦ ¦ ¦ da'1)).
One has
-е(Х) фХ)(аа,...,ап)= (-1
Let
Then
aij =ao(da1---da'-1)
¦ ¦¦ da"-l)an.
duj = (da° ¦ -da''1) e» (
+ (-1)' a^da1 • -da' ¦ ¦ dan-l)an
+ (-1)" aa(dai---da>-i) a>(da?+i ¦ • da").
Thus for j € {1,..., n — 1} one has
(-1У'-1 ifiia^da'-'
e(A)(-iy ^(<fa0---
= (-1)"-'
ij-l)ai(dai+l--da"))-
в").
Taking into account the cases j — 0 and j = n gives the desired result.
Let us now determine ф", ф" - ф € Bn(A, A") such that
"a0 а"-1).
so that 92 = ± <p. Since Л0! = 0 there
Г>" - е(А) V"A)(a° о") = bi^li ^
Let « = Воф and write « = #i + «2 with A»x = 0, 92 € CJ^
exists ф-i € С""' such that #i = Oi/>i where пфг - ф\ - e(\) ф$.
Parallel to lemma 6 one checks that D о b = b1 о D and hence DFVi) = V9\- Let ф" = ф — Ьф1. As
D = В0Ь + b'B0 we get пф = VВаф = Ъ'в^ + Ь'в2 hence пф" = b'B2 = ? Ъ'<р. Finally since bip = 0 one has
As an immediate application of this lemma we get:
Corollary 24. The image of S : HC"-'(A) —¦ HC"+l(A) is the kernel of the map I : HC"+l(A) —
Я"+1(Л-4*)-
This is a really useful criterion for deciding when a given cocycle is a cup product by HC2(C), a question
which arises naturally in order to determine the dimension of a given class in H*(A)- In particular it shows
that if V is a compact manifold of dimension m and if we take A = C°°(V), any cocycle г in HC"(A)
(satisfying the obvious continuity requirements (cf. section 2)) is in the image of S for n > m = dimV.
Let us now prove the second important lemma:
Lemma 25. The obvious map from (ImBn Ker J>)/6(Im B) to (Ker В П Ker Ь)/Ь(КетВ) is bijective.
Proof. Let us show the injectivity. let <p € ImBnKerft, say <p € Z"+l(A), and assume <p € 4(KerS). Then
lemma 23 shows that <p and So = 0 are in the same class in #C*+l(.4) and hence <p € b(lmB).
Let us show the surjectivity. Let ip € Zn+l(A,A"), B<p = 0 and ф € C"(A,A"), ф - e(X) фх = Ва<р. As
above one gets ВаЬф = Ba<p- This shows that ip' = tp - Ьф € Z"+l(A) since Dtp1 = Bobtp1 + 6'Bov' = 0. Let
159

us show that Вф g bC?~2. Since ф — eW ФХ — Вг>Ьф one has b'Воф = 0. One checks easily that б'2 = О
and that the V cohomology on C"(A,A") is trivial (if b'ipi = 0 one has b'ip1(a°) ..., e", 1) = 0 i.e. tp\ — Vtp^
where
Ы°° a"-1) = (-1)"-' Vi(a0,... ,a"-\l)).
Thus Воф = b'O for some в € С1 and Вф = АЬ'в = ЬАв € ЬС^'2.
Thus since C"~2 = ImB one has Bi/> = *B#i for some 0! € С" i.e. t/> + 60i € Ker В and Ьф ? 4(Кег В).
As <p — Ьф g ^"+1 this ends the proof of the surjectivity.
Putting together the above lemmas 23, 25 we arrive at an expression of 5 : HC"~l(A) -> HCn+1(A)
involving b and B:
More explicitely, given <p € Z"~'(,4) one has у € Imfl, thus ^> = Вф for some t/>> and this determines
uniquely Ьф € (Ker 4 П Ker B)/6(Ker B) = HC+1(A). To check that Ьф is equal to n(J+1) S*p one chooses
ф as in proposition 15:
da'-1) a>(d«> + 1 ¦ ¦ ¦ da")).
As an immediate corollary we get:
Theorem 26. The following triangle is exact:
H-(A,A')
HC-(A) Л. HC'(A)
Proof. We have already seen that Im 5" = Ker Л By the above description of 5" one has Ker 5" = Im B.
Next В о I = 0 since В is equal to 0 on C\. Finally if <p € Z"(A,A") and B<p € B^~l, Bip - ЬВв {от some
ве С"'1 so that
ip + Ьв в Ker В П Ker 4 С Im / + 4(Кег В)
by lemma 25. Thus Ker В = Im/.
We shall now identify the long exact sequence given by theorem 26 with the one derived from the exact
sequence of complexes
0 — Сл -~ С — С/Сл — 0.
Corollary 27. The morphism of complexes В : C/C\ —<¦ С induces an isomorphism of Hn(C/Cx) with
HCn~l(A) and identifies the above triangle with the long exact sequence derived from the exact sequence of
complexes 0 _ C> — С —• C/Cx -* 0.
Proof. This follows from the five lemma applied to
160
H"(CX) - H"(C) -. H»(C/CX) -
Together with theorem 21 we get:
Corollary 28. a) Two cycles with characters n, т^ are cobordani if and only if Sn = Sib in HC(A).
b) One has a canonical isomorphism
M'(A) О C = H'(A).
Af(C)
c) Under that ,somorph,sm the canomcal filtrat.on F» H' (A) corresponds to ike filtration of the left side by
the dimension of the cycles.
Proof, of b). Both sides are identical with the inductive limit of the system (HC"(A), S).
Let us now carefully normalize the two differentials du d* of the double complex С so that the map S is
given simply by dirfj1. This is done as follows:
a) C"-m = С—ЧАЛ'), 4n,m€ I,
b) for v € C'm, dlV ={n-m + l)b<p€ C"+lm;
c) for <p € C"", d2<p = jij B<p € C"'m+1 (if n = "•¦ the latter is 0).
Note that did2 = -</2di follows from Bb = -ЬВ.
Theorem 29. a) The initial term E2 of the spectral sequence associated to the first filtration FPC =
is equal to 0.
b) Let FiC = ?m>, C"m be the second filtration, then H'(F'C) = HC"(A) forn = p-2q.
c) The cohomology of the double complex С is given by
E
and
tf"(C) =He"{A) if n is even
tf"(C) = Hadi(A) if n is odd.
d) The spectral sequence associated to the second filtration i, convergent: it ™~^*о
graded ? f H'(A)/F^H'(A) and it coincides with the spectral sequence associated with the exact couple.
In particular its initial term E? is
Ker(/oS)/Im(/oS).
Proof, a) Let us consider the exact sequence of complexes of cochains 0 - ImB - Ker В - J^l^T
0 where the «.boundary is 4. By lemma 25 the first map: ImB - Ker В becomes an .s
cohomology, thus the Ь cohomology of the complex Ker B/ImB is 0.
161

C"m, satisfy dtp = 0, where d =
?. By a) it is («homologous in FqC
b) Let <p e (F4C)P =
to an element ф of CT>-«-». Then dtp = 0 means ф G Ker 4 П Ker B, and ф ? Imd means t/> € 4(Ker S). Thus
using the isomorphism
(Кег4ПКегВ)/6(КегВ) = ЯС-2»(.4) (lemma 25)
one gets the result.
to HT(F1~1C)
c) By the above computation of 5 as rfirfj1 (lemma 23) we see that the map from
is the map S from ЯС"!д(^() to #C-J*+3(.4); thus the answer "is immediate.
d) The convergence of the spectral sequence is obvious, since Cnm = 0 for m > n. Since the filtration of
H"(C) given by H"(F'C) coincides with the natural filtration of H"(A) (cf. the proof of c)), the limit of
the spectral sequence is the associated graded
J2 Fq H'"(A)/F'I+1 H"(A) for n even.
(where 6'<p(a°, a") = J2"=1 f(a", ¦¦¦ ,<Ha')i.. • ,a") for all a' G ^). This is the natural extension of the
basic formula of differential geometry dx = dix + ixd, expressing the Lie derivative with respect to a vector
field X on a manifold.
c) Homotopy invariance of H*(A). Let A be an algebra (with unit), В a locally convex topological algebra
and f G Z"(B) a continuous cocyde (cf. appendix B). Let pt, i e [0,1], be a family of homomorphisms
p, : A —> В such that
for all a € A , the map i € [0,1] -> p,(a) 6 В is of class C1.
Then the images by S of the cocycles p^ip and p\<p coincide. To prove this one extends the Hochschild cocycle
Vj#V on В ® C" ([0,1]) giving the cobordism of ip with itself (i.e. ф(/",Г) = /„' f°df' , V/' e ^([0,1]))
to a Hochschild cocycle on the algebra (^([O, 1],B) of C'-mpas from [0, 1] to B. Then the map p : A -*¦
С'([0, 1], В), (p(a))t — Pt(a), defines a chain over A and is a cobordism of p^tp with p*y>. This shows that if
one restricts to continuous cocycles, one has
Y^ f НМ(А)/Р1+1 НМ(А) for n odd.
«
It is clear that the initial term E? is Ker fofl/ImloB. One then checks that it coincides with the spectral
sequence of the exact couple.
We shall end this section with several remarks.
Remarks, a) Relative theory. Since the cohomology theory HC(A) is defined from the cohomology of a
complex (Сд, 6), it is easy to develop a relative theory HC*(A, B), for pairs A —•¦ В of algebras, where зг is
a surjective homomorphism. To the exact sequence of complexes
0 — CJ@) С С"(Л) — C"(A,B) = Cn(A)/Cn(B) — 0
corresponds a long exact sequence of cohomology groups.
Using the five lemma, the results of this section on the absolute groups extend easily to the relative
groups, provided that one also extends the Hochschild theory Hm(A,A*) to the relative case.
b) Action ofH'(A,A). Using the product V of [Car-E]
Hn(A,Mi) ® Hm(A,Mi) —¦ Hn+m(A,Mi ®a M2)
one sees that H'(A,A) becomes agraded commutative algebra (using А®д А = A, as A bimodules) which
acts on H"(A,A*) (since A ®.< A' = A*). In particular any derivation 6 of A defines an element [S] of
ff'(-4,-4). The explicit formula of [Саг-E] for the product V would give, at the cochain level
(<pV6)(aa,al,...,an+1) = <pF(an+1)aa,a1 a") , V<p
One checks that at level of cohomology classes it coincides with
.
= гГAУ ^(A» *») 6
n -\- 1 ¦*—'
J = l
With the latter formula one checks the equality
«V = (Jo В)Fф<р) + «#((J о в)<р) in H"+1(A,Am)
162
da"+1)) , 4<p € Zn(A,A').
d) Excision. M. Wodzicki ([Woi]) has characterized the non unital algebras which satisfy excision in
Hochschild and cyclic homology by the very simple property of ff-unitality: An algebra A (over C) is
Я-unital iff the 4' complex is acyclic.
163

Ш.2. Examples
The results of section 1.7) such as theorem 26 give a very powerful tool to compute the cyclic cohomology
of a given algebra A by tying it up with the standard tools of homological algebra available to compute
Hochschild cohomology from projective resolutions of A viewed as an A bimodule. We shall illustrate this
by two examples a) 0). In example 7) we describe the result of D. Burghelea for the cyclic cohomology of
the group ring A — СГ of a discrete group Г.
a) A = С°°(К), V a compact smooth manifold
We endow this algebra A = C°°(K) with its natural Frechet space topology and only consider continuous
multilinear forms on A (cf. Appendix B). Thus the topology of A is given by the seminorms Pn(/) = Sup
|o|<n
|d°7| using local charts in V. As a locally convex vector space C°°(V) is then nuclear so that topological
tensor products are uniquely defined and for any integer n the n-fold topological tensor product:
is canonically isomorphic to С°°(К").
In particular the algebra В = A§>A° is canonically isomorphic to С°°(К X V). Thus A viewed as an A
bimodule corresponds to the B-module given by the diagonal inclusion:
Д : V — V x V , A(p) = (p,p) Vp € V.
Using a projective resolution of the diagonal \nV xV one gets:
Proposition 1. [Co, 6] Let V be a smooth compact manifold and A the locally convex topological algebra
). Then
a) The following map <p —* Cv is a canonical isomorphism of the continuous Hochschild cohomology group
Hk(A, A*) with the space Vt of k-dimensional de Rham currents on V:
(Cv, fdf1 Л • • • Л dfk) = 1/fc!
, /'<'>, •.. /'<*>)
V/0,...,/*
b) Under the isomorphism С the operator I°B : Hk(A,A') -» Hk~'(A,A') is the de Rham boundary for
currents.
We refer to [C016] for the proof. The analogous statement for the algebra of polynomials on an affine variety
is due to Hochschild Kostant Rosenberg [Ho-K-R].
In particular we recover from this proposition the de Rham complex of the manifold V without any appeal
to the commutativity of the algebra A which is used in a crucial manner in the construction of the exterior
algebra. Thus for instance if we replace in proposition 1 the algebra A by the algebra Mq(A) = Mq(C)®A of
matrices over A we bose the commutativity and the exterbr algebra but the cohomology groups Hk(A,A'),
the operator 7, В ... still make sense and yield, using Morita invariance of Hochschild cohomology ([Car-E])
the same result as for к = 1. The formula (a) is no longer valid and the Hochschild cocycle (class) associated
to a current С € X>i is now given by the formula:
V/' € C°°(K) , y? € M,(C).
let us now compute the cyclic cohomology HC(C°°(V)):
164
Theorem 2. Let A be the locally convex topological algebra C°°(V). Then
1) For each k, HCk(A) is canonically isomorphic to the direct sum
Ker4(C Vt) ф tft_3(V,C) Ф Ht_4(V,C) В ¦ ¦ ¦
(where H,(V,C) is the usual de Rham homology ofV and b the de Rham boundary).
2) H"(A) is canontcaly isomorphic to the de Rham homology H.(V,C) (with filtration by dimensions).
Proof. 1) Let us explicitely describe the isomorphism. Let f € HCk(A). Then the current С associated to
l(ip) is given by
(С/Ч1 л ¦ ¦ ¦ лdfk) = l/i! ? v(f°,/'<")_...,/•<*>).
<T€Sk
It is closed (since B(I(<p)) = 0), so that the cochain
P(f,f\-- -,/*)= (C,f>dfl A .--Ad/*)
belongs to Z\ (A). The class of tp — Tp in HCk (A) is well determined, and is by construction in the kernel of
I. Thus by theorem 26 there exists ф ? HCk~2(A) with Si/> = <p —ip, and ф is unique modulo the image
B. Thus the homology class of the closed current СA(ф)) is well determined. Moreover the class of ф — ф
in HCk~2{A) is well determined. Repeating this process one gets the desired sequence of homology classes
~ 00
uij € .fft_2j(K,C). By construction, tp is in the same class (in HCk(A)) as C+ Yl S'wj (where for any
j=i
closed current Uj in the class one takes
This shows that the map that we just constructed is an injection of HCk(A) to Ker b ® flt_3( V, С) ф • ¦ • ф
The surjectivity is obvious.
2) In 1) we see by the construction of the isomorphism, that S : HCk(A) —> HCk+2(A) is the map which
associates to each С € Ker b its homology class. The inclusion follows, Note that in this example the spectral
sequence of theorem 26 is degenerate so that its ?2 term is already the de Rham homology of V.
Theorem 2 shows in particular that the periodic cyclic coliomology of C°(V), with its natural filtration,
is the de Rham homology of the manifold V. One should not however conclude too hastily that cyclic
cohomology gives nothing new in this case of smooth manifolds. We shall now see indeed on the example of
the fundamental class [S'} of the circle Si that its image by the periodicity operator:
Sk[Sl] € HC2k+l(C°°(S1))
extends to the much larger algebra CfS1) of Holder continuous functions of exponent a > sjijrj, i.e. of
functions / such that:
\f(x)-f(v)\<Cd(x,y)-
where d is the usual metric on S1.
To get a nice formula for a cyclic cocycle rt ~ S*[S'] which extends to С we shall write it as a cocycle on
Cf (R) but a similar formula works for S1.
Proposition 3. a) The following equality defines a cyclic cocycle Tt € HC2k+1(Cf(R)):
165

л/1 /"+i)=/a«d)/1('2_?(*D) -
X2 - X1
b) 77ie restriction ofrk to C°°(R) is equal in #C2t+1 to ct5*TU, where ro is <Ae homology fundamental class
o/R:
ondct =BtiJ
- 1) ... 3.1>.
The multiple integral makes sense since each of the terms ^'^'Jzi'i-'i—' >s O(W ~ x'~x\"~l). One then
checks that rt is a cyclic cocycle and that b) holds.
Thus, using a similar formula for Sk[Sl] € HC2k+1(C"(S1)) we see that we get an explicit formula, for the
pushforward «/'•[¦S1] without perturbing the map ф : S1 —• V, Holder continuous of exponent a > ^jlpy
<ms4(/V--,/"+1) = м Ф'/",г /',-¦¦, Ф' /2t+1)
V/°, ¦ ¦ ¦,/2t+1 €C°°(V), with <l>'f = /оф.
This gives a cyclic cocycle ф.^1] € HC3k+1(Ca'(V)) whose formula does not use any smoothing of the
Holder continuous map ф.
The naive formula:
Js*
does not make sense unless the functions g' = ф" f' on Sl belong to the Sobolev space W1'2 governed by
the finiteness of
where g is the Fourier transform of g. Thus it does not make sense for the ф' f, f € C°°(V) when / is only
Holder of exponent a < k-
Of course in the periodic cyclic cohomology of C°°(V), the above cocycle coincides with 5*i/>i[5'1] where ф'
is smooth and homotopic to ф.
Remarks 4. a) Let as above A = C°°(V) and assume, to simplify, that the Euler characteristic of V
vanishes. Let then X be a non vanishing section of the vector bundle on V x V which is the pull back by
the second projection pr-, : V x V —> V of the complexified tangent bundle Tc(V), and which satisfies:
For (a, 6) € V x V close enough to the diagonal, X(a,b) coincides with the real tangent vector expf'(a),
where exp6 : Tb(V) —»¦ V is the exponential map associated to a given Riemannian metric on V.
Let then Et be the complex vector bundle onVxV which is the pull back by the second projection pr^ of the
exterior power /\kT?(V). The contraction ix by the section X gives a well defined complex of C"(V x V)
modules, i.e. of C°°(V) bimodules:
V,Ei)
— 0
where n = dim V and Д is the diagonal V ^ V x V. This gives an explicit projective resolution M' of the
Л bimodule A, and a proof of proposition 1 ([C016]).
166
Let then Mt — (-4®-4°)®.4et be the standard resolution of the bimodule A, with the boundary
bk : Mt -* Mk-i
given by the equality:
**(l®ai® ¦ -® at) = (a! ® 1)® (<J2® •• -®at)
t-i
+ (-1)* A ® a?)® (a,
Then an explicit homotopy of the resolutions is given by:
F : M' — M
¦••®at
(-Fu)(aAx> x*) = (Х(х\Ь)Л--/\Х(хк,Ь) , ш(а,Ь))
Vu€X't = C°°(V2,Et) , Va.i.x1,...,!* € V.
Working out explicitely the homotopy formulae yields for any given cyclic cocycle ip € ifC'(Cco(V)) explicit
closed currents uij of dimension q — 2j such that
b) Let W С V be a submanifold of V, Г : C°°(V) -. C^fVV) the restriction map, and 0 — Ken* ->
C°°(V) -* C°°(W) —> 0 the corresponding exact sequence of algebras. For the ordinary homology groups
one has a long exact sequence
where the connecting map is of degree —1.
Since HC" is defined as a cohomology theory, i.e. from a cochain complex, the long exact sequence
-. HC(C°°(W)) ^ HC(C°°(V)) ^ HC'I(C"'(V),CCO(W)) Л HC+\C°°(W)) — •¦¦
has a connecting map of degree +1. So one may wonder how this is compatible with theorem 2. The point
is that the connecting map for the long exact sequence of Hochschild cohomology groups is 0 (any current
on W whose image in V is zero, does vanish), thus \m(d) С IC|-'(C"(ff)).
d) Only very trivial cyclic cocycles on C°°(V) do extend continuously to the С algebra C(V) of continuous
functions on a compact manifold. In fact for any compact space X the continuous Hochschild cohomoiogy of
A = C(X) with coefficients in the bimodule A' is trivial in dimension n > 1. Thus by theorem 26 the cyclic
cohomology of A is given by HC^"(A) = HC°(A) and HC*"+\A) = 0. This remark extends to arbitrary
nuclear С algebras.
/3) The cyclic cohomology of the non commutative torus A = Ae, 9 € R/Z
Let A = ехр2т>'#. Denote by 5(Z2) the space of sequences (апт)„ m€Za of rapid decay (i.e. (|n| + |m|)'|anm|
is bounded for any q € N).
Let Ae be the algebra whose generic element is a formal sum
product is specified by the equality U2U1 = \UiU?.
We let r be the canonical trace on As given by:
167
>' where (anm) G S(Z2) and the

г(Е а„ U") = а@,0) where for v = (п!,п2) € Z2 we let G" = U?1 U%*.
For в € Q this algebra is Morita equivalent to the commutative algebra of smooth functions on the 2-torus.
Thus in the case в € Q, the computation of Я* (Ав) follows from theorem 2.
We shall now do the computation for arbitrary в. The first step is to compute the Hochachild cohomology
H(Ae,Ag), where of course Ae is considered as a locally convex topological algebra (using the seminorms
pq(a) — Sup(l + |n| + |m|)'|an,m|). We say that в satisfies a diophantine condition if the sequence |1 — Л"|—г
is O(nk) for some k.
Proposition 5. [Co!6] a) Let в <? Q. One has Н°(Ав,А'в) = С.
b) If9?Q satisfies a diophantine condition, then H> (Ae ,A"e) is of dimension 2 for j' = 1, and of dimension
1 for j = 2.
c) If в ? Q does not satisfies a diophantine condition, then Я1, Я2 are infinite dimensional non ffausdorff
spaces.
(Recall that by theorem 2, H'(Ae,A's) is infinite dimensional for j < 2 when в 6 Q.)
At this point, it might seem hopeless to compute the periodic cyclic cohomology H*(Ae) when в is an
irrational number not satisfying a diophantine condition, since the Hochschild cohomology is already quite
complicated. We shall see however that even in that case, where H'(Ae,A%) is infinite dimensional non
Hausdorff, the homology of the complex (Hn(Ag,A^), IoB) is still finite dimensional.
¦Proposition 6. [C016] (в i Q). One has НС°(Ав) = С and the map
is an isomorphism.
(Thus in particular any 1-dimensional current is closed.)
Theorem 7. [CoN] a) For all values of в, H (Ae) =s C2 and HoM(Ae) = C2.
b) Let т be the canonical trace on Ae- The following cyclic cocycles <plt <p-> give a basis of Haii(Ae) =
Н\Аа,А'вIЛш(иВ) : Vj(xa, г1) = ф%(х1), Vr' € A».
c) One has H'v(Ae) = H2(Ae); it is a vector space of dimension 2 with basis St (t the canonical trace) and
the functional tp given by
?.(*°,*1,*2) = Bxi)-1 r(*°(«i(*')M*2)-M*1)«i(*J)) V*' €Ae.
In these formulas, Si, S? are the basic derivations of Ae : Si(U") —2riniUl/, Sj(U") = ViriniU".
j) The cyclic cohomology of the group ring СГ, Г discrete group
Let first G be a compact group and X a topological space gifted with a continuous action of G. Then the
equivariant cohomology Hq of X is defined as the cohomology of the homotopy quotient Xq = Xxg EG,
where EG is the total space of the universal G-principal bundle over the classifying space BG. In particular
#ё(А') is in a natural manner a module over Hg(pl) = H'(BG).
For G = Sl, the one dimensional torus, BSl = Poo(C) and Hgt(pt) is a polynomial ring in one generator of
degree 2.
The formal analogy between cyclic cohomology and S'-equivariant cohomology is well understood from the
equality ([Co22])
168
ВЛ = BSl
where Л is the small category introduced in [Соз2] which governs cyclic cohomology (cf. appendix A) through
the equality
HC(A)=Ext\(A\Ci)
where the functor A —> A^ from algebras to Л-modules (appendix A) gives the appropriate linearization of
the non abelian category of algebras. For group rings, i.e. for algebras (over C) of the form:
where Г is a discrete (countable) group, the following theorem of D. Burghelea ([Bu]) yields a natural Sl-
space whose 5'equivariant cohomology (with complex coefficients) is the cyclic cohomology of A. Recall
that the free bop space Ys of a given topological space У is the space MapE1, Y) of continuous maps from
Sl to У with the compact open topology, this space is naturally an S1 space, using the action of S1 on S1
by rotations.
Theorem 8. ([Bu]) let Г be a discrete group, A = СГ its group ring.
a) The Hochschild cohomology H'(A,A") is canonically isomorphic to the cohomology H*((BT)S', C) of the
free loop space of the classifying space о/Г.
b) The cyclic cohomology HC'(A) is canonically isomorphic to the S1 -equivariant cohomology #5,((ВГM',С).
Moreover the isomorphism b) is compatible with the module structure over H"(B\) = H'(BSl), and under
the isomorphisms a) b) the long exact sequence of theorem 26 becomes the Gysin exact sequence relating
H' to я;,.
As a corollary of this theorem we get the computation of the cyclic cohomology of СГ in terms of the
cohomology of the subgroups of Г defined as follows:
1) For any g € Г let Cg = {h € Г, gh = hg) be the centralizer of g.
2) For any g € 7 let Ng = Cg/gz be the quotient of Cg by the (central) subgroup generated by g.
Let then (Г) be the set of conjugacy classes of Г, let (Г)' С (Г) be the subset of classes of elements g G Г of
finite order and (Г)" its complement.
By construction the groups Cs, Na only depend upon the class g € (Г) of g.
Corollary 9. ([Bu])
1)Я*(СГ,(СГ)*)= П H'(C3,C)
H'(Ng,C)-
Moreover the structure of ЯС*(С) module on ЯС*(СГ), i.e. the operator 5, decomposes over the conjugacy
classes and is the obvious one for the finite ones g € (Г)'. For the infinite ones the operator S is the product
by the 2-cocycle ш3 € H^(NS,C) of the central extension:
2)ЯС(СГ)= П (Я*(ЛГ„С)®#С-(С))х
о -* г -. с.
¦N,-
where we used the inclusion Z С C, and the map H2(N3,Z) —* H2(Ng,C). In particular the infinite
conjugacy classes j € (Г)" may contribute non trivially to the periodic cyclic cohomology Я*(СГ) (cf. [Bu]).
We shall now give the details of the proof of theorem 8 (cf. [Bu]). First, with A = СГ, the associated cyclic
vector space C(A) (cf. Appendix A) is the linear span of the following cyclic set (yn,dj,,si,tn):
169

<U(9o,9b
- ,9») = (9o,--,9i9i+i>-->9r.) 6 Yn-i , for 0< i < п- 1
n-i)eУп-ь
«'„(90. ¦¦¦,»«) = (So,- ¦ ,9i.l.9.+i,..-,9»i) 6 Уп+i
М9о,---,9а) = (9n.9o.---.9n-i) 6 Yn.
Thus it follows trivially that the Hochschild (resp. cyclic) cohomology H'(A,A') (resp. HC'(A)) is the
cohomology (resp. 5l-equivariant cohomology) of the geometric realization [Y| of У
where the cyclic structure of У endows \Y\ with a canonical action of S1 (cf. Appendix A).
As
onfiguration space Cs» (Г) is generated by the following open sets:
i h,gi,---,9k)= ¦|a;Supp(a)CU7J , J[a(e) = g, I
where the Ij are open intervals in S1, the (jj belong to Г, and the product J~[ а(в) is the time ordered product
(i) () values of a at #i < 92 < ••• < "fc where Supp(a) П Ij = {9i,...,9t}. The natural
homeomorphism h : \Y ll h th doition:
a($i) •• а(вк) of the values of a at #i < 92 < ••• < "fc where Supp(a) П Ij = {9i,..,t}
h : \Y\ —* Cs>(T) is denned as follows. At the set theoretic level one has the decomposition:
where DegYo is the union of the images of the maps s'n_1 : У„_1 —» Yn and Int(An) the interior of the n
simplex Д„ = {(A0i=o,....n ; A; > 0 , ? A,- = 1}.
Then the image by h of ((gi), (A;)) € (Yn\Degyn) x Int Д„ is the configuration with support contained in
the set {0, Ao, Ao + Ai,..., Ao + • • • + An-ij and such that
«@) = 3a . «(Ao) = 9i,--.,« I 2_,л' I = 3* ' f°r • < "•
V о
Note that since (yO 6 yn\DegYn one has j, ф 1г for i ф 0 but one may have 90 = lr in which case the
cardinality of the support of the configuration a is equal to n.
It is then not difficult to check that h is an ^-equivariant homeomorphism:
h:|Y|-.Cs.(r).
Thus theorem 8 follows from:
170
Proposition l'O. [M15] [Эеб] The configuration space Csi(F) is Sl-equivariantly weakly homotopy equivalent
to the free loop space (ВГ)Я .
More explicitely let I С Sl be the open interval S'\{0}, then С/(Г) is canonically homeomorphic to ВТ, the
above weak homotopy equivalence associates to a configuration a ? Csi(F) the loop /? = j(a),
/3(9) = restriction to I of cuR, e С,(Г) ~ ВГ , V0 6 Sl
where Re is the rotation Rn(t) = t + B V< 6 S1.
The classifying space ВГ is the geometric realization |X| of the simplicial set X given by
i,-- ,9n) = (92,---,9n)
^(9i, ¦ ,9..) = (9b---,9n-i)
The natural homeomorphism С/(Г) с; |Х[ associates to a configuration a on I = ]0,1[, with suppa =
{*i,*2, ¦ ,«n}, *. < *.+i <*(*;) = 9>> tlle following element of (A'n\DegXn) x Int Д„:
To prove proposition 10 one compares the two fibrations
Г
i
пвг
5|(Г) — С,(Г)
i i
sr
where for a G Csi(F) its restriction res(a) to / determines a up to the value a@) 6 Г. In the second
fibration пВГ is the loop space of ВГ. The vertical arrows С/(Г) —» ВГ and Г —> ПВГ are weak homotopy
equivalences and thus so is
i:Cs.(r)-,(Sr)s'.
S) Cyclic cohomology of
Let V be a smooth manifold and Г a discrete group acting on V by diffeomorphisms. In this section we shall
describe the cyclic cohomology of the crossed product algebra A = Cf(V) X Г. The analogues of the above
results of D. Burghelea are due to V. Nistor [Nis] and to J.L. Brylinski and Nistor [Bry-N]. We shall describe
an earlier result [С01З] namely that the periodic cyclic cohomology H"(A) contains as a direct factor the
twisted cohomology groups H*(VT, C). We are taking here the notations of section II.7. Thus Vr is the
homotopy quotient V Хг ЕТ and г is the real vector bundle on Vr associated to the (Г-equivariant) tangent
bundle TV of V. The algebra A is the convolution algebra of smooth functions with compact support on
V x Г and the convolution product is given by:
171

A)
(/i/2)(z.<0= Ц /1A,Si) /2A91,92)-
We let U. be the element of A given, when V is compact, by U,(x,k) = 0 if 9 ф к U,(x,k) - 1 if 9 - *-
When V is not compact this do<5 not define an element of Л but a multipher of A. In all cases, any element
/ of A can be uniquely written as a finite sum:
B)
and one has the algebraic rule-.
(
Since the homotopy quotient Vr is the geometric realization of a simplicial manifold (cf. Appendix A) we can
d ibe th^wstedlohomology Щ(УГ) аз the cohomology of a double complex (Ml^ • 5) More
explicitely we can view the space ВГ as the geometric realization of the simplicial set еГ where (еГ)„ _ I
and
4(90,.--,9») =
D)
The group Г acts on the right on both V and еГ and by [Bott] theorem 4 5 the Mwisted cohomology
ff'IV Cl is the cohomology of the bicomplex of Г invariant simplicial r-twisted forms on V x еГ. Now a
fwiiild form onV isTsame thing as a smooth de Rham current: to the twisted form, one assocates the
smooth current C:
E) C(«)
(for any differential form a with compact support such that deg(a) + deg(u) = dim V).
It will be more convenient to describe the double complex (C\dud2) computing Щ(УТ,С) using currents
on V rather than twisted forms. This is done in detail as follows:
We let C"-« - {0} ""less .. > 0 and -dimI/ < m < 0. Otherwise we let C"-« be the space of totally
Inflymmetnc maps 7 : Г"" Z n_m(V), from Г"» to the space n_m(V) of de Rham currents of dimension
—m on V, which satisfy:
We have used the right action of Г on V to act on currents. The «boundary dx : C"'m - С"*1'™ is given
by
о
172
The coboundar'y d2 : C"m -» Cnm+1 is the de Rham boundary:
We can summarize the above discussion by:
Proposition 11. The twisted cohomology H^(Vr,C) is naturally isomorphic, with a shift of dimension of
dimV, to the cohomology of the above bicomplex (C* ,di,d2).
This holds irrespective of the regularity imposed on the currents and we shall perform our constructions with
arbitrary currents. It is important though, to note that the filtration of the cohomology of the bicomplex
given by the maximal value of n-m on the support of cocycles G„,т), does depend on the regularity. Let us
illustrate this by an example. We let V = Sl, Г = Z and the action of Г on V is given by a diffeomorphism
(p G Diff+fS1). We choose (p with Liouville rotation number and not C°° conjugate to a rotation (cf. [Her]).
The homotopy quotient Vp is the mapping torus, quotient of V x R by the diffeomorphism ip:
(9) $(x, s) = (tp(x), s + 1) ?i ? V = S1 , Vj ? R.
It is by construction a 2-torus which fibers over ВГ = R/Z. Its cohomotopy group я-'(Уг) has another
natural generator given by continuous maps Vr —> V = S'¦ We want to compute the corresponding cocycle
in the bicomplex (C, dltd2). (Here the group Г preserves the orientation so that we can ignore the twisting
by r. Thus one has a canonical map 7г'(Кг) —> #T'(Vr, C)). If we use arbitrary currents the corresponding
cocycle is easy to describe, it is given by the following element of C°'°:
A0)
7o € ^o is the unique Г-invariant probability measure on V = Sl.
Of course this measure is a 0-dimensional current and one has dtj = din = 0-
Since ip is not C°° conjugate to a rotation the current 70 is not smooth and we now have to describe a smooth
cocycle 7' in the above bicomplex belonging to the same cohomology class. For this we let 7J G По be any
smooth 0-dimensional current such that {I, j'o) = 1. It is not <p invariant but the equation ipj'o — f'a = cf 7J
can easily be solved with т[ a smooth 1-dimensional current on V = Sx. This yields the desired smooth
cocycle 7'. As we shall see, while 70 gives rise to a trace on the crossed product algebra A = C?°(V) ХГ,
the cocycle 7' gives rise to a cyclic 2-cocycle on A, with the same class in periodic cyclic cohomology.
We shall now describe in full generality a morphism Ф of bicomplexes from (C*, d\, d2) to the F, B) bicomplex
of the algebra A. This will give the desired map from H"(Vr, C) to the periodic cyclic cohomology of A and
will make full use of the (b,B) description of the latter (section 1). Our construction of the morphism Ф
works for smooth groupoids for which the maps r,s are etale but the case G = V ХГ has interesting special
features due to the total antisymmetry of the cochains 7(90,- •• ,9n), 9. G Г. To exploit this, let us introduce
an auxiliary graded differential algebra C. As an algebra, С is the crossed product С — В >оа Г where В is
the graded tensor product:
(И)
B= Л*A/)®Л*(СГ')
utomorphisn r , „a _tiJ „
Г on A*(V) commuting with the differential and satisfying:
A2)
«!.„(/)(*) = !(xg) V/ G Cf(V) , x?V , 9
173

The action a2
ion a2 of Г on Л*(СГ") preserves the subspace СГ' = Л'СГ" and is given by the equality:
A3) aXgSk = 6k,-i-6ri Vj.fcGT.
Since the action a of Г on 5 preserves the (bi)grading of В the crossed product С = й>зГ has a canonical
bigrading. We shall write the generic element of С as a finite sum:
С = ?>,?/,, bg?B.
Г
We endow the algebra В with the differential:
A4) d(w <g> e) = du ® e Vu; G A'(V) , e 6 Л*
where do; is the usual differential of forms. (Thus we have endowed Л* with the O-differential.)
Finally we define the differential d in the algebra С by:
A5) d(b u,) = {db) u, + (-i)ei ь и, s, Vbes.ser.
Lemma 12. (C,d) is a graded differential algebra.
Proof. By construction С is a bigraded algebra, thus it is enough to show that the two components d',d"
of d given by
A6) d!(b Ug) = (-\)db b UgSg, d"(bUg) = (db)Ug V6eB,S6r
are derivations of С such that d'2 = d = d'd" +d"a" = 0.
It is clear that d" is a derivation and that d = 0. To check that d' is a derivation one has to show that for
91,92 G Г,
This follows from ?/„ «„ Ug, = Ug,g, <*,-.(«„) = У,1Й(«„Л - «,,) by A3).
Since <52 = 0 one has d'2 = 0. Finally d'd" = -dV from A6).
Let now 7 G C"'m be a
by:
A7) 7(<
cochain in the bicomplex (C ,du d2). We associate to j a linear form у on С defined
« ® *„¦¦ A.) = <«.7(i.ei.-.ft.» ^er'"er"(V)
A8) 7F U,) = 0 unless g - lr and 6 € >rm(l/) ® Л".
The relation between the coboundaries di,d2 of С (see G) (8)) and the derivations d',d" of С is given by:
Lemma 13. Lei у 6 C"m.
a) y{aia2 - (-ijaa.ea, в201) _ (_1)Э«, (d^)(Ol rfa2) Vaj € С
174
b)
Va € С
Proof, a) We can assume that aj = bj Ug> with bj G B, and that jig2 = 1. Then a\a2 = 6i a?1F2)
and (_\)ва,а*, a^ai _ (_ j^ao,aoa j2 a,FJ(J1) = a,3Fi aSlF2)) using the graded commutativity of B. Also
ai d^ = (—\)d" 6i aj,F2)Ejj. Thus (with 6 =6i aj,F2)) it is enough to show that for any g G Г,
A9) 7F-a,F)) = (-l)"+m(dT7)F^) V6GB.
One can assume that 6 = ui®Sgt ... <S,. with^i G Г , ы G A'm(V). Onehasa9F) = ali9(u)®a2](r(#9l ...6g.)
and:
*»!»-¦ *»»»-' •• ¦*».«-' -V1 *ft»-' ¦•¦*».»-' -*»!»-' *»-' ¦¦ A»*-1 -•••-*»¦»-' 'or' -A—i»-' *»-'•
Thus using F) and the definition of di G) we get A9).
b) follows from A7) and the definition of d? (8).
Note that since d = d" on В the equality b) still holds for d instead of d".
We can now give a general construction of cyclic cocycles on Л = C?°(V) ХГ ([Coi3]):
Theorem 14. a) The following map Ф is я morpkism of ike bicomplex {Cm,d\,d2) to the F,5) bicomplex
of A: For у G С"'"', Ф(у) is the n — m + \ linear form on A given, with t = n — m+ 1, 6j/
Ф(Т)(аЛ ...,*<) = А„,,
where An,m = Jt+i^i-
€ A,
b) ГАе corresponding map of cohomology groups gives a canonical inclusion Ф» : K*(Vr,C) —* H*{A) of
H*{Vp, C) as a direct factor of the periodic cyclic cohomology of A = C%°(V) ХГ.
Proof, a) Let us first compute 6(ФG))(а:0,... xt+l). The Hochschild coboundary of the functional (a:1) —»
g(dx' + l ...dxl x° dx1 ...dx') gives {-!)> y(aj x'+l - x'+1 aj) where aj = dx'^ .. .dx1*1 x° dxx...dx'.
Thus by lemma 13 a) we get
B0)
= (-1)' An,m
Since d\ = 0, (di7)~ is a graded trace on С by lemma 13, and we can rewrite B0) as:
B1)
dx1 ...dip
Using da:' = d'a:' + d"xk and considering the tenns in which d! appears n + 1 times in the product:
*°(dV + d'V)... (d'i:'+l + dV+1)
we get:
175

B2) Ь ФG)(«°, •.., r'+1) = (n + 1) А„,т (dl7)~ (a:0 dx1... dxt+1).
Since (dij)~ is a graded trace on С one has
*(dn)(x0 z'+1) = An+1,m (t+i) (d17)~ (r° rfi1 ... dr<+l)
so that using B2) we get <?(</i7) = b ФG).
Let us now compute В ФG). One has:
I
So ФG) (x°,.. ..г') = А„,т ^(_iy«-i) ^(rfjj . . mdxl-i dxo j^i-iy
о
(For instance for I = 2 one gets three terms: 7((/i° dx1) - ^(dz1 da:0) + 7(dr° dx1).) Thus one has:
Ф(Т) (*° i') = A+ 1) А„
.. .dx'-1 dx° . ..
By lemma 13 b) we thus have:
B3) В ФG) (х° »'"») = (*+l) А„,т^(-1)«-1'('-Я (d27)~ (dj ...dx'-1 x° dr1 ...d**).
a
Thus ВФG) = Ф(Л2т).
b) We shall describe in section 7 7) in the context of foliations a natural retraction A : H*(A) —> H*(Vr, C)
using localization. The conclusion follows from the equality А о Ф, = id.
Remark 15. a) In the special case V = {pt} the above construction gives a cycle (C, d, y) on the algebra
A = СГ, for any group cocycle 7 G Z"(T, C) represented by a totally antisymmetric right invariant cochain
у ¦ Г"*1 — С with di у = 0. The algebra С = (Л"СГ')хГ is a non trivial quotient of the universal
differential algebra ПСГ used in section 1.
b) Let us explain the analogue of the above construction (theorem 14) in the general case of smooth groupoids
G such that r, s are etale maps. The bicomplex (C, d\, di) of proposition 11 is now the bicomplex of twisted
differential forms on the simplicial manifold Mr(G) (Appendix A) which is the nerve of the small category
G. We describe it in terms of currents, so that C"<m is the space of de Rham currents of dimension —m on
the manifold:
B4)
The first coboundary di is the simplicial one,
B5) dl = (-1)™ S (-l)i d]
wheredoGi,---,7n) = G2.---,7n).d;Gb---,7«) = Gi,.--,7;7j + i 7»), dnGi, • ¦ ¦ .7n) = Gi, • ¦ •, 7—i)-
Note that we pull back currents in formula 25 which is possible since we are only using etale maps.
The second coboundary d2 is the de Rham d' as in formula (8). We restrict to the normalized subcomplex
of currents which vanish if any jj is a unit or if 71.. .у„ is a unit.
Let us now describe the analogue of the bigraded differential algebra (C,d',d") of lemma 12.
176
As a linear space the space Cn'm of elements of С of bidegree (n, m) is the quotient of the space of smooth
compactly supported differential forms of degree m on G^n+1\ by the subspace of forms with support in
{GО1 ¦ ¦ ¦ On); 7j is a unit for some j ф 0}. The differential d" is the ordinary differential of forms. The
product and the differential d1 are given as follows:
,7n,, • ¦ • ,7n,+na) = 51 Ul^70 71.1-1.7) Л w3
77'=7»,
•, 7n,+n,) +
B6)
J=0
7T'=7,
UlG0' ¦¦¦•Tj-1.7.7' 7n,-i)Au2Gn, 7m+n3)
where we have used the etale maps r,s : G —> G*0' to identify the corresponding cotangent spaces and
perform the wedge product
B7) (d'aj)Go,. . .,7n+i) = 0 unless 70 is a unit, and (d'ui){y0, ¦ ¦ ¦ ,7»+i) = wGi> •• ¦> 7n+i) if 70 « a unit.
Let с G C"'m ba a cochain in the bicomplex (C.di.dj). We associate to с the linear map con C"m obtained
from the pushforward of the current с by the map:
Then lemma 13 holds but with the important difference that part a) only holds for a2 € C(Ol°' = C™(G).
This is the price to pay for the loss of the total antisymmetry of cochains. The maps Ф is defined as in
theorem 14 and this theorem still holds because its proof only used the above weaker form of lemma 13. We
shall see in section 6 many concrete examples of cyclic cocycles on C%°(G) constructed using Ф.
To conclude this section we refer the reader to the work of Nistor [Nis] and Brylinski [Bry-N] for the complete
description of the cyclic cohomology of crossed products.
177

III.3. Pairing of cyclic cohomology with К theory
Let A be aunital (non commutative) algebra and Ko(A), K\(A) its algebraic/f-theory groups (cf. [Mil]). By
definition Kt>(A) is the group associated to the semi-group of stable isomorphism classes of finite projective
modules over A- Also Ki(A) is the quotient of the group GL^A) by its commutator subgroup, where
GLm A) is the inductive limit of the groups GLn(A) of invertible elements of Mn(A), under the maps
x Ol
'~ 0 lj-
In this section we shall define by straightforward formulae a pairing between HC*"(A) and Kq(A) and
between НСоЛА(А) and Ki(A). For an equivalent approach see [Кагз].
The pairing satisfies {S<p,e) = (<p,e), for <p G HC(A), e G K(A) and hence is in fact defined on H'(A) =
НС (А) ®нс- С As a computational device we shall also formulate the pairing in terms of connexions and
curvature as one does for the usual Chern character for smooth manifolds.
This will show the Morita invariance HC'(A) and will give in the case A abelian, an action of the ring
ЩА) on HC'(A).
Lemma 1. Let (p G Z"(A) and p,q G Proj Мь(А) be two idempotents of the form p = uvf q — vu for some
u,j? Mic(A). Then the following cocycles on В = {x G Mt(A), xp = px = r} differ by a coboundary
ф?(а°,.. . ,a") = (tp#Tr)(ya°u,... ,va"u).
Proof. First, replacing A by Mt(A) one may assume that 4 = 1. Then one can replace p, q, u, v by ? ,
, , and hence assume the existence of an invertible element w such that wpw~x — q,
и = pw~x = w~xq, v = qw = wp I take w = \ ). Then the result follows from proposition 1.8.
Recall that an equivalent description of /\'o(.4) is as the abelian group associated to the semi-group of stable
equivalence classes of idempotents e G Proj Mic(A).
Proposition 2. a) The following equality defines a bilinear pairing between Ka(A) and HC*V(A) : ([e],[<p]) =
(т!)-1(у>#Тг)(е,...,е) fore G Proj Mk{A) an <p G Z$m(A).
b) Onehas{{c],lSv]} = ([e],{<p]}.
c) Let if (resp. ф) be an even cyclic cocycle on an algebra A (resp. Б), then for any e G Ko(A), f G Kq(B)
one has:
(e®f ,
Proof. First if tp G B2xm(A), tpitTr is also a coboundary, tp-#Tr = Ьф and hence (tp-#Tr)(e,..., e) =
Ьф(е,. .. ,e) =52 (—1)' Ф{е, ¦•¦)*) = Ф(е> ¦. • ,e) = 0, since фх = —ф. This together with lemma 1 shows
that (cp#Tr)(e,... ,e) only depends on the equivalence class of e. Since replacing e by does not
change the result, one gets the additivity and hence a).
b) One has S<p(e,...,e) =? $(t(dt)>-1 t(de)n~'+1) and, since e2 = e,one has edee = 0, e{deJ = {defe,
i = i
so that
178
S<p(e e) = (m+i)p(e e).
c) Follows from b).
We shall now describe the odd case.
Proposition 3. a) (Ae following equality defines a bilinear pairing between К\(А) and HCaM(A):
(H,M) = ^?2-r(t + i)-1
(^#Tr)(»-' - 1, u - 1, u - 1,..., и - 1)
where tp G Z%(A) and и G GLk(A).
b) One*(u([«],[Sy]) = (M,M).
c) Let ip (resp. ф) be an even (resp. odd) cyclic cocycle on an algebra A (resp. B) then for any e G Ko(A),
и G Ki(B) one has:
([e® u + (l-e)® 1] , <р#ф) = {e,<p) (и,ф).
Proof, a) Let A be the algebra obtained from A by adjoining a unit. Since A is already unital, A is
isomorphic to the product of A by C, by means of the homomorphism p:(a,X) —> (e + Al, A) of A to A x C.
Let $ G Zl(A) be defined by the equality
?((a°,A°),...,(an,An)) = <p(a°,..., a") , Ч(а\У) ? А.
Let us check that 6? = 0. For (a0, A0),... ,(a"+l, An+I) G „4 one has
?(a°, A0),..., (a*, A')(oi+1, Ai+1),..., (an+1, A"+1)) = ^(e0,..., а'а'+1,..., a"+1)
Thus
Ыр((аа, Aa),... , (a"+1, An+
Now for u G GLi(A) one has
= \°<p(a\
el <xn) = 0.
where п = (и, 1) G A G C. Thus to show that this function x{u) satisfies
X(nv) = x(") + X'(") fOT u,v ?GLi(A),
one can assume that <p(l,a°,..., a") = 0 for a' 6 A, and replace x by
Now one has with U =
V= Г °
[0 v\
X(uv) = (VitTr)(U-\ ?/,..., U~\U),
x(u) + X(v) =
Since U is connected to V by the smooth path
\ V,..., V~\ V).
179

_ Ги Ol [sin( -cosil fl 0] Г sin( cosil
"'"'" lj [cosi sin( j [0 v\ [-cos( sinij
it is enough to check that
Using (сГ1)' = —и, lu'tu^1 the desired equality follows easily. We have shown that the tight hand side of 3
a) defines a homomorphism of GLk(A) to C. The compatibility with the inclusion GLt С GLe is obvious.
To show that the result is 0 if <p is a coboundary, one may assume that 4 = 1, and, using the above argument,
that <p = Ьф where ф G CJ, фA, a0,... ,a"~2) = 0 for a' G A (One has Ъ$ = (Ьф) foe ф € CJ.) Then
one gets 6^(u~',.. .,u~', a) = 0.
b) The proof is left to the reader.
Remark. The normalization 2 a) of the pairing between Ko and HC1 is uniquely specified by conditions 2
b) c). The normalization of the pairing between K\ and НСоЛЛ is only specified up to an overall multiplicative
constant A, independent of n, by conditions 3 b) c). Our choice C a)) is, up to the choice of the square root
of 2i, the only one for which the following formula holds:
(«Л», <р#ф) = («,*>} (»,Ф)
where the product Л ; A'i(.4) x A'i(S) —> Ka(A®B) is defined in the context of pre C*-algebras (cf. chapter
IV section 8). To check the above formula one just needs to know that if e G C°°(T2, P\(C)) is a degree 1
map of the 2 torus to Pi(C) С Мз(С) then r(e,e,e) = 2т» where г is the cyclic 2-cocycle on C°(T2) given
by: r(/°, /', /2) = /T2 /° dfl Л dp, V/> 6 C~(T2).
Let H'(A) — HC*(A)®hc(c) С be the periodic cyclic cohomology of A.
Here ЯС*(С), which by corollary 1.13 is identified with a polynomial ring C[a], acts on С by P(<r) >-* P(l)-
This homomorphism of ЯС(С) to С is the pairing given by proposition 2 with the generator of A'o(C) = Z.
By construction H'(A) is the inductive limit of the groups HC"{A) under the map 5 : HC"(A) —>
HC"+2{A), or equivalently the quotient of HC(A) by the equivalence relation p ~ S<p. As such, it
inherits a natural Z/2 grading and a filtration1.
F"H'(A) = Im HCn(A).
Corollary 4. One has a canonical pairing between H*"(A) and K0(A), and between НоЛЛ(А) and Ki(A).
The following notion will be important both in explicit computations of the above pairing (this is already
clear in the case A = C°°(V), V a smooth manifold) and in the discussion of Morita equivalences.
Definition 5. Let A —> fl be a cycle over A, and ? a finite projective module over A. Then a connexion V
on S is a linear map V : ? —> ? ®л U1 snch that
+ ? ® dp(x) ,Ч?&?,х?А.
Here ? is a right module over A and il' is considered as a bimodule over A using the homomorphism
p : A —* fi° and the ring structure of Q*. Let us list a number of obvious properties:
Proposition 6. a) Let e G End,»(?) be an idempotent and V a connexion on ?; then ? —> (e ® 1)V? is a
connexion on e?.
180
b) Any finite projective module ? admits a connexion.
c) The space of connexions is an affine space over the vector space Hom>E,
d) Any connexion V extends uniquely to a linear map of ? — ? ®a П into itself such thai
V(?®u>) = (V?V+?®dw , V? ?? , ш 6 П.
Proof, a) One multiplies the equality of definition 5 by e ® 1 (on the left).
b) By a) one can assume that ? = C* ® A for some k. Then, with (&)j=i t the canonical basis of ?, put
Note that, if 4 = 1 (for instance), then A <8u П1 = рA)П1 and Va = p(\)dp(a) for any a G -4 since A is
unital. This differs in general from d), even when p(\) is the unit of п°.
c) Immediate.
d) By construction ? is the projective module over fl induced by the homomorphism p. The uniqueness
statement is obvious since V? is already denned for ? 6 ?. The existence follows from the equality
?n ® du =
for any
and ы G fi.
We shall now construct a cycle over End,»(?). We start with the graded algebra Endn(^) (where T is of
degree к if T~?> С ?'+k for all j). For any T G Endn(?) of degree 4 we let S(T) = VT - (-\)kTV. By
the equality d) one gets
v(?w) = (v^v + (-i)des ?<ь for fef.uea,
and hence that S(T) e Endn(?), and is of degree 4 + 1. By construction 6 is a graded derivation of Endn(?).
Next, since ? is a finite projective module, the graded trace / : fl" —» С defines a trace, which we shall still
denote by J", on the graded algebra Endn(?).
Lemma 7. One has J S(T) = 0 for any T G Endn(?) of degree n — 1.
Proof. First, if we replace the connexion V by V = V + Г, where Г G Нош^(?,5®лП1), the corresponding
extension to ? is V = V + Г, where Г ? Endn(?) and is of degree 1. Thus it is enough to prove the lemma
for some connexion on ?. Hence we can assume that ? — eAk for some e G ProjMfc(^4) and that V is
given by 6 a) from a connexion Vo on Ak. Then using the equality S(T) = eSo(T)e for T G End? С End^o
(?o = Ak), as well as
60(T) = 60{eTe) = 60(e)T + S(T) + (-l)eT TS0(e),
one is reduced to the case ? = Ak, with V given by 6 b). Let us end the computation say with 4=1. Let
e = />A). One has ? = ей, Endn(?) = епе, S(a) = e(da)e. Thus/?(a) = J(d(eae)-(de)a-{-l)da ade) = 0.
Now we do not yet have a cycle over Endv4(i") by taking the obvious homomorphism of End,»(?) in Endn(?),
the differential 8 and the integral J. In fact the crucial property <S2 = 0 is not satisfied:
Proposition 8. a) The map в = V2 of ? to ? is an endomorphism: в G Endn(?) and S2(T) = вТ-Тв for
all T G Endn(?).
b) One has \[?],[т}) = ^/С™, when n is even, n = 2m, where [?] G K0(A) is the class of ?, and т is the
character ofu.
181

Proof, a) One uses the rules V(i^) = (Vi))w + (—l)d<fs-; ^du and d2 = 0 to check that )
b) Let us show that J вт is independent of the choice of the connexion V. The result is then easily checked
by taking on_? = eAk the connexion of proposition 6. Thus let V = V +Г where Г is an endomorphism of
degree 1 of ?. it is enough to check that the derivative of / 9J" is 0 where 0, corresponds to V, = V + (Г.
Also it is enough to do it for t = 0. we get:
As (зт *«),=„ = TV + УГ = «(Г) one has
= m J«((Г Г) = 0.
Thus, while S2 ф 0, there exists в 6 Q! = Endn(?) such that
62{T) = ет-те , vr e a1.
We shall now construct a cycle from the quadruple (п1, S, 0, J").
Lemma 9. Let (fi',c5,0,/) be a quadruple suck that Of is a graded algebra, S a graded derivation of degree
1 of П' and в е П'2 satisfies
6{в) = 0 and 62(u) = вш-шв for ш € ft'.
tlien one constructs canomcaly a cycle by adjoining to Q' an element X of degree 1 with dX = 0 such that
X2 = 0, uiXu2 = 0, Vw, e П'.
Proof. Let П" be the graded algebra obtained by adjoining X. Any element of Q" has the form ш =
шц +uii2X + Xui2i + A'w22.Y, uiij e п'. Thus, as a vector space, п" coincides with М2(п'), the product is
such that
['  Г 1 Г 1 n 1 Г ' '
1 2 J ~ [1 2 J [Ов \ [] 
and the grading is obtained by considering X as an element of degree 1; thus [ыу] is of degree к when шц
is of degree t, w12 and u2i of degree к - 1 and w22 of degree к - 2. One checks easily that П" is a graded
algebra containing П'. The differential d is given by the conditions du - S(ui) + Xu - (-l)d«8" шх for
uEil'C П", and rfX = 0. One gets
1
0 -6
1 0
1 2 ] ,_iyl«gu, [1 2 1 ГО 1]
1 2 J l > [1 2J [-в О]'
one checks that the two terms on the right define graded derivations of Q." and that d2 = 0.
Finally one checks that the equality
/ - ' ! de w f
J 1- - J П J 2
defines a closed graded trace.
Putting together proposition 8 a) and lemma 9 we get:
182
Corollary 10. Let Л —* fi be a cycle over A, S a finite projective module over A and A1 — End^(?). To
each connexion V on S corresponds canonicalty a cycle A' —> Q' over A*-
One can show that the character r' ? Z"(A') of this new cycle has a class [r'] 6 H"(A') independent of
the choice of the connexion V, which coincides with the class given by lemma 1. One can easily check a
reciprocity formula which takes care of the Morita equivalence.
Corollary 11. Let А, В be unital algebras and ? an А, В bimodule, finite projective on both sides, with
A = Ende(?), В = End,»(?). Then HC"(A) is canomcally isomorphic to HC'(B).
Finally when A is abelian, and one is given a finite projective module ? over A, then one has an obvious
homomorphism of A to A' = End>(?). Thus in this case, by restriction to A of the cycle of corollary 10 one
gets:
Corollary 12. When A is abelian, HC(A) is in a natural manner a module over the ring Ko(A).
To give some meaning to this statement we shall compute an example. We let V be a compact oriented
smooth manifold. Let A = С~(^) and fi be the cycle over A given by the ordinary de Rham complex
and integration of forms of degree n. Let ? be a complex vector bundle over V and ? — С*°(У, Е) the
corresponding finite projective module over A = C°° (V). Then the notion of connexion given by definition 5
coincides with the usual notion. Thus corollary 12 yields a new cocycle r G Z"(A), A = C°°( V), canonically
associated to V. We shall leave as an exercise the following proposition.
Proposition 13. LetWb be the differential form, of degree 2k on V which gives the component of degree Ik
of the Chern character of the bundle E with connexion V : uk = 1/fc! Thice (вк), where в is the curvature
form. Then one has the equality
¦=ъг
where шк € Z?~2t(.-4) и given by
шк , v/ e A =
and where г is the restriction io A = (^"(V) of the character of the cycle associated to the bundle E, the
connexion V, and the de Rham cycle of A by corollary 10.
Of course in this example of A = C°°(V) the pairing between cyclic cohomology and Ka(A) gives back the
ordinary Chern character of vector bundles.
As a more sophisticated example let us consider example 2, /3), i.e. the irrational rotation algebra. The
smooth subalgebra Ae С Ад is stable under holomorphic functional calculus (appendix C) and thus by the
result of Pimsner and Voiculescu ([Pi-V2]) one has K0(A») —Z2. An explicit description of finite projective
modules over At was given in [Co^] and shown by M. Rieffel [Ri4] to classify up to isomorphism all the
finite projective modules over A». We have already seen it in a slightly different guise in section II.8/3).
Let (p, q) G Z2, q > 0 be a pair of relatively prime integers (p = 0 being allowed). Let us construct a finite
projective module ? over Ae as follows.
We let 5(R) be the usual Schwartz space of complex valued functions on the real line and let two operators
Vu V2 on 5(R) be defined by:
where г = ? - в and e(s) = expB;n.s) V* e R
183

One has of course:
Next, let Л" be a finite dimensional Hilbert space and wi, tu2 be the unitary operators on К such that:
wiwi = e(p/q)wiw2 , w\ = w\ = 1.
In other words К is a (finite dimensional) representation of the Heisenberg commutation relations for the
finite cyclic group Z/gZ, and thus it is a direct sum of d equivalent irreducible representations.
It is clear now that in the tensor product ? = S(R) <8> K, one has
(V2<g>io2)(Vi ®wi) =I(Vi ®№!)A/2 ®w2) A = е(в).
We turn ? into a right At module as follows:
The above equality shows that this is compatible with the presentation of A». It is easy to check that for
any a ? Ae, the element ?a still belongs to ? using for instance the general properties of nuclear spaces.
By [Coi2] the obtained right module over Ae is finite and projective. In fact:
Theorem 14. ([R14]) Let ? be a finite projective module on An then either ? is free: ? = Ar for some
p > 0, or ? is isomorphic to S(R) ® A' with the above module structure.
let us now recall (theorem 2.7) that Hew{Ae) is a vector space of dimension 2 with basis Sr and (p, where r
is the canonical trace of A$ and where the cyclic 2-cocycle (p is:
V{x0,x\x2) = B*i)-1 T(x"(Si[x1)S2(x2)-SJ(x1)Si(xi)) V*' ? A,
where S\,S2 are the natural commuting derivations of A3. The pairing of A'o with Sr is the same as with
the trace r, and given by the Murray and von Neumann dimension:
{r.,)
To compute the pairing of /v0 with <p let us use the following 2 dimensional cycle with character <p.
As a graded algebra П" is the tensor product of Ae by the exterior algebra Л*С2 of the two dimensional
vector space C2 = Cej +Ce2- The differential d is uniquely specified by the equality:
d(a® a) = c5i(a)®(ei Л a) +?2A)® (e2 Л a)
for any a G Ae, a G Л*С2.
Finally the graded trace: U2->C is:
x Л e2) -. Btti)- ' r(a) € С
A connexion (definition 5) on a finite projective module ? over Ae is thus given by a pair Vi, V2 of covariant
different! als,satisfying:
V,(«a) = (Vj
and the curvature 0 is easily identified as
, a?Ae
(ViV2-
i Л«г2).
184
Proposition 15. ([Со[2]) Let ?pq = S(R,K) be as above, then the following formulae define a connexion
V on ?:
where ? = p/q ~ в ¦
The curvature of this connexion is constant, equal to — %z± ® (e1 Л е2) and by proposition 8 b) the value of
the pairing is:
1
(ld5) = (p— aq) = q.
Corollary 16. Let (p ? HC2(At) be the above cyclic 2-cocycle, then
(K0(A,),<p)cZ.
This integrality result will be fully understood and exploited in chapter 4 section 6.
Finally we remark that (for в ? Q) the filtration of H'"(At) by dimensions is not compatible with the lattice
dual to Ka(Ae), since the 0 dimensional class of г does not pair integrally with Ka. In the next section we
shall compute the pairing of К theory with cyclic cohomology in the case of group rings.
185

III.4. The higher index theorem for covering spaces
Let Г be a discrete group acting properly and freely on a smooth manifold M with compact quotient
M = М/Г. Any Г invariant elliptic differential operator D on M yields by the parametrix construction
detailed below а К theory class:
Ind(D) = A'o(ftr)
where 72.Г is the group ring of Г with coefficients in the algebra 72. of matrices (aij)i,jeN with rapid decay:
Sup ik f Kj|<oo VMeisi.
In this section we shall extend the Atiyah Singer index theorem for covering spaces [Ats] pSinj], which
computes the pairing of Ind(D) with the natural trace of 72Г, to the case of arbitrary group cocycles ([Co-
Mi]). More specifically given a normalized group cocycle c?Z'(f,C) we have seen (section 1 a)) that there
is a corresponding cyclic cocycle rc on the group ring СГ given by the equality
тс(д°,...,дп) = 0 if
¦¦/?"
Tc(g0,...,g") = c(g\...tg") if g«gl ¦ ¦ ¦ g" = 1.
Since the following functional is a trace on 1Z:
Trace(a) = ^jT ац
the cup product rc# Trace is а к dimensional cyclic cocycle on 72.Г and, for к even, it makes sense to pair it
with Indp D. Let us define carefully the latter index.
a) The smooth groupoid of a covering space
Let as above Г act freely and properly on the smooth manifold M. Let us endow the manifold G, quotient
of M x M by the diagonal action of Г, with the following groupoid structure
G<0) = M/T = M
r(x,y) = x? M , s(x,y) = t/G M VJ.yGM
(Г, y)o(jf, г) = («. г) V2, jj, F G Af.
Since (af,y) = (xg,yg) Vz, у G M, j6F, the last equality above is sufficient to define the composition of
composable elements. One checks that G is a smooth groupoid. When Г = tti(M) acts on the universal
cover M of M then G is the fundamental groupoid ([Sp]) of M, i.e. the groupoid of homotopy classes of
paths in M. Note that while G<0' = M is compact the groupoid G is not compact if Г is infinite.
Let J = C^°{G) be the convolution algebra of the smooth groupoid G (cf. chapter II) and let us, by fixing
the volume form on M associated to a fixed Riemannian metric, ignore the ^ density bundle.
Let then A be the convolution algebra of distributions T G ^"'(G) which are multipliers of J i.e. satisfy
G) V/eC~(G).
Any such T € A is characterized by the corresponding Г invariant operator on C™(M), with Schwartz kernel
the distribution 71. In particular we have a canonical inclusion:
C°°(M) С A
186
of the algebra of smooth functions on M as Г-invariant multiplication operators on Cf(M) and thus as a
subalgebra of A. Now let D be a Г-invariant elliptic differential operator:
D :
Cf(M,
where the Ej are the Г-equivariant vector bundles on M corresponding to vector bundles Ej on M. The
above inclusion С°°(М) С A allows to induce them to finite projective modules ?j on A. Moreover the
existence of a parametrix for D modulo the ideal J = Cf(G) yields (cf. section II.9):
Proposition 1. The triple (S\ ,??, D) defines a quasi-isomorphism over the algebras J = CJ°(G) and А Э J-
We refer to [Co-Mi] for the detailed proof.
We shall then let Ind(D) G A(C|°(G)) be the index associated to the quasi-isomorphism (?b?2,?>) by
proposition II.9.2.
The groupoid G has two important properties, the first is that as a small category it is equivalent to the
discrete group Г. This will allow us in 0) to relate C™(G) to the group ring 72.Г. The second is that, as
a smooth groupoid, it is locally isomorphic to the trivial groupoid M x M (cf. Chapter II), this allows to
compute the pairing of Ind D with cyclic cocycles G).
/?) The group ring 7JJ
Let It be as above the ring of infinite matrices with rapid decay. Let us construct a canonical map:
playing at this algebraic level the same role as the A' theory isomorphism:
coming from the strong Morita equivalence C*(G) ~ C"(V) associated to the equivalence of groupoids G ~ Г.
The following lemma is the algebraic counterpart of proposition II.4.1.
Lemma 2. Lei the algebra С°°(Л/) 0 СГ ad on the right on the vector space C™(M) by:
f G C°°(M), g G Г, x G M with x e M the class ofx. Then Cf(M) is a finite projective module over
In fact it is useful to describe explicitely an idempotent E G Mn(C°° (M) <8> СГ) and isomorphism of right
modules: U e Ноти(?, ЕВ") where we let В = C°°(AQjg> СГ and S = C™(M) be the above module. Let
thus {Vi)i=i< irt be a finite open cover of M, fti :Vi —• M be local continuous sections of the projection
,r : М -^ М
and (xi)i=i ч be a smooth partition of unity in M subordinate to the covering (K).=i »• We may assume
that the x\ axe smooth functions as well.
Then one defines module homomorphisms U G Home(f, B"), U* G Home(B", ?) as follows:
9j) = a9'1 /?;(*)) «,1/2(i) Vi G М,э G Г, j G {I n} and € G C?(M) = ?¦
187

Vi € M, tt(i) = г, т) ей".
And where g(f3j(x), x) is the unique g ? Г such that
цх = 0j(x) , r = tt(?).
One checks directly that both С and U" are B-module morphisms and that U'U = idc ¦
Thus the product:
e -uv e м„(В)
is an idempotent and С gives an isomorphism:
s~eb".
The components Ejt of the matrix E G М„(В) are easily computed and given by:
where /?,¦ f}^1 is a short hand notation for the 1-cocycle with values in Г associated to the covering Vj and
sections /3j.
The above isomorphism U : ? ~ Ев" is an isomorphism of B-modules and thus a fortiori of СГ modules
where СГ is the subalgebra 1 ® СГ of C°° (Af) ® СГ.
Thus the homomorphism 9, 0(T) = С/ГС* transforms Г invariant linear operators on CJ°(M) in СГ endo-
morphisms of EB". Since В = C°°(A/) ® СГ an arbitrary СГ endomorphism S of B" is given by a matrix
(S4)i,,-=i,....n «here S4 € Endc(C">(M)) ® СГ.
Now let Km be the algebra of smoothing operators on C°°(M), then one has:
Proposition 3. The homomorphism в maps the algebra C%°(G) to Мп(Им ® СГ).
Indeed it is given explicitely by the equality:
в(к)ц(х,у,д) = хЛ*УП ХЛУI'2 ЧЫ*)>9Ыу)) V* 6 C?(MxT M),x,y? М,д ? Г
which shows that each e(k)jj(g) is a smoothing operator.
Proposition 4. The induced map:
ЩС? (G)) -» Ka(HM ® СГ)
is independent of the choice of(Uj,f3j,Xj)j = i,..,,n-
Indeed, though the choice of the B-module morphism U € Нотв(?, B"), involved in the construction of в,
is not unique, we know that two such choices U, U' are equivalent. In fact, at the expense of replacing n by
2n we may even assume that there exists W 6 М„(В), invertible, such that
U' = W°V.
Then в'(Т) = We(T) W~x VT € C™(G), where W is a multiplier of the algebra М„(ПМ ® СГ). Thus we
get the equality of Й» and в', : A'0(C~(G)) —• К0(Т1м ® СГ). Finally the manifold M has disappeared since
one has:
Lemma 5. Let M be a compact manifold, dim M > 0. Then there is a canonical class of isomorphisms:
р-.Ъм ^K
188
unique modulo inner automorphisms ofTL.
Indeed the algebra Иц,, depends functorially upon the pair L2(M) D C™(M) of a Hilbert space and a
dense (nuclear) subspace. Moreover it is easy to check, using for instance an orthonormal basis of L\M) of
eigenvectors for a Laplacian, that the above pair is isomorphic to the pair ?2(N) D 5(N) where 5(N) is the
subspace of ?2 of sequences of rapid decay:
Sup n*|on| <oo V* 6N.
Putting together proposition 4 and lemma 5 we get a well denned К theory map:
Definition 6. Let D be a Г-invariant elliptic differential operator D : C™(M, Ei) —» C™(M, B2). Then its
equivariant index is Indr(D) = j(Ind(D)) € А'(ТгГ)
Note that it is not possible to define Indr(D) as an element of А'о(СГ) since the latter group is too small in
general and in particular for torsion free groups.
7) The index theorem
Let now с ? if'(Г, С) be a group cocycle, with q = 2k and even integer. The cup product r,;#Trace = (pc is
a well defined cyclic cocycle in the algebra 7ъГ and thus we can pair it with the group Kq{TZ-T) as described
in details in section 3. The following result computes the pairing <pc (Index D) in terms of the principal
symbol <?d of D, the classifying map:
ф : M — BY
of the Г-principal bundle M over M and the usual ingredients of the Atiyah Singer index theorem, namely
the Todd genus Td(TcM) of the complexified tangent bundle of M.
Theorem 7. ([Co-Mi]) Let T be a (countable) discrete group acting properly and freely on a manifold M
and D a Г-invartant elliptic differential operator on M. For any group cocycle с G Z2t(I\C) one has:
(rc#Trace , Ind(D)> = l (^.)t щ; (ch(<rD) Т<1(ТсМ)ф'(с) , [ГМ])
where M = М/Г, ф : M -» BY is the classifying map, aпdф'(c) the pull back of the class of с in Я2'(ВГ,С).
We refer to [Co-Mi] for the proof which relies on the local isomorphism of the groupoid G of the covering
space with the groupoid M x M thus reducing the statement to an index theorem for germs of cocycles near
the diagonal on the groupoid M x M, i.e. to Alexander Spanier cohomology. (Note that we use the pairing
as defined in proposition 3.2 which eliminates the term k\ of [Co-Mi].)
As a special case of theorem 7 one gets that the equivariant index of the signature operator Dsign on M,
paired with rc#Trace gives (up to a numerical factor) the Novikov higher signature:
Signc(M, ф) = (ЦМ) U ф'(с),[М]).
Thus if we knew that the equivariant index:
Indr(Dsign) € Л'о(ТгГ)
is a homotopy invariant of the pair (Л/, (/>), then the Novikov conjecture would follow.
But the only thing we know, is, thanks to the index theorem of Miscenko and Kasparov (cf. section II.4)
that:
189

Lemma 8. The image of \ndr(Dsign) m K0(C"T) by the morphism UX —> К ® C"(V) is the Miscenko
Kasparov С Y-signature, and is homotopy invariant!
We refer to [Co-Mi] for the details of the proof. We are thus confronted with the crucial problem of extending
the К theory invariant:
<p : К0(ПГ) — С
given by <pc(x) = (тс#Тгасе , x), to the larger C* algebra К ® C"(T).
We can formulate the following corollary of theorem 7 and lemma 8:
Corollary 9. Lei Г be a discrete group and с е 22*(Г,С) a group cocycle. Then if the pairing ipc with the
cyclic cocycle тс extends from I<o(TlT) to Ka(K <S> C'(Y)), the pair (Г, e) satisfies the Novikov conjecture.
As we mentioned already in the introduction this extension problem will occupy the second part of this
chapter. We shall first describe what can be done directly for discrete groups.
190
III.5. The Novikov conjecture for hyperbolic gi-oups
We shall see in this section that (.he extension problem for A" theory invariants given by group cocycles
(corollary 4.9) can be solved for Gromov's word hyperbolic groups, thus proving the Novikov conjecture for
this large class of group ([Co-Mi]).
a) Word hyperbolic groups
There are (cf. [Gro3]) several equivalent, definitions of word hyperbolic groups Г. We shall use the definition
in terms of the hyperbolicity of the (discrete) metric space (F,d) where d is the left invariant distance
d(gi,92) — ^(sf1^), Vji,i/2 6 Г and ( is the word length. II. is relative to a finite system /СГ of generators
of Г, with F = F~l, and is defined as the length of the shortest word in the generators representing g:
Definition 1. [Сгоз] Let (X,d) be a metric .s/>rtte. It is hyperbolic iff for some xq ? X, there exists 6q > 0
suck that the following inequality holds:
j:.u),6{y,-)) -6„ Vx.;/,.-e X
where 6{x,y) = \(tl{x,xn) + (/(;/, i-0) - d{x.;/)).
Fig.2. Hyperbolic space
It then follows that the same hokls for any xa e ,V (wil.li the constant 2SB) so that the choice of i0 is irrelevant.
The simplest example of hyperbolic metric spaces besides the line and the trees, is given by complete simply
connected Riemannian manifolds with sectional curvature niajorized everywhere by —г < О.
Definition 2. [Сгоз] Lei Г he a finitely generated discrete group. Then Г is hyperbolic iff the metric space
(Г,с/) (associated to a finite set of generators F ofV) is hyperbolic.
This definition does not depend upon the set. of generators, at least in the case we are interested in, namely
that of finitely presented groups. Any finitely presented group Г is the fundamental group iT|(P) of a 2
191

dimensional cell complex associated naturally to the presentation of Г. Then Г is hyperbolic ([Gr]) iff the
following isoperimetric inequality is satisfied by a smooth connected neighborhood V of P С R", re > 5.
There exists С < oo smcA that every smooth simple curve S ire V which is contractible in V bounds a smooth
embedded disk DcV such that:
Area D < С Length 5.
Finally the third equivalent formulation of hyperbolicity is that the metric space (Г, d) looks treelike when
seen from very far distance ([вгоз]). Nevertheless one should not conclude too hastily that hyperbolic groups
are one dimensional objects and for instance:
Proposition 3. ([Сгоз]) For every finite polyhedron Vo one can find an aspherical polyhedron V with word
hyperbolic fundamental group Г = iri(V) such that dim V — n = dim Vo and V admits a continuous map
f : V —> Vo which is injective on the cohomology H*(Vo). Furthermore, if Vo is a manifold, then one also
can choose V a manifold, such that f : ff*(Vo,Q) —» ff*(V, Q) sends the Pontryagin classes of Vo to those
ofV.
In order to get familiar with definition 2 of hyperbolic groups let us use it to estimate the number of
decompositions g = Jij2 of a given j 6 Г, such that l(gi) + l(g2) < l(g) + cst. Thus we fix a system of
generators F of Г, let ? the corresponding length function on Г and 60 the constant, provided by definition
2, such that:
her
Л 6 Г.
Lemma 4. a) Let gug2 ? Г, n; = e{gi) and let t be such that ({gi
Si>92 ? Г suc'1 '*"' ff'i?2 = 9U2, ?(g'i) = f(ffi) — (, and g\ — g[k with
= %,) + %2) -2*. There exist
b) Let 6 < 00. There exists С < oo such that the number of decompositions g = g\g? of any g ? Г, with
l(g\) fixed and ((gi) + ?{g2) < ^(j) + 6, is majorized by C.
Proof, a) Note first that I < inf(r>i, П2) with the above notations, so that n; — t > 0. Writing g = g\g? as
a product of ((g) elements of F we can define g\ as the product of the n, — I first elements and p2 as 'he
product of the last n2 — t elements. One has:
Щд'ия) = e(g[) + t{g) -
2«(Л, 9) =
Thus the hyperbolicity of Г yields:
i) +
= 2n, - 2<.
Hence
than
i,j'i) > 2(ni -<) - 26O and d(ff,, g\) < t + 26o- Thus it = (j'l)
has length less
b) Let j =
one has d(j
]j2 with ?(ji) = >»i and l(g) = ((g\) + ?{дг) — It as above. By hypothesis 11 <b thus as in a)
,j'i) < * + 26O and the result follows.
/3) The Haagerup inequality
In his work on the approximation property for С algebras ([H6]) U. Haagerup proved an important estimate
for the norm of the convolution operator A(i), x e СГ, acting in the Hubert space <?(Г) where Г is a free
192
group. This estimate was then extended to word hyperbolic groups in [Jol] [Harp]. We shall give its proof
below using the above lemma 4.
Theorem 5. ([Jol] [Harp]) Let Г he a word hyperbolic group, g -> %) the word length. Then for some
к < oo and С < 00 one has for arey x 6 СГ:
'2
( Y
where »k{x) = ? A + %))" |*„|2 , i = ? x,g.
Wr J
Here X(x) is the operator in ^2(Г) of left convolution by x:
Writing x = Y,l("' where i^n) = 0 unless t(g) = n we just, need to find an inequality of the form:
(*) II A(x-<">) ||< P(n) || z<"> ||2
where P is a polynomial in n and || у ||2 is the B norm. (One then has || A(i) ||< ? || A(i<")) ||< ? P(n) ||
We can thus fix n ? N and assume that xg = 0 Vj 6 Г, l(g) ф п. Let us then consider the orthogonal
decomposition:
C2(D = н =e Hm
where, with ?,,jsr the canonical orthonormal basis of ^2(Г), the subspace Hm is the closed linear span of
the г,, t(g) = ra. Let A(i) = {Tm^m:i)mi^N be the matrix of operators:
corresponding to A(i). The triangle inequality shows that Traim3 = 0 unless |rai — ra2| < n (since kh^ = Ab
C(lc) = n, ?(hj) = im implies |m, — m2| < n). Thus in order to prove (*) it is enough to find a polynomial
P(n) such that:
\\Tm,,m, ||< P(n) || 1 ||2 Vm,,mj.
Let us fix m, 5 > 0. Let { be such that ?(g) = 0 unless l(g) — ra. One has:
For J ? Г, f(j) = m + n ~ 2q, let g[,g'2 ? Г be such that д\д'2 - g, t(g\) = n- q, l(g'2) = m - q. Then
by lemma 4 we can, for any decomposition д = i/ij2, <?(ffi) = n, t(gi) — ">, find it ? Г, with ji = j',t,
'(^) < 9 + 2*o- Then jo = k~lgi,. Thus the Schwartz inequality gives
193

Adding up these inequalities we get
2< c2 QT |
where С is the constant of lemma 4 b) for 6 = ЧЬц.
Remark 6. The above estimate of the convolution norm || X(x) \\, x 6 СГ in terms of the word length
norms щ(х) fails in general and in particular for any solvable group with exponential growth. Indeed for any
amenable group the trivial representation is weakly contained in the regular representation and it follows
that for x € СГ, xg > 0 Vg e Г, one has
But the inequality ^,xg <C Uk(x) is easily violated for any С, к when Г has exponential growth. We refer
to [Jolj for a general study of groups for which theorem 5 is valid.
7) Extension to C^(Y) of К theory invariants
Let Г be a word hyperbolic group and [c] g ЯР(Г,С) a group cohomology class (assume p even > 0). By
[Groe] let с e ZP(T,C) be a bounded group cocycle in that class, i.e. |c(ji,... ,дь)\ < С Vj,- € Г. Let
тс е ?'(СГ ® TV) be the corresponding cyclic cocycle on the group ring СГ ® 7t = 71Г of Г with coefficients
in the ring H. of matrices of rapid decay (cf. section 4).
Theorem 7. ([Co-M]]) The К theory invariant rc : KO(TIT) —> С extends to the К theory of the С algebra
To prove this theorem we shall show that the cyclic cocycle rc extends by continuity to a cyclic cocycle on
the closure С of КГ under holomorphic functional calculus in С*(Г)®С Any element x of С;(Г)®ДС yields
a matrix [xij) of elements of С*(Г) and we can thus define:
1/2
(For i ? C'(V) the components xg ? С are well defined, r9 = (xsg,sg) so that uk(x) = (?A +t(gJk)\xg\'1I/'i
is unambigous.)
Let us show that for any element x of the holomorphic closure С of TIT in С*(Г) ® AC one has:
Nk(x) < oo V* ? N.
We shall construct asubalgebaB of C*(Y)®K, which contains Cr®ft, is closed under holomorphic functional
calculus (Appendix C) and such that Nk(x) < oo for any x ? В and it € N.
To this end, we let Д, resp. D, be the (unbounded) operator in <?2(N), resp. B{Y), defined by:
A«j = jSj U ? N), rep. DS, = \g\8, (</ 6 Г).
Next, we consider the (unbounded) derivations d = adD of ?(?2(Г)) and Ъ = ad(D®I) of ?(f2(r)®f2(N)),
and set:
B={i6C;(r)®jC|3'(i).(/«iA) is bounded V* ? N}.
First we claim that В contains CT0TZ. Indeed, if x = A(j)®S, with g ? Г, Л the left regular representation
and S ? ft, then !Tfe(r)o(/ ® Д) = Э*(А(г)) ® 5Д; but both дк(\(д)) and 5Д are bounded.
Secondly, since {x ? С" (Г) ® АС|ю(/ ® Д) bounded} is evidently a left ideal, В is a subalgebra, in fact a left
ideal in
194
Вте = П Domain dk.
fc>0
Now Вте is stable under holomorphic functional calculus and, therefore, so is B.
Finally, let us check that Nk(x) < oo, Vi ? В and it ? N. As DS, = 0, one has
(J*(x)oi ® д)(«« ®«,-) = i E Di (lij *е:) ® *'
= > E i/'ij(s)«»®«
Thus, there exists С < 00 such that
СГ
therefore,
E
which implies JVfe(i) < 00, Vi e N.
To conclude the proof of theorem 7 we just need to show that the cyclic cocycle rc# Trace on СГ ® H.
is continuous for the norm Nk, к large enough, and thus extends by continuity to C. It is clear from the
definition of Nt that rc# Trace is continuous with respect to Nk if тс is continuous with respect to 14. The
continuity of tc with respect to ft, к large enough, follows from theorem 5 and the boundedness of the group
cocycle с (its polynomial growth is sufficient) ([Jol]). Indeed one has:
тс(х°,х\...,х')= Y, ЛЛ
where yJ'(j) = |a.J(s)| e С Vj e Г. Thus
i E «
»o-9» = l
\rc(x\ x\..., x')\ <\\ с || {y°*yl* ..
/ p
°(!'0) =H c II C" П "
\ 1
as a corollary of theorem 7 and corollary 4.9 one thus gets:
Theorem 8. ([Co-Mi]) Let Г be a word hyperbolic group then Г satisfies the Novikov conjecture.
Remark 9. 1) For another subsequent proof of this result cf. [Co-G-M].
2) The proof of the above theorem has been extended by C. Ogle to show that even in the presence of torsion
for Г the analytic assembly map (chapter 2 section 10) is rationally injective ([O]).
195

III.6. Factors of type III, cyclic cohomology and the Godbillon-Vey invariant
Let (V,F) be a transversally oriented codimension one foliation. The Godbillon-Vey class GV € #3(V,R) is
a 3 dimensional cohomology class, the simplest non trivial example of Gelfand-Fuchs cohomology class. It
is defined as follows: As (V, F) is transversally oriented there exists a nowhere vanishing smooth 1-form ш
on V which defines the foliation by the equality:
Ft = {X?Tt{V);ur{X) = Q} Vi € V.
The integrability condition of the bundle F С TV is equivalent to the existence of a 1-form a on V such
that:
One then checks the following:
Lemma 1. ([Go-V]) The 3 form a A da is closed and its cohomology caiss GV ? H3(V,R) is independent
of the choices of и and a once (V, F) is given.
Indeed, fixing и first and changing a to a', the equality (a' — а) Л ы = 0 shows that a' == a + рш for some
p?C'x(V), then
а' Л da' — а Л da + d(pu> A a).
This shows that the class of а Л da is independent of a. If one replaces w by w' = /ui, / ? C°°(V,R') then
one has
Thus the class of а Л da is independent of the choice of a/ and is hence closed.
The cohomology class GV ? H3(V, R) is not rational in general, in fact when V is a compact oriented 3
manifold, the Godbillon-Vey invariant:
(GV,[V))eR
can assume any real value ([Thu2])-
With a little more work one can define GV as a 3 dimensional cohomology class:
GV ? H3(BG,R)
where BG is the classifying space of the holonomy groupoid G of (V,F) (cf. [На(|2,з] and 7) below).
In this section we shall prove that the Godbillon-Vey class yields К theory invariants:
for the C" algebra of the foliation, parametrized by elements z ? W(M) of the flow of weights of the von
Neumann algebra M of (V, F). These invariants enjoy two properties:
1) ?«„(,) = <Pz VA 6 R+ , г ? W(M) where вх ? Autiy(M) is the flow of weights.
2) tpi is related to GV ? H3(BG,R) by the equality:
4>Mx)) = (cft.(x), GV) Vi 6 K.,T(BG)
where ц is the analytic assembly map (chapter 2 section 8).
196
As an immediate corollary of the existence of such A' theory invariants it follows that if GV ф 0 (as an
element of H3(BG,R)) then the flow of weights W{M) admits a (normal) invariant probability measure
(theorem 21) [С01З] and the von Neumann algebra M has a non trivial type III component ([Hu-K]]).
The existence of tpz follows from the interplay between a) the transverse fundamental class of (V, F), which
is a certain canonical cyclic cohomology class [V/F] ? HCx(Cf(G)) encoding the 1-dimensional oriented
transverse structure, and /3) the canonical dynamics 6(t) ? OutM = AutAf/InnAf of the von Neumann
algebra M of the foliation, given by the modular automorphism group of M (cf. chapter 1 section 7).
The point is that in general [V/F] fails to be invariant under the evolution 6(t) but its second time derivative,
jir[V/F] does vanish and the first time derivative ji[V/F] yields by contraction: is ji[V/F], a cyclic 2-
cohomology class
Ujt[V/F] ? HC\C?(G))
whose associated A' theory invariant, once extended to C*(V, F) yields <p\.
The relation: (for any x ? K,T(BG))
expresses a deep interplay between nou commutative measure theory and Gelfand-Fuchs cohomology.
Let us now enter in this in more details, beginning with a technique to extend К theory invariants:
tp : K{A) — С
from a subalgebra А С Л of a C* algebra to
tp : K(A) -> C.
a) Extension of densely defined cyclic cocycles on Banach algebras
Let Л be a unital С algebra, then the traces т on A are the only interesting continuous cyclic cocycles on
A ([C016]). As a rule the relevant cocycles are only defined on a dense subalgebra:
AC A
and are not continuous for the C* algebra norm of A.
In this section we shall give a relative continuity condition on cyclic cocycles on A which will ensure that
the К theory invariant:
tp : K(A) — С
associated to the cyclic cocycle, extends to K(A).
Let us begin by giving a simple condition implying that two (continuous) traces r0, r, yield the same map:
If we were to consider r0, т\ as elements of HC°(A) then a sufficient condition would be: r0 — T\ ? В(Кегб)
(cf. section 3) i.e. the existence of a Hoclischild 1-cocycle ф ? Hl(A, A") such that Вф — та — т\, or in other
words, with r = ro — л :
i/>(l,a) -ф(а, 1) = т(а) Va ? A.
As the Hoclischild cohomology H1(A1A') always vanishes ([H3]) the above condition is of no use and only
yields trivial homologies between traces.
197

As we shall see this is no longer the case if we allow ф to be unbounded. A Hochschild 1-cocyclet/i g Z1(A, A')
is a derivation from the algebra A to the bimodule A* of continuous linear forms on A and the notion of
unbounded derivation from A to A* is perfectly meaningful. Let us recall the standard terminology:
Given two Banach spaces Bi, B2 an unbounded operator T from 3\ to B2 is given by a linear subspace
Dom(T) С В, and a linear map T ¦ Dom(T) -* B2.
One says that T is densely defined when DomT is dense in Bi, and that it is closable when the closure of its
graph:
graph(T) =
6 Bi x B2 ; « € DomT}
is the graph of an unbounded operator T, the closure of T. The adjoint T* of a densely denned operator T
is the unbounded operator from B2, the dual Banach space of B2, to BJ denned by:
DomT* = {17 ? B; ;3c < 00 with |(T?, >7)| < с ||? ||
for any ? ? DomT}
G™ ij, 0 = A7, T?) V17 ? DomT' , i ? DomT.
The adjoint V is densely denned provided T is closable.
If Bi is a Banach algebra and B2 a Banach Bi-bimodule:
IIi«iliilII li
Then an unbounded operator from Bi to B2 is called a derivation iff: Dom6 is a subalgebra of Bi and
6(ab) = 6(a)b + a6(b) Va,6eDomi5.
One then has the following standard result:
Lemma 2. Lei 6 be a densely defined and closable derivation from Bi to B2, then its closure b is still a
derivation, and its domain Dom6 is a subalgebra of B\ stable under holomorphic functional calculus.
Proof. One has a ? Domi iff there exists я„ ? В,, with an —> a, b{an) convergent. Then 6(a) = Iimi5(an).
It follows easily that Dom? is an algebra and ? a derivation. To prove the stability under holomorphic
functional calculus of Dom6, let us first show that if о ? Dom6 is invertible in B\ then a~l € Domi5.
As Dom7 = A is dense in Bi, there exists b ? A with || 1 — a6 ||< 1 || 1 — 6a ||< 1, hence it is enough to
show that if a e Л, || a |l< 1 then A - a) ? A. This is clear since ? ak -* A - a), and 6 I T ak ) is
norm convergent, because || 6{ak) ||< к \\ а \\k~l || 6(a) || for any it ? N.
Applying this to the resolvent (a —A), A ? SpectrumB|(a) one sees that Domjis stable under holomorphic
functional calculus. This still holds if we replace A and Bi by Mn(A), Mn(Bi) (using the derivation
6® id : A ® Mn -> B2 <8> М„).
Let us now return to homology between traces and prove the following ([С01З]).
Proposition 3. Let 3 be a umtal Banach algebra, 6 a densely defined derivation of В with values in the
dual space B* (viewed as a bimodule over В with (aipb,x) — (ip,bxa) Va, x,b g В , <p g B*) and assume
that the unit \в belongs to the domam of the adjoint 6" of 6, then:
a. т — 6'A) is a trace on B.
b. The map of A'o(B) to С given by т is equal to 0.
Proof, a) One has т(ху) = (xt/,<5"(l)) = {6(xy), 1) = F{x),y) + (%), 1) = т(ух) Чх,у ? Dom<5.
198
b) The equality F(x),y) + (S(y),x) — т(ху) for x, у ? Dom<5 shows that Dom<5 С Domi5', with b"(x) =
—6(x) + xt for any x ? Dom6. This shows that 6 is a closable operator from the Banach space В to the
dual Banach space B". Let 6 be the closure of 6, then the domain of 6 : A = Dom6, is a subalgebra of В
and Ь is a derivation from A to B*. Moreover (lemma 2) the inclusion А С В gives an isomorphism
K0(A) - A'o(B)
(cf. appendix C). Thus it is enough to show that the map from Ka(A) to С given by т is equal to 0 which
follows from the equality г = Вф where ф ? Zl(A,A*) is given by:
ф(ао,а1) = {а°,Ца1)) Va°, a1 ? A.
Let us illustrate proposition 3 by a very simple example.
Let [i be a Radon measure of 0 total mass on the connected compact manifold V and let us express the
homology between the current fi and the current 0 in C* algebra terms:
Example 4. If ft(V) = 0 there exists a densely defined derivation 6 from the С algebra A = C(V) to its
dual A* such that 1д belongs to the domain of the adjoint 6* and6*(\) = [Л.
Proof. Choose a Riemannian metric on V and assign in a Borel manner a geodesic path 7rPl? : [0,1] —*¦ V
to each pair p, q of elements of V. Assuming for simplicity that ц is real, let ^ = j*+ — р~~ be its Jordan
decomposition, and put for f,g ? C"(V):
Then the equality (*(?),/) = т(/,д) gives a densely denned derivation 6 from A to A' (considered as a
bimodule over A) and 6*A) = fi+(K)/i since
hg) = J(g(p) - я{ч)) Ф+Ы d»-(q) = »+(V) Jgdp.
Let В be a Banach algebra. We shall now show how to construct maps from K\(B) to С using instead of a
trace a homology (in the sense of proposition 3) between the trace т = 0 and itself:
Definition 4. Lei В be a Banach algebra. By a 1-trace on В we mean a densely defined derivation 6 from
В io B" such that
) Vr,y?Dom,5.
Proposition 5. Let 6 be a \-trace on 3, then:
a) 6 is closable.
b) There exists a unique map of K\C) to С such that, for any и ? GXn(Dom6) one has:
Proof. We can assume that В is unital with 6A) = 0.
a) Any skew symmetric operator from В to B* is closable [Kat].
b) We can assume that 6 is closed, let A be its domain. By lemma 2, any element of A which is invertible in
В is invertible in A and the same holds for Mn(A) С М„(В). Thus, since the open set Afn(B) of invertible
elements in Mn(B) is locally convex, two elements u, v of GLn(A) which are in the same connected component
of GLn(B) are connected by a piecewise affine path in GLn(A).
199

On such a path < —> ui the function /(() = (u, ^(uj)) is constant, since its derivative is: — (u,~l щ
u,-\6(uO> +(«Г1,*(«1)) = 0 since (и~\б(щ)) = -{6(щ*),щ) = (u, 6(и,)щ1,щ). The result then
follows.
We shall now give non-trivial examples of 1-traces on C* algebras. The simplest example is to take a one
parameter group at of automorphisms of A, and an a-invariant trace т on A. let then D be the generator
of (at), i.e. the closed derivation
Щ") =}un \ (<».(*) - *)•
The equality x ? DomD —> 6(x) ¦=¦ D(x)r ? A' defines a 1-trace on A and the map of Ki(A) to С given
by this 1-trace coincides with the one defined in [Co]. We shall now give other examples of 1-traces, not
of the above form (the algebra A will have no nonzero traces in some examples) and use them to prove
the non-triviality of K^(A) for crossed products A — C(Sl) xf of S1 by a group of orientation preserving
homeomorphisms. (Note that we shall use the reduced crossed product so that the result we get is stronger
than for the maximal crossed product).
Theorem 6. Let Г be a countable group of orientation preserving homeomorphisms of S1 and A =
G(S') хГ be the reduced crossed product C" algebra. Then the canonical homomorphism i : CIS1) —* A is
an injection of Ki(C(S1)) = Z m I<i(A).
Proof. On В = C(S'), we have a natural 1-trace obtained as the weak closure of the derivation 6, with
domain C^iS1), which assigns to each / ? Coa{S1), the differential if viewed as an element of B". (This
uses the orientation.) This weak closure of 6 is easily identified using distribution theory, as the derivation
(also noted 6) with domain BV{S1), the space of functions / e CIS1) with bounded variation, i.e. such
that df is a measure, 6(f) = df. A function / is of bounded variation iff the sums ?^|/(e»+i) — f{xi)\
are bounded when the finite subset (ii)i=i m of S1 varies with x; in the same order on S1 as i/m. It
is then clear that if <p is any orientation preserving homeomorphism of S1, <p* : CIS1) —* C(S') leaves
Dom6 = BV(S1) invariant and that 6(foip) — <pF(f)). Using this Г-equivariant 1-trace on CIS1) we shall
now construct a 1-trace on A. Let Л = {a ? A ; a =J2 ag U3 with ая ф й only for finitely many g g Г and
г
<Jj ? BV(S') Vj e Г}. For any a ? A, let 6(a) € A' be the linear functional given by
(«,%))= /"^i, «/(<*.»,-') Vie A
J г
Here, for any g g Г, da3-i is a measure on Sl and g(dag-i) is its image under the action of g ? Г on S1.
For any x ? A, z = ^ xg Ug, xg is (in a faithful representation) a matrix element of i so that one has
|| xg ||<|| x || where the latter is the norm in the reduced crossed product. Thus 6(a) € A*. Let us check
that 6 is a 1-trace. For a,b ? A one has:
ri 9 as
For a, b, с ? A one has:
= -/?»•
{ab,6(c)) - (a,6(bc)) + (ca,6(b))
)" bSl)(go gi)dc}, - I Y, a"
J JOSUS = 1
200
/ Y, SoG/o")"' Oj,(ff0ffl)<ttn
909193=1
Y J a«° 3o(db3,)(go л)' cg,
9o9i9s = l
oiK1)" «».(Ло fti)rf*»3 = 0.
Thus 6 is a 1-trace on Л, and for any unitary u ? BV(S1) with winding number equal to 1 one has:
(u-\<5(u)) = I и'1 du = 1iw.
This shows that u" is a non trivial element of K\{A) for any n g Z, n ф 0.
In a similar manner we shall now construct a 1-trace г on the C" algebra C(V, F) of a transversally oriented
codimension 1 foliation and detect non trivial elements of К\(C*{V, F)).
Let thus (V, F) be such a foliation and G its holonomy groupoid. Let T С V be a submanifold of dimension
one of V which is everywhere transverse to F and intersects every leaf of (V, F). We do not assume that T
is compact or connected, so that it always exist. We shall work with the reduced groupoid:
GT = {7 e G ; rG) 6 T , s(y) ? T)
which is equivalent to G and is a one dimensional manifold (we assume that it is Hausdorff and refer to
[CoiO] for the general case). One has by construction G^) = T and the maps r,s from Gt to G^ are etale
maps. We shall now construct a 1-trace on the C" algebra A = C*(Gt) which is strongly Morita equivalent
to C*(V, F) = C'(G) and hence has the same К theory.
Proposition 7. Let G be a l-dimensional smooth groupoid such that G^ is one dimensional and that G and
G<°) are oriented with r and s orientation preserving. Then the following equality defines a cyclic 1-cocycle
on Cf(G) which is a 1-irace on C;(G)
= /
Jg
G)-
Proof. Let us first check that г is a cyclic cocycle on C^°{G). The map 7 —> y~1 is an orientation preserving
diffeomorphism of G thus r(/°, f1) = -r(/1, /°) V/J ? GC°°(G).
The product in Cf(G) is given by:
Since the source and range maps are etale we can use them to identify the bundles T'G, s'(T*(T)), r"(T"(T))
over G. We then have the following equality:
in the space of 1-forms on G. To prove this one may assume that the supports of fj are so small that both
r and s are diffeomorphisms of neighborhoods of Support fj with open sets in T. The required equality is
then just the Leibnitz rule for d. The equality Ът = 0 follows from the above and the equivalence for 71, 72,
73 6 G between 717273 ? G(o) and 727371 ? G<0>. Finally to show that т is a 1-trace on C'(G) one just
needs to show that for any /, ? CJ°(G) the linear functional
/ 6 Cf(G) - / /G) df^-,-1) = L(f)
Ja
201

is continuous for the С algebra norm of C*(G) which is straightforward using the left regular representation
ofC;(G)inL2(G).
As an immediate corollary we get a way to detect the fundamental class of V/F in К theory. The latter class
makes sense for any transversally oriented foliation and yields [V/F]* ? K,(C*(V, F)) with the same parity as
the codimension q otF. To define [V/F]* choose an orientation preserving transversal embedding ф : Dq —• V
where ГУ is the q dimensional open disk, and consider ф\@,) ? K,(C*(V,F)) where 0, ? K*(Dt) is. the
canonical Bott generator of K'(Dq) = Z. One then checks (Chapter 2 section 8):
Lemma 8. The class ф\(р,) ? K.(C*(V, F)) is independent of the choice of ф.
We shall denote it by [V/F]' ? K,(C'(V F)). Using propositions 5 and 7 we get:
Theorem 9. Let (V, F) be a tmnsversally oriented foliation of codimension 1 then [V/F]* is a non torsion
element of Ki(C*(V,F)).
Proof. Let as above T Q V Ъч л transverse 1 dimensional manifold such that the holonomy groupoid G of
(V,F) is equivalent to Gt¦ Then C*(G) is strongly Morita equivalent to C*(Gt) and under this equivalence
the element [V/F]* ? Ki(C*(V,F)) becomes j.(a) where j : Ca{T) С C'(GT) is the canonical inclusion of
Ca(T) in C"(Gt) and a is any element of I<i(C0(T)) associated to the Bott generator of one of the (oriented)
connected components of T. As the restriction of the 1-trace r of proposition 7 to C0(T) is the fundamental
class of this oriented manifold the answer follows from proposition 5.
Remarks 10. 1) We shall see in section 7 that theorem 9 is true in any codimension but the transverse
fundamental class in cyclic cohomology, i.e. r, is much easier to analyze in codimension 1.
2) If В is a non unital Banach algebra, proposition 3 still holds if one replaces the equality r = S*(l) by.
т{ху) = {6{x),y) + (%), x) Vx, у ? DomS.
3) Let Л be a non unital C* algebra, т a densely defined semi-continuous weight which is a trace: r(x*x) =
т(хх') Vx ? A. Then using 2) above one can show that if 6 is a densely defined derivation from A to A*
such that
a) DomS П Dom1/2 {r) is dense in A, where Dom]/2('') = {x ? А, т(х'х) < oo}
b) (S(x), y) + (<5(t/), x) = т(ху) Vi, у ? DomS П Dom1/2(r)
then т defines the 0-map from Ko(A) to C.
This happens if there exists a one parameter group of automorphisms $t of A such that тов% = е' т Vt ? R.
Thus if <p ? A' is a K.M.S. state for a one parameter group 0t then the associated trace r on the crossed
product A — A Xt, R (cf. chapter 5) is always homologous to 0.
4) The conclusion of Theorem 6 does not hold when Г fails to preserve the orientation of S\ in fact if
[u] € K\Dl) is the generator, then 2i.[u] = 0 in Kt(A).
We shall now extend proposition 5 to higher dimensional densely defined cyclic cocydes on Banach algebras.
Let В be a (not necessarily unital) Banach algebra. Recall (section 1) that to any n + 1 linear functional т
on an algebra A we associate the linear form f on Q™(,4) given by.
?(a° da> da2 da") = т(а°,al,... ,o") Va' ? A.
This gives a meaning to ?((xv rfa')(i2 da2)(x3 da3) ¦ ¦ ¦ (i" da")), for a', x' ? A, but of course one could also
define it directly by a formula. Thus, for n = 2, ?((*' do')(i2 da2)) = r(xl, oli2, a2) - rfx'a1, X2,a2). The
crucial definition of this section is the following:
202
Definition 11. Let В be a Banach algebra. By an n-tracn on В we mean ann + l linear functional т on a
dense subalgebra A of В such that
a) r is a cyclic cocycle on A.
b) For any a' ? A i = 1,..., n there exists С = Ca, „» < oo such thai:
\т((хг dai)(x2da*)--(x" da"))\ < С || x1 || • ¦¦ || xn || Vi'el
Note that the conditions a), b) are still satisfied if we replace A by any subalgebra which is still dense in B.
Examples 12. a) Let V be a smooth manifold (not necessarily compact). Let A = Co(V) be the C* algebra
of continuous functions vanishing at oo. Recall that a de Rham current С on V of dimension p, is a linear
functional on the space Cf{V,f\r T*-V) of differential forms of degree p on V, which is continuous in the
following sense: for any compact А" С V and family ua ? Cf (V, N> T^(V)) with Support ua С A' Va
converging to 0 in the Ck topology (on the coefficients of the forms ша) one has С(ша) —> 0. In other words
С is of order к when for anyui ? CZ°(V,AP T?{V)) the linear functional / ? C"(V) -» C(fu) is continuous
in the Ck topology.
Let then С be a closed current of dimension p and order 0 on V, put
Let us check that it defines a p-trace on the C* algebra Ca(V). Its domain Cf(V) is a dense subalgebra of
C0(V) and one easily checks the cyclic cocycle property of т using the closedness of С One has
a1 -.¦x'da?) = СЦхЧа1) Л . ¦.
= C(xl ¦ • x"dal Л ¦ • • Л da").
Thus, since da1 Л ¦ Л da? = и belongs to C$°(V, Лр T^(V)) there exists, as С is of order 0, a Ca,...„, < oo
such that:
[ЦхЧа1- x"da")\<Cai...af П\\х>\\.
One checks that the map tp of Theorem 13 from Ki(Co(V)) = K'(V) to С is given, up to normalization, by
where ch is the usual Chern character: K*(V) -> H'(V,R) and [C] 6 #,(V,C) is the homology class of the
closed current С. Since there are always enough closed currents of order 0 to yield all of H.(V, C) we have
not lost any information on the Chern character ch e ? #*(V, R) in this presentation.
b) Let Д be a locally finite simplicial complex, A" = |Д| the associated locally compact space. Let us
construct enough p-traces on the С algebra A = Со(|Д|) to recover the usual Chern character, as in a). Let
7 = Yl Ai Si be a locally finite cycle of dimension p (i.e., the Si are all oriented p-simplexes of Д and the \t's
are complex numbers, with by = J^ A; s, = 0). Put
-(/"....,/*) = ?
where the f> ? CC(X) have the following property (cf. [Su6]): On each simplex s of Д the restriction of/
is equal to the restriction of a C00 function on the affine space of s. This space Cf (Д) С Со(Х) is a dense
subalgebra and one checks as in a) that г is a p-trace on Co(X). Again the map ip of Theorem 13 from
K'C(V) to С is given by.
<p{x) = {ch x, Ы)
where ch is the usual Chern character and [7] is the homology class of 7 in the homology of locally finite
chains on X, dual to the cohomology with compact support H*(X).
203

с) Let (A, G, a) be а С dynamical system, i.e. A is a C" algebra on which the locally compact group G acts
by automorphism ag ? AutA Vj ? G. Assume that G is a Lie group and let т be an a-invariant trace on
A. Let fi be the graded differential algebra of right invariant differential forms (with complex coefficients)
on G- By construction Q is, as a graded algebra, identical with Лс T^(G) the exterior algebra on the dual
of the Lie algebra of G. Let ? ? HP(Q') be a p-homology class in the dual chain complex Q*. Considering t
as a closed linear form on Q?, put:
<r(i°,...,x") = (r®t)(x°
x> e Л°
Here, Ax is the dense subalgebra of A formed of elements x ? A for which g —> а„(х) is a smooth function
from G to the Banach space A. The differentials dx belong to the tensor product algebra: Л°° ® fi, and are
defined as follows. Let X,- ? LieG be a basis of the Lie algebra of G, ui' ? (LieG)* the dual basis, Si g Der(A)
the unbounded derivations of A given by (<3x, a3K=l one takes:
One checks that т ® t is a closed graded trace on the differential algebra A"> ® fi and it follows that a is a
cyclic cocycle on the algebra Л°°. Let us show that it is a p-trace. For fixed a1,..., ap ? A, the expression
^(r'da1 • • ¦ xpdap) is a finite linear combination of terms of the form:
where y> e Ax is of the form St(a'). Since by hypothesis т ? A* one has ^(x'y1 ¦ ¦ ¦xpyp\ < С ||
r'y1 ¦ • xpyt ||< С || i1 || ¦ ¦ ¦ || x" || thus one gets the conclusion.
Applying Theorem 13 below one recovers the Chern character that we introduced in [CoB], from K(A) to
Я*(Я) (which is dual to Я.(Я)).
The above examples are easy instances of the following general construction of К theory invariants which
will be used extensively in /3) and in section 7.
Theorem 13. Let т be an n-trace on a Banach algebra B. Then there exists a canonically associated map
tp of Ki(B) (i=nA)) to С such that:
a) Ifn is even and e ? ProjAf, (Domain r) then
?>([«]) = (r#Tr)(e e).
b) // n is odd and u Е GLq (Domain r) then
tp([u]) = {T#Tr)(u~l, u, u'1 u,..., u, o).
We refer to [С01З] for the proof. Note that by [С01З], т extends uniquely to an n-trace with domain the
closure of Domr under holomorphic functional calculus in B. In particular if т and r1 agree on a common
dense domain they define the same К theory invariants: ip = <p'.
Remark 14. Combining the construction of n-traces in example b) with the known results on the Chern
character, we see that if X is a locally compact space coming from a locally finite simplicial complex all
maps from K(C0(X)) to С which are additive come from n-traces on A — Co(X).
At tins point it would be tempting to define for arbitrary n a homology relation between n-traces on an
arbitrary Banach algebra extending that given by proposition 3 and work out an analogue of homology
in this context. We shall however see that this would be premature, because it would overlook a purely
non-commutative phenomenon which prevents some very natural densely defined re-cocycles on С algebras
to be n-traces, for n > 2.
204
As a simple example take A = С*(Г) the reduced С algebra of the following solvable discrete group Г. We
let Z act by automorphisms on the group Z2 using the unimodular matrix a = \ , and let Г = Z2 >JO Z
be the semi-direct product. Now consider the group cocycle с ? Z2(I\R) given by the equality:
= "l Л a"
g,- =(»,-, nj) € Г.
(We represent an element of Г as a pair (»,n) where v ? T? С R2 and n ? Z, the Л is the usual exterior
product, with values in A2R2 ~ R.)
Given any discrete group and group cocycle с which is normalized so that c(pi,..., gn) = 0 if any j,- = 1 or
Sl... gn = 1 one gets (section 1) a cyclic re cocycle on the group ring Л = С(Г) by the equality:
Thus here we get a cyclic 2-cocycle on A = С(Г), and thus a densely defined cyclic cocycle on A = С*(Г).
One has:
xl(ga)
2, Js) -
Let a1, a2 ? С(Г). If for fixed ji and g3 the function of g2:
would easily get the desired estimate:
с{9192,9з) - c(9i, 9з) was bounded one
However, this precisely fails in our example, since with gi - (»i,0), дз = (vo, 0) we get, for 92 = @,n):
Moreover, fixing such a choice of гц, г>2 and identifying Г with a subgroup of the unitary group of С(Г) С
С* (Г) we get
T(x1dalx2da.2) = Vi Ла"(»2)
for i1 = й'/«""sf1, a1 = (»i,0), x2 = Л", a2 = (o2, 0) with h - @, 1) ? Г. This shows that т is not a
2-trace.
Since, as one easily checks, there is for each pair a1, a2 of С(Г) а constant Cai,a, such that
the obstruction to т being a 2-trace comes from the non-commutativity do1!2 ф x^da1. The analysis of
this lack of commutativity will occupy us in section 7 iu the special case of the cyclic cocycle on Ca(V) хГ
coming from the Г-equivariant fundamental class of the manifold V on which the discrete group Г acts by
orientation preserving difleomorpliisms. (The above 2-cocycle on C(Z2 Xa Z) is a special case of this more
general problem, since it is exactly the equivariant fundamental class of the 2-torus dual to Z2 on which Z
acts by a).
/3) The Bott-Thurston cocycle and the equality GV = h^[
Let as above (V, F) be a transversally oriented codimension one foliation and GT the reduction of its holonomy
groupoid by a transversal T. Since GT is equivalent to G we shall in fact start with an arbitrary one
dimensional smooth groupoid G as ill proposition 7 above, and leave the translation in foliation language to
th d
the reader.
205

To a choice of a smooth nowhere vanishing 1-density p on the one dimensional manifold G^0' corresponds
a faithful normal weight on the von Neumann algebra of the regular representation of G; (i.e. in foliation
language on the von Neumann algebra of the foliation (chapter I)). This weight tpp is given on Cf(G) by.
M/) = /
Л;<°)
The modular automorphism group <rt of ipp leaves the subalgebra Cf (G) globally invariant and is given by
the following formula which one can check for instance by verifying the KMSi condition for the pair <pp, at
(cf; chapter V)
Klf /G) V7 6 G , / 6 Cf(G)
where the modular homomorphism 6 : G —» R+ is given by the ratio of the following 1-densities on G:
6 = r'p/s'p.
We shall let I = log 6 be the corresponding homomorphism from G to R.
Let now т be the cyclic 1-cocycle on Cf(G), corresponding to the transverse fundamental class [V/F] in
the foliation case (proposition 7).
Proposition 15. a) The following equality defines a 1-trace on C*(G) with domain Cf(G):
't (Л /') =Jim I (г(«г,(Л. «•.(/')) - r(f, /'))
b) The one trace г is invariant under the automorphisms <rt ofC*(G).
a) Let D be the derivation of Cf(G) given by Jim ±(<r«- 1). Then т (/°, /') = r(D/0,/') + r(/°,D/') =
Ja
/ (T) () () (() / /G-1) /'G) )
Ja Ja Ja
where dl is the 1-form on G which is the differential of the real valued function (. One checks that for fixed
/' ? C?(G), the linear form L, L(f") =r G0,/1) is continuous on C;(G) so that a) follows.
b) Since S(y) = Hi'1)'1 the above formula:
-(/",/') = / Л7-1) /'G) dt(y)
Ja
is obviously invariant under <Tt-
We shall now define in general the contraction of a cyclic n-cocycle by the generator D of a one parameter
group of automorphisms fixing the cocycle and check that the n-trace property ш preserved.
Lemma 16. Lei <p be an n trace on the C" algebra A, invariant under the one parameter group of automor-
automorphisms (<7<)teu with generator D. Assume that Dom^ П DomD is dense in A. Then the following formula
defines an (n + 1) trace on A : ф — id<P:
We already saw this formula in section 1 Remark b). Ir clearly defines a Hochschild cocycle. The invariance
of <p under <Tt implies that ф is cyclic. The n+l trace property is straightforward using Dom^JП DomD as
a domain (cf. [С01З] for details).
206
When ifi is a 1-cocycle the formula for ф = iDtp can be written as
Let then y=rbe the time derivative of the transverse fundamental class (proposition 15), and let us compute
iDtp. One has
0,/1)- / /Yr1 /'G) <tfG)
Ja
and (Df)(y) = t(f)f(j). Thus the computation is straightforward and gives:
/47.) /2Ы cGi,7s)
where 0G1,72) = ^G2) <tfGi) - ^G1) ^^G2) is the Bott-Thurston 2-cocycle, a 2-cocycle on local diffeomor-
phisms of a 1-manifold with values in 1-forms on that manifold. The latter cocycle is the Godbillon-Vey
3-dimensional cohomology class in another guise and this is reflected in the following:
Theorem 17. Let fi : K.r(BG) —> K(C'(G)) be the analytic assembly map where G = Gt is the reduced
holonomy groupoid of the transversally oriented codimension 1 foliation (V, F). Let ф =z iD т be the 2-trace
on C*(G) given by 15 and 16. Then for any x ? K*iT(BG) one has:
Here GV is viewed as a cohomology class GV E H3(BG,R) and the twisting K.T in the К homology
introduces a shift of parity so that it is A'JiT(BG) which by the Chern character сЛ. maps to H^d(BG).
The proof of this result is a special case of the general index theorem of section 7 below. We refer to [С01З]
for a direct proof. We shall now localise the above A" theory invariant to the flow of weights W{M) of the
von Neumann algebra M of the foliation.
7) Invariant measures on the flow of weights
Let as above (V, F) be a codimension one transversally oriented foliation and Gt = G be the reduced
holonomy groupoid. The von Neumann algebra of the foliation (Chapter I) is the same, up to tensoring by
a factor of type /„,, as the von Neumann algebra M of the left regular representation of Cf(G) in ?2(G).
The flow of weights of the von Neumann algebra of the foliation (cf. Chapter V) is the same as the flow
of weights W(M). It is thus given by the dual action в of the Pontrjagin dual RA = R^. restricted to the
center Z(M х„ R) of the von Neumann algebra crossed product of M by the modular automorphism group
Ct = <rf' of section /3). Since the latter group <rt leaves the subalgebra Cf(G) С М (and its norm closure
C*(G)) globally invariant, we can use the С algebra crossed product:
in order to investigate the center of M х„ R. By construction the С algebra В is weakly dense in M X, R =
N, and the center Z(N) will appear from the following notion.
Definition 18. Let В be а С algebra, r a l-irace on В and 6 : В —> В' the corresponding unbounded
derivation, so that rfx0,!1) = (i°,6(i1)) VzJ e Domr. Then r is anabelian iff the domain of the adjoint
S" : B" -> B' contains the center Z{B") and 6*{z) =0 Vz e Z(B").
Recall that the bidual B" of a C" algebra is a von Neumann algebra so that its center makes good sense.
The term anabelian is motivated by the following trivial point:
If В is commutative, any anabelian 1-trace is 0.
207

In other words such 1-traces can exist only in the non commutative context.
Proposition 19. 1) Let r be an anabelian \-trace on the С algebra B, then for any z ? Z(B") the •
following equality defines a \-trace on B:
Ti(r°,r') = (s°, *«(*')> VaJ' ? Domr.
2) For any и ? A''(B), the map z ? Z(B") -> {tz,u) is a normal linear form on Z(B").
Proof. 1) Since B" acts by multiplication on its predual B* the element z6(x1) g B* makes sense and
z6(xl) = 6(x1)z since z ? Z(B"). It is clear that tz is Hochschild cocycle, let us check that tz(x°,xx) = Щ
-гДх1,!0) Vx> 6 Domr. One has т2(х> ,x°) = {x* ,z6(x0)) = (i'z,6(io)> = M
where the last equality follows from the hypothesis on r:
(z,<5(t/)) = 0 VyeDomr.
Finally {z,x°6{x1)) = {zxo,6(x1)) = {x°,z6(x1)) = r,(i°,i').
2) By construction (u~\z6(u)) is a normal linear form in the variable г S B".
Next, let us take В = C;(G) >ь R and let ф„:
фа : K0(C'{G)) -» А',(В)
be the Thom isomorphism in К theory (Chapter II appendix C). Let ф be the 2 trace on C*(G) related to
the Godbillon-Vey class by theorem 17.
Theorem 20. There exists an anabelian l-trace т on В = C;(G)xvR such that:
1) r is invariant under the dual action @л)леи'
2) т corresponds to ф by the Thom isomorphism, i.e.
{т,Фа(у)) = (Ф,У) Vy 6
Before we explain the proof of this theorem, let us deduce from it the following corollary.
Theorem 21. ([С01З]) Let (V, F) be a codimension 1 transversally oriented foliation, M its von Neumann
algebra, (W(M),9\) its flow of weights.
a) If the Godbillon-Vey class GV e H3(BG,R) is not zero (which holds ifGV ? H3(V,R) is non zero) there
exists a №x invariant probability measure in the normal measure class on W(M).
b) The conclusion of a) holds if the Bott-Thurston 2-cocycle ф of theorem 11 pairs non trivially with
K0(C'(V,F)).
Corollary 22. [Hu-K,] If GV ф 0 then the von Neumann algebra M has a non trivial type III component.
The corollary is immediate since if M is semifmite its flow of weight is given, up to multiplicity, by the
action of R^. by translations on L°° (R^^-) which admits no invariant probability measure in the Lebesgue
measure class.
Assuming theorem 20 the proof of theorem 21 if the following. First it is enough to prove b) since, by theorem
17, if GV ф 0 in H3(BG, R) then there exists an element x ? K0(C'(V,F)), in the image of the analytic
208
assembly map, such that (х,ф) ф 0. Next, by 20 2) and the Thom isomorphism (Chapter II appendix С
one gets an element u s Ki(B) such that:
Let us then consider the normal linear functional L on the center Z(Mx,, R) given by:
Vze
invariant so that:
But since the action 9\ in pointwise norm continuous in the parameter A, it follows that it acts trivially oi
A'i(B) and we see that {тдх(г), и) = (гг, w), i.e. that
Laffx = L VAeR;.
It now remains to construct a l-trace on В fulfilling the conditions 1) 2) of theorem 20. By Chapter II
proposition 5.7 one has В = C^{G\) where the smooth groupoid G\ is associated to Gand the homomorphism
6 : G —* R; as follows:
g, = Gxr;, g(,0) = g<0' x r;
'¦G, A) = (rG), A) , SG, A) = (sG), A<5G)) ; V7 e G , A 6 R;
Gb Ai)G2, A2) = G1 072, Ai) for any composable pair.
In the foliation context, this means replacing the foliation (V,F) by a foliation of codimension 2 on the
total space of the R; principal bundle over V of transverse densities (Chapter I section 4). The dual action
(^a)a€R* , 0A € A.utC^(G\), is given by the obvious automorphisms of Gt:
The anabelian l-trace г on В = C'(Gi) is given by the formula:
r(.f°Jl)= [ /°G-')/'G)"G)
where и is the following 2-form on the 2-manifold G\:
where I = log 6 as above and A is the variable A S R^. Since и is invariant under 9л one checks 20
1). Since %-') = -e(j) one checks that r(/',/°) = -r(/0,/') V/J e Cf(Gi). The cocycle property
'Gi ° 72) = A71) + A72) implies the cocycle property for ы, so that т is a Hochschild 1-cocycle. To show
that the one trace r is anabelian one proves that any element,- of the center of the von Neumann algebra
generated by В in L2(G\) is given by a function /G), which is in LX(G\) and satisfies:
Jut is zero almost everywhere.
We refer to [С01З] for the detailed proof which can be taken as an exercise. Finally the statement 20 2) is a
special case of the following general fact (applied to r and 0a).
209

23
(В в) be а С dy»am,cal system and г а в ,nvar,ani 1-irace on B. The fo»oW,ng
ЙЛ = Bx,R, — ^r the *., «Con.-
(/"(О.WH
С В X» R-
/or Я 6 СГ (
2) ?ei W : A4(B) -> Ka(Bx,R) be the Thorn «omorphtsm,
action. Then for any у 6 Ki(B) one has:
, and D = Tt<rt be ike ,e»«.«. /
wAere ix>? и Me contraction ofrbyD (Lemma 16).
We refer to [Co,3] for the simple proof.
210
III.7. The transverse fundamental class for foliations and geometric corollaries
Let (V, F) be a foliated manifold and г = TV/F the transverse bundle of the foliation. The holonomy
groupoid G of (V, F) acts in a natural way on r, by the differential of the holonomy. Thus every у : x —* у
у (Е G determines a linear map:
Mr) : rx — Ty.
It is not in general possible to find a Euclidean metric on r which is invariant under the above action of G.
In fact we have already seen in chapter I that in the type III situation there does not even exist a measurable
volume element,
», 6 Л* rt , q = Codim F
which is invariant under the action of G. The non existence of a holonomy invariant measurable volume
element is equivalent to the outer property for some ( e R of the modular automorphism group of the von
Neumann algebra of the foliation.
Here we have to take care of the full information on /1G) (not only its determinant and not only in the
measurable category). To that effect we shall construct the analogue of Tomita's modular operator Д and
modular morphism Д Д. Let g be an arbitrary smooth Euclidean metric on the real vector bundle r.
Thus for ? 6 тх we let j| ? \\g= ((C,?)»I'2 be the corresponding norms and inner products and drop the
index g if no ambiguity can arise.
Using g we define а С module ? - ?g on the C* algebra C'(V, F) of the foliation, as follows. Recall that
C* (V, F) is the completion of the convolution algebra C™(G, Q1/2) defined in II.8. The algebra Cf(G, П1'2)
acts by right convolution on the linear space C^°('^2 ())
A)
(U){7)=
wherey=r(T).
Endowing the complexified bundle re with the inner product associated to g and antilinear in the first
variable, the following formula defines a C™(G,n''2) valued inner product:
B) I
for anyf,Tj6Cf°(G,nI/2®r*(rc)).
One then checks that the completion ? — ?s of the linear space Cf(G7Cl1^2 <8> r*(rc)) for the norm:
\\(\\=(\\{U)\\C4V,F))U2
becomes, using formulas A) and B), а С module over C*(V,F). The above construction did not involve
the action of G on r. We shall now use this action Л to define a left action of <^°(G,n'/2) on ? by the
equality:
C)
(/0G) =
AGi)
V/ 6
The analogue of the modular operator Д then appears in the following:
Proposition 1. For any f 6 Cf(G,u1/2) the formula C) defines an endomorphism \(f) of the C" module
? whose adjoint A(/)* is given by the equality
211

WTO(i)= f /*(ti)
Jg»
where /#(T) = /(Г) Д(т), «»<< ДЫ 6 End(rc(r(T)),
The proof is the same as [С01З] Lemma 3.1.
This proposition shows that, unless the metric on r is G-invariant, the representation A is not a * represen-
representation, the nuance between A(/)* and A(/*) being measured by Д. In particular A is not in general bounded
for the C* algebra norms on both Endc-(v,f )? and C*(V, F) Э Cf(G, fi1/2). However:
Lemma 2. The densely defined homomorphism A is a closable homomorphism of C* algebras.
(cf. [С01З]). In other words the closure of the graph of A is the graph of a densely defined homomorphism.
Then gifted with the graph norm || x ||л=|| x \\ + \\ A(z) || the domain В of the closure A of A is a Banach
algebra, which is dense in the С algebra A = C*(V, F). The C* module ? is then а В — A bimodule.
The information contained in this structure is easy to formulate when the bundle r is trivial over V, with
say a basis (?>)i=i q of orthonormal sections. Then ? is the С module A' over A and the closable
homomorphism A : A —* Mq(A) = Endx(?) corresponds to the coaction of G?(</,R) on A associated to the
homomorphism:
h:G-+GL(q,R).
It is only for q = 1 that such a coaction can be simply described as an action by automorphisms of the
Pontryagin dual group.
To complete the description of the transverse structure of the foliation we shall define transverse differenti-
differentiation. First note that the above construction of ? as a bimodule applies equally well to any G-equivariant
vector bundle E over V endowed with a (non equivariant) Euclidean structure. The domain В of the left
action of A on ? depends of course upon the choice of E. At the formal level the tensor calculus is still
available but one has to be careful about domain problems. Thus, if we let O» be the bimodule associated
to transverse differential forms of degree j, it is the С module completion of
and one has a densely defined product, compatible with the bimodule structure:
[и Л и')
и;G,)Л71«'GГ17)
One still has ш/Ли/ =ш Л /и/ V/ 6 C™(G, П1/2) but it is no longer true that w Ли' = ±ш' Ли since this
does not even hold for j = к = 0, the algebra Cf(G,Ql/2) being non commutative.
Transverse differentiation is not quite canonical and requires the harmless choice of an horizontal distribution
Я, i.e. of a vector bundle Я С TV such that for each x 6 V, ttx is a supplement of the integrable bundle
Fx. Of course we do not require the integrability of Я and one can for instance take Я — F1- for a fixed
Riemannian metric on V. We refrain from using the word connection to qualify Я (even though it will be
used as a connection on the bundle L2(L) over V/F) in order to avoid confusion with connections on the
bundle r over V/F.
212
Elements of the algebra C"(G, П1'2) are j densities, i.e. sections over G of the bundle r*(?2J/2) ® «'(П1'2)
where Qj is the bundle over V whose fiber at x is the space of maps:
p : hhFx — С , p(Xv) = |A|1/2p(t;) VA 6 R
(with к = dim-F).
Thus to differentiate elements of C™(G, П1'2) we first need to know how to differentiate a | density p 6
C°°( V, ?ij/2). Let us first define transverse differentiation for sections of the bundle ЛгF* ® A'r' over V, so
that,
dH ¦¦ Coo(V,ArF*®A'r-)-.Coo(V,ArF*®A' + 1r*).
Using the decomposition T = F + H we get an isomorphism jx of the exterior algebras:
г; -> лг;
v.
Letting j : C°°(V, AF' ® Лг') -=> C°°(VKT') be the corresponding isomorphism, we define, dHu for
и 6 C°°(V, ArF' ® Л'т*) as the component of type (r, s + 1) of j~4(jM), where d(j(ui)) is the differential
of j(u) 6 C°°(V, Л^Г+*'Т*) on V. Note that unlike d#", which depends upon the choice of Я, the component
of type (r + 1, s) of j~ld(j(ui)), i.e. the longitudinal differential of и is canonical, once (V, F) is given.
Next, as any | density p 6 C°°(V,Qp) can be written, at least locally, in the form p = /|ш|'/2, where
/ 6 C°°(V) and ш 6 C°°(V,(AkF)'), к = dim.F, we can define its transverse differential dgp unambigously
by the formula:
И <Ы/М1/2) = (<W)MI/2 + /И1'2 ^dHu/u).
(More explicitely if we work in local coordinates around x ? V, we can take a domain of foliation chart of
the form R* x R«. Then the i density p is of the form:
p(t,u) = r(t,u) Idu1 A- -Adu^1'2.
Given a vector field A'(() = X'(t) ^7 on the transversal R*, let then X be its unique horizontal lift:
X(t,u) = X'(t) & + &(t,u) &.
Then (dnp)!p evaluated at ((, u) on the vector X(t) 6 тт is:
Let us now consider the smooth groupoid G. The map ж = (r,s) : G -^* V x V is an immersion (with in
general a fairly complicated range: the equivalence relation: x and у belong to the same leaf). Given 7 6 G,
1 : x — у and X e ту = г"(т)-, let „Y e Я„ be the horizontal lift of X and let X' 6 Hx be the horizontal
lift of h(y)~lX € rTI i.e. the transverse vector at x corresponding to X by holonomy. Then the vector
(X,X') 6 7V(T)(V x V) belongs to t,(T7(G)) and it thus determines a unique tangent vector Хб T-,(G).
We can thus define dHf 6 CC°°(G, r"(r)) for / ? CC°°(G) by:
(P) (dHf)(X) = l
Let us now state the main properties of transverse differentiation:
213

Proposition 3. Let (V, F) be a foliated manifold, Я an horizontal distribution.
1) The following equality defines uniquely a derivation dH from Cf(G,Ql/2) to the bimodule Cf(G, П1/2® %
dH(r'(p)fs'(p)) = r'(dHp)fs'(p) + r'(p)(dHf)s'(p) + r'(p)fs'(dHp)
for p 6 C~(V,fij/2), / 6 C?(G).
2) The derivation in extends uniquely to a derivation of the graded algebra C~(G, Q1'2® г*(Лт*)) such thai
for any w 6 C°°(V, At') and f 6 Cf(G, fi1/2) one has:
In 1) and 2) the algebra structures are of course given by convolution on the groupoid G and the exterior
product in Лг*.
Due to the lack of integrability of the subbundle Я of TV, it is not true in general that d2H — 0. We shall
however overcome easily this difficulty using lemma 9 of section 3 and the fact that dj, is an inner derivation
of C™(G, П1/2 <8> г*(Лт*)). Let us first consider d?H acting on differential forms on V:
for any pair of horizontal vector fields X,Y 6 C°°(V, H), where (pF,pT) is the isomorphism T —> F<Bt given
by H. The following equality then holds for arbitrary forms:
du = cIlu + dtfuj — Bui
where the last term stands for contraction by — $. Then computing the component of type (r, s + 2) of d2ui
for ш of type (r, s) one gets:
djfu; = (dL6 + OdL)u Vw.
The operator dLt) + 8dL just involves longitudinal Lie derivatives and thus there exists a (unique) element,
0 6 CC-°°(G, П1/2<8> r'(AV))
with support contained in G*0' and whose action on C°°(V, Пр ) is the Lie derivative db& +0dz,.
Lemma 4. /or any u 6 Cf(G,fi1/2 ® г'(Лг')) one /iai d^u; = вЛы-шЛ0. Moreover duQ = 0.
Indeed one knows a priori that d2^ is a derivation and the equality is easy to check using proposition 3.
By lemma 9 of section 3 we thus get a differential graded algebra ЙС|0о whose elements are two by two
matrices with entries ыц ? Cf(G,fi1/2 ® г*(Лг*)). The product is given by:
and the differential d by:
wu 121 f 1 f» l
u2i w22j [0 0J [u;^! u;22J '
0 -e
1 о
214
We have thus.gained the equality d2 = 0 while keeping the previous properties of Cf, iu available.
Let us now define, with q = CodinvF, the integral of forms w 6 C™(G,fi1'2® Л'г*) as an immediate
adaptation of the measure theory Lemma 1.4.2.
Lemma 5. Lei (V,F) be a transversally oriented foliated manifold. The following equality defines a closed
graded trace on Cf(G,ul12 ® г*(Лг*)), dH:
t(ui) = /
= 0 if degw jt q).
Here the restriction of ш 6 C™(G,fi1/2® г*(Л«т*)) to the subspace G<0' С G yields a section of UF ® Л«г*
on V and has a canonical integral over V using the transverse orientation. Thus one has т(ы2 Ли/]) =
(_1)«1«з T(bJi Aui) and T(dHu) = 0.
Let then г be the extension of r to ПС|ОО given by:
Proposition 6. The triple (fi,rf,r) is a cycle over the algebra Cf(G,n'/2).
We shall call this cycle the fundamental cycle of the transversally oriented foliation (V, F). In particular it
yields acyclic cocycle of dimension q, <pH on C™(G,Q1/2), which depends on the auxiliary choice of the
horizontal distribution H but whose cohomology class is independent of this choice.
Definition 7, Let (V, F) be a transversally oriented foliation of codimension q. Then its transverse funda-
fundamental class is the cyclic cohomology class [V/F] e ЯС (Q°(G, ul/2)) of the above cycles.
0) Geometric corollaries
The above construction of the transverse fundamental class [V/F] of a transversally oriented foliation yields
a cyclic cohomology class on the dense subalgebra Л = Cf(G, п1/2) of the C*-algebra C"(V, F) of the
foliation and hence a corresponding map of Л'-theory:
K(A) Л С.
We have solved in [С01З] the problem of topological invariance of this map, i.e. of extension of ip to a map:
K(C'(V,F)) Л С.
Let us now state several corollaries of the solution of this problem ([Coj3]).
Theorem 8. Let (V, F) be a (not necessarily compact) foliated manifold which is transversally oriented. Let
G ~ Graph(V, F) be its holonomy groupoid and ж : BG —+ STq be the map classifying the natural Haefiiger
structure (q - CodimVj, г the bundle on BG given by the transverse bundle of (V, F). Let V. С H'(BG,C)
be the ring generated by the Pontrjagin classes ofr, the Chern classes of holonomy equivariant bundles on
V and w'(H'{WO,)).
For any Р6Ж then exists an additive map <p of K, (C*( V, F)) to С such thai:
<p(li(x)) = (Ф ch(z), P) V* &K.r(BG).
Here K.T(BG) is the geometric group as defined in chapter II proposition 8-4, ch is the Chern character:
K.,T(BG) —* H,(Bt,St) where г is the bundle on BG corresponding to the transverse bundle of (V,F),
and Ф : H.(Bt, St) — H.(BG) is the Thorn isomorphism.
215

An important step in the proof of this theorem is the longitudinal index theorem for foliations (chapter 2,
theorem 9-6).
We shall now state several corollaries.
Let (V, F) be a transversally oriented foliation of codimension q of a connected manifold V.
Let W be the O-dimensional manifold consisting of one point and / : W —» V/F the map from W to V/F
associated to a leaf and A'-oriented by the transverse orientation of (V, F). Then the triple (W, y, f) where у
is the class of the trivial bundle on W is a geometric cycle for (V, F) (cf. chapter 2), and the corresponding
A'-theory class /!(y) ? A, (C*(V, F)) is independent of the choice of the leaf. We shall denote this class by
[V/F]' and call it the orientation class in A-theory.
Corollary 9. Let (V, F) be a transversally oriented foliation of codimension q, then the class [V/F]" м о
поп torsion element of Kt(C"(V, F)).
Note that V need not be compact. When (V,F) is not transversally orientable, with V connected [V/F]'
is a 2-torsion element (see [Tor] for relevant computations). We now pass to two other corollaries which are
purely geometric, i.e. the C"*-algebra C*(V, F) disappears in the statement. Its role is to allow to integrate
in А^-theory in two steps 1) Along the leaves of the foliation (this provides under suitable К orientation
hypothesis a map from A'*(V) to K,(C"(V,F))) 2) Over the space of leaves (this provides a map from
K.(C'(V,F)) to C).
That the composition of these two steps is the same as integration (in A-theory) over V is a corollary of
theorem 8.
It follows from [Li] [Ros] that if the bundle F is a Spin bundle which can be endowed with a Euclidean
metric with strictly positive (lower bounded by ? > 0) scalar curvature (it makes sense since F is integrable)
then the longitudinal integral of the trivial bundle does vanish (i.e. the К theory index of the longitudinal
Dirac operator is equal to 0). Thus the integral over V of the trivial bundle does vanish or in other words
A(V) = 0. Note that here only F is assumed to be Spin so that A(V) is not a priori an integer. More
precisely:
Corollary 10. Lei V be a compact foliated oriented manifold. Assume that the integrable bundle F С TV
is a Spin bundle and is endowed with a metric of strictly positive (> e > 0) scalar curvature. Let 7J be
the subring of H*(V,C) generated by the Ponirjagin classes of r — TV/F, the Chern classes of holonomy
equivanant bundles and the range of the natural map: H'(WO,) ->¦ H"{V, C). Then (A(F)ui , [V]) = 0,
Vw € V,; where A(F) is the A genus of this Spin bundle.
When F = TV i.e. when the foliation has just one leaf, this is exactly the content of the well-known vanishing
theorem of A. Lichnerowicz ([Li]).
As an immediate application we see that no spin foliation of a compact manifold V, with non-zero A genus:
A(V) ф 0, does admit a metric of strictly positive scalar curvature.
Proof. The projection V —> V/F is К oriented by the Spin structure on F and hence defines a geometric
cycle x 6 A',,T(BG). The argument of [Ros] shows that the analytical index of the Dirac operator along the
leaves of (V, F) is equal to 0 in K,(C"(V, F)), so that one has /i(x) = 0 with ц the analytic assembly map
(chapter 2.8).
Let f : V —> BG be the map associated to the projection V —+ V/F. There exists a polynomial P in the
Pontrjagin classes of r, with leading coefficient 1, such that
Ф о ch(i) = f.(A(F) U [V]) U P 6 H.(BG).
Thus, since р(х) = 0, the result follows from Theorem 8.
216
Corollary Hi [Bau-C] Let (V, F), (V, F') be oriented and transversally oriented compact foliated manifolds
Lei f : V —> V be a smooth, orientation preserving, leafwise homotopy equivalence. Then for any element
P of the ring К С H"(V,C) of Corollary 10 one has:
((f'C(V')-C(V)), PU[V])=0
where C(V) (resp. (V)) is the L-class ofV (resp. V').
7) Index formula for longitudinal elliptic operators
The main difficulty in the proof of theorem 8 is to show the topological invariance of the cyclic cohomology
map tp : K(A) -» С, A = CJ°(G,Q1/2). We refer to [Co^] for the proof. Here we shall explain how to
compute the pairing: .
A)
(<J>(c),Ind?>) 6 С
of the cyclic cohomology of A with the index Ind(O) 6 Ka(A) of an arbitrary longitudinal elliptic operator
D, as denned in section II.9 a). The result is stated quite generally in [Coj] theorem p.888 but we shall
make it more specific, using the natural map:
B)
: h;(bg)-> h"(a)
constructed in section 2 7) theorem 14 and remark b).
It is worthwile to formulate the general result in terms of invariant elliptic operators on G-manifolds where
G is a smooth groupoid (cf. section 11.10 a)). We shall assume that G is etale, i.e. that the maps r,s are
etale. This suffices to cover the case of foliations since the holonomy groupoid G of a foliation is equivalent
to the reduced groupoid Gt, T a suitable transversal, and the latter is etale. We let Ф be the morphism of
bicomplexes constructed in section 2 7) from the bicomplex of twisted simplicial forms on the nerve of G to
the (b, B) bicomplex of the algebra A = Cf(G).
When formulated in terms of the (b,B) bicomplex the pairing between cyclic cohomology and A* theory of
proposition 3.2 is given by the following formula:
C)
where (^2n) is a (b,B) cocycle, i.e. btf2n
Equivalently if (^2n) is a cocycle in the (
n+2 = 0 Vn, and e is an idempotent.
bicomplex of theorem 1.29 the pairing reads:
D)
-±,е е).
(We shall come back to this point in great detail in chapter IV section IV.7, the replacement of e by e — ^,
due to E. Getzler, improves the original formula.)
Let now P be a proper G-manifold (definition 1 of section 11.10). Thus to P corresponds a contravariant
functor from the small category G to the category of manifolds and difFeomorphisms. To each x 6 V = G*0' it
assigns the fiber «~'({z}) of the submersion a : P —> G*°>. The right action p —> pay gives a diffeomorphism
from Py to Px for 7 € G, ^G) = y, 5G) = x. To keep things simple we shall assume that the discrete groups
G% = {7 € G ; r(y) = s(y)} are torsion free for any x 6 G<0'. Then the properness of P implies that G%
acts freely on Px.
217

Let О be a G-invariant elliptic differential operator on the G-manifold P, i.e. a family Dx of elliptic
differential operators on Px which is smooth on the total space P and is invariant under the action of G.
When the quotient manifold P/G of P by the action of G, is compact, the constructions of section II.9 a)
or of section 4 a) of this chapter, yield a well defined К theory class:
E) vlnd(O) 6 Ka{C?(G) ® K).
Finally the К theory class of the principal symbol со of D yields as in section 11.10 a) a well defined
element, [crp] of the geometric group K^piG). Since G is etale and torsion free the latter group is equal to
the г-twisted /f-homology group A'.r(SG). We can now state:
Theorem 12. Let G be a smooth etale groupoid without torsion. Let D be a G-invariant elliptic differential
operator on a proper G-manifold P with P/G compact. Let с be a cocycle of total degree 2q in the hicomplex of
simplicial currents on the nerve of G and Ф(с) the associated cyclic cohomology class in the (b, B) bicomplex
H C~(G)®7J ГЛп
We recall (section 2 7)) that in the bicomplex (C* ,di,di) of simplicial currents one had C'm = 0 unless
n > 0, — dimG<0' < m < 0. In particular the total degree: n + m can vary from — dimG(°' to +00.
Note that с 6 H*(BG) pairs with ch,([cj)\) € H,T(BG). Let us explain briefly why theorem 12 reduces to
4.7 in the case of discrete groups: G = Г. Let с be a group cocycle, с 6 Z2'(f, C). Then by section 2 7) one
gets:
F) Ф(с) = ге.
But the cyclic cocycle rc of 4.7 is here viewed as a cocycle in the (b, B) bicomplex. The point then is that
the formula C) for the pairing of F, B) cocycles with A' theory accounts for the strange numerical factor
q\/Bq)\ of theorem 4.7 (cf. [C-M] for a discussion of this numerical factor).
Let us now specialize theorem 12 to the case of foliations. We let (V,F) be a foliated compact manifold.
In section a) we have carefully described the cyclic cohomology class on the algebra .4 = Cf(G,П1'2)
corresponding under the natural Morita equivalence to the fundamental class of reduced groupoids, Gt-, T
a transversal. The same procedure applies to map the periodic cylic cohomology H"(Cf{Gr)) to H'(A).
Combined with the construction of the morphism Ф of section 2 a), we thus obtain a canonical map:
G)
() Ф. : irT{BG)^H-(A),
where we have used the equivalence of smooth groupoids G ~ Gt to replace BGt by BG.
Applying theorem 12 to the Gt manifold P = {7 e G ; «G) € T} one obtains:
Corollary 13. ([Соь]) Let (V, F) be a compact foliated manifold, D a longitudinal elliptic operator, an
IndO e K0(A), A = Cf (G,Q1/2) tls analytical index (cf. 11.9 a)). Then for any cohomology clas
6 #2«+dimT(BG) one has:
IndO e K0(A), A = Cf (
и 6 #2«+dimT(BG), one has:
(Ф.(ш),1пс\О) =
(w,ch,(aD)).
To end this section we shall explain, in this context of foliations, how to construct a left inverse X of the map
Ф. ([Co])
218
(8) X: H'(A)-* H"T(BG).
In fact we shall only describe the composition of this map with the pull back: H*(BG) —*¦ H*(V). When
one varies V without varying V/F one gets the desired information. The idea of the construction of \y.
(9) Xv : H'(A) - H'T(V)
is to exploit the focal triviality of the foliation as follows: For each open set U С V one lets A{U) be the
algebra associated (by C™(-, Q1'2)) to the restriction of the foliation to U. If U\ С Ui one has an obvious
inclusion:
A0)
Thus one obtains for each n a presheaf Г„ on V by setting:
A1) Y"(U)=C"(A(V)~,A(V)~),
the space of n + 1 continuous linear forms on A{?0"", ^he agebra obtained by adjunction of a unit to
The construction of the (b, B) bicomplex being functorial it thus yields a presheaf of bicomplexes: (T^n~m\btB).
Choosing a covering U — (Ua) of V sufficiently fine so that the multiple intersections are domains of foliation
charts, we get a triple complex;
A2)
; b,B,6
where 6 is the Cech coboundary.
Moreover there is an obvious forgetful map ф from the periodic cyclic cohomology H*(A), A = A(V) to the
cohomology of the triple complex A2). So far we have just localized the periodic cyclic cohomology of A in
a straightforward manner. The interesting point is that the cohomology of the triple complex A2) is easy
to compute and is directly related to the resolution of the orientation sheaf of the transverse bundle т by
transverse currents. More precisely one defines another triple complex (Г', d\, di, d%) as follows: For each
к € 0,1,..., Codim F, let ?lt be the sheaf of holonomy invariant transverse currents. Thus for any open
set U, Qt(U) is the kernel of the longitudinal differential in the space C~°°(Cf, ACodimr-t r*) of generalized
sections of the exterior power of the dual of the transverse bundle. If V is a domain of foliation chart we are
just dealing with currents on the space of plaques U/F. We let:
A3) Г'"'™'" = ^ф^ JIn_m (Uo П ... П Up)
and the coboundaries are di = 0, di = de Rham coboundary, d$ = Cech coboundary.
The construction of section or), involving the additional choice of the subbundle H С TV transverse to F,
applies with minor modifications to yield a morphism of triple complexes:
A4) {T',dudi,d3) -1 (Г,Ь,В,6)
which defines an isomorphism in cohomology independent of the choice of H. One thus obtains the desired
construction of Xv as:
A5) Av=0,-lo</>,
since the cohomology of (V,di,di,d3) is almost by construction equal to H"(V). Indeed the sheaves Qt
provide a resolution of the orientation sheaf of r. One can in particular reformulate the index formula
(corollary 13) in terms of A v ([Соь]). One can also compare the natural filtration of H" (A) by the dimensions
of cyclic cocycles with the filtration of H'(V) given by the above resolution of the orientation sheaf of r.
219

Appendix A. The cyclic category Л
The definition (section 1) of the cyclic cohomology functors HC" from the category of algebras to vector
spaces is simple and direct. It is however important from a conceptual point of view to obtain these functors
as derived functors from the simplest of them, HC°, which to an algebra A assigns the linear space of traces
on .4, i.e. of linear forms г such that:
т(ху) = т(ух) Vz,y&A.
This is done in this appendix (cf. [C022] [Lo]) using a natural functor from the (non abelian) category of
algebras to the abelian category of Л-modules described below. In some sense this appendix analyzes the
clockwork underlying the algebraic manipulations of section 1.
a) The simplicial category Д
We first review some standard notions ([Lo]).
Let A be the small category whose objects are the totally ordered finite sets:
[n] = {0 < 1< 2... < n} , n 6N,
and whose morphisms are the increasing maps (where increasing means non decreasing).
Definition 1. Let С be a category. A stmplicial object in С is a contravariant functor from Д to C.
Such a functor X is uniquely specified by the objects Xn corresponding to [n] and by the morphisms
di '• Xn —»¦ Xn~i, 0 < 1i < n; Sj : Xn —+ Xn+i, 0 < j ' < n which correspond to the following elements 6,-, try of
Д. For each n and i ? {0,1,..., n], <5" : [n — 1] —+ [n] is the injection which misses 2 while a? : [n 4- 1] —*¦ [n]
is the surjection such that Oj(j) = Cj(j + 1) = j.
The conditions fulfilled by the morphisms d,-, Sj of С in a simplicial object are obtained by transposing the
following standard presentation of Д.
Proposition 2. The morphisms <x™, <5J generate Д which admits the following presentation:
6jSi = M;-l for i < j
CTjCTj = <Ti(Г, +1 for i < j
Vjbi = <5iCTj-i if i< j > In if i = j or i = j + 1,
i5i_l<Jj for i > j + 1.
Let us now describe the geometric realization |X| of a simplicial set X. One lets Д be the following functor
from the small category Д to the category of topological spaces. For each 1», Д(п) is the standard n-simplex:
{(U) 6 R"+1 ; 0 < (,• < 1 , E U = 1}. To ^ corresponds the face map:
<5;((<o, -.., «n_i» = (*o, -.., <i-i, 0,«,-,..., «„-0
and to <Tj the degeneracy map:
<Tj((t0, . . . ,<„ +
= (U, ¦ ..,tj-uii+
220
The geometric realization |A'| of a simplicial set X is then the quotient of the topological space X хд Д
U Xn x Д" by the equivalence relation which identifies (z,a,(y)) with (a'(at), y) for any morphism a
Д. This continues to make sense when X is a simplicial topological space (cf. [Lo]).
With these notations we can give a formal definition of the classifying space ВС of a small category С. Е
definition the nerve of С is a simplicial set Mr(C). The elements of Мг„(С) are the composable n-upli
of morphisms belonging to C, while the faces di (resp. degeneracies Sj) are obtained using composition <
adjacent morphisms (resp. the identity morphism). More precisely, with Л,...,/„ composable morphisn
one lets:
,•..,Л.) = (/2,..-,Л.) ; 4(Л,...,/„) = (Л,....Л/.+1,¦..,/„) for 1 < i < n- 1
and
Definition 3. The classifying space ВС of a small category С is the geometric realization of the simplicia
set Mr(C).
This applies for instance to discrete groups Г viewed as small categories with a single object. Taking саге о
topological categories (cf. [Lo]) it also applies to smooth groupoids as described in chapter II.
/?) The cyclic category Л
For n 6 N we let Z/n + I be the cyclic group, quotient of Z by the subgroup (n + 1)Z. We identify Z/n + 1
with the group of (n + l)-th roots of unity by the homomorphism:
е„ : Z/n + 1 - S1 = {X e С , |A| = 1},
en(k) - expB7rii/n + 1).
We endow S1 with its usual trigonometric orientation and let for X,/t 6 S1, [A,/i] be the closed interval from
A to f.t. We say that a map ф : Z/n + 1 —*¦ Z/m + 1 is increasing iff for any A, /i 6 Z/n + 1 the image by ф of
the interval
[A,/i]nZ/n + l = {A,A + l.A + 2, ...,it} С Z/n + 1
is contained in the interval {ф(\),ф{/л)] nZ/m + 1.
The cyclic category Л is the small category with one object An for each n 6 N and with morphisms
/ € Hom(An, Am) the homotopy classes of continuous increasing maps from Sl to S1, of degree 1 and such
that:
<рA/п + 1) С Z/m + 1.
For A 6 Z/n + 1 the value of ^(A) e Z/m + 1 is independent of the choice of tp in the homotopy^lass / and
gives a well defined increasing map / : Z/n + 1 —> Z/m + 1. One thus obtains a functor / —» / from Л to
the category of sets.
Proposition 4. a) Let n, m € N and ф : Z/n + 1 —> Z/m + 1 be a non constant increasing map. The» there
exists a unique f ? Нот(Л„,Лто) such that f = ф.
221

b) Let фь e Hom(An,A,n) be the constant mtip with range {k}. Their ait n+ 1 elements f 6 Нот(Лп,Лт)
such that f = i>k-
Proof. Composing ф with a rotation we may assume (in additive notation) that ф@) = 0- Then for any
Jl < h ? {0,1 «} one lias ф(Ц) € @,^'(J2)] С {0,1 га} so that ф is a non decreasing map of
-[0 < 1 <-¦¦<"} to {0 < 1 < ... < m]. When ф is not equal to 0 it determines uniquely the homotopy
class of y, <p = Ф- When ф is constant, equal to 0, one has the choice of the interval [en(j),en(j + 1)] mapped
onto S1 by ip, кр = ф.
Fig.3. / 6 Пот(Л5, Л3), f(j) = 1 V/ ? Z/6Z, / is constant on [4,3]
and equal to 1; it makes a complete turn on [3, 4]
We see in particular that Aq is not. a final object in the small category Л and Пот(ЛП)Ло) has cardinality
n + 1. Any element / € Иот(Л,г,Лт) is characterized by the (possibly empty) following intervals of S1:
f-\j)= n 9-1 {j} . jeZ/m + l.
This follows from proposition 4 a) if / is not. constant ami from Л b) if / is constant.
A collection (^j)jez/m+i °f (possibly empty) intervals comes from an / ? Нот(Л„, Лт) iff they satisfy:
1) If /, ф 0 then /j = [A, fi] for some A,,/. e Z/n +1.
2) The Ij П Z/n + 1 form a partition of Z/» + 1.
3) If/,- = [A,/i] ^0, /j + i = ... = /г_! = 0, /t = [Y,//] ^ Й then V =/j + 1 in Z/ti + 1.
This follows directly from proposition 4. One has:
Ij П Z/n + 1 = Г l {Л Vj e Z/»i + 1,
and if/ is not constant, /~' {j} determines J~l(j). We shall identify the category Д of section a) with
a subcategory of A as follows. To each It ? Нот^([л], ['»]) we associate, the corresponding increasing
map: j —> /i(j') from Z/n + 1 = {0,1,...,»} to Z/ш + 1 = {(). ] m). This specifies uniquely an
element A, 6 Нот(А„,Лт) such that A» = /i provided that. Л is not constant. To handle the constant map
h(i) = к Vi we just need to specify the interval: /1.71 (k) = [0, n].
¦2-22
Proposition'5. [Co22] a) The functor * : Д —> Л identifies A with a subcategory of A.
b) Any f 6 Нот(Лп,Лт) can be uniquely written as the product A.fc where A 6 Д and к is an automorphism.
In other words Л = AC where С is the subcategory of Л with the same objects and with morphisms
the invertible morphisms of Л. By proposition 4, С is a groupoid which is the union of the cyclic groups
Aut(An)~Z/n + l.
Proof, a) For each n let 0n 6 ]n/n + 1,1[ and an — expB)ri0n). By imposing the condition:
to the continuous increasing degree 1 maps involved in the definition of Нот(Л„,Лт), we obtain a subca-
subcategory of Л. One easily verifies that this subcategory is the image of A by * and that the functor / —> /
composed with * is the identity of Д.
b) Let / 6 Нот(Л„,Лт) and tp be a representative of/. There exists a unique interval / = [еп(;'),е„(У+ 1)]
of S1 such that am € <p(I)- Moreover / does not depend on the choice of tp 6 /. Thus there exists a unique
i 6 Aut(An) such that the interval associated to / о i is [е„(п), 1]. By a) one then has / о к~' 6 Д. This
proves the existence and uniqueness of the decomposition.
For each n let rn g Aut(An) be the generator of the cyclic group such that ?n(j) = j — 1 V/. By a slight
abuse of notation we let <5/, <4 (instead of <5j*, cr;.) be the canonical generators of Д (cf. proposition 2) viewed
as elements of A, By propositions 2 and 5 the <5", ст" and г„ generate A.
Proposition 6. ([Lo]) The relations of proposition 2 together with the following ones give a presentation of
A.
г„ Si = <5j_i г„_! for 1 < i < n , тп 6a = <5n
C7{ = (Ti_i Г„+1 fOT 1 < i < П , Г„ СТО — "n
= In-
We refer to [Lo] for the proof. In [Lo] J. Loday takes the above presentation as the definition of A. This
lead him to many interesting generalizations of the cyclic category involving dihedral, symmetric and braid
groups ([Fied-L]). Transposing the relations of proposition 6 one obtains a description of cyclic objects in a
category in terms of the morphisms d"n,s4,(n corresponding to the above generators of A.
Definition 7. Let С be a category. A cyclic object in С is a contravariant functor X from A to С
This definition differs only superficially from the one given in [C022] which used covariant functors. We shall
indeed describe a canonical contravariant functor / —» /* from A to A which gives an isomorphism Лор ~ A.
The composition of any covariant functor X with • is then a cyclic object. Given / 6 Hom(An, Am ) we define
/* 6 Нот(Л^, Л„) by the intervals Jt = (/*)-'(*)> * J Z/n + 1 constructed as follows. If /(* - 1) ф f(k),
we let Л = [f (к - 1) + 1, /DI (i.e. the interval [em(/D - 1) + 1), em(f(k))} in S1). If /(* - 1) = /(*) we
223

let Jt = 0 if the representatives y> of / are constant in the interval [4 - l,i] and Jh = [/(fc - 1) + l,/(fc)]
otherwise (cf. fig.4).
Fig.4. This pictures / g Hom(Au,A-), / is constant equal to j on the interval Ij.
The adjoint /* 6 Hom(Ar,An) is constant equal to ( on the interval Jt
Proposition 8. a) The intervals (Jthei/n+i satisfy conditions 1) 2) 3) and uniquely define the element
f" e Нот(Л„,,Л„) such that /—'(*) = Jh-
b) The map /->/" is аи isomorphism of the opposite category Ac|> with the category A.
The proof of a) is straightforward. To understand b) without, any tedious check one compares, for an arbitrary
monoid Г, the following covariant and contravariant. functors from A to the category of sets:
Covariant
wit.li
= П «<
When /~'(j) = в we let bj = 1 6 Г, otherwise f~l(j) 's a wc" specified interval: / '(j) П Z/m + 1 -
{г, г + 1,...,»} and the product агаг+1 ...a, is well defined. (Note that when /"'(j) contains Z/m + 1 we
need more than f~'(j) П Z/m + 1 to specify the endpoints.)
Contravariant
Y,, = Г"+1 Vn € N , У(Я : Г'"+1 — Г",
У(Я(«о,.-.,а„,) = (*о Ь„) with bj = П "<•
/i
224
Again there is a small ambiguity in the notation and we mean bj = Y[ at, where the Jj are the intervals
used in proposition 8.
The covariance and contravariance of the above functors shows that / —» /* is a contravariant functor and
that Y(f) — X(f') V/. That • is bijective is obvious since we described two equivalent ways of labeling
all collections of intervals satisfying conditions 1) 2) 3) above.
7) The Л-module A*1 associated to an algebra A
Let Л be a unital algebra over a field fc. Let us define a covariant functor from A to fc-vector spaces as
follows. For each n 6 N we let:
А^ = _4*n+1 = .4 <8> .4 (g>... <8> .4 (n + 1 terms).
For each / 6 Hom(An, Am) the action Д of / is given by:
where у*' = fj xl. (We use the same conventions as in /3).)
J-'O)
00
This covariant functor Л' defines a fc(A) module whose underlying vector space is ?2 A®^"+1\ and which
n=0
we still note by A** ¦
By composition with the contravariant functor * : A —+ A of proposition 8, one obtains a cyclic vector space
C(A), whose faces, degeneracies, and permutations are given by:
rf(io ® xi ® ... 81 in) = (x0 ® ... ® ijii + i ® ... ® «„) 0<i<n-l
xn)
Depending on the context it is more convenient to use the fc(A) module A*1 or the cyclic vector space C(A).
We let fc11 be the trivial i(A) module.
Proposition 9. 1) Let т be a trace on A then the equality
т'(ха ® ...®x")=r{x°x1 ...x")
defines an element т' о/Нотцл)(Л^,^).
2) The map r —* r' is an isomorphism of the linear space of traces on A with Homjfc(A)(-4 , У*).
The proof is straightforward.
The above discussion of i-algebras works with minor changes for rings and we let A*1 be the Z(A) module
associated to a ring .4. The categories of i(A) modules (or Z(A) modules) are abelian categories and we use
the standard notation Ext" for the derived functors of the functor Horn (cf. [Car-E]). By proposition 9 the
functor Algebra A — HC°(A) = {traces on A] is the composition of Hom4(A)(-,l:1) = Ext°(A)(-,fc11) with I).
The higher cyclic cohomology groups HC11 are just the derived functors Extn as follows:
225

Theorem 10. ([Co22]) For any k-algebra Л the cyclic cohomology groups HCn(A) are canonically isomor-
phic to
This theorem yields in particular obvious definitions of the cyclic cohomology functors for rings: ^^
and of the bivariant theory for algebras, as Ext^A)^^^1')- A simple non trivial example of element of
Endfc(A)(.4'l,.4'1) is provided, for A = СГ a group ring, by the projection on a given conjugacy class (cf.
[Lo]).
For the proof of theorem 10 (cf. [Сог2]) one constructs a projective resolution of the trivial Z(A) module 2*.
One gets in particular:
Theorem 11. [Co22] a) The ring ExtA(Zll,Z'1) is the polynomial ring Z[a] where the generator a is of degree
2.
b) The classifying space В A of the small category A is BSl = Poo(C).
The Yoneda product with the generator a 6 Ext^Z'.Z") defines the S operation of cyclic cohomology
(section 1) in full generality and justifies the normalization of S in section 1.
The equality В A — BS1 conforts the analogy between cyclic cohomology and S1 equivariant cohomology.
7) Cyclic spaces and S1 spaces
The inclusion of Л as a subcategory of Л (proposition 5) shows that any cyclic object in a category С is in
particular asimplicial object in C. Let us apply this to cyclic sets X (or more generally to cyclic topological
spaces (cyclic spaces for short)) and denote by \X\ the geometric realization of the underlying simplicial set
(resp. space). One has
Proposition 12. [Bur-F] [Good] [Lo] Let X be a cyclic space. Then \X\, the geometric realization of the
underlying simplicial space, admits a canonical action of S1. One obtains in this way a functor from the
category of cyclic spaces to the category of Sl spaces.
We refer to [Lo] for a detailed proof. We shall briefly describe the S1 action. First the forgetful functor,
which to a cyclic space assigns the underlying simplicial space, has a left adjoint: the induction functor F
from simplicial spaces to cyclic spaces. For any simplicial space У (viewed as a right Д-space) the cyclic
space .F(Y) (viewed as a right Л-space) is
F(Y) = Y хд Л
where Л is viewed as a left Д-space and a right Л-space. Using the equality Л = AC of proposition 5 one
describes more concretely the cyclic space F(Y) as follows:
Fn(Y) = У„ х Си Vn e N,
the right action of / € Нот(Л„,Лт) being given by:
(у.)/ (г)
where h € A, c' 6 С„ and he' = cf (using proposition 5). The geometric realization |F(Y)| of the underlying
simplicial space is homeomorphic to (У| x S1 using the following canonical maps p : |.Р(У)| —* )У) and
q : \F(Y)\ -* Sl. Recall (cf. a)) that \F(Y)\ is a quotient of U Fn(Y) x Д" and use the obvious action of
С„ on Д" to map -Р„(У) х Д" = У„ х С„ х Д" to У.хД". One then checks (cf. [Lo]) that this map p is
compatible with the equivalence relations yielding p : |-Р(У)| —» |У|. Next q is just obtained by functoriality
of .F using the constant map from У to the trivial simplicial set: {pt} and the equality |F(pt)| = S1.
226
All of this holds for any simplicial space У and we shall simply write
~ \Y\ x S1.
When X is a cyclic space then its cyclic structure yields a morphism p of cyclic spaces from FIX) to X
given by the right action of Л on X:
Р:ХхлА-*Х , (yj)^yf.
With the above identifications this yields the desired action of Sl on \X\:
\p\ : \F(X)\ = \X\ x S1 - \X\.
We refer to [Lo] for a clear and detailed proof.
Let X be a simplicial set and Z(X) the corresponding simplicial abelian group. It is then straightforward
that, letting Z = 2(pt) be the trivial simplicial abelian group, one has ExtJ^j(Z(X),Z) ~ Hn(\X\,Z) (resp.
Тог*(л)A,1(Х)) ~ Н„(\Х\,1)) where \X\ is the geometric realization of X (cf. for instance [Lo] theorem
6.2.2). Let us now assume that X is a cyclic set and let us endow \X\ with the S'-action given by proposition
12. The following theorem is a natural generalization of theorem 11:
Theorem 13. ([Bur-F]) One has canonical isomorphisms:
where Z(X) is the cyclic abelian group associated 1o the cyclic space X.
A similar statement holds for an arbitrary ring of coefficients and the Gysin sequence relating the equivariant
S'-cohomology H"sl corresponds to the Gysin sequence relating the Ext functors for Z(A) and Z(A) modules
whose theorem 1.26 is a special case ([Соэ2]).
Problem 14. While the resolution used in the proof of theorem 10 gives a concrete way of formulating
Ext5(A)(-,Z) for arbitrary cyclic abelian groups, one is missing a similar description for the bivariant theory
and in particular for ExtJ(A)(-^ll>;^'') where .4 and В are i-algebras, fc a field.
227

Appendix B. Locally convex algebras
We shall briefly indicate how section 1 adapts to a topological situation. Thus we shall assume now that
the algebra .4 is endowed with a locally convex topology, for which the product .4 X .4 —> .4 is continuous.
In other words, for any continuous seminorm p on .4 there exists a continuous seminorm p' such that
p(ab) < p'(a) p/(b), Va,6 6 A. Then we replace the algebraic dual .4* of A by the topological dual, and
the space C™ (A, A* ) of (n + l)-linear functional on A by the space of continuous (n + 1 )-linear functional:
tp e C" if and only if for some continuous seminorm p on A one has
Иао,...,а")|<р(а°)...р(а"), W e A.
Since the product is continuous one has b<p (E C"+1, V^ SE C™. Since the formulae for the cup product of
cochains only involve the product in A they still make sense for continuous multilinear functions and all the
results of section 1 apply with no change.
There is however an important point which we wish to discuss: the use of resolutions in the computation of
the Hochschild cohomology. Note first that we may as well assume that .4 is complete, since C™ is unaffected
if one replaces A by its completion, which is still a locally convex topological algebra.
Let В be a complete locally convex topological algebra. By a topological module over В we mean a locally
convex vector space M, which is a B-module, and is such that the map F, ?) —> b? is continuous (from
В x M to M). We say that M is topologically projective if it is direct summand of a topological module
of the form M' = В<ЬтгЕ, where E is a complete locally convex vector space and ®T means the projective
tensor product ([Grot!]).
In particular M is complete, as a closed subspace of the complete locally convex vector space M'.
It is clear then that if M\ and Mi are topological B-modules which are complete (as locally convex vector
spaces) and p : Mi —* Mi is a continuous B-linear map with a continuous C-linear cross-section s, one can
complete the triangle of continuous B-linear maps
Mi
1 / V\
M i Mi
for any continuous B-liuear map / : M —> Mi-
Definition 1. Lei M be a topological B-module. By a (iopological) projective resolution of M we mean an
exaci sequence of projective B-modules and B-linear continuous maps
M ?-M0 ^-Mi Ь-Mi <
which admits a C-linear continuous homotopy s* : Mi —* Mi+i
bi+i Sj +Si-i bi = id , Vi.
As in [Car-E] the module A over В = A®, A" (tensor product of the algebra A by the opposite algebra .4°)
given by
(a® ba) c-acb , a,i,c6 Л
admits the following canonical projective resolution:
1) М„ = B®*En (as a B-module), with En = A®, ¦ ¦ -ёцА (п factors);
2) ? : Ma —> .4 is given by t(a ® i°) = ai, a, b e .4;
228
+(-1)" A®
3) bn(l®ai® ¦¦•®а„) = (ai® l)<g>(e»2<8> ¦¦•®а„ + ?
а°)®(а!® ¦¦¦®on_i).
The usual section is obviously continuous:
sn((a ® *°) ® (oi ® • ¦ ¦ ® а„)) = A ® 6°) ® (о ® ai ® • • • ® а„)).
Comparing this resolution with an arbitrary topological projective resolution of the module .4 over В yields:
Lemma 2. For every topological projective resolution (M", in) of the module A over Б = А®-,А°, the
Hochschild cohomology H"(A,A') coincides with the cohomology of the complex
HomB(M°,A') $¦ RomB(M\Am) — ¦ • •
(where Horns means continuous В linear maps).
Of course, this lemma extends to any complete topological bimodule over A.
We refer to [Helem] for more refined tools in this topological context.
229

Appendix C. Stability under holomorphic functional calculus
Let Л be a Banach algebra over С and A a subalgebra of А, Л, and A be obtained by adjoining a unit.
Definition 1. A is stable under holomorphic functional calculus iff for any a € A and any function f
holomorphic in a neighborhood of the spectrum of a in A one has /(a) € A.
The first important result is that if A is dense in A the above property is inherited by the subalgebra Mn(A)
of МП(Л) for any n.
Proposition 2. If the dense subalgebra A of the Banach algebra A is stable under holomorphic functional
calculus then so is Mn(A) in Mn(A) for any n S N.
We refer to [Sch] for a detailed proof. One can use (cf. [Bosi]) the following identity in Мг(Л) as a substitute
for the determinant of matrices:
1 -ai2(o22 - a2iaf11al2)'
0 1
in a suitable neighborhood of the identity in Mi(A). Let А С A be stable under holomorphic
functional calculus.
In particular one has GLn(A) — GLn{A) П М„(А), hence if we endow GLn{A) with the induced topology
we get a topological group which is locally contractible as a topological space. We recall the density theorem
(cf. [At3], [Kar2]).
Proposition 3. Lei A be a dense subalgebra of A, stable under holomorphic functional calculus.
a) The inclusion i : A —> A is an isomorphism of Ko-groups
i. : K0(A) - K0(A).
b) Let GLoo(A) be the inductive limit of the topological groups GLn(A). Then i» yields an isomorphism,
230
CHAPTER IV
QUANTIZED CALCULUS
The basic idea of this chapter, and of 11011 commutative differential geometry ([C016]), is to quantize the
differential calculus using the following operator theoretic notion for the differential:
Here / is an element of an involutive algebra A of operators in a Hilbert space Л, while F is a selfadjoint
operator of square one (-F2 = 1) in 7i. At first one should think of / as a function on a manifold, i.e. of A
as an algebra of functions, but one virtue of our construction is that it will apply in the non commutative
case as well.
Since the word quantization is often, overused we feel the need to justify its use in our context.
First, in the case of manifolds the above formula replaces the differential df by an operator theoretic ex-
expression involving a commutator, which is similar to the replacement of the Poisson brackets of classical
mechanics by commutators (cf. Chapter 1 section 1).
Second, the integrality aspect of quantization (such as the integrality of the energy levels of the harmonic
oscillators ([Dir] and Chapter V section 10)) will have as a counterpart the integrality of the index of a
Fredholm operator which will play a crucial role in our context (sections 5, 6 and proposition 2 below).
We shall also see (appendix D) how the deformation aspect of quantization fits with our context.
Before we begin to develop a calculus based on the above formula, we need to specify the required properties
of the triple (A,H, F), or eqnivalently of the representation of A in the pair (H, F). The following notion of
Fredholm representation or equivalently of Fredliolm module is due to Atiyaii [At2] Miscenko [Misi], Brown
Douglas Fillmore [BDF] and Kasparov [Kas-].
Definition 1. Let A be an involnlive algebra (overC). Then a Fredholm module over A is given by:
1) An involutive (possibly degenerate) representation тг of A in a Hilbert space 7i.
2) An operator F = F', F2 = 1, in H such that:
[F, тт(а)] Is a compact operator for any а € A.
Such a Fredholm module will be called odd. An even Fredholm module is given by an odd Fredholm module
(H, F) as above together with a 2/2 grading 7, 7 = 7*, -f2 = 1 of the Hilbert space H such that:
231

This operator P is (up to the factor i) equal to the Paneitz' operator P = Д2 + rf*{2Ricci - 4/3r}d ([)
already known to be the analogue of the scalar Laplacian in 4-dimensional conformal geometry.
Equation 7 uses the volume element dv so that P itself is not conformally invariant, its principal symbol is:
(8) G4 (P) (
which is positive.
The conformal anomaly of the functional integral
/
П dX(x)
is that of (det-P)'2 and can be computed (cf. [B-O]). The above discussion gives a very clear indication
that the induced gravity theory from the above scalar field theory in dimension 4 should be of great interest,
(_J_. ¦ - -* — - - * - — -
in analogy with the 2-dimensional case.
276
IV.5. Fredholm modules and rank one discrete groups
As a rule the construction of Fredholm modules over поп commutative spaces is a generalization of the
theory of elliptic operators on a manifold. Let Г be a discrete group and Л = СГ the group ring of Г. Then
the corresponding non commutative space is the "dual" Г of Г and elliptic operators on
"multiplication" operators when described in the Fourier space ? (Г).
examples of Fredholm modules over СГ, or eqmvalently of Fredholm
Г correspond to
iace ?2(Г). In this section we shall thus construct
representations of Г.
over the reduced C'-algebra of the free group which already appears in [Pi-Уз] and in the work of J. Cuntz
[Cui], and whose geometric meaning in terms of trees was clarified by P. Julg and A. Valette in [Ju-V].
Fig.5. Tree
Now let Г be an arbitrary free group, and T a tree on which Г acts freely and transitively. By definition T
is a 1-dimensional simplicial complex which is connected and simply connected. For j =¦ 0, 1 let TJ be the
set of j-simplices in T. Let p ? T° and <p : -TA\{/>} — Tl be the Injection which associates to any q € T°,
9 7^ Pi the only 1-siinplcx containing q and belonging to the interval \j>><i] (Fig.5). One readily checks that
the bijection (p is almost equtvariant in the following sense: for all у G Г one has <p{gq) — Q<pi,4) except
for finitely many q's. Next, let H+ = f2(T°), W" - ^-(T1) ф С. The action of Г on T° and T1 yields a
C7(r)-module structure on ?'2{Tj), j = 0,1, and hence on W* if we put
Let P be the unitary operator P : IF+ — // given hy
277

Pep = @,1) , Ре, = ?„(,, V? ф р
(where for any set X, (ex)x^x is the natural basis of l?(X)). The almost equivariance of ip shows that:
[о р-^Л
, is an even Fredkolm module over
Л = С;(Г), and A= {a,[F,a]€Cl(H)} is a dense subalgebra of A.
b) The 0-dimensional character ofGi, F) is the canonical trace г on С"(Г), т(^адд) = <ц.
Proof, a) For any g G Г the operator gP — Pg is of finite rank, hence the group ring СГ is contained in
A = {a€ С,*(Г), [F, a] € Cl(H)}. As СГ is dense in С;(Г) the conclusion follows.
b) Let us compute the character of (H, F). Let a € A. Then a - P-'aP € Cl(H+), and
^ТгасеG^[^,о]) = Trace(o - P~laP).
Let r be the unique positive trace on A such that f E3 a?ff) — ai i f°r any element a = ^ as5 of СГ, where
1 G Г is the unit. Then for any «6,4 = С"(Г), a — r(a)l belongs to the norm closure of the linear span of
the elements g e Г, д ф 1.
Since the action of Г on T1 is free, it follows that the diagonal entries in the matrix of a — r(o)l in P(T')
are all equal to 0. This shows that for any a € A one has,
Trace(o-p-1oP) = r(o) Trace(l -P~1\P) = T(a).
Thus the character of {H, F) is the restriction of г to A, and since г is faithful and positive on C"(T) we
get:
Corollary 2. ([Pi-Уз]) Let Г be a free group. Then the reduced C*-algebra С*(Г) contains no поп trivial
idempotent.
Proof. By Corollary 1.9 one has г(Л*о(Л)) С Z, which shows that if e is an idempotent, which can be
assumed selfadjoint, then r(e) S 2 П [0, 1] = {0,1}. Thus e 6 {0,1} since г is faithful.
Remark 3. The proof of lemma 1 shows that for any a G СГ the quantum differential da is a finite rank
operator. Let then (СГ)~ be the smallest subalgebra В of С'(Г) containing СГ and having the following
property for any n e N:
xe м„(В)пмп(с;(Т))-1 »i? м„(В)-1.
One easily checks that for any a € (СГ)~, the operator da is of finite rank. It is natural to conjecture that
the converse holds. This can be proved when the free group Г is equal to Z. In this case the algebra A is the
algebra of rational fractions with no pole on S1 С Л (С) and the conjecture follows from a classical result of
Kronecker (proposition 1 of section 3).
We shall now pass to Fredholm representations due to Miscenko [Misi] for discrete subgroups Г of semisimple
Lie groups G.
Let thus G be a semisimple Lie group, H С G a maximal compact subgroup, and let X = G/H be the
corresponding homogeneous space over G endowed with a G invariant Riemannian metric of non positive
curvature. Let q — dim G/H and p : H —• Spinc(<() be a lifting of the isotropy representation to the Spin,,
covering of SO(q). Let then S be the associated G-equivariant hermitian vector bundle of complex spinors
on X = G/H. It is 2/2 graded when q is even.
Given two points x,y € Л', х ф у, we let u(x, у) е TX(X) be the unit tangent vector at x tangent to the
geodesic segment from x to y.
278
operat
(FS)(x)=fa,a)t(x) V*€a
where jk means Clifford multiplication by the vector u.
Note that (H, F) has the same parity as q = dim G/H. One has F2 = 1 since each a(x, o) is a unit vector.
To check that [F, b] is compact for any b ? С"(Г) it is enough to do it for elements J € Г, i.e. to check that
gFg~l — F is compact. But gFg~1 is given by the same formula as F with the point a g X replaced by да.
As the orbit a is a discrete subset of A' and as A' has non positive curvature, one has:
|| я(х, о) - я(х, да) ||—» 0 when х —. оо , х е а,
and the compactness of gFg~l — F follows.
The above proof shows that with A = СГ С С*(Г), the above Fredholm module over A is p-summable iff
the following holds:
Vtf S Г , ]Г || u(i,o)-«(i,ja) ||p<oo.
When G is of real rank one, the symmetric space X = G/H has strictly negative sectional curvature:
к < —e < 0. Thus the comparison theorem shows that || u(x, a) — я(х, b) |[ decreases, for fixed a, 6 € X, like
exp(— ed(x,a)) where d is the geodesic distance in A'. Since the number of elements of the orbit a in the ball
B(a,R) = {x e X; d(x,a) < R] is bounded by a constant times the volume of B(a,R), i.e. by Сехр(АЯ),
we get:
Proposition 5. Let Г be a discrete subgroup of a semisimple Lie group G of real rank one. Then the
Fredholm module (H,F) of Proposition 4 is finitely summable over A = СГ.
Let us assume that Г is torsion free, so that it acts freely on X. Then the above construction of Gi, F) can
be reformulated using only the following map A from Г to the unit sphere S9~l of R?:
A(s) = u(x0,j-1a)eTIO(X) = R«
where xq € a is a base point in the orbit a.
Indeed, given a map A from Г to Sq~1 we can let Г act by left translations in the Hilbert space H =
<?2(r) ® S(R'), where S(R') is the Spin representation of Spin(g), and we can define an operator F by the
equality:
(*) (F()(9) = МШ vs e г , « e n.
The only condition required for G{,F) to be a Fredholm module is then the following:
(*•) VS e Г || \(gk) - \(k) Ц-. 0 when к _ op in Г.
Proposition 6. Le( Г 6e a discrete group and A : Г —. S'~' a mop /rom Г to the unit sphere ofW satisfying
(**). Then let H = ?2(Г) ® S(R?) 6e Me tensor product of the left regular representation of Г by the trivial
representation in S(R'). ie( f be jiuen ij/ (*). Tften (H, F) is a Fredholm module over С?(Г). Л is p
summable over СГ г/f Me following holds:
279

оо УЛ€Г.
9€Г
For g°,..., gn € Г let us compute the operator
9a{F^\..\F,g»\.
Since gFg~l is given by Clifford multiplication by X(g~1-), we get the following formula:
(да1Р,д1}...[Р,дпЩд) = ф(дЩ(да...д"Г> g) VS € Г
where w is the following map from Г to Cliff с (R?):
ш(д) = (Цк^д) - Цк^д)) .,. (Цк^д) - Цк~1д)) Уд € Г
where k3- = д° ... д> е Г.
We thus get:
Lemma 7. With the notations of Proposition 6 and with n of the same parity as q, let Gi,F) be n + 1
summable. Then its n-dimensional character is given on СГ by
r(g°,...,g") = 0 if да...дп ф\
г(<Д...,»") = А„ ? traceGu(S)) if j°...j"=1.
(with An os in definition 1.1).
Here the trace is the trace on Cliffc(R') coming from the representation in S(R') and the Z/2 grading 7 is
used only in the even case. The computation of г is particularly easy when n = q with say q even. Let us
then introduce the function from Гп+1 to С given by
<7(*o, ii, ¦ ..,*„) = traceG(A(fc0-') - A^J)) ... (Цк~^) - Щ~1)).
Since n = q one gets;
*(k0, ...,*„) = 2'/2 .'^(A^-1) - A(tf')) Л ... Л (A(i-i,) - Aft-1))
where we identify A?Rfl with R using the orientation given by 7. In other words, up to a numerical factor,
cr is the oriented volume in R' of the simplex with vertices the A(?r'), j = 0, 1,...,}. Thus one has bo~ = 0
where:
»+i v
F<7)(*„,...,*„+,)= ?(-1И ^(io,..., Jfcy,...,Jfcn+i).
0
The same equality holds for every translate of a,
(go-)(k°,...,kn) = o-(g-1k0 j-'ifc") Vfc'" e Г , S € Г,
and hence for the sum:
280
By construction o- is thus a cocycle in the complex (C,6) of Chapter 3 section 1 which defines group
cohomology, so that the following equality defines a group cocycle c:
As 5 is invariant under left translations, one has:
Thus for so ... г„ = 1 one gets:
Lemma 8. Let q be even and (H,F) be q + 1 summable. Then on СГ the character of(H, F) is the cyclic
cocycle rc associated by Chapter 3.1 example 2, c) to the group cocycle с obtained from the sum cf = ^2д„,
with
„(ко,..., kg) — Oriented volume of the simplex in H4 with the vertices A(jby-1).
We have been a bit careless in the above proof in using
instead of Tr'(fg°[F, g1}... [F, gn]) = jTracefi^^, g°].. . [F, gn]) This question however does not arise in
the following examples where Gi, F) will be q-summable.
In general, given a Lipschitz proper map a : Г —. R« from a finitely generated discrete group Г with word
metric to the Euclidean space R4, one has a natural pull back map: ([CGM])
from the cohomology with compact supports in R' to the group cohomology with real coefficients. Indeed
let u be a (closed) differential form of degree (,u? Cf(W,Ag), /e,w = 1, and let
a{9o, ¦ ¦ ¦
, ¦ ¦ ¦, 9,) = /
o,•••.«, €Г
where s(a(^t)) is the (/-simplex spanned by the ct(gi) G R?. Then the following sum defines a Г-invariant
element of С, with ba = 0 (cf. Chapter 3 section 1):
Note that the sum over Г makes good sense since for fixed gt € Г it involves only finitely many non zero
terms. The class of 0 in tf«(r,R) will be noted:
Next, given a discrete subgroup Г С G of a semisimple Lie group, we can apply the above procedure to any
of the maps:
ар:Г-^Тр(С/К) , cp(g) = exp;' (Г'(p))
where К is a maximal compact subgroup of G, X = G/K is endowed with a G invariant Riemannian metric
of non positive curvature, and expp is the corresponding exponential map.
281

The non positivity of the curvature ensures that ap is a contraction, and it is proper by construction. The
resulting element <*p[R'], q = dim(G/A"), is independent of the choice of p, and we shall call it the volume
cocycle on Г.
Theorem 9. Lei #' be ihe q dimensional Hyperbolic space, and Г С SO(q, 1) be a discrete group of
isometrtes of ff'. Let ("H,F) be the Fredholm module overC"(T) given by Proposition 4.
a) Gi, F) is p-summable overCT iff the Poincare series is convergent at p
where d is ihe hyperbolic distance.
b) Gi, F) is always q summable over СГ, and its q~dimensional character is given by ihe cyclic cocycle rc,
where с is the volume cocycle on Г.
Proof, a) The law of cosines for hyperbolic triangles:
ch a = ch b ch с — sh b sh с cos a
shows that || u(gxo, a) — u(gxo, goa) |J is of the order of exp(—d(gxo, a)) when g varies in Г with go € Г fixed
Thus [F,g0] S С Vjo G Г iff J]r*'D'«) < oo, and a) follows.
b) The Poincare series is always convergent for p > q — 1 (cf. [Su2]) since the volume of the ball of radius
R in #' grows like exp((q — 1)Л). By Lemma 8 the computation of the character follows from the equality
a =¦ [R'] where a is the (Alexander Spanier) g-cocycle on R' given by:
<»¦(&>,.•¦,?,) = gdim. volume of s I ' j
where s (flfv) is the simplex spanned by the vertices j]|v G S'. We refer to [C016] for a similar compu-
computation.
By Chapter 3 section 5 Theorem 7 we know that for any bounded group cocycle с on a word hyperbolic
group Г, the cocycle rc extends by continuity to the pre C"-algebra A which is the closure of СГ in С*(Г)
under holomorphic functional calculus. By construction the character r of the (/-summable Fredholm module
{7i,F) is well defined on A. It is thus natural to ask if the equality of Theorem 8 b) still holds in HC(A).
For complex hyperbolic or quaternion hyperbolic spaces the degree of summability of G1, F) is (in general)
strictly larger than q, and so one should expect that 8 b) is replaced by the same equality in periodic cyclic
cohomology.
(C).
282
shall then show, y
geometry in understanding stable numerical quantities, that is, quantities invariant under small variations
of the natural parameters, which appear in the quantum physics of solids, The first examples of topological
invariants associated with Schrodinger's equation are due independently to a mathematician, S. Novikov
[Noi], and a physicist, D. Thouless [Tho]. The use of non commutative geometry, which makes it possible
to eliminate the rationality hypothesis of [Noi], is due to J. Bellissard [Bel2]. We shall follow his work in
sections 0) and 7) below.
a) Elliptic theory on Tj
We already met the non commutative torus Jj in chapter И from the Kronecker foliation (section 8 /3)) and
in chapter III where we computed its cyclic cohomology (section 2 /3)).
Let us recall that the topology of T; is given by the C-algebra As generated by two unitaries UltU? such
that
U2Ui = XU1U2 A = expBir»0),
while the smooth structure of Tj is given by the subalgebra A» of A»:
A, = {? anm(J?U? ; a € S(Z2)}
where .S(Z2) is the linear space of sequences of rapid decay on Z2:
(|n|* + Н*)|а„,т| bounded for all к > 0.
It is easy to check that Ae is a pre C*-algebra since it is stable under holomorphic functional calculus in A$.
We have seen (theorem 14 of section II1.3) the classification of smooth vector bundles on T^, and in order to
avoid unrevealing notational complications we shall choose one of them and expound the theory of elliptic
operators acting on sections of this particular vector bundle. This space of sections is (cf. loc. Cit.) the
Schwartz space:
? = S(R)
of smooth functions ( on R whose derivatives are of rapid decay. The right action of As on ? is given (cf.
Loc. Cit) by:
(f ' С/2)(л) = cxpBwis)?(s) V{ € S{R) , s € R.
Note that while the presentation of A) only uses A = exp2iri'0, i.e. the class of в in R/Z, the action of Ae
on iS(R) involves now an explicit choice for в, which we take in the interval ]0,1].
By proposition 15 of section Ш.З, the vector fields on T; given by the derivations 6j, j = 1,2 such that:
SjiUt) = 0 if k^j, 6j(Uj) = 2« Vj
lift to the smooth sections ? 6 ? as the following covariant derivatives:
(Vi«)(s) = -2t» "- i(s) , (V2f)» = dt/ds V? € S(R).
283

The exact form of these operators Vj, V2 is not relevant here since we shall be interested in general operators
D in ? = S(R), of the form:
where the sum is a finite sum, while the coefficients СО|/з are operators of order 0 on ?, i.e. are elements of
) V<*,/J.
In the commutative case (say в = 1), the above operators О give exactly all the ordinary differential operators
on the sections of the bundle (for в = 1 the corresponding bundle is a non trivial line bundle). We shall now
see that in the general case these operators D can still be treated in the same way as in the commutative
case, and that the elliptic theory is available for them. Before we proceed let us note that an operator D as
above is an arbitrary element of the algebra of operators on S(R) С L2(R) generated by the following ones:
1) The operator of multiplication by s,
(TOW = s «W V»€R
2) The operator d/ds of differentiation
3) The operator of multiplication by e«(s) = exp B?ri J)
4) The operator Д of finite difference:
In fact, more precisely the algebra End^«(?) of endomorphisms of ? is naturally isomorphic toyl»/, ff = 1/Й,
with generators Vi,V2
=exp2«0'V1l/2
given by the formulae:
(V2()(s)=exp
Thus one is also allowed infinite sums like Ylan,mV"V™ for a e <S(Z2). They include in particular arbitrary
smooth functions /, with period в, acting by multiplication on 5(R).
By analogy with the commutative case we let:
Definition 1. a) For n G N, we say that D is of order < n iff'
b) The symbol oforder n, an(D), is defined as the map from Sl to Ae' given byo-(-qitrfo) = Yla+e=n Cc/i
Viji.ift €R with Vi + Щ - 1.
c) We say that D is elliptic tffa(t]ltj]2) is an invertible element of Aw for any (m,m) € S1.
Note that since Ae- is stable under holomoiphic functional calculus in the C*-algebra A)/ which is its norm
closure in the operators in C'(R), one could equivalently replace c) by the condition:
c') o-(t)\,tJ) is an invertible operator in L-(R) for any G71,1J) € Sl.
284
We can now state the analogue in our case of the main results of the classical theory of elliptic differential
operators.
Theorem 2. ([Coi2]) Let D = J2 С„|/3 V^Vj be elliptic in the above sense. Then:
a) The space Ker D - {( € ?2(R); D? = 0} is finite dimensional.
b) Any ( € L2(R) such that D( = 0 belongs to S(R).
Our next result is an index formula, that is, an explicit formula for the Fredholm index of D\
IndexD = dim Ker D - dim Ker D'.
(Here D' is the Hilbert space adjoint of D in L2(R). One could equivalently consider the codimension of the
image of D instead of dimKer(?>").)
Our index formula will involve a specific cyclic cocycle on the algebra C°°(Sl ,Af) of symbols. We have
already computed in chapter 3 the cyclic cohomology of A%: and found as generators of ЯСеуеп the following
cyclic cocycles: td,t2:
1)
r0
= ao,o Va
(^2)
so that To is the canonical trace on At'. Equivalently r0 is the Murray and von Neumann trace on the
(hyperfinite) type Ih factor generated by the operators Vb V2 in ?2(R).
2) r2 is the character of the following cycle on Ав' (cf. chapter 3 section 3 proposition 15). One takes
Q" = A)' <S Л"С2, the tensor product of A& by the graded algebra which is the exterior algebra of C2. The
differential d is given by:
d(a ® a) = 6[(a)ei Л a + <52(a)e2 Л a Va € Af , a € Л*С2
where e,, j = 1,2 is the canonical basis of C2 and where, as for A»,
S'^Vt) = 0 if j ф к , b'j(Vj) = imVj ; j = 1,2.
Finally the closed graded trace / : Q2 —• С is given by
/ a® (ei Ле2) = ro(a)
Thus the formula for ro is:
By Corollary 16 of chapter 3 section 3 one has for any element E of Ка(Ав') that
where q € 2 is uniquely determined by the equality
ra(E) = p + qO" p,qeZ and в" = V - [в1].
Let then p = [Sl] be the fundamental class of Sl, i.e. the cyclic 1-cocycle on C^fS1) given by
or equivalently the de Rharn differential algebra and integral. We then obtain two natural cyclic cocycles
and 7^ on the algebra of symbols:
285

B = C°°(S1,A,.)=C°a{S1)§A,..
We thus let n = рфта and r3 = p#r2. They are defined on the algebraic tensor product C°°(Sl) <& A),
which is all we need for our symbols ^Cc/j^f^f. They extend however by continuity to C°°(Sl,A$i) =
C'"(S1)®As'- A straightforward calculation shows that:
a) nOctri) = /sl г„ (<7„(<) ? <7,(()) dt
b) тз(со, "it "¦г.с'з) = /5i Л>(<го <^i Л d<72 Л da$)dt
where we define da\ Л dai Л d<T3 as the element of Cx>(Sl,Ae') given by the map:
where г(тг) is the signature of the permutation ?r of {1,2,3}, while the three derivations д\, д?,дз are given
respectively by:
and
@з<т)(О = -?¦ a(t) , for any t e S1 , a € C™^1, A')-
at
In order to make these formulae more explicit, let us write any element a of A$' as a Laurent series in V\\
where /„ S C°°(R/0Z) is for each rj a periodic function of period 0, /„ = 22 "„m V2m. Of course the algebraic
rule is that of Hie crossed product:
with obvious notations and a(g)(s) = ij{s + 1) Vs € R, V3 e C°°(R/0Z).
With this notation, the normalized trace r0 is the integral of /0 over one period, i
Similarly, we can write an arbitrary element a e CC°(S1,A)') as a Laurent series:
where each <jn is a doubly periodic function <7n(t,$); t G Sl, s G R/^Z. Then the derivations d\,di,l
given by.
(<V)n (t,s) = e~<rn(t,s)
OS
(дз<г)п (t,s) = ^(rn(t,s).
One then gets the formulae:
286
a1) nWa,<rl) = i/o'/s. го(Иаз^)о((,«) Л rf»
b') ^((r0,^1,^2,^3) = }/0*/5. M^dcr' Ada2 Adcr3H(t,s) dt ds.
We can now state the analogue of the Atiyah Singer index theorem for the elliptic operators D:
Theorem 3. ([Coi2]) Let D = ? C<,,pVf V§ 6e etfiplic, onrf <
a+0<n
symbol. Then
dimKerD-dimKerD* = iBjri)~2 тз(т~1, a, a'
be its principal
1 п((т-1,а).
We thus obtain an index theorem for the highly non local operators D — ?] Cap V" V^ on the real line. As
in the classical case of differential operators on manifolds, this index theorem has two important corollaries:
Corollary 4. a) //IndexD > 0 (Леге exist non trivial solutions f € <S(R) of the equation Df = 0.
b) For any invertible a ? C"(Sx, As*) the following quantity is an integer:
One obtains in particular an explanation for the integrality of r2 : (r2, Ko(A$')) С Z. But we shall come
back to that point in detail at the end of this section.
We shall now give non trivial examples of elliptic operators D of the above form. Our first class of examples
will only require qualitative information about the periodic functions ft g C°°(R/Ze) involved in its formula
which is:
1
(Df)(s) = s f(s) + ]T gk(s) f'(s + k) V/ € 5(R).
it=-i
Here the <д are real valued functions. We assume that — 1 < go < 1, and that 0 < g\ < 1, and that the
operator T:
(Tf)(s)= ]T 9k(s)f(s + k)
fc=-l
is selfadjoint, which means that f/_i(s) = gi(s — 1).
The qualitative behaviour of the pair of functions g-i,go € C°°(R/#Z) is the following (fig.). We assume
that on the circle R/#Z there is an interval I with I + I = J disjoint from /. We let К (resp. L) be the
interval from I to J (cf. fig.) (resp. from J to /). Then the conditions are:
A' ¦
\go(s + 1) + ga(s)\ < 1 Vs €
- 1/4,1
V* € /.
As soon as the intervai / is chosen, these conditions are easy to fulfill and we could have used strict inequalities
everywhere to get a non empty open set in C(R/#ZJ. We now have:
287

Corollary 5. Let jo.S-i € C°°(R/eZ) be two functions satisfying the above conditions, and let ffi(s) —
g-its + 1). Then the operator D, (Df)(s) = s f(s)+ ? <M«) f(» + k) V/ € 5(R), w elliptic in йе
t=-i
sense of definition 1. Its index is equal to
IndexD = 1 + 2[l/0]
шЛеге [1/Й] is Йе integral part of l/в. The equation Df = 0 admits at least 1 +2[l/#] linearly independent
real solutions f € -SfR)-
We thus get an existence theorem as well as a regularity result (theorem 2). Since the conditions on g_ltga
are open, they can be fulfilled by analytic functions and even by trigonometric polynomials. The function
E(l/0) is discontinuous when 0 € Ы~* С ]0,1]. The reason why this does not entail a contradiction is
that when в~1 gets close to an integer it becomes more and more difficult to find an interval / in R/9Z
such that / and 1 -r \ are disjoint, this being of course impossible for в~1 € N. The proof of corollary 5 is
a straightforward application of theorem 3 or equivalently of the computation of r? on the Powers Rieffel
idempotent ее* € At' (cf. [Pow^Ri^Co!?]).
One can construct many examples of elliptic operators with non trivial indices using simply the formula:
(Df)(s) = sf{s) + (Tf')(s)
where the operator T € Ae> is selfadjoint and invertible. This implies that D = — j|j Vi + TVj is elliptic
with principal symbol in the same class as:
<r((i,b) = « +(-<T.
The index formula of theorem 3 reduces in that case to
IndexD = ^ij| r,[E. f.\ E) - r,,(( 1 - 2E))
where E is the spectral projection of T belonging to the interval [0, +x>[,
Е=\[аМ(Г).
One has E € At' since this algebra is stable under holomorphic functional calculus. The use of the Chern
character r? in order to label the gaps of selfadjoint operators has been very successful in the hands of J.
Bellissard, and we shall deal in detail with tliis in 7) of this section. Perhaps the simplest non trivial example
of a selfadjoint invertible T G As1 as above is the Peierls operator:
(Tf)(s) = f(s + 1) + f(s - 1) + 2 cos (Ц^\ f(s) + A f(s)
which, when 9 is a Liouville number, is invertible for any A outside a nowhere dense Cantor set К [B-S]
on which the index will change discontiiiiiously. We refer to [Мои] [CEY] [BS] for more information on the
operator T and gap labeling. In [CEY] the following (-/-analogue of the binomial formula is successfully used
to show that the spectrum of T is a Cantor set when 0 is a Uouville number:
whenever VU = \UV with A a scalar (here A = expBiri0')), and where the Gaussian polynomial (?)
given by:
\k)x
-A)"
288
As a corollary of the gap labelling of the Peierls operator and of theorem 3 one gets the following result.
Corollary 6. Let в be a Liouville number, N an integer. Then there exists A € ] — 2, 2[ such that the
following difference differential equation on R admits at least N linearly independent solutions f g S(R):
oS^+X^ f'(s)=0.
We shall now pass to the application of the integralita of the cyclic cocycle 7^ to the Quantum Hall effect,
due to 3. Bellissard.
C) The Quantum Hall effect
Fig.6. Hall current
Let us describe the experimental facts pertaining to the Quantum Hall effect, starting with the classical
Hall effect which goes back to 1880 ([Hal]). One considers (fig.6) a very thin strip 5 of pure metal and a
strong magnetic field B, uniform and perpendicular to (.he strip. Under these conditions elementary classical
electrodynamics shows that particles of mass m and charge e in the plane 5 must move in circular orbits
with angular frequency given by t.he "cyclotron frequency"
ыс = е B/m.
Let us view the charge carriers in S as a two dimensional gas of classical charged particles with density N
and charge e. Then if an additional electric field E is applied in the plane S (cf. fig.l) there is a drift of the
above circular orbits with velocity E/В in the direction perpendicular to E. The resulting current density
j perpendicular to both E and В is such that, in the stationuary state the resulting force vanishes, i.e.
NeE+jAB = 0.
In practice the charge carriers are scattered in a time which is short with respect to the cyclotron period,
so that the observed current is mostly in the direction of the electric lield E. It does however have a small
component in the perpendicular direction, which is the Hall current given using the above formula by:
j = NeB AE/\B\2,
289

which shows that \j\ = (Ne/B)\E\. In other words, the Hall conductivity <тд, i.e. the ratio of the Hall
current to the electric potential is equal in first approximation to a linear function o{ N:
сгя = Ne/B.
As early as 1880 ([Hal]) Hall observed the above drift current j and showed that the sign of the charge e
may be negative or positive depending on the metal considered. This was the first evidence of what is now
understood as electron or hole conduction.
In the regime of very low temperatures, T ~ 1° Л', the effects of quantum mechanics become predominent,
and can with a gross oversimplification be described as follows. First the two-dimensional gas of charge
carriers, say of electrons, has a one particle Hamiltonian given by the Landau formula:
H = {p-eAJ/2m
where p is the quantum mechanical momentum operator and A is a (classical) vector potential solution
of rot(A) = B. Also m is the effective mass of the charge carrier. It is immediate that the operators
Kj = Pj — eAj satisfy the commutation relation:
[7Y,, ЛЧ] = ih e В
while H — K2/2m. Thus the energy levels of the charge carrier have discrete values which are all integer
multiples of Planck's constant times the cyclotron frequency:
En = nh
Each of these "Landau levels" is highly degenerate, due to the translation invariance of the system, and can
be filled by ~ eB/h. charge carriers per unit area.
A naive argument combining the filling of Landau levels with the drift velocity E/B thus allows to expect
in the quantum regime a Hall current density given by
j = — x e x n— = n( e2/h) E
n. a
where n is the number of filled Landau levels and j is in a direction perpendicular to the electric field. In
particular this implies that the Hall conductivity crH is an integer multiple <гя = n e2/h of e2/h, provided
the Fermi level is just in between two Laudau levels. This argument however does not, in any way, account
for the existence of the plateaux of conductivity which were discovered experimentally by K. von Klitzing,
G. Dorda and M. Pepper ([Kl-D-P]). In their paper with title:
New Method for High-Accuracy Determination of the Fine-Structure Constant
Based on Quantized Hall Resistance
the above three authors exhibited the quantization of the Hall conductivity thus giving the possibility of
290
determining the fine structure constant Q = fj with an accuracy comparable to the best available methods.
Fig.7. Plateaux of conductivity
We cannot end the above account of the experimental results without mentioning the fractional Hall effect
found by D.C. Tsui, II. Stormer and AC. Gossard. They observed in heterojunctioii devices with high
electron mobility that, besides the plateaux corresponding to integral multiples of e2/h there are other flat
regions for the Hall voltage which correspond to rational values of h/e2 un with mostly odd denominators.
7) The work of J. Bellissarcl 011 the integrality of ац
A first explanation for the integrality of 077 on the plateaux of vanishing direct conductivity (cf. fig.7) was
given by Laughlin in 1981 [La] using the gauge invariance of the one electron Hamiltonian together with a
special topology of the sample. Then Avron and Seiler put (.he argument of [La] in rigorous mathematical
form assuming that, on the plateau, the Fermi level (cf. below) belongs to a gap in the spectrum of the one
particle Hamiltonian. This approach is however unsatisfactory in as much as it uses special topology of the
sample and also does not account, for the. role of localised electron states, tied up with disorder, which imply
that the plateaux cannot correspond to gaps in the spectrum of the one particle Hamiltonian. To explain
this more carefully we need to introduce anotlier parameter than (he charge carrier density N. It is called
the Fermi level ft, and plays the role of a chemical potential. In the approximation of a free Fermi gas, the
thermal average of any observable quantity A at inverse temperature /3 = 1/kT and chemical potential ft is
given by:
(.4)p,,. = Гт^ щ Tracev(/(tf).4)
where Tracev denotes the local trace of the operators in (he finite volume I'cS and where / is the Fermi
weight function:
In general the Fermi level /i is adjusted so as to give the correct value to l.he charge carrier density:
291

Note that the right hand side N@,/i) of this formula is in the limit of 0 temperature, i.e. 0 —* +00,
depending only on the spectral projection E^ of Я on the interval ] — 00, fi\, and is thus insensitive to the
variation of fi in a spectral gap.
We shall now explain the results of J. Bellissard on the integrality of <тя- They are directly in the line of
the argument of Thouless, Kohimoto, Den N'rjs and Nightingale [Tho-K-N-dN], who investigated the case of
a perfectly periodic crystal with the hypothesis that the magnetic flux in units ft/e is rational, an obviously
unwanted assumption. Their argument shows clearly that the origin of the integrality of an is not the
shape of the sample, but rather the topology of the so called Brillouin zone in momentum space. When the
magnetic flux is irrational this Brillouin zone becomes a non commutative torus T2,, and the integrality of
ац is the integrality of the cyclic 2-cocycle r2 of section a) on the projection E^.
We shall first explain why this is true and then why, taking inpurities into account, the integrality will
persist. This will in particular eliminate the unwanted assumption that /1 belongs to a gap of the one
particle Hamiltonian, using the full force of the integrality result (corollary 9) of section 1 of this chapter.
The case of a periodic crystal.
Let us take as a model of metallic strip S the plane R2 with atoms at each vertex of a periodic lattice Г С R2.
The interaction of these atoms with the charge carrier, let us say the electron, thus modifies the one particle
Hamiltonian to:
Я = На + V , Ho = (p - eAflim
where the potential V is a Г-periodic function on R2. The whole set-up is invariant under the group Г of
plane translations belonging to Г, so that we should get a corresponding projective representation of Г on the
one particle quantum mechanical Hilbert space "H. We should normally write 7{ as the space of L2 sections
of a complex line bundle L on R2 with constant curvature, but this just means that, viewing 7{ as ?2(R2),
the correct action of the translation group is given by the unitaries, called magnetic translations:
U(X) = exp i(p - eA) ¦ A' VA 6 R2.
For X G Г this unitary commutes with Я, but due to the curvature the U(X) do not commute with each
other. For the generators ei,e2 of Г we get the commutation relation:
U2Ui = XULU2 ; A = exp 2тгг0
where Uj — U(ej) and where 9 is the flux of the magnetic field В through a fundamental domain for the
lattice Г, in dimensionless units. The role of the rationality of в in the Thouless et al paper thus appears
clearly, since we know that when в is irrational the von Neumann algebra W in "H of operators which have
the symmetries Ut, I 6 Г:
w =
=т
is the hyperfinite factor of type Il^a namely Яо^ ,
In other words, if we investigate the operators which obey the natural invariance of the problem, we are not
in a type / but in a type JJoo situation. From the measure theory point of view the Brillouin zone is of type
//. Moreover the canonical trace г on the factor W is given, using an averaging sequence Vj of compact
subsets of R2, by:
A)
r{T) =
-L Tracev(T),
292
so that this part of the thermodynamic limit has a clear interpretation.
Since we need to understand the topology of the non commutative Brillouin zone, we need a C*-algebra
А С W of observables for our system. By construction any bounded function /(#) belongs to W, and in
view of the formula giving the statistical average of observables, it is natural to require that A contains /(#)
for any / 6 Co(R). The obtained algebra so far is commutative and is too small to perform the computation
of the Hall conductivity. For that purpose we need another observable which is the current j associated to
the motion of the charge carrier. This current is a vector, given classically by
where X is the position of the charge carrier. Thus in quantum mechanics we have:
B)
J = ei/h
where it is understood that both sides are pairs of operators, i.e. given by their components on a basis of
R2.
with Xj the multiplication operator by the coordinate.
To understand clearly why J is invariant under the symmetries Ut, I 6 Г we can rewrite the formula B) as:
C)
J -ejh (д«,(Я)),=о
where the group (R2)A dual of R2 acts by automorphisms a,, s 6 (R2)A, on the von Neumann algebra W
by:
a,(T) = e"x T e~"x VT 6 W.
We thus can take J as an observable (except for the trivial fact that since J is unbounded, just as for Я we
need to use f(J), f 6 C0(R2)).
But in order to compute the Hall conductivity we also need to turn on an electric field E and see how our
quantum statistical system reacts. This means that we replace the time evolution given by Я, at(a) =
ettH a e~ttfI, by the time evolution associated to the perturbed Hamiltonian:
H' = H + eE ¦ X,
or equivalently by the differential equation:
jt a't(a) = i/h[H,a]+ e/h j{ atE(a).
This makes it clear that the smallest C-algebra of observables appropriate for the computation of the Hall
conductivity, besides containing f(H), f 6 C0(R) and f(J), f 6 Co(R2), should be invariant under the
automorphism group <*, of W. In fact (cf. [Bel]) it is not difficult to see that the C*-algebra A generated
by the a,(f(H)) does contain the functions of the current, and is thus the natural algebra of observables for
our problem. On А С W we have the (semifinite semicontinuous) trace r coming from the von Neumann
algebra W (formula A)) and the automorphism group (a,) with generators the derivations:
We thus get a densely defined cyclic 2-cocycle on A given by the formula:
293

E) r2(oo,oi,o2) = r(aoF1a162a2 - S2al6la2)).
By construction r2 is a 2-trace (cf. chapter 3 section 6 definition 11) so that it pairs with K0(A). Now the
crucial fact is the following formula, known as the Kubo formula, for the Hall conductivity <rH in the limit
of O-temperature, and assuming that the Fermi level /i is in a gap of the Hamiltonian Я.
Lemma 7. [Bel] If/i ? Spec Я then the Hall conductivity аи is given by:
e2 e2 1
when Ец is the spectral projection of H corresponding to energies smaller than the Fermi level /i.
Thus we see that the integrality of cjjy (in units of e2/h) is implied by the integrality of the cyclic cocycle
r2 on the C*-algebra A. In the example at hand one can show that 75 is integral, as a corollary of chapter
3 section 3 corollary 16. But we want a conceptual reason for this integrality which still survives in more
difficult circumstances. This will be obtained from corollary 9 of section 1 by the construction of a Predholm
module over A whose Chern character is equal to To.
Let us describe this even Fredholm module. The Hilbert space representation of A is just given by two copies
ofW,
with the Z/2 grading given by 7 = .
The representation of .4 is thus a 6 A — о ® 1 = " ° . The operator F, F = Г, F2 = 1 in H' is given
о и
и- о
where U is the operator in ?2(R2) of multiplication by the function ti(xi,X2) =
metric and orientation on R2.
uses a Euclidean
Theorem 8. [Bel][Coi2] The above triple Gi',F,f) is a B,oo) summable Fredholm module over the C-
algebra A, and its 2 dimensional character is equal to т?.
The proof is the same as that of proposition 5 and lemma 7 of section 2. It is interesting to relate more
precisely the above construction with the construction of section 2 and of chapter 3 section 4 (i.e. the index
theorem for covering spaces). We shall briefly mention how this is done. First, with the notations of section
HI.4, we take M — R2 while the lattice Г acts by translations on M with quotient the two dimensional
torus M = R2/I\ Let then G be the fundamental groupoid of this covering (cf. Ш.4). Then the C"*-algebra
A above is contained (and coincides with, in the generic case) in the C"*-algebra: C*(G,u) of the smooth
groupoid G twisted by the 2-cocycle w on G associated to the line bundle L on R2. (One has a canonical
homomorphism p : G ~<¦ R2 with p(x, y) = x — y, and u = p'c where с е Z2(R2, J7(l)) corresponds to the
Heisenberg central extension of R2 given by the magnetic field B.)
The strong Morita equivalence of C(G) with С""(Г) of III.4 adapts here to yield the strong Morita equiva-
equivalence:
C(G,u) ~ A»
294
.here в comes from the 2-cocycle с/Г 6 H2B\U(l)) = U(l). Under this equivalence and as in III.4 the
cocycle r2 corresponds to the 2-cocycle v on A, of corollary 16 section III.3, which proves its integrality.
Next the construction of the Fredholm module of theorem 8 is a special case of the following variant of
proposition 3 of section 2. Let M be a compact Riemannian manifold^ith non pos.tive curvature and 0 a
closed-form on M. Let then Г = ъ(М) be its fundamental group, M the universal cover of M w,th the
lifted metric so that Г acts by isometries on M. Let L> a Hermitian line bundle on M with compatible
omecTon V and curvature equal to the lifted form 0. This defines a 2-cocycle „on the fundamental
°oupo d G of the covering M. The value of „ on a pair (x, jj), (», z) of composable elements of G » given
gently by the parallel transport ш L around the geodesic triangle T with vert.ces X,y,z, or as:
u(z,y,z) = exp2« / 0.
Jt
By construction the C'-algebra A = C(G,») acts in the Hilbert space П = L2(M,L) of square integrable
sections of the line bundle L, by the formula:
where for 7 = (x,y)/T ? G, U(f) is the parallel transport in L along the geodesic from у to x in M. Let
then о ? M be a base point and S the Spin representation of Spin(TaA/). The following equality defines, as
in 2.3, a Freholm module over A:
a) H' = H ® 5
b) For any / 6 A = C'{G,u) one has k(i ® s) = ifc? ® s, V? 6 Ti, s 6 5
c) F is the Clifford multiplication by the vector-valued function u(x, o) which to x E M associates the unit
vector at a pointing towards x.
The degree of summability and the computation of the character are handled in the same way as in section
2. We shall now close this parenthesis and go back to the Quantum Hall effect.
Real samples and localised states.
When the Fermi level /» lies in a gap / of the spectrum of Я the (O-temperature limit of the) charge carrier
density ЛГ as given by formula above, is insensitive to the variation of fi. In real samples, due to the presence
of a small amount of impurities, the dependence of Af on fi is never of this kind, and the ratio of Inf^rp
to Sup^j- can be as big as 1/3. This implies that the parameters /1 and N are equivalent parameters
for real samples. It however introduces an essential new difficulty: since the spectrum of the energy Я is
now connected, say equal to [0,+oo[, it is no longer true on the plateaux of conductivity that the spectral
projection E,, belongs to the C*-algebra of continuous functions f(H), f 6 Co(R), since the characteristic
function of] — 00, /1] is not continuous on Spectrum(ff).
The solution of this difficulty is quite remarkable (cf. [Bel]) and deserves the attention of the reader.
The qualitative idea from physics is that while the impurities eliminate the gaps in the spectrum of Я,
they however fill these gaps mostly by electron states which ate localized and do not participate to the
conductivity. This is why one observes experimentally the plateaux of the conductivity as a function of the
Fermi level. This qualitative idea is put on rigorous mathematical ground by [BGMS] using [FS] and [MS] for
realistic models. Now the meaning of the localization of the electron states with energies in a small interval
]/j — e, /1 + s[ around the Fermi level p is the following:
Lemma 9. If ц lies m a gap of extended states of H then the characteristic function Ец(Х) = {1 if A <
ft ; 0 */ A > fi) is quasicontinuous on Spectrum Я.
We are using here a terminology of function theory which we already met in section 3 when we developed
our calculus on S1. Thanks to proposition 2 of that section we can adopt the following general notion of
quantum calculus.
295

Definition 10. Let A be а С-algebra and CH,F) a Fredholm module over A. Let W be the von Neumann
ibra weak closure of A. Then an element f ? W is called quasicontinuous iff
Definition 10. Lei A be а С -algebra and (H, F) a Fredholm module over A. L
algebra weak closure of A. Then an element f ? W is called quasicontinuous iff
[FJ]€IC
(i.e. is a compact operator).
It goes without saying that elements of A sire called continuous, this referring to the case when A is commu-
commutative so that A = Co(X), with X = Spectrum A. In the case of lemma 9 the C-algebra A is the algebra of
functions /(Я), / 6 Co(R), in the Hilbert space i2(R2), and the Fredholm module is the same as in theorem
8 above. Thus we have:
W = L2(R2) <g> C2 = ?{+ 0 И".
The representation of f(H) is by f(H) ® 1 = ^^ ,.°„. , and the operator F is F = ° ^ as
L u /(H)J L^ UJ
above.
We shall see that, exactly as above, the Hall conductivity for /i in a gap of extended states is given by the
pairing between the К homology class of (W, F, 7) and the spectral projection Е„, and is hence an integer :
the Fredholm index of the operator E^ UEp (cf. proposition 2 of the introduction of this chapter). As above
this equality relies on proposition 8 of section 1, i.e. the computation of the character of G{',F, 7).
Let us now describe how this is done, taking into account the random parameter w. This parameter w belongs
to a probability space (О, Р) which is the configuration space for impurities. To the translation invariance of
the system corresponds an action T of the translation group R2 by automorphisms of the probability space
(fi, F), such that the magnetic translations U(X) satisfy:
U(X) Hw U{Xy = ЯТ(А-)Ш Vw 6 П , A' 6 R2.
There is a natural topology on О given by the weak topology on ?(L2(R2)) pulled back by the map:
u»l/, from П to С(П) К, = (Hw + j)-1.
Moreover it is harmless to take the weak closure V'(O), and hence to assume that О is a compact space. Then,
the analogue of the C*-algebra C'(G,ui) of the periodic case, is the twisted crossed product C'-algebra:
.4 = С(п)хт,с R2.
Recall (cf. chapter II appendix C) that any / 6 CC(R2,C(Q)) defines an element of A which we denote by
J fx ux dX,
with the following algebraic rules:
«Л- / ul- = / ° Tx' VAT 6 R2 , V/ 6 C(fi)
ux «r = c(X, Y) UA-+V- ЧХ, Y 6 R2
where с е Z'(H2, GA)) is the Heisenberg 2-cocycle.
By construction every p&Q yields a unitary representation тгр of A in Z2(R2) given by:
(fx) ЩХ) dX
where U(X) is the magnetic translation operator (I), while np(f) for / 6 C(fi) is the multiplication operator
in L2(R2) by the restriction of/ to the orbit of p:
296
(*„(/)«) Р') = /(Г-г(Р)) «(V) VY 6R2 , «6 ?2(R2).
The C*-algebra norm on A is given by
= sup 1ЫЯЦ,
n
and CC(R2, C(O)) is a dense involutive subalgebra of A (cf. chapter II appendix C),
Let г be the trace on A dual to the probability measure P. It is semifinite and semicontinuous, and for any
element f = J fx ux dX, f 6 CC(R2, C(Q)), one has
r(f'f)= f [ \KX,p)\2 dX dp.
Jx Ja
If we assume ergodicity of the action T of R2 on (fi, P), this trace r coincides, using the ergodic theorem,
with the trace per unit volume in almost all the representations irp:
r(f) =J^1 Щ Tracev(Tp(/))-
Next, the dual action (chapter II appendix С proposition 4) of R2 = R2 on A is given by:
c, (J fx ux dx\ = J fx e"x ux dX
V/6 CC(R2,C(Q)), Va6R2.
As in the periodic case, we get a densely defined cyclic 2-cocycle on A by the formula:
MfoJi.h) = r(foFifi 6?h - «2/, «!/2))
where Sj is the derivation Sj — {djat,)s-0.
This formula continues to make sense when the /j are elements (in the domain of <Sj) of the von Neumann
algebra W weak closure of A in the (type II) representation associated to the trace r, and are such that:
TF;{f)' «,(/))< OO.
The analogue of lemma 7 above is now:
Lemma 11. ([Bel]) If ft is in 0 gap of extended slates, then the Hall conductivity <тц is given by:
where E^ is the spectral projection of И corresponding to energies smaller than the Fermi level.
Moreover, the integrality of r2 on Е„ follows exactly as above using the Fredholm modules over A given, for
any p 6 П, by the representation ;r,, Q 1 of A in W = ?2(R") ® C2, the same operator F = ,,, n , and
the Z/J2 grading 7 = . The only new ingredient is that one computes the integral over О of the
characters of these Fredholm modules over A to obtain r2. The remarkable fact is that the resulting index
formula, (which implies the integrality):
;— T-,(E,>, В„,ВИ) =
2xi
a.e. on Q
continues to hold for the quasicontinuous projections involved in lemma 11 (cf. [Bel]).
297

Clearly all the above discussion of the Quantum Hall effect is done in the approximation which neglects
the mutual interaction of the electrons which is supposed to be responsible for the fractional Hall effect (cf.
section /?)). There are tentative explanations of the latter effect (cf. [Fr] and the literature there) based on
conformal field theory, which in particular yield the odd denominators 2k + 1. In order to adapt the above
discussion to the interacting case it is necessary to extend the finite dimensional tools used above (cyclic
cohomology and finitely summable Fredholm modules) to the infinite dimensional situation of an indefinite
number of particles, i.e. of quantum field theory. This extension will be done in the next sections.
We should insist however on the infrared nature of the Quantum Hall effect or of the determination of
the fine structure constant, as opposed to the ultraviolet nature of the Fredholm modules associated to
supersymmetric Quantum Field theories in section 9. In the Quantum Hall effect it is the differentiable
structure of the Brillouin zone, i.e. of momentum space, which is playing a role.
_ _ p(),
к the dimension of the leaves. The C"-algebra of the foliation acts in L~(L) by the unitary representation
x, of section 11.8 a), where x is a chosen origin x ? L. The operator F in L2(L) (g> S is given by Clifford
multiplication:
where u(y, x) 6 %(L) is the unit tangent vector to the geodesic from x to у in L.
298
IV.7. Entire cyclic cohomology
The results of section 1 on the characters of Fredholm modules are limited to the finite dimensional case by
the hypothesis of finite summahility of the modules (definition 3 of the introduction). This hypothesis does
not hold in examples coming from higher rank Lie groups (section 5) or quantum field theory, but is replaced
by the weaker 0-summability condition (definition 4 of the introduction). Exactly as cyclic cohomology was
dictated as the natural receptacle for the characters of finitely summable Fredholm modules ([C016]), the
fl-summable condition (in its unbounded form, cf. section 8) dictates, as a receptacle for the character, the
entire cyclic cohomology ([C02O]). We shall first explain this theory and its pairing with К theory. The
construction of the character ([Co2u][.]LO]) will be done in section 8.
a) Entire cyclic cohomology of Danacli algebras
Let A be a unital Banach algebra over C. Let us recall l.he construction (chapter III section 1) of the
fundamental F,5) bicornplex of cyclic cohomology. For any positive integer n EN, one lets Cn(A,A*) be
the space of continuous n+ 1 linear forms ф on .4. For n < 0 one sets Cn = {0}. One defines two differentials
6, В as follows:
b:C" -»
(~\)' ф(а° a'aJ + 1 «" + ')
(Ьф)(а°,. .. , o"+1)
о
2) В : С" -* С", (Вф) = АВоф, where
(ВоФ)(а°,.. . ,а"-1) = ФA, а",.. ., а"'')-(-!)" Ф(аа,. .. .а",1) Чф 6 С",
By lemma 19 of section 1 chapter :! one has Ir = D- = 0 and ЫЗ = —Bb, so that one obtains a bicomplex
(Cim;di,d3), where C"'" = C"~'" for any 11,in 6 1,
1)Л;Г,,„_С,>+1.,„;^ = __:С-
„ ^c,.,m + l
/\
в
• Г "
г
/
X
с2
/
с1
\
\
с'
ь
А
с3
\
с6
/
в
cs
bN
\
с"
А
4-"
ь
\
\
с1"
/
-•
с'
\
\
Fig.8. Bicomplex
•299

The main lemma (chapter 3 section 1 lemma 25) asserts that the b cohomology of the complex KerB/ImB
is zero, so that the spectral sequence associated to the first filtration has the E2 term equal to 0. Since the
bicomplex C"'m has support in {(n,m),(n + m) > 0} this spectral sequence does not converge in general
when we take cochains with finite support, and by loc. cit. theorem 29, the cohomology of the bicomplex
when taken with finite supports, is exactly the periodic cyclic cohomology H*(A). If we take cochains
with arbitrary supports, without any control of their growth, then by the above lemma 25 we get a trivial
cohomology. It turns out, however, that provided we control the growth of ||0m|| in a cochain @2n) or
@2n+i) of the F, B) bicomplex, we then get the relevant cohomology to analyze infinite-dimensional spaces
and cycles. Because of the periodicity Crt'm —> Qn+Um+i ш tne bicomplex (b, B), it is convenient to just
work with
Cev =
6 C2" Vn 6 N1}
and
C°dd = {@2n+i)»6N , 02„+, 6 C2tl+1 Vn 6 N}
and the boundary operator д = dt + d2 which maps C™ to C°dd and C°dd to Cev. We shall enforce the
following growth condition.
Definition 1. An even (resp. odd) cochain @2„)„ем 6 C (resp. @2o+i)neN 6 СоАЛ) is called entire iff
the radius of convergence "/52 ||<^2п||(г"/п!) (resp. 52||<fc>n+i|| z"/n\) is infinity.
Here for any m and ф 6 С"", the norm ||0|| is the Bauach space norm:
||0||= sup{Wa°,....am)|; |И < 1}.
It follows, in particular, that any entire even cochain @2„) 6 Cev defines an entire function F^ on the Banach
space A by
}ф(х) = ^(-\Г ф2п(х,...,х)/п\.
Lemma 2. If ф is an even (resp. odd) entire cochain, then so is (rfi
Proof. For фт G Cm one has \\Ьфт\\ < {т + 2)\\фт\\ and \\Вофт\\ < Щфт\\, I|AAi<M < 2т\\Фт\\, thus
the conclusion.
Definition 3. Let Л be a Banach algebra. Then the entire cyclic cohomology of Л is the cohomology of the
short complex:
C?(A) ~С
of entire cochains in A.
We thus have two groups H'y(A) and Н°М(А). There is an obvious map from H(A) to H,(A), where H(A)
(chapter 3 section 1) is the periodic cyclic cohomology of A. We also have a natural filtration of H, by the
dimensions of the cochains, where @2,,) say, is of dimension < к if ф2„ = 0 Vn, 2n > k. However, unlike
what happens for Я, this filtration does not, in general, exhaust all of Hc. In fact it exhausts exactly the
image of H(A) in H,{A).
Let us now compute #, for the simplest case, i.e. when A = С is the trivial Banach algebra. An element
of CJ" is given by an infinite sequence (А2„)„ем, Л2л 6 С such that J2 |А2п|(^"/п!) < o° for any z and,
300
similarly for Cfdd. The boundary д = di + d2 of (А2„) is 0, since both 6 and В are 0 on even cochains. For
modd and0 6C, 0(a°,... ,om) = Aa°...am one has
„0 „m-l-, _ 1„.\„°
(Ьф)(а° am + 1) = Aa°...am + 1 , (Вф)(а°,..., am) = 2mAo° .. .am
thus
(d10)(ao,...,am+1) = (m+l)Aao...a">+1 , (d20)(a°,.. ., o1-1) = 2Aa° .. .a"-1.
So the boundary fl(A) of an odd cochain (А2„ + ,) is given by d(AJn = 2nA2n_i + 2A2n+b Thus, д(Х) = О
means that A2n+1 = (-l)"n!A,, and hence is possible only if A = 0, for A 6 C?«d Moreover, for any
(A2n) 6 СГ, the series <т(А) = E~(-l)"( W"!) * convergent and <r(A) = 0 iff A 6 дС?л- Thus we have
Proposition 4. One has Н?Л(С) = {0}, and Hf(C) = С «iU isomorphism given by
In order to relate entire cyclic cocycles with an infinite-dimensional version of the cycles of chapter 3 section
1 we need the following normalization condition:
Definition 5. We shall say thai a cocycle (ф2п) (resp. @2,, + iV is normalized iff for any m one has
ВоФт = — АВоф„,.
<*)
In other words, the cochain ВаФт is already cyclic: Вафт 6 C?" ', so that A/т)Л(Во0т) = Вцфт. Only
the normalized cocycles have a natural interpretation in terms of the universal differential algebra OA.
Lemma 6. For every entire cocycle there is a normalized cohomologous entire cocycle.
We refer to [Co20] for the original proof. It has been simplified by Cuntz and Quillen [Cu-Q].
Remark 7. a) The above definition of entire cyclic cohomology and its pairing with К theory (cf. section
6) below) adapt to arbitrary locally convex algebras Л over С as follows: A cochain (t?>2n) (resp. (^гп+i)) is
called entire iff for any bounded subset ? С Л there exists С = Cs < oo with:
|(p2n(a°,...,o2")| <CS n! Vaj 6 S Vn 6 N.
Replacing S by A~lS for A > 0, one gets that for any bounded subset S of Л and A > 0 there exists С = Сед
such that:
|^(o°,...,o2")| <C A2" u! Va> eS , Vn 6 N.
b) Let A be an algebra over C. Then endowed with the finest locally convex topology, A is a locally convex
algebra, so that its entire cyclic cohomology is well-defined by a). The bounded subsets of A are convex
hulls of finite subsets S. Thus a cochain ip is entire iff for any finite subset ? of A there exists С = Cs < oo
such that:
|i?(a",...,<J2")| <C n! Va-> 6 S , Vn 6 N.
c) We have used the (rfi, d?) bicomplex of chapter 3 section 1 theorem 29 so that S = d\d^ , and the pairing
of A'o with Щ" is given by the function fv whose existence fixes the growth condition. The trivial change
to the F, B) bicomplex is summarized in the following table:
301

F, В) bicomplex
6 : С" -» C"+1
В :С" -*С"~1
<№+.)
llvsnll <СХ А2"/п!
||^2n+i|| ^ C\ A"n/n!
Ц-1)-^№ („...,„
(^ь^з) bicomplex
di~{n+ 1N
^ n
V>2n = Bn)! (^>2„
^2n + l ^ Bn + 1)! ^2n + l
HV'Snll <СЛ A2" n!
HV^n+lll < C\ A2" n!
/?) Infinite dimensional cycles
In chapter 3 we took as a starting point for cyclic cohoniology the notion of cycle of dimension n, given by
a graded differential algebra (Q,d) and a homogeneous linear form J of degree n such that
1) /Uiui = (-1)*1*3 fu-tui Vuj 6 Пк> , j = 1,2
2) / du = 0 Vw 6 П"-'.
In this section we shall show that in order to handle the infinite dimensional case one just needs to replace
the homogeneous conditions 1) 2) by the single mhomogeneous condition:
3) /(Ш1ш2-(-1I1*2 Ujui) = (-1L' Jduidu2 V4?Q'i.
We shall first work purely algebraically, and then formulate the growth condition of entire cocycles in terms
of П.
We shall say that a linear form /* on a differential graded algebra (O,d) is even (resp. odd) iff /i(w) = 0
Vu> 6 О*, к odd (resp. к even).
The following is the infinite dimensional analogue of proposition 4 chapter 3 section 1.
Proposition 8. [C02O] a) Let A be an algebra over C, (П, d) a graded differential algebra such, that fi° = Л
and fi an even (resp. odd) linear form on О satisfying 3):
dw2) Vwj 6 П*'.
Then the equality:
D)
V2»(o°, • ¦ ¦ ,«2") = At, n(aodal... do2") Vo, 6 ><
302
А„ = (-1)" Bn - 1)... 3.1, defines a normalised cocycle (tp?n) in the (di,</2) bicomplex (resp.
() = *'„ /i(), with \'n = (-1)" 2пBП - 2).. .4.2Л
b) Let (tpin) (resp. (ip^n+i)) be a normalised cocycle in the (di,d2) bicomplex. Then the following equality
defines an even (resp. odd) linear form /i on the universal differential algebra QA such that /1 satisfies 3)
and 4):
E) li((aa + al)dal...da2n) = K4^(ao,...,a2n) + c,(Bo (P2n)(o1 a2"))
(resp. /i = A'~'(^2n+i +a Ba tpin+\)) Vo» 6 -4, A 6 C.
Proof, b) Let us check the even case. We let ф2п - К1 fin so that Ba ф2п is cyclic (use definition 5) and
one has:
Во Ф2п = Ь ф2п-Ъ Vr>.
We shall first show that for any a E A, da belongs to the centralizer of the functional /1 defined by E). The
equality
rtdatfa1 ... da2"-1)) = (-IJ" f((dal... da2"' ^da)
follows from the cyclicity of Ваф2п. One has Ваф2п = Ьф^п-2 so that ЬВ0ф2п = 0, and also ВаЬф2п = О,
since Ъф-in is cyclic. Thus, the equality Bob +b'Bo = D (chapter 3) entails that:
ф2п(а°,...,а2"-1,а)-{-1J" ф2п(а, а°,..., a2"'1) + (-1J" B0^»(eo°,o1,... .o2") = 0,
i.e. that
rfdaiaOda1 ... da2'-1)) = (-IJ" ^((a^o1 .. .da2""')*»)¦
Thus, it follows that any du belongs to the centralizer of /1. Let us now show that
/i(ow — ui) = /*(da dui) Vo?j4. (*)
With ы = a°do1 . .. da2n one has
li(wa) = i,(aa(dal ... do2")a) = Ф2п(а°, a1,..., a2", o2no)
-02n(ao,o1,...,a2n-V",a) + ...+
+ (-1У 02n(a° a2"-Ja2"-J + 1,... ,0) + ... +
Thus
/j(wo-aw) = 6i/.2n(o°,o1,...,o2",o) = Bo Vn+2(a°,... ,o2",a)
= /i(du da) = —]i(da du).
Finally, we just need to check that if и, = о du is of degree i, with о 6 Л, and w2 6 О is of degree k2, one
has 3). Since du is in the centralizer of /1 we have
/i(o dw u2 - dW(w3 a)).
Using (¦) we get
Ы0Ы1) = /4° du u2- (-1)*1' ига|
303

The above proof shows that, conversely, any functional /i on QA which is even (resp. odd) and satisfies 3)
defines an even (resp. odd) normalized cochain (г/>2„) (resp. (r/>2n + i)) such that bV>m = Вофт+i for any m,
by the equality
<Ma° am) = p(aa dal...dam).
Thus, since QA is the universal differential graded algebra over A, we get a).
Let us now assume that A is a Banach algebra, and endow the universal differential algebra QA with the -si
norms (ГАгуз!)
where || ||* is the projective tensor product norm on
Qk — A® A® ...® A.
Theorem 9. The equalities D) E) of proposition 8 establish a canonical bijection between normalized entire
cocycles on A and linear forms on QA satisfying C) and continuous for all the norm \\ \\r.
Using proposition 8 the proof is immediate.
The natural topology on QA provided by theorem 9 is not the projective limit of the normed spaces || ||r,
r —> oo as in [Агуз]. It is the inductive limit, for r —> 0,
Urn (QA,\\\\r).
For each г > 0 the completion of QA for the norm || ||r is a Banach algebra Or, and for r' < г the natural
homomorphism: Qr —> Qr/ which is the identity on QA, is norm decreasing. Let then:
Q,(A) = lim Qr(A) , r > 0.
By construction ?le(A) is a locally convex algebra and the homomorphism:
o
given on each Пг by the augmentation J^ и„ ^шо of QA, is continuous.
Proposition 10, 1) A linear form /i on QA is continuous for all the norms || ||г, г > 0 iff it is continuous
on Qt(A).
2) Lei J = Kers, e : Qe(A) —• A. Then any element ш ? J is quasinilpotent, i.e. Al — и is inverlible in
QS(A) for any А ф 0.
Proof. 1) is clear.
oo
2) One has uffl, for some г > 0 and ы =J2 ш„, ш„ 6 Q"A with E г" ||ш„||» < oo. Replacing r by r1 <C r,
l
one can assume that ЦА^!^. < 1, and the answer follows since Or* is a Banach algebra.
Finally QZ(A) is by construction a Z/2 graded differential algebra, since the differential d of Q(A) is continuous
for all the norms || ||r. The range of d is contained in the ideal J of proposition 10.
304
¦y) Traces on QA, ?A
In this section we shall reformulate theorem 8 of section /?) in terms of a deformed algebra structure on QA
which yields the Cuntz algebra QA ([C]). This procedure is similar to the one used by Fedosov [F] in the
context of manifolds. The general deformation procedure is:
Lemma XI. ([Co-C]) Let (Q,d) be a differential Z/2 graded algebra and A 6 C. The following equalities
define an associative bilinear product on Q = Qev + Oodd :
u\ -\ ui = UJ1U2 + A u\ du^ if ^
ij\ -д W2 = bj\u>2 if Ui G OeV , W2 € Q-
It is easy to check directly (cf. [Co-C]). The corresponding algebra is equal to О for A = 0 and is independent
of A for A ^0.
The Z/2 grading of П, i.e. the involutive automorphism
extends to a Z/2 grading of the deformed algebra by the equality:
<тл(ш) = (-l)^" (u-A du).
One checks directly that <rA is an involutive automorphism of the deformed product. We then have the
following natural interpretation of condition 3) of section /3) above, in terms of the deformed product:
Lemma 12. ([CooO]) Let Q,d,-\ be as in lemma II and a\ be the Z/2 grading of the algebra (O,-a). For
any odd linear form т on (О, ¦д.о'а) let r be Us resiriciion to OQdd extended as 0 on Qev. Then tke map
т —* т is a canonical bijectton between odd traces on (Q, \, (T\) and odd linear forms on (O,d) such that:
C')
,Ы1) = \ A2 (-1)''
Proof. Let first т be an odd trace on (П, -\) and т the corresponding linear form on Q. To check C') one
can assume that ui\ 6 fiodd, шг е Qev ¦ Then the equalities
= ^ т(ш -<
for ш = Ш1 dui, imply that
— Auj) —
A2
" 2
which gives the equality C').
Conversely let /t be an odd linear form on П satisfying C'). Then define the linear form r on (П, a) by:
Vu;
Vw 6
By construction /j vanishes on Qev and is the restriction of r to Qodd, thus /1 = f. Let us check that
ro(rj = -r. One has for ш ? fiev,
305

сг\(ш) =и — A dw , т(сг\(ш)) = т(ш — A du) = — /i(dw) — A ii(dui) = — — n(du>) = —т(ш).
For u 6 nodd one has сгл(ш) = -w + A du, т(о\(ы)) = -t-(w) + A r(du) = —т(ы) since d2 — 0.
Using C') one checks in a similar way that r is a trace.
The above lemma 12 has an analogue in the even case. One introduces for that purpose the algebra ?\ which
is the crossed product of (П, л) by its Z/2 grading automorphism o\:
?x = (О,-л)»1„А Z/2.
We endow Sx with the dual Z/2 grading ox- One has o\(F) = -F, д\(ш) = ш Vu € Q, where F, F2 = 1 is
the element of the crossed product associated to the generator of Z/2.
The direct analogue of lemma 12 is then ([C02O]):
Lemma 13. Let?\, o\, F be as above. For any odd linear form т on?\ let r be the restriction ofu —> t(Flj)
to П" extended by 0 on nodd. The map r —¦ r is a canonical bisection between odd traces on ?\ and even
linear forms on (O, d) satisfying condition C') of lemma 12.
The proof is similar to that of lemma 12.
Notice that the above lemmas have been proved in the case of arbitrary Z/2 graded differential algebras,
so that we can now apply them to the differential algebra Q,(A) of section J3) and theorem 9. To match
conditions C) and C') we shall take A = \/5, and for reasons which will soon become clear we shall denote
by Qc(A) the Z/2 graded algebra obtained from (П,(A),d) by lemma 11. Similarly we let Qe(A) be obtained
as above as the crossed product of Q,(A) by its Z/2 grading о, <т2 = 1.
Both Qe and Qe are locally convex algebras and theorem 9 together with lemma 12 (resp. 13) yields a
canonical bijection between continuous odd traces r on Q, (resp. Qt) and normalized odd (resp. even)
entire cocycles on the Banach algebra A.
It remains now to identify the universal algebras Qe(A), Q?(A), and their topology.
Let us first proceed purely algebraically and denote by QA the Z/2 graded algebra obtained from (QA,d)
using the deformation of lemma 11 for A = s/2. Also we let QA be the crossed product of QA by its Z/2
grading a.
Proposition 14. ([Co-C]) Let A be an algebra over C.
a) The following pair of homomorphisms from A to QA give an isomorphism of the free product A *c Л with
QA
Pi(a) = aeu°A Va? A ; />2(a) = a - v/2 da Va 6 A.
The Z/2 grading o~ of QA is the automorphism which exchanges pi{a) with Pi(a) for any a ? A.
b) The following pair of homomorphisms from A and Z/2 to QA give an isomorphism p' : ,4*cC(Z/2) —* QA'.
p\ (a) = a 6 п°А Va€A; p'2(n) = F" Vn 6 Z/2.
The Z/2 grading a' of QA satisfies о' о p\ = p\, a-'(F) = -F.
We shall first use the above isomorphisms to translate in terms of QA and QA the equality D) of proposition
8. As in [О12] one lets, for a 6 A, qa be the difference:
306
qa = Pi(a) - p2(a) 6 QA.
Also we shall omit pi when no confusion can arise, i.e. identify A as the subalgebra Pi(A) of QA.
Using lemmas 12 and 13 we translate proposition 8 as follows:
Proposition 15. a) Lei т be an odd trace on QA. Then the following equality defines a normalized cocycle
(() bil
(<P2n+i) I»
bicomplex:
p (
Let т be an odd trace on QA. Then the following equality defines a normalized cocycle (^2n) in the
b)
bicomplex:
VaJ 6.4
A;; = Bn- l)Bn-3)...3.1.
Let us also mention the following variant of a). The algebra QA = QA ~Xa' Z/2 which is the crossed product
of QA by its Z/2 grading a\ is isomorphic to M2(QA). It has a very simple presentation, as in 14b). Indeed
it is generated by a copy of A and a pair of elements F, у of square 1 such that:
c)
~fU =: 07 Vo g A , jF = -F-f
and these relations constitute, with -y2 = F2 = 1, a presentation of QA.
Then 15a) can be written as follows.
Corollary 16. Let r be a trace on QA Then the following equality defines a normalized cocycle in the
i.dj) bicomplex:
2n+i (a0, ..., <r"+1) = n\ r GF a°[F, a1)... [F, a2n+l]) W 6 A.
In [Co-C] J. Cuntz and the author give the generai form of odd traces on both QA and QA, so it might
appear at first sight that such traces are easy to construct and are not interesting from a cohomological point
of view. It turns out, however, that the explicit construction [Co-C] is the translation of the triviality of the
first spectral sequence of the F, B) bicomplex (chapter 3). Thus, when A is a Banach algebra, this explicit
construction becomes incompatible with continuity and does not exclude the existence of nontrivial traces.
Let Л be a Banach algebra. We endow the algebra QA with the locally convex topology inherited from
the inductive limit topology of HA (theorem 9) by the deformation isomorphism (lemma 11). We let QCA
correspond in the same way to П..4. Thus QA С QtA is a subalgebra of QCA and its topology is the
restriction of the topology of QeA. Continuous (odd) traces on both algebras correspond bijectively to each
other by restriction and extension by continuity.
The crossed products QA, QA and QSA, QCA inherit their topologies in a canonical way since they are
crossed products by finite groups. Theorem 9 reads now as follows:
Theorem 17. Let A be a Banach algebra. The equalities of proposition 15 and corollary 16 establish a
canonical bijection beiween continuous odd traces on QtA (resp. Q,A, resp. QCA) and entire normalized
odd (resp. even, resp. odd) cocycles on A.
307

In the statement one can use indifferently QA or QeA, The structure of the locally convex algebra QCA is
very similar to that of QtA as described in proposition 10. First one also has the augmentation morphism;
t:QcA-*A.
It is given by the same formula as s : Oe A —* A, and the fact that it is a homomorphism is insensitive to the
deformation of the product (lemma 11). In terms of the difference map q on Q,A one has e о q = 0.
Proposition 18. Let J = Ker ?, ? : Qe A—> A. Any element x ? J is quasinilpotent, i.e. A1 — x is inveriible
in QtA for any А ф 0.
Thus QeA is a quasinilpotent extension of A. This fact plays a decisive role in the construction of the
character in section 8 below.
S) Pairing with A'o(A)
The computation (proposition 4) of the entire cyclic cohomology of С yields a natural pairing of Hf(A)
with Ko(A) for any Banach algebra A. We developed it in [C02O] for the normalized cocycles (definition 5)
which by lemma 6 is not a restriction on the pairing.
Lemma 19. Lei (Фзп)п?Ы be a normalized eniire cocycle on A. Then if ф ? Im3 С C*v, one has
о
/от any idempoient e 6 A.
Proof. Let (V»+i) 6 Cfdd be such that дф = ф. Thus for each n,
ф2п = 2n *1/>2„_1 + Jj^y Вф2„+1.
Now since ф is normalized, Воф
is cyclic for any n. Let
7J" is cyclic so that
ВпЬф2п-1 = тг-
in
2 t
Вф2п+г(е,..., e).
One has, since e2 — e, that
an = (Ь'Воф2п+1){е,... ,e) = ({D - В0Ь)ф
= {Оф2п+1)(е,..., e) = 2ф2п+1(е,... ,e
Also
Thus
so that
е,..., e) = 2r> - an_i + а„,
' e) =
о
308
Next, let q ? N and Aq — Mq® A— Mq(A) be the Banach algebra of q x q matrices over A. For any ф 6 С™,
let фя be the natural extension фя — Tr # ф of ф to Af?(A) (cf. chapter 3 section 1), i.e. by definition
фяA1° ®a° p™ ®am) = Trace(^° ...цт) ф(а°,... ,am),
where ц> 6 Mq(C) and a> 6 A.
Lemma 20. 1) For any entire even (resp. odd) cochain (ф2п) (resp. (фъ>+\)) on A the cochain (ф2п) (resp.
(ф2„+1)) on A is also eniire.
2) The map ф —* фя is a morphism of the complexes of eniire cochains.
Proof. 1) One has an inequality of the form \\фя\\ < ?m||^|| for ф 6 С", hence the answer.
2) cf. chapter 3 section 1.
Theorem 21. Lei ф = (<^2n)n6M be an eniire normalized cocycle on A, and F+,
n\ n
the corresponding entire function on M^IA). Then the restriction of F^ to the idempoienis e = e2, e ?
defines on additive map: Ko{A) —> C. The value (ф, [е]) of F^(e) only depends upon the class of ф
Proof. Replacing A by A and ф?„ by ф?п,
one can assume that each ф-г„ vanishes if some x', г > 0, is equal to 1. We just need to show that the
value of F4, on e ? Proj МЯ(А) only depends upon the connected component of e in Proj Mq(A). Since
the map ф —> ф9 is a morphism of complexes, we can assume that 9 = 1. Then let t —> e(() be а С map
of [0,1] to Proj(A). We want to show that {d/dt)Ft(e(i)) = 0. One has (d/dt)(e(t)) = [a(t),e{t)\, where
a(t) = (l~2e(t))(d/dt)e(t). We just need to compute (d/d^Fii,^)) fort =0, and we let e = e@), a = o@).
We have:
<fo»(e, ¦¦¦.[<»,e], .-.e).
Thus, by lemma 11, in order to show that the above derivative vanishes, it is enough to prove that the
following cocycle (</>'2n)neN is a coboundary.
Let
Using the equality Вфгп-1 = А9?„_2, where
one checks that for any n one has:
d\
309

In terms of the topological algebra QtA (cf, proposition 10), this pairing has a remarkable expression:
Theorem 22. [Co20] Let т be a continuous odd trace on QCA. Then the map of K0(A) to С given by
theorem 21 and the entire even cocycle associated to т is obtained by the formula
e 6 ProjA -+t\F-
Proof. Up to an overall normalization constant, the entire cocyle ф associated to г has components ф^„
given by
-Ы<Л •.. У") = (-1)" 2""Bn - 1).. .3.1 T(Fa°q{al).. .q(a2n)).
Thus, the answer follows from the formula giving F^.
Remarks 23. 1) The normalization condition for the cocycle ф in lemma 11 and theorem 13 can be removed
(cf. [Ge-S]) by the following minor modification of the formula of lemma 11:
4" ¦)
2) When A is a C"-algebra then ? A has a natural (non complete) C"*-algebra norm which defines a stronger
topology than the one used in theorem 14. There are however interesting continuous traces on ?A for the
C*-norm, used in [Co-Mi].
3) The above pairing and theorem 21 apply without change to the case of arbitrary algebras over C, using
remark 6 to define entire cyclic cohomology in that generality.
e) Entire cyclic cohomology of S1
By theorem 26 of chapter 3 section 1 the periodic cyclic cohomology H"(A) of an algebra A with finite
Hochschild dimension n is exhausted by the image in H"(A) of the cyclic cohomology groups HC(A),
q < n. In order to obtain a similar result for entire cyclic cohomology (either using remark 6 or assuming
that A is a Banach algebra), one needs to construct a homotopy (Tt, h > n of the Bar resolution, with a
good control on the size of o~k for large k. We shall do this here for the algebra A of Laurent polynomials
and compute its entire cyclic cohomology (as defined in remark 6). This will in particular give the analogue
of theorem 21 for the odd case. We shall then explicitly compute the map from entire cyclic cohomology of
function algebras A over Sl to the de Rham homology of S1 and interpret the formulae as deformations of
the algebra Q,A.
Let us first recall that given an algebra A, the standard Bar resolution of the bimodule A over В = A®A°
is given by the acyclic chain complex (Mt,b)
bk(l ® ai
® 1)® (a2 <g> .
(-lO 1 ® a,
...® ak
+ (-1)* (l®oj)® (о, ® ...® ot_,) Vaj eA,
which suffices to define bt as a B-module map from Mt to Mt~i (cf. chapter 3).
In the topological context (cf. chapter 3 appendix B) the above tensor products are тг-tensor products of
locally convex vector spaces.
310
Let us take for A an algebra of functions in one complex variable г. Since A is commutative, we can ignore
except for notational convenience the distinction between A and A0, and consider any element / 6 Mk —
A®A°® A®k, as a function of к + 2 variables:
We do not yet specify the domain of the complex variables nor the regularity of /, but one can have in mind
the case of Laurent polynomials A — C[z, z].
Let us define a S-module map an : Mn —> Mn+\, n > 1 by:
{<Т„ /)(Z,Z°,ZU ...,*„+,)= (-1)" f(z,Z°, *!,..., zn) +
(_1)n+1 ?2±i_L?! (f(Zizo,zu...,zn_u:n)-f(z,z°,z1<...,zn_1,za)).
Lemma 24. For n > 1 one has 6n+i cn +(Tn_i bn = idn.
For g G Л4п one has:
(-1У g(:,za, zu...,z,-, .-,-,..., *„_,) + (-!)" j(*,*0,*,, ...,*„_,, z°).
Using this equality one computes {bn + i ""л + Vn-i bn)f and gets the desired equality.
Theorem 25. Lei A = C^,;] be ike algebra of Laurent polynomials. Then iis (algebraic) entire cyclic
cohomology is given by
with generators given by the cyclic cocycles,
MI) = J fU) dz ,
We shall sketch the proof and at. the same time introduce useful notations for the general case. We first
specialize the general homotopy a of lemma 24 to the bimodule A" over A which yields linear maps:
а„ : C"+l(A,A') — C"(A,A")
such that an b + b an_i = id Vn > 1.
The transposed map a'n : A9" _ д®("+" js given by:
(«UK.-о,..., ;,,+i) = (-1)" /(*>,.... z«)
+ (-!)»+' '""+1 ~ -'" (f(zo,...,z^uzn)-f(zo,...,zn.uzo)).
Given a cocycle (we take it odd for clenuiteness) (^2t+i)teM in the (b, B) bicomplex (chapter 3 section 1)
one gets a cohomologous cocycle (vifc + ibe* w't'1 Vst+i = 0 Vi > 0 by adding to <p the coboundary of the
cochain (ф2к)к?Ы whose components are given using the homotopy a by:
311

ф-, = a <p3 + aBa <ps + ...
These formulae are standard homotopy formulae for cocycles with finite support in any bicomplex, and the
only difficulty we have is to show that they continue to make sense for entire cocycles which have arbitrary
supports. Since we are dealing with the algebraic form of entire cohomology (remark 6) and use the F, B)
bicomplex instead of the equivalent (di, ^2) bicomplex, the growth condition on our cochains is (cf. table or):
"For any finite subset S С A there exists С = Cs such that:
\<P2k+i(a°,..., a2^1)! < С (*!)-' W € E."
Given a finite subset S of the algebra A of Laurent polynomials, we let rf(E) be the maximal degree of
d
elements of S, so that any / € E can be written as ]T /,• zj.
The estimate 011 the homotopy a which allows to complete the proof of theorem 25 is the following:
Lemma 26. a) Lei f ? .4®(ra+1) be a Laurent polynomial of degree at most d in each variable Zj, then
an B' f E Л®("+1) has the same property.
b) Lei || ||, be the I1 norm on A®" for any n,
Then if the degree of f is less lhan d, one has:
IK BS /Ц, <Bd+2I1/11,.
Proof. Since the cyclic permutations do not change the degree, it is enough to prove a) for a'n Bj /• One
has:
(Bj /)(*o, ...,*„+,) = /(г, *„ + ,)-(-1)" /(го,..., г„)
so that:
(«Ui B'o /)<*o, ¦ ¦ ¦, --„+,, г„+2) = (-1)"+1 (Bj /)(го,..., zn+1)
The conclusion is clearly true for (BJ /)(z0 г„+,), and we just need to deal with the other term. Also
we may assume that / is of the form:
/(го --„) = Л(г0,..., г„_,) Jj with \q\ < d.
We then just have to evaluate the Laurent polynomial:
For q > 0 we get:
312
For q < 0, q = —p, we get:
(zn+2 - z0) (<+1 *o ' + <++\ *;¦ + ¦ ¦ ¦ + Vli 4) •
In both cases we check that the degree is less than d, which proves a). We also check that the I1 norm of
the right hand side satisfies the inequality b).
We can now complete the proof of theorem 25. First, the formula
ergent. Indeed, given a finite subset S of A, there exists by hypothesis С =
such that
Thus, taking for S the monomials Л zq, \q\ < d, it follows that for any Laurent polynomials fj of degree
less than d one has:
Using lemma 26 and the equality В = ABOt we thus get:
|*>jm+jn.i ((Ba)'- (/o,...,/2t+i))| < CKi (m + ky
for any /у ё A of degree less than d.
Taking Л small enough thus gives a constant Cd such that:
|((«à ?>2m+3t+i) (/0,.. ¦, /3t+1)l <
i!) П ll/jlli
for any fj ^ A of degree less than d.
This shows the desired convergence and also that (-фгк) is au entire cochain.
As an easy corollary of theorem 25 we get for any locally convex algebra A a pairing between K\(A) and
Н^Л(А). Indeed any invertible element u e GLn(A) determines a homomorphism of the algebra A of
Laurent polynomials:
/>„ : A — Mn(A),
and the pull-back />* <p of any odd entire cocycle ip on A is cohomologous to a multiple A T\ of т\. One sets
(«,?>) = A.
The explicit formula for A follows from the above proof and was also obtained by [Cu-Q]:
Corollary 27. Lei A be a locally convex algebra. Tlie following equality defines a pairing between K\(A)
and ЩМ(А):
Vu g СЬ,(Л) С Мд(Л) »»d а?(</ normalized cocycle <p i« (Ae (b,B) bicomplex.
The notation <?' was introduced above in lemma 20.
313

We shall now go much further and compute explicitly, given an entire cocycle (<P2n+i) on A = CZ as in
theorem 25, the entire cochain ф = (i>2n) such that:
<P2n + l = Ь ф2п + В ф-2п+2 Vn > 1.
The initial formula for фи which we have to simplify is
4>2k = a ?>2i+i + ct(Ba) <p2h+3 + ¦¦¦¦
= 1 for j ф 1.
In order to simplify the computation we shall assume that
(*)
„+i(/° /2"+1) = 0 if s
This is a weak normalization condition. At the level of chains, i.e. of A®^n+1\ it means that we can ignore
any function /(z°,..., г") which is independent of some z', j> ф 1. This simplifies the formula for the map
aln Bq of lemma 26, which gives now:
0,. ..,2n+1) - (-1) гп+1
i,... ,гп_ь го)
for any /еЛв<"+1>.
Thus a'n Bq is essentially a divided difference, and the iterates (a' B')" will invoke iterated divided differences
which are known to satisfy remarkable identities.
The computation is now straightforward, and the result is best formulated in terms of the algebra QA of
section 7). We recall that the latter algebra is generated by A and two elements F,7 with F2 = ~f2 = 1,
70 = a-f Vo ? A, -yF = -F7.
Because of the weak normalization condition (*) above, the distinction between A and A is unnecessary, so
that the unit of QA is now the unit of A.
One then lets r be a trace on Q?A and (v>2n+i) the cocycle in the (b,B) bicomplex given by:
?>2,. + l(aV.-,a2"+1)=*,, rfrF a^F.a1]. ..[F,cr"+1]) W &A
where t~l = Bn + l)Bn - 1) .. .3.1. One gets, with и the generator of A:
Lemma 28. With ip = (tpin + i) assoczaied to the trace r on QA as above, the cochain ф = (ф2п) such that
4>2n+\ = Ь i>2v. + В 4>in+2 Vn > 1 is given by ф->п = а' (<?2„-ц - А в7п+\) where:
...[7F,f"+4u+\[F,u]))
where d^n(A) = 2"n („A - 4A2)"+* d\.
Note that one has used the possibility of applying the Laurent polynomials /] ё A to any invertible element
of an algebra, and in particular to u + \[F, u] which is invertible in QtA.
This formula fits with our quantized calculus where the quantum differential (cf. introduction) is given by
the graded commutator with F, or equivalently 7 times the commutator with yF. Thus f(u + X[F, u]) plays
the role of /(u + A "du"). The following formula is the only one needed in the proof of lemma 28:
¦ 314
7 [jF, /(« + A[F, «])] = BA) (/(« + A[F, «]) - /(« - A[F,«])).
It relates the quantum differential of /(u + A "du") to the difference quotient Iiu+X "du")-/(u-x "ju">
The proof of this formula is straightforward for Laurent polynomials, or more generally for f(u) = (u — z)~1
for which both sides are easy to compute.
We shall now interpret the formula of lemma 28 using a natural deformation of the algebra QeA to an
exterior algebra over A.
Let us define for A € [0, | [ an endomorphism <тл of QtA by giving its value on the generators as follows:
a) <rx(u) = u + \[F,u]
b) <rx(F) = F
One checks easily that <гд{7)" — 1 a"d that <r\(j) commutes with стд(и) and anticommutes with F.
Moreover one checks that, the ffj's form a semigroup:
o-\ о o-y — ax» with 2A" = ——.
1 +4AA'
With these endomorphisms we can rewrite for any / € A,
[yF, /(« + \{F,u])] = A - 4A2)-1/2 ax (-f[F, /(«)]),
so that the formula for flsn + i (lemma 28) simplifies to:
(rj{/o)(rj([F]/Ib ,.[F,/2"+']7F)) dX.
Since r is a trace one can replace the endomorphisms <Гд by the inner conjugate ones, or,, given, for s € R by
a,(u) = -(« + F11F) + e— Uu - FuF)
a.(-r) = 7 , a,(F) = F.
One checks indeed that with s =¦ -5 log(l - 4A-) one has
a,(x) = Z~' ffj(i) Z, Vx e Q,A
with Z, = ch I + F sh k , 2A = th t.
Thus the above formula can be written:
92n+i(/°,...,/2"+I) =2"" *»/"*"¦ ((^ e.(/o)) «, ([F,/']... [F,/2n+1] 7
(One has to check that the terms in Jj Z, do not contribute, but this is easy.)
We thus get the following reformulation of lemma 28.
Lemma 29. Let (а,),ея be the one parameteT group of automorphisms of Qe(A) given by ct,(u) = %(u +
FuF) + e-' i(u - FuF); «,G) = 7, a,(F) = F Vs e R. Let then <p = (y2n+i) *e (Ae entire cocycle on A
associated to a trace r on Q,A, and ф = (ф?„) 1he entire cochain given by theorem 25:
315

Then one has fan — ос i/4n+i where:
where S = (jL a,)t_0 is the derivation associated to a,.
Given the above formula for 02n+i and the equality:
1p2n = Ct (<p2n + l '
of lemma 28, the proof of 29 follows using the equality:
-1] S(P)
2» + l -oo
?>2n + l(/ , • ¦ • , f-n"T ) — у I ds(T о or, J
о Jo
[FA?)] [FJi+1]...[FJ2n+1]j if e л
which is a consequence of the vanishing of lim a, ([F, /]) V/ € A. The next step is to compute a r/4n+i.
One uses for this the identity:
where u is the canonical generator of A and f: = (z - u)~' where г is a formal parameter. Moreover using
the equality:
it follows that
[F.fA [F,u] Л = 2*(Л) -[F,A]F «(Л) [F,u] Д + [F Л] «(«) Л = F S(f.) + 6(f,)F.
These identities suffice to express the value of ф'7п+1 on a'(f° ® ... ® f2"-1 ® Д) as a linear function of
A, so that one can then replace A by an arbitrary function /2" and obtain the following formula for i>in,
4 Фк + 4l
V--,/2") = «n Г
J о
ds(r0 a,)(JT(-iy 7F
Г) [F /J+1]... [F, f2"] - 7 /V, /']¦¦• [F, Z2"] F
316
We are now ready' to conclude that the homotopy formula of theorem 25 is a special case of the following
general proposition involving now an arbitrary algebra A (which is no longer an algebra of functions in one
variable).
Proposition 30. Let A be an algebra, т a trace on QA, and S a derivation of QA such thai 6(-f) = 0.
a) Lei ^2n+i(/0 f2n+1) = („ r (jF f[F, f1] ... [F,/2"+1]) 1f> € A and i>\n(fu f2n) =
in (E (-1У r {lF P>[F, f1]... [F, P-1] S(P) [F, P}... [F, f2"]) - r G f°[F, fl] ... [F, f"'1] F 6(f»))
Then for any n one has:
b) Lei VL(/°.-.-./2") =E ("I
V/' € А. ГАеп Ьф2„ = В ф1„ = 0 Vn.
We have used the short-hand notation 6 V211+1 to mean:
The proof of this proposition is straightforward. We can now conclude this long computation by the following:
Theorem 31. Lei A — C(Z) be 1he algebra of Laurent polynomials, <p = (<p2n+i) «" entire normalized
cocycle on A, and т the corresponding trace on QA. Let (ф->п) be the entire cochain
Then i>in = /~ ds (^i,,(s)+ Ф1„(в)) where 0i,,(s) is given by proposition 30 applied to the trace то a, on
QA and the derivation 6 = (j; «»M_0-
We assume that <p satisfies the weak normalization condition that <p?n+i(f°,..., /2n+1) = 0 if some P = \
for j ф 0.
317

IV.8. The Chern character of #-summable Fredholm modules
In this section we shall extend the construction (of section 1 of this chapter) of the Chern character in
A' homology to cover the infinite dimensional situation of 0-summable Fredholm modules. Recall that a
Fredholm module (H, F) over an algebra A is 0-summable iff for any a ? A one has:
[F,a]eJ1/2
where J1'2 is the two-sided ideal of compact operators T in H such that pn{T) = О ((logn)'2) for n —> со.
This is the content of definition 4 of the introduction.
We shall first establish (theorem 4) the existence of an unbounded selfadjoint operator D in It such that,
1) SignD = F
2) [D,a] is bounded for any a g A
3) Trace {e~D') < oo.
Condition 2) means that the pair Gi,D) defines а Л' cycle over A (section 2 definition 11).
Definition 1. А К cycle (H,D) is в-summabk iff Trace(e-D') < oo.
In the present section we shall construct the Chern character of 0-summable K cycles as an entire cyclic
cohomology class:
Ch(H,D) eH"(A)
with the same parity as Gi, D), i.e. even iff we are given a Z/2 grading 7 commuting with any a g A and
anticommuting with D.
The first construction was done in [C02O] using the algebras QA, QA of section 5 and an auxiliary quasinilpo-
tent algebra of operator valued distributions with support in [0,+oo[. Another construction, but of the same
cohomology class, was then done by [Л LO] whose formula for the entire cyclic cocycle is simpler and easier to
use than our formula ([C026]). There are however important features of the construction which are appearant
only in the original approach, such as normalization of the cocycles and the use of the super group R1'1.
We shall thus expound both approaches (sections /?) and 7) below). In sections 9 we shall concentrate on
examples of 9-summable A* cycles coming from discrete groups and supersymmetric Quantum field theory.
a) Fredholm modules and A' cycles
Let H be a Hilbert space, and let
= 0(logn)-1/2 n —00}
J = {Тек ; Li,l(T)
By construction one has T g Jl/- <S> |T|2 g J.
Lemma 2. Both J and J1/- are two sided ideals of operators, and Jl^J1^ = J.
Proof. The inequalities M,,(TiT2) < ||Ti|| /«„(Г2), Mn(TiT2) < Pn(Ti) ||T2|| for any n € N, and:
/<»+m(T, + T2) < )in(T1) + /jto(Tj) Vn, ro e N
(section 2) show that both J and J1'2 are two sided ideals. The inequality
< ^(Ti) iim(Ti) Vn.raeN
318
shows that Ji/'2J1/- с J and the conclusion follows.
The natural norm on the ideal J is given by:
N
The sum Yl C°gn) 1 is equivalent to the integral logarithm
Jo bgu
and it is better to define:
This yields the following natural notations:
J = Li(H) , J1/2 = Ы
As for any normed ideal, one has:
Lemma 3. Lei A be a C'-algebra and (H,F) a Fredholm module over A. Lei A = {o g A ; [F,a] g
'^}- Then A is stable under holomorphic functional calculus.
It follows that when CH,F) is 9-summable over A, i.e. when A is norm dense in A, then A and A have the
same К theory.
We shall now prove the following existence theorem.
Theorem 4. Lei A be а С -algebra, Gi,F) a Fredholm module over A, and А С A a couniably generated
subalgebra such thai:
[F, a] g LiinD) Va g A.
Then there exists a selfadjoini unbounded operator D in.7i such thai:
1) SignD = F
2) [D, a] is bounded for any a g A
3) Trace (e~D') < °°-
Proof. Since В = {T g C(U) ; [F,T] e Lil/-(H)} is selfadjoint and stable under holomorphic functional
calculus, we can enlarge A and assume that it is generated by a countable group Г of unitaries, u g Г, which
contains F. (One has [F, F] = 0 so that F g B.) Using Г we shall define an averaging operation of the form:
where the weight function p is such that:
a) p(u) > 0 Vu g Г
b) sup {p(v)~l p(uv) ; v e Г} < со Vu g Г
r
319

The existence of such a weight function p on a group Г is inherited by any quotient group Г" = h(T) using:
P'(u)= Y, p(v) v«er'.
h(v) = u
It is also inherited by any subgroup Г" С Г.
This existence is easy for say the free group on 2 generators, taking p(g) = e~^''s' where l(g) is the word
length of g e Г and 0 is large enough to get c). Thus it follows for the free group with countably many
generators, which is a subgroup of the free group with 2 generators. Using quotients it then follows for
arbitrary countable groups.
We shall thus fix a weight function p on Г satisfying a) b) c). The operation M from C(H) to C(H) is linear,
positive (i.e. M(T) > 0 if T > 0) and quasiinvariani under Г:
uM(T) u < А„ M(T) VT>0
where Au € R+ depends only upon u g Г. We shall normalize p by J^ P{u) = 1> so that M(l) = 1. Also
since F € Г, we сац assume that M(T) commutes with F for any T (replace M by ^(M + FMF)).
We shall now conjugate the linear operation M by the nonlinear transformation / defined for T positive and
bounded by
/(T) = exp(-r-').
(We take of course /@) = 0.)
The inverse transformation f~l is defined for 0 < T < A < 1 by:
Г'(Т) = -(bg Г)"'.
This non linear transformation is opemtor monotone (cf. [P]) i.e.
0 <T, <T2 < 1 ^ rl(Ti)< f-l{T2),
since it is a composition of the operator monotone functions log and — 1/x.
For any T e C(H) , T > 0, we let:
It is by construction a positive bounded operator.
Next, let us consider the quantum analogue of a Riemannian metric, gfltf dx** dx", for our algebra A as being
given by a positive operator in Ti of the form:
where the x*1, (i?N generate A and дц„ is a positive matrix of elements of A. For our purpose we shall
choose the x" to be unitaries «** generating the group Г, and take positive scalar coefficients g,,,, > 0 such
that
is a convergent series in ЫСН), which is easy to achieve (taking e.g. gL1 = 2"*1 \\[F, и*1]* [F, «''Щ^,')-
The operator G will play the role of the Green's function, so that our choice of operator D will be specified
by the equality:
320
We shall now prove that D = F{&(G))~1/:! has the required properties. We have to be careful to insure that
Kere(G) = 0. For this let us consider the subspace of %:
¦Ho = П Ker([F, u]»).
By construction Ho is Г-invariant (and hence F invariant since F ё Г). On Ho one has [F, u] = 0 Vu 6 Г.
Thus the Fredholm module (H, F) decomposes as a direct sum of the trivial part (Ho, Fo), with Fo = F\Ho
commuting with any о € A, and of a Fredholm module (H1, F') for which Ho = 0. Let us assume that
Ho = {0} and proceed to show that Ker9(G) = 0. By construction Ker f(T) = KerX for any T, so that
with T = G one has Ker f(G) С П Ker[F, u*]. Next,
. KerAf(/(G))C П Ker([F, «"]»)= П Ker([F,u]«)
since the u" generate Г. Thus we get Ker6(G) = KetM(f(G)) - {0}.
Next, we shall show that for any и € Г the operator
и e(G)-1 и -e(G)
is bounded. One has и 6(G) и = /"'(и M(T) u~l). Thus
и e(G) «"' = -\oS(uM(T) u).
So the answer follows from the general inequality, valid for any T > 0,
A;.1, M(T) < и M(T) u < Ao M(T)
and the operator monotony of the logarithm:
\og(M(T))~ log(Vi) < Iog(u M(T) и) < 1о8(Л<(Г)) + log(Au).
It follows immediately that
is bounded for any и ё Г, and hence that since Г generates Л linearly,
||[e(G)-l/2,o]| <oo We A.
Thus F[e(G)~1/2, a] is bounded for any a e A.
Since D = F e(G)~1/>2, in order to prove 2), i.e. that [D,a] is bounded for any a g A, it remains to show
that:
is bounded for any of the generators x/l of Г appearing in the formula for G.
For this it is enough to check that 8(G) > AG for some A > 0. But /"' M(f(G)) > f'\p{l) /(G)) =
(G-1 - log^l))-1 > AG for A-1 = 1 - ||G|| loSp(l).
Finally we need to prove 3). Since we can replace D by an arbitrary scalar multiple of D without altering
1) and 2), it is enough to show that with the above construction one gets:
e~D* 6 С for some finite p.
Since e~°3 = M(f{G)), it is enough to show that
f(G) 6 С for some finite p.
321

But G € Li(U) so that ^„(G) = O((logn)-1) < c(logn)-1 for some finite c, which shows that A»n(/(G)) <
On Wo the same construction applies and is much easier, starting with Go a strictly positive element of
Li(H0).
Remarks 5. a) The above proof yields in fact also the condition 4): [Д2, a] is bounded for any a & A. This
condition implies that [|?*|,a] is bounded for any ogA
b) Let (H, D) be a 0-summable К cycle over A such that [\D\, a] is bounded for any a e A. Let F = SignB.
Then [F,a] € Ы1'2(П) for any a e A.
c) The above proof does not yield а К cycle such that Trace (e~^D') < oo for all /? > 0. In fact by lemma
6 of appendix С there exists a constant С < со such that for any unitaries щ,
&oo {";} < С sup ||[Д2,и(]||
where one assumes Trace (е~°2) < oo. Here kx is the obstruction ([Voi]) to the existence of quasicentral
units relative to the Ma<;aev ideal ?*"•':
In general, given an arbitrary normed ideal J of compact operators (cf. appendix C), the obstruction kj to
the existence of quasicentral approximate units is defined [Voi], for any finite subset X of C(H) by
MA')=Liminf \\[X,h]\\j
where ft runs through finite rank operators 0 < ft < 1, while ||[A',ft]||/ = Sup ||[x, h]\\j.
i€A-
The results of D. Voiculescu [Voi] on quasicentral units relative to the Ma?aev ideal show that in general
one has
This gives a lower bound to the values of 0 such that Trace (e~^D ) < со.
0) The super group R1'1 and the convolution algebra С of operator valued distributions on
[0,+oo[
In this section we shall construct a convolution algebra С of operator valued distributions and a trace r on
С which will be the natural receptacle for representations of the quasiuilpotent extensions Qe, Q, of section
7 theorem 17.
We let 7i be a Hilbert space. By an operator valued distribution we mean a continuous linear map T from
the Schwartz space S(H) (with its usual nuclear-space topology) to the Banach space C(H) of bounded
operators in 4. Thus, there exists by hypothesis a continuous seminorm p on S(R) such that ||X(/)|| < p{f)
V/ g 5(R). We let С be the space of operator-valued distributions T which satisfy the following properties:
1) Support (T) С R+ = [0,+oo[ .
2) There exists r > 0 and an analytic operator-valued function l(z) for z ? С = U sU, where U is the disk
*>o
with center at 1 and radius r, such that
a) t(s) = T(s) on ]0, +oo[ , (i.e. T(f) = //(«) t(s) ds V/ e Cf (]0, oo[))
b) the function denned for p ? ]l,oo[,
322
ft(p) = sup \\t(z)\\p
is majorized by a polynomial in p for p —> oo.
Inb), the norm ||<(^)||p is the Banach space norm of the Schatten class ??(?{). In particular we see that <A) e
?'CH) is a trace class operator. The operator-valued analytic function t(z) is, of course, uniquely determined
by the distribution T, and we shall use the abuse of notation T(z) instead oft(z). Two distributions Tb T2 € С
such that Ti(z) = Тг(^) Vz € ]0, +oo[ differ by a distribution with support the origin, of the form E at 6^,
where ak e C{H), and 6^ is the ib-th derivative of the Dirac mass *o at the origin.
The condition 2) essentially means that T takes its values in operators of suitable Schatten class so that the
quantity Trace T(l) is well defined.
All operator-valued distributions on R with support {0} belong to ?, in particular the products So x id,
<50 x id of the Dirac mass at 0, or of its derivative, by the identity operator in C(H). To lighten the notation
we shall simply write <5o, S'o-
Lemma 6. a) Let T € C. Then the derivative V - (d/ds)T also belongs to С
b) Let T e C. There exists an integer q and S € С such thai T - 5('' has support {0} and thai
sup sup ||S(z)||p < со,
P (l/)C/
where V = {z € С, |г - 1| <r].
Proof, a) By definition, T'(f) = -T(f') V/ € 5(R), so that V is an operator-valued distribution satisfying
property 1). Let r and U be as in 2) for T, and let r' = r/2. Then by Cauchy's theorem the operator T'(z)
for z e A/P) U', U' = {г € C, \z - 1| < r'}, is of the form /„eA/(,)t, T(v) dp(u), where fi has total mass less
than 2p/r. Thus
sup ||Г'( = )Пр < — S"P \\T(z)\\T,
геО/pW r 2?(l/p)t/
which proves that T1 satisfies property 2).
b) By hypothesis, there exists С < со and q € N such that, with the notations of 2), ft(p) < Ср*. Let Tk be,
for к = 0,1,..., the operator-valued analytic function in С — U sU, defined inductively by T0(z) - T(z)
and Tk+1(z) = /,' Tk(n)du. For z € (lfp)U one has
dt
where
bt(p) = sup ||Т„(г)||р
(since \\Tk(u)\\p < \\Tk(u)\\p, for p' < p).
Thus we see that hk is of the order of pq~k for к < q, and that ft, is of the order of logp, while h,+1 is
bounded. Then let 5 be the operator-valued distribution given by
S(f) = J f(s) Tq+i (s) ds V/ € 5(R).
It is well denned, since ||T,+1(s)|| is bounded on [0,1] and by a polynomial for large s. By construction the
q + 1-th derivative of S agrees with T outside the origin, thus the conclusion.
323

We can now show that С is an algebra under the convolution product, which at the formal level can be
written
= I Ti(u
Jo
) T2(s - и) du.
More precisely, given / € S(R), one can find о„,6„ g S(R) such that the restriction to ] — l,oo[ x
] - 1, oo[ of the function (s,u) —> f(s + u) is given by the convergent series E an ® bn. Then (Ti * T2)(/) =
S ^K) Т2F„)
Lemma 7. 7/Ti, T2 e ? then Ti * T2 e ?.
Proof. By lemma 6 one can assume that T; is given by
Ti(f)= Г f(s)Ti(s)ds,
Jo
where Ti(s) is an analytic operator valued function in С = U (tU), U = {z € C, \z — \\< r), and where
C, = sup sup \\Щг)\\р < oo.
p (Up)u
Then let T(z) = Jo'Ti(Az) T2((l — A)z)i ^A. I' is by construction an analytic operator-valued function
defined in C. One has, for z e (l/p)U, that
where
so that by Holder's inequality.
||Г,(А.-)Т2(A-.
Thus we get, for any z <
— V , A-А)ге —
Pi P2
P Pi P2
C2.
\\T(z)\\p < f \\Tx(\z) T2((l - A).-)||p |z| dX < \z\ C, C2.
It follows that T(z) defines an element of C, and it coincides with the convolution product Ti * T2.
Let A = S'o be the derivative of the Dirac mass at 0. One has A g C, and as an operator valued distribution
A has a natural square root, (for the convolution product) given by
But this square root does not define an element of the algebra C, since (when dim('H) = oo) it fails to satisfy
condition 2) above, because the identity operator does not belong to any Cp. We thus need to adjoin the
square root Л1'2 of Л to C, and for this we consider the algebra С of pairs (Tq,Ti) of elements of С with
product given by:
(To, TO * (So, S,) = (TbSo + ATiS,,T0S,
where we write T0S0 for To * So, etc. Since A belongs to the center of ?, one checks that the above product
turns С into an algebra. This algebra С contains С (by the homomorphism T —> (X, 0)), and the central
element A1/2 = @,(S0) so that every element of С is of the form A + BX1'2 with А, В ё С.
Lemma 8. The equality t(T0,Ti) — trace (Tj(l)) defines a trace т on the algebra C.
Proof. By condition 2) we know that Ti(l) belongs to ?' so that the trace is well defined. Since
(Tb.Ti) * (So,Si) = (TbSo + ATiSi.ToSi +TiS0),
it is enough to check that T —¦ trace T(l) is a trace on the algebra ?. The proof of lemma 7 shows that for
Ti € ? of the form TJ(/) = / /(s) Tt(a) ds, with
sup sup ||T((z)||p < 00,
•p zz(\lp)U
one has:
trace (Ti ¦ T2)(s) = trace (T2 * Ti)(s) Vs € U.
Thus for any power of A one has
trace ((A* T, *T2)A)) = trace ((A1 T2 *T,)A)).
Now by lemma 6, to show that trace (Si * S->)A) - trace (S2 * ¦?i)(l), we can assume that Sj = Xk' Ту +
Uj, where Tj is as above and Uj has support the origin. Thus, we just need to check, say, that trace
(A*1 Til/2)A) = trace (U-, A'1 Ti)(l), which follows from trace (ab) = trace (bo), a € C(H), 6 e ?'(«).
Let us recall that the Hopf algebra Я of smooth functions on the super group R*1'1' is given as follows: as
an algebra one has:
Я = C^fR1'1) = C°°(R) ® A(R),
the tensor product of the algebra of smooth functions on R by the exterior algebra A(R) of a one dimensional
vector space. Thus every element of Я is given by a sum/+<;?, where /,j6 C°°(R), ?2 = 0. The interesting
structure comes from the coproduct Д : Я —¦ Я ® Я which corresponds to the super group structure; being
an algebra morphism it is fully specified by its value on C°°(R) С Я and by ДD) =4®l + l®4l one has:
(Д/) = Д0(Л + До(/') i ® «, where /' = ? f(s) and
До : C~(R)
is the usual coproduct,
D)
Ao(f)(s,t) =
Equivalently, the (topological) dual Я" of Я is endowed with a product which we can now describe. Every
element of Я* is uniquely of the form (To,^), where To.T! e Co~°°(R) are distributions with compact
support on R, and:
(f + gZ,(To,T1))=To(f)+Tl(g).
The product * on Я* dual to the coproduct Д is given by:
E)
F)
Lemma 9. The product * on H* is given by:
(To,Ti)*(So,Si) = (To * So + 6'0 * T, *Sl,T0*S1 +T, * So).
Using ?2 = 0 this follows from formula D). Thus we see that the algebra С is really a convolution algebra
of operator-valued distributions on the supergroup R'11', thus clarifying the relations between our formulae
of op
and supersymmetry.
324
325

7) The Chem character of A" cycles
Let A be a locally convex algebra, and let (H, D) be a 0-summable A" cycle over A (cf. Definition 1 before
section a)). In this section we shall construct the Chern character,
ch.(H,D)e Щ(А)
as an entire cyclic cohomology class for A.
For this purpose we may as well replace A by the Banach algebra A of bounded operators:
A = {a e C(H) ; [D,a] is bounded}
which we can endow with the norm p(a) = \\a\\ + \\[D, a}\\. By theorem 17 of section 7, the normalized entire
even (resp. odd) cocycles on A correspond to odd traces on the algebra Qe(A) (resp. Qc A). By proposition
18 of section 7, QCA is a quasinilpotent extension of A. Thus to construct ch,(H, D) as a normalized cocyde
requires a trace on a suitable quasinilpotent extension of the algebra A. We shall use, for that purpose, the
convolution algebra С of section /?) and the trace r given by lemma 8. The representation of A in H, i.e.
the inclusion А С ?(%), yields a natural homomorphism from A to C, given by
p(a) = a 60 Чае A.
We shall now extend, using D, the homomorphism p to a homomorphism of QtA to C, when the К cycle is
even. In the odd case we shall get a homomorphism
At the formal level, if we denote as above by A1/2 the formal square root of A = 6'0, the formulae for p and
p' are given by:
p(F) = (D + f Xl'2)(D2 + X)-1'-
pl(F) = \u- о] ¦ u ={D + ¦'
(even case)
(odd case)
The next two lemmas are useful for both even and odd cases.
Lemma 10. One has, with s = Re г > 0, p e [l,oo[ ,
Proof. One has
To prove the second inequality it is enough to show that the operator norm of ||Д e't'/
1/v/s. But this follows from the inequality x е"'*/2'*2 < l/y/s for x real and positive.
Lemma 10 shows that we can define an element N of С by the equality
N(f) =-LJ f(s) -L e-">2 ds, fe 5(R).
326
is bounded by
The integral makes sense since the operator norm of 1/y/s e sD is integrable near the origin. We shall,
however, also need to define the distribution DN which is formally given by
/(s) -L D e-«D3 ds_
By lemma 10, one has an analytic operator-valued function,
defined for Re г > 0, and such that sup ||(?W)(z)||,, is of the order of p, p —<• oo. However, since the
operator norm of (DN}(s) is of the order of l/s, s —<• 0, and is not integrable, we have to be very careful in
the definition of the distribution DN.
Lemma 11. a) The Laplace transform of the distribution N is given by /0°° N(s) e~'x ds = (D2 +X)-1/2.
b) There exists a unique element of C, noted DN, whose Laplace transform is equal to D(D2 + A)-1'2. One
has (DN)(s) = D(N{s)) for any s > 0.
Proof, a) Follows from the equality
Jo Vй
b) The uniqueness is clear. Let us prove the existence. One has
= - Г
ж Jo
= 1A _
dp
) Г
Jo
dp.
Now D{D2 + 1)-1/2 is the Laplace transform of the element of ? given by D(D2 + I)'2 *0- Thus we just
have to show, using lemma 10, that /0°° D(D2 + A + p)'1 (D2 + 1 + p)-1 p~^2 dp is the Laplace transform
of an element of C. But D(D2 + 1 + /»)"' (D2 + X + p)'1 is the Laplace transform of D(D2 + 1 + p)-1
e-'(D*+p)^ an(j jt is enough to check that the operator norm of
T(s) = Г D(D2 + 1 + p)-1 e-^D'^ p-1'2 dp
Jo
is integrable near s = 0. One has:
Jo
since \\D(D2 + 1+/»)"'|| < A + P)~4"~¦ Thus since
Г
Jo
+ p)-
dp<(Z- logs) = O(\ log *|)
when s -> 0, we see that the operator norm of T(s) is integrable near 0. The same estimate works for the
С norm ||T(z)||p for z e A/pW, and shows that T 6 C.
Let us now treat the even case.
Lemma 12. The equality F = (DN,-/N) defines an element of С of square So.
327

Proof. By lemma 11, the element DN g С is well defined. Since 7 anticommutes with D, DN anticommutes
with jN, so that the square is given by F2 = ((DNJ + АЛГ2,0). Now the Laplace transform of (DNJ
is (Lemma 11) equal to D2(D2 + A), and that of XN2 is X(D2 + A). Thus, the Laplace transform of
(DNJ + \N2 is equal to 1 and we get F2 = (So,0).
We can now state the main technical result of this section:
Theorem 13. ([C02 0]) There exists a unique continuous homomorphism p of Qt(A) to С such thai:
p(a) = aS0 4aeA
We refer to [C02O] section 5 for the proof of the continuity of the homomorphism/). The convolution algebra
С was defined so as to make these estimates straightforward. Let г be the trace on С given by lemma 8.
Corollary 14. ([C02O]) Lei (%,?>,7) be an even в-summable К cycle on a locally convex algebra A. Then
the following equality defines a normalized even entire cocycle (<^2r»)n€N on A:
<ры{а° ") = Г (n + §)r (p (F a"[F,a1]... [F,a2"})) Va> e A.
Note that <p is here a cocycle in the (di,d2) bicomplex. The corollary follows from theorem 13 and theorem
17 of section 7.
The odd case is treated in the same way. One defines p(F) as the 2x2 matrix . g Mi(C) where:
U = (DN, iN)
with the above notations. Also 7 = g Mi(C).
Theorem 15. ([Co20]) There exists a unique continuous homomorphism p' of Qt(A) to M?(C) such that:
p(a)=aS0 Vaevl, p(F) =
As above one gets the following corollary.
Corollary 16. Let ('H, D) be an odd в-summable К cycle on a locally convex algebra A. Then the following
equality defines a normalized odd entire cocycle (<р2п+\)п^ы on Л:
?>2n+i(a°, ..., a2n+1) = n! t' pf G F a"[F, a1] ... [F, a2n+1]) 4a' 6 A
where t' = r® Trace on M2(C) = С ® Л/2(С).
We can now adopt the following definition:
Definition 17. Lei (%, D) be a в-summable К cycle on the locally convex algebra A. Then its Chern
character ch.CH,D) is the class of the cocycle <p of corollaries 14, 16 in H*(A).
6) The index formula
Let A be an involutive algebra and (H,D,-f) an even A" cycle over A. Then as in proposition 2 of the
introduction of this chapter, the A' homology class of CH,D, 7) determines an index map:
- Z,
328
which can be defined as follows.
Let e € Proj M,(A) be an idempotent, e e Mq(A), and let (Hq,Dq,jq) be the К cycle on Mq(A) given by:
H, =H®C , Dq = D ® 1 , 7, = 7 ® 1
with the obvious action of A ® M,(C) in Hq.
Proposition 18. a) The operator e D+ e from e %+ to e %~ has finite-dimensional kernel and cokernel.
b) The following equality determines an additive map tp from Kq{A) to 2.;
4>([e\) = Index(e ?>+ e).
The proof and the index map are the same as in proposition 2 of the introduction. The operator F = Sign(?>)
in H determines a (pre) Fredholm module on A with the same index map as D.
We shall now show that, when the К cycle (%, D, 7) is 0-summable, the index map tp is given by a formula
in terms of the Chern character ch,{4,D,y) (definition 17).
Theorem 19. [C02O] Lei A be a locally convex algebra and CH,D,y) an even в-summable К cycle over A.
Then for any x g K0(A) one has:
We use here the pairing between A' theory and entire cyclic cohomology of theorem 21 section 7.
Theorem 19 is the infinite-dimensional analogue of proposition 8 of section 1.
To prove it one uses the invariance of the index and of the Chern character under a homotopy of the operator
D. This allows one to reduce to the situation where D commutes with the idempotent e, in which case the
ti f th th i th MKn Sing fmula:
This allows one to ee o e
ertion of the theorem is the McKean Singer formula:
Index (D+) = Trace G e-D").
Indeed the component <po of the entire cocycle (<p?n) of corollary 14 is given by:
f>o(a) = Trace G a e~D ).
The explicit homotopy between D and an operator Di which commutes with D is given by:
D, = D + ii , * = -e[D, e] + [D, e]e.
Finally, the homotopy iiivariance of the character follows from the next proposition applied to the algebra
В = С with trace т of section /3), to the homomorphism p of theorem 13, and to the elements Ft of С
corresponding to D, (cf. [Co20] for the technical details).
Proposition 20. Lei p : A —> В be a homomorphism of algebras, with В uniial, and lei т be a trace on B.
Let Ft € B, i ? [0,1], be a C1 family of elements of В with F,2 = 1V1? [0, 1].
a) For every I g [0,1], the following equality defines a cocycle in the (d\,d2) bicomplex of A:
<p2n(a° a2") = 2-nBn-iy.[r(Ftp(aa)[Fl,p(a1)]...[Ft,p(a2")]) W € A.
b) Let for n > 0 and i € [0,1],
329

Wa° «ta+l) = 2-Bn-l)« Bп)-^(-1У r(F, p(a°)[Fs,P
l
(djdt Ft) [F,,p(al)] ... [F1,/J(a2"+1)]) Va' e A.
The
for all te [0,1].
a) follows from proposition 15 b) of section 7 and b) from a straightforward computation.
e) The JLO cocycle
We have defined above the Chern character:
ch.(H,D)e Щ(А)
of К cycles over locally convex algebras (definition 17), and proved the index formula (theorem 19). However
except for the component of dimension 0: '
<?o(«) = Trace G a e'0*),
the higher components of the normalized entire cocycles given by corollaries 14 and 16 are given by compli-
complicated formulae.
Motivated by examples of 0-summable A' cycles arising in Supersymmetric quantum field theory Jaffe
Lesniewski and Osterwalter ([JLO]) obtained a much simpler cocycle, which was then shown ([Co26]) to be
cohomologous to the normalized cocycle of corollaries 14 and 16.
Theorem 21. (JLO) Let D,D) be a в-summable К cycle over a locally convex algebra A. Then the
following formulae define an entire cocycle (in the (b,B) bicornplex).
a) Even case:
e A.
dsQ...dS2n
l,»,>0
Trace (a0 e-'°D* [D,al] e-"°2... [O,a2n] е-"»°2 [D,a2"+1] e-"-+lD')
Let us prove this using the algebra С of section 0) and the trace r0 on С given by:
ro(T)= Trace (T(l)) VT e С
Let us treat the odd case. The JLO formula is, in our language:
330
Trace G a0 e'' [D, a1] e —D* ... [O, a2"] <¦-'- .°2 [Да2"] е—'°2
b) Odd case:
In this formula A is the element 6'0 of C, but it is convenient to think of it as the free variable of Laplace
transforms, which converts the convolution product of С into the ordinary pointwise product of operator-
valued functions of the real positive variable A (*).
The cocycle property of <p : bip2n-1 + B<?2n+i = 0, can be checked directly using the following straightforward
equalities:
¦ ¦ ¦ [D,a2n] b;+A of the algebra ?, so that the cocycle property follows
One gets indeed that:
for the element T = -[D,a°]
from:
Finally, it is straightforward to check that the JLO cocycle is an entire cocycle, the point being that it
involves the commutators [D,a'\ in a very simple manner (cf. [JLO]).
Theorem 22. ([Co36]) Let CH,D) be a 9-summable К cycle over an algebra A. Then the JLO cocycle is
cohomologous, as an entire cocycle on A, with the Chern character of definition 17
The proof of this theorem is less straightforward than it looks, and is done in details in [C026].
We have now at our disposal two equivalent cocycles for the Chern character. One is normalized and comes
from a natural homomorphism p of Q,A in С (theorem 13). The other one, i.e. the JLO cocycle, is given
by a beautifully simple formula. The first formula involves the supergroup R1'1 (cf. section /3)). The second
(JLO) formula was found in a supersymmetric context, in which the operator Я = ?>2 is the generator of
time translation while О is a "supercharge" operator. In the next section we shall treat two main examples
of 9-summable A' cycles coming from 1) Discrete subgroups of higher rank Lie groups 2) Supersymmetric
quantum field theory.
(*) One cannot however permute the Laplace transform with the trace, since an operator like а0 йа'+^ [D, a1]
... [O, a2n + 1] Di + X is in general not of trace class for A a scalar, when D is only 9-summable.
331

IV.9. 0-summable K cycles, discrete groups and quantum field theory
The examples of sections 3, 4, 5, 6 of Fredholm modules were limited to the finite-dimensional, i.e. finitely
summable, situation. Thus for instance, in section 5 the rank of the Lie group G had to be equal to one, to
insure the finite summability condition. In this section we shall give two families of examples of 0-summable
Fredholm modules, (in the form of 0-summable К cycles) arising from discrete subgroups of Lie groups of
arbitrary rank, and from quantum field theory.
a) Discrete subgroups of Lie groups
Before we proceed and discuss the natural 9-summable A" cycles associated to discrete subgroups of Lie
groups, we prove, in order to show the need for the infinite dimensional theory, a general result showing that
the C*-algebra С*(Г) of a discrete non amenable group Г does not possess any finitely summable К cycle.
Theorem 1. [C02I] Let T be a non amenable discrete group, and C*{T) be the reduced C*-algebra о/Г. Let
i^H,D) be a Hilbert space representation of Cy(Y) with an unbounded selfadjoini operator D with compact
resolvent in H such that:
{a e СГ*(Г) ; [D,a] bounded} is dense in СГ'(Г).
Then for any p < 00 one has Trace((l + ?>2)~p) = +00.
Proof. Let к be an integer, and assume that Trace((l + D2)~k) < 00. Then for ? > 0 let T(e) =
A + e D2)-k (Trace(l + ? D2)-")-1.
One has T(e) € ClCH), T(e) > 0, Trace T(e) = 1. Moreover the C1 norm of commutators tends to 0 with
Indeed, to prove this one can assume that [D, a] is bounded, in which case the proof follows from the Holder
inequality and the simple estimate:
for ?-0,
which is straightforward using the equality
[a,(\
)'1} = {\ + e
e D[D,a]
D2)'1 [Да] s D(\ + e D2)'
and the bound: \\el/2 D(\ + e О2)-'\\ < 1/2.
Let then C2 be the Hilbert space "H&H of Hilbert Schmidt operators in "H, and let {(г) = T(eI/2 6 C2 for
? > 0. The Powers Stormer inequality [Pow-S] shows that as ? —¦¦ 0,
||а{(?)-«(е)а||з-0 Va € СГ'(Г).
Let then ж be the representation of Г in C" given by:
(, - Я ( -Г
€ Г , V{ € С2
One has Ще)\\ = 1 V? > 0; \\*(g) ((e) - i(e)\\ — 0 for ? — 0 and any g 6 Г. This shows that the
representation ж of Г in C2 weakly contains the trivial representation of Г which is a contradiction with the
non amenability of Г since ir, as a representation of the reduced C*-algebra of Г, is weakly contained in the
regular representation of Г.
Let us now pass to a general construction of 0-summable A' cycles.
332
Proposition 1i Let Г С G be a countable subgroup of a semisimple Lie group G, and let X = G/K be
the quotient of G by a maximal compact subgroup K, endowed with its canonical G-invariant Riemannian
metric. LefH be the Hilbert space of L2 differential forms on X, on which Г acts by left translations. Then
the pair (H, D) is a в-summable К cycle over A = СГ, where D = dr + (dT)", dr = e~rv d eT*, and lip
is the Morse function on X given by the square of the geodesic distance to the point К 6 X. (One assumes
тфО.)
The proof is simple and works in general ([Co2l]), but in order to get a very explicit formula for the operator
H = D2, making clear that e'"" is of trace class for any /3 > 0 and allowing further computations, we shall
make the further hypothesis that G is an analytic subgroup of SL(n,R). One can then choose the faithful
matrix representation of G so that С = G and К = G П O(n, R). Then X imbeds as a totally geodesic
subspace of the Riemannian space P(n,R) of positive invertible matrices, by the map fi:
= я я'
6 X = G/K
where the metric onP = P(n,H) is given by:
||Ф/Л||2= Trace ((p-1?J)
for any differentiable path p(() in P.
Given p € X С P, the geodesic distance d(p, 1) of p with 1 € X is the same in X or in P, and is equal to
the Hilbert Schmidt norm of logp:
= d(p, IJ = Trace ((logpJ).
y arc length, is t —> p"' where
g —> pl1+l) or equivalently t —> p
For any differential form ш on X, vie let a"(ui) be the operator (in H) of exterior multiplication by ш:
The geodesic from 1 to p, parametrized by arc length, is t —> p"' where a = rf(p, I). The gradient of <p is
the tangent vector at t = 0 to the path ( —> pl1+l) or equivalently t —> p+ ( plogp 6 P.
Its adjoint a(u) is given by contraction iu by w.
One has obviously, with ш = dip;
dr = d + т а* (ш)
D = dr + d'T =
Let us first compute the commutator of D with a group element я acting on X as an isometry, and check
that it is bounded. One has
[D,g\ = g({g-1 Dg)-D),
where the differential form as is equal to g'ui — ui.
Thus ||[?>, j]|| = sup ||a»(p)||, and we just have to check that this is finite. Now g'u = d(g'ip) and
g"ip(p) = ip(gp) = h d{gp,lf = 1 d(p, «r^l)J- Thus the tangent vector at p 6 X corresponding to
«j(p) € T;{X) is
expj'Ol-exp;1^-1^)
where expp : TP(X) — X denotes the exponential map of the Riemannian manifold X.
333

Since exp is a contraction one lias:
It follows that if Г is any subgroup of G, then [D, a] is bounded for any element a of СГ.
We shall now compute Я = D2. By [Wit] it is the sum of three terms:
2 = (d+d"f
\\dipf
where (d + d*J is the Laplacian Д on X, \\d<p\\ is the operator of multiplication by the scalar function
||((fy>)(p)||2 = 2<p(p) = trace((logpJ), and where the operator T is the following operator of order 0
Г = ЦУ^)„, [в;,а„]
where Vdtp is the covariant differential of dip, viewed as an endomorphism of TP(X) and a* (resp. а„) is
exterior (resp. interior) multiplication by elements of a basis.
Let us compute Vdip and check that its norm is of the order of d(p, 1) when p tends to oo in X. Since X
is a totally geodesic submanifold of P = P(n,R), we just need to do the computation in P. Given p 6 P
we can identify TP(P) with the linear space S(n,R) of symmetric matrices, using the isometry p~1/2 -p/2,
Then a straighforward computation gives:
The eigenvalues of 6 in S(n, R) are the differences ^(A,- — Xj); j = 1,..., n where the A,- are the eigenvalues
of log p. Thus the eigenvalues of (Vd<p)p are the '}T*ll\\] hj = 1,... ,n. Since th(x) —*¦ 1 as x —> со one
gets an inequality of the form -^ < \x\ +c and hence;
||(Vdvj)r|| =
as p^ coin P.
The two operators in H given by A — (d +d')- and г Т +т2 \\dtp\\2 = г Т +2r2 <p are bounded below since
(p(p) = d(p, IJ. Thus the Golden Thomson inequality of [Ar4] applies and gives for any /3 > 0:
Trace(e-"D3) < Trace (е-"д е-«^+:<г»„)) < oo.
The operator e-f(.-rT+2r'v)_ (T ^ q), is of order 0, with norm behaving like e-Av><» as p -^ oo.
By construction the К cycle (H, D) is even, with the Z/2 grading given by the degree of differential forms:
Let us now assume that Г is a discrete subgroup of G. Then H is in a natural manner an A — В bimoduk
where Л = СГ as above is the group rmg of Г while the algebra В is the algebra of Г periodic Lipschitz
bounded functions on X:
8 = {fe Ct(T\X) ; ||V/|| bounded}.
It acts by multiplication operators in H — L2(X,/\"):
and this action commutes by construction with the action of Г.
Proposition 3. Let Г be a discrete subgroup ofG.
334
a) (%, D,j) is ав-summable К cycle over the closure С under holomorphic functional calculus of А® В in
the C'-algebra Сь(Г\Х,С;(Г)), where С;(Г) is the reduced C'-algebra о/Г.
b) For any a 6 A, b 6 В one has:
Proof. For any / e В the commutator [rfT,/] is equal to [d,f], and is hence bounded since / is a Lipschitz
function. Thus [D,f] is bounded for any f e B. Moreover both [dT,f] and [d'r,f] are Г invariant operators,
so that [D, f] commutes with Г for any / 6 B.
As the algebras А, В commute with each other this proves b).
Next, since Г is discrete in G, its representation in L2(X, Л*) is a direct integral of representations which
are quasi equivalent to the regular representation off (they are equivalent when Г is torsion free). Thus the
representation of A ® В in 7i extends to a representation of the C*-algebra:
сь(г\х,с;(Г)).
Since the algebra {T € CCH);[D,T] bounded} is stable under holomorphic functional calculus in C(H) we
get a).
By construction С is a pre C*-algebra. It contains the closure under holomorphic functional calculus of СГ
in С*(Г). Similarly it contains B, which is by construction closed under holomorphic functional calculus
(but not dense) in Сь(Т\Х).
The index map of the К cycle Gi, D,f) thus yields a bilinear map:
This map is easy to compute for elements of А'(С*(Г)) which are in the range of the analytic assembly map
(chapter II):
fi:
Theorem 4. For any x S А'.(ВТ) and у 6 A'(B) the index of(D,j) evaluated on /i(x)igij/ 6 K(C) is given
by:
<рЫх)®у) = (сЛ.(х),сЛ*(у)>
where ch.(x) 6 H.(BF) ~ H.(F\X) is the Chern character of the К homology class x, and ch'(y) €
Н"(Г\Х) is the Chern character of the К theory class y.
Note that сЛ.(х) is a homology class with compact support in Г\Х, so that it pairs with the cohomology
class (with no support, restriction) ch"(y).
3 that every Г equivariant Hermitian vecto b
determines canonically a K cycle CHe,De,7e) over С*(Г) which can be concretely described as follows.
HE =L2(X,A- ®E)
is the Hilbert space of L2 forms on X with coefficients in E in which Г acts by left translations.
335

is given by the same formula as D = dT +d*T but with the covariant differentiation V of sections of E instead
of the differential d.
TEu = (-1)' ш Vw € L2(X, Л* <g> E).
We can now state the following corollaries of propositions 2, 3 and theorem 4.
Corollary 5. Lei Г С G be a discrete subgroup of a semisimple Lie group, and A the pre С-algebra closure
under holomorphic functional calculus о/СГ in С*(Г).
a) The К cycle {"He, De>7e) over A is 9-summable.
b) Let ifiE 6 Hf(A) be the Chern character of (He, Db,-/b)- Then for any x 6 K.(BT) one has:
(<pE,»(x))=(ch.(x),ch'(E)).
The proof of proposition 2 is easily modified to give a).
By theorem 4 the index map ф determined by the К cycle ("We. &e>7e) satisfies:
ф(ц(х)) = (ch«(x),cIi*(E)) Vx 6 Л'.(ВГ).
Thus the answer follows from theorem 19 of section 8.
Let us now combine corollary 5 with the higher index theorem of chapter 3 section 4.
Proposition 6. Ltt G, Г and ("We, De-Je) be as above. There exists a unique elenient 0 € Неу(ВГ,С) such
that the associated cyclic cohomology class: r@ 6 //Cev(CT), has the same pairing with ц(Ко(ВГ)) С КоG1Г)
as the entire cyclic cocycle ipe- One has moreover:
0 = ch E.
In particular if we take for E the trivial one dimensional vector bundle, i.e. for De = A we get that, on the
image of ц, the entire cyclic cocycle <p = cA*("W, D, j) pairs in the same way as the normalized trace r:
т:Сг"(Г)-С
а, Я I = ae Va = ? a, g € СГ.
/
The equality of т with <p on р(К\(ВГ)) is not sufficient to conclude that г(А'о(С*(Г))) С Z for Г torsion-free,
since we do not know that /i is surjective.
The conclusion r(/vo(C; (Г))) С Z would follow if we could show that the entire cyclic cocydes ip and r on A
are cohomologous. We shall take a first step in this direction by proving that (p is cohomologous to a simpler
entire cyclic cocycle ф whose component of degree 0, i/>0, is equal to т when Г is torsion free. In fact this
simplification will be obtained for any of the entire cocycles ^e, which will all be in the range of a natural
map Ф from closed differential forms on M = V\X to entire cyclic cocycles on A.
We shall assume, to simplify the discussion, that Г is cocompact and torsion free, so that M is a compact
manifold. The general case is not more difficult but requires heavier notation.
Let 5 be the Spinor bundle on X with its natural G equivariant structure, and let V* be the spin connection.
Let S be the Hilbert bundle over M whose fiber at each p € M is the Hilbert space:
336
off2 sections of the restriction of the bundle 5 to the Г orbit Jr~'(p) С X above p. (We denote by jr • X —> M
the quotient map.)
To the connection V* corresponds a natural connection V on the Hilbert bundle S over M, uniquely deter-
determined by the natural isomorphism:
L2(M,Z)~L2(X,S).
More precisely, for any smooth section { of S on M, one lets (' be the corresponding section of S on X and
one has:
(Vy ()(P) = (V^ C)DU,)=P € ?p
for any tangent vector У € TP(M) lifted as Y, € T,(X) for any }6I, t(j) = p.
Next, let T be the unbounded endomorphism of the Hilbert bundle ? over M given by:
where 0 denotes a fixed origin in X and, for any q 6 X, c(q,0) denotes the operator in S, of Clifford
multiplication by the unique tangent vector at q whose exponential is 0.
We shall assume that X is even dimensional and use the superconnection Z = 7V + T, where 7 is the
Z/2 grading of the Hilbert bundle E (cf. [Qi] for the notion of superconnection on a Hilbert bundle on a
manifold).
The algebra С*(Г) acts by endomorphisms of the bundle ? since Г acts by translations on each of the fibers
One then checks that the hypothesis of the following proposition is fulfilled by the action of the pre C*-algebra
асс;(Г).
Proposition 7. [Co:>5] Let M be a compact manifold, (?,7) a Z/2 graded Hilbert bundle on M equipped
with a superconnection Z = jV + T, and let A be a subalgebra of the algebra of endomorphisms of the bundle
(?,7) such that:
a) For every 06Л, the commutator 6{a) = [Z, a] is a bounded endomorphism of the bundle ?.
b) The operator-valued differential form exp(— 0 Z~) is iracable for any 0 > 0.
Then the following formula defines, for any closed even smooth form u> on M, an entire cocycle in the (A, B)
bicomplex of A:
y#
#n(a° a2") = / / шЛ ТгасеGО° е-'02' #(а')
Let us apply this proposition in the above context, and denote by Ф the corresponding map, Ф(ш) = <p",
from H"(M) to Я*(Л). The computation of the character c/i.(WE, De, Je) = 4>E 's partially achieved by
the following result.
Theorem 8. With the above notation one has
cIi.CHe.De^e) =Ф(и)
where ш = A{M)chE, where A(M) is the A genus of M ([Mi-S]).
337

/3) Supersymmetric quantum field theory
The apparent incompatibility of quantum mechanics and special relativity (cf. for instance [Fey]) is resolved
only by the theory of quantum fields, i.e. by the quantization of a mechanical system with infinitely many
degrees of freedom.
As we shall see by working out in full details a concrete example, this infinity of the number of degrees
of freedom implies, at least in supersymmetric theories, the infinity of the cohomological dimension of the
algebras of local observables. One expects in general that, provided an infrared cutoff is given, i.e. space is
assumed to be compact, the following holds:
A) The supercharge operator Q in the vacuum representation Hilbert space "H, Z/2 graded by Г = (—1)*^
(where Nj is the Fermion number operator) defines a 9-summable A'-cycle over the algebras U(O) of local
observables.
B) The index map given by Q, i.e. the map from Ka{UL{O)) to Z, (UL(O) = {A € U(O);[Q, A] bounded})
given by the index of Q with coefficient in а К theory class, fails to be polynomial, in the following sense:
Definition 9. Let A be an algebra (over C), and ф : h'o(A) —> С an additive map. Then -ф is polynomial
iff there exists a cyclic cocycle т 6 HC"(A) such that:
ф(е) = (т,е) V«eA',(i).
We shall only prove A) and B) in an explicit and non interacting example, the N = 2 Wess Zumino model
on a 2 dimensional cylinder space time. The results of [J-L-W] on constructive quantum Field theory prove
A) in the interacting case. They should allow to prove B) in the interacting case as well.
Before we embark on the description of the model, we shall work out a toy example with finitely many
degrees of freedom.
This example is a slight variation of the construction of proposition 2. We let E be a finite dimensional
Euclidean vector space over H, and ip anon degenerate quadratic form on E. Then let "H = L2(E,A* Ec)
be the Hilbert space of complex, square integrable differential forms on E. We let D = dv + (rf^)* where
dv — e~* d e*\ d being the exterior differentiation.
Writing <p(x) =
k ^> xh
0, in a suitable orthonormal basis of E, with xi = (z, et-), it is immediate
l
to compute the spectrum of D and check that it is Bn, oo) summable, n = dim-E1. Moreover the index of D,
relative to the Z/2 grading 7 of 7i given by (—1)' on Л* Eq, is equal to (—1)"-, where n_ is the number of
negative eigenvalues A, < 0 of ip.
Let к < n, {!,... ,(k 6 E, щ,..., nk 6 E' be such that:
(?,%)€ 2jtZ V«,j = l,...,i.
To each (,i we associate the unitary Ui in "H given by translation by {< g E. To each ly we associate the
unitary Vj in H given by multiplication by the periodic function
The above condition shows that these 2n unitaries commute pairwise. As in proposition 3 they all have a
bounded commutator with D, i.e. belong to the algebra Ad = {T € C(H)\ Ту = jT, [D,T] bounded}.
We let [Ui Л U2 Л ... Л Uk Л К Л ... Л 14] € I<o(AD) be the A' theory class obtained as a cup product of the
above commuting unitaries. There are two equivalent ways of describing it.
The first is to consider the compact manifold M = T2k, which is a torus of dimension 24, and the hoino-
morpbism p from C°°(M) to Ad given with obvious notations by:
338
р(щ) = Ui , P{vj) = Vj
where (ab ..., et,»,,..., »i) are the natural S'-valued coordinates on M. Using the orientation given by
this ordering of the coordinates, one gets a well defined class /3 6 K°(M) = K0(C°°(M)), that of the Bott
element, whose Chern character is the orientation class:
One then defines:
ch(j3) =[M]€ #2'(M,R).
Л ... Л Ut Л V1 Л ... Л Vi] = р.A3) € Ka(AD).
The second is to use the cup product in algebraic theory, which yields an element of К^(Ао) whose image
by Bott periodicity is the above element. (Note that Ad, being stable by holomorphie functional calculus,
satisfies Bott periodicity.)
The computation of the index of D with coefficient in the К theory class [Ui Л ... Л Ut Л Vi Л ... Л Vi] is
given by:
Lemma 10. The index of D relative to the Z/2 grading 7 and the К theory class [f/]A.. .f\Ut^ViA.. .AVt]
is equal to
(-1)"- B*)-* det((u,7,,-))eZ.
The proof is a straightforward application of the Riemann-Roch theorem, but we shall give the details of the
identification of D with а д operator, since it throws light on the previous example of proposition 3.
Let us assume, to simplify the discussion, that the matrix ({{j, ty))i,j is not degenerate. Then the dimension
of the linear span F of the {,-'s (resp. F' of the fy-'s) is equal to k, and E is the linear span of F and
(Пх = {( e E; ((, щ) = 0 v> = 1 *}.
Thus there exists a lattice Гс?, i.e. a discrete cocompact subgroup of E, such that:
& € Г , Vi = 1,..., * ; ijj € Г1" Vj = 1 к
where the dual lattice TL is {щ 6 E', (rj,T) С 2irZ).
As in proposition 3, we let the group ring СГ act in "H by translations, and C(E/V) act in "H by multiplication
operators. These two commutative and commuting algebras generate a C*-algebra A in Ti, whose spectrum
S is naturally the torus:
5 = Г x E/T = Е"/Тх x E/Y.
One has ||[?>,/]|| < 00 for any / € C°°(S), and moreover the image of p : C°°(M) —> Ad is contained in
C°°{S). This shows that e = [?/, Л ... Л Ut Л Vx Л ... Л Kb] = />„(/?) can be viewed as an element of K°(S),
namely the pull back of /3 by the projection p : 5 —> M which is the transpose of p.
We shall now endow 5 with a complex Kaehler structure and a holomorphie line bundle L, so that the
К -cycle on C°°(S) given by CH,D,j) is isomorphic to the Ъ operator with coefficient in L. First, given an
element (X, Y) of E x ?", one gets a corresponding vector field on S, i.e. a derivation &(x,Y) of A:
Н,№ , I + U, dx f
for any function / on E/T and any g € Г.
The Schwartz space S(E) is a finite projective right module over C°°(S) for the action of C°°(S) given by:
= f(p) t(p-g)
339
Г ,

A connection V on the corresponding complex line bundle L on S (so that S(E) ~ C°°(M, L) as finite
projective modules) is now given by:
(Эх О(Р) + i(Y,
One checks that V(x,y) satisfies the Leibniz rule with respect to the derivation 6(x,Y)i and also that it is
compatible with the metric on L given by the following C°° (S)-valued inner product on S{E):
The curvature of the line bundle L is given by the formula:
г , g' 6 г1-.
E'
where aj(ai,a2) = (У^Хя) — (Y2, Xi) is the canonical symplectic form on the tangent space E X E* of S.
Next, let the quadratic form <p on ? be expressed, as above, as <p(p) = ? ^ ^j((p>cj)eJ where (Cj)^!^..^
is an orthonormal basis of the Euclidean space E. We let e' be the dual basis of E', so that (p,?j)e =
(p, e-*) Vp € E. Then the operator dv is given by:
where aj is the operator of exterior multiplication by e, in Л ?с, so that their representation is characterized
by the commutation relations:
Let us consider the R linear isomorphism of E x E" with C" given by z,- = ±(я,- +(i/A;) t/y).
Then ^=- = ^e .,д eJ j as a derivation of A, and using the connection V on the complex line bundle L we get:
We endow the manifold S with the complex Kaehler structure given by the above isomorphism of its real
tangent bundle with the trivial bundle with fiber С and standard inner product. We then have a natural
isomorphism of 4 = L2(E, A"EC) with L2(S, A0'* ® L) where A0'* is the exterior algebra bundle over the
complex manifold 5. This isomorphism U is the tensor product f/k ® Uj of the isomorphisms Ut, : L2(E) ~
L2(S, L) and Uj : A0'* ~ A'Ec- Here Uj comes from the identification of Ec with C" given by the basis ej,
while f/k extends the canonical isomorphism S(E) —>C°°(S, L). The next lemma is now clear:
Lemma 11. Under the isomorphism U : H —> L2(S,A°'" ® L) the action of A = C(S) becomes the action
by multiplication operators, the operator dip becomes the di operator and the Z/2 grading 7 becomes (—1)*
on A0'* ® L. The К cycle (W, D, 7) и isomorpkic to the К cycle c?t + Cl)' °n S.
Let us, as a first application of this lemma, compute the index of D with coefficients in [U\ Л t/г Л ... Л ft Л
V, Л ... Л Vt] = е, i.e. let us prove lemma 10. Since e = p*(/3) where p : S —> Af is the transpose of the
homomorphism p defined above, and /3 is the Bott К theory class, /3 € K°(M), we can apply the Riemann
Roch theorem (cf. [Gii]) and get:
Index(A.) = {Td(S) ch{L) ch(e),[S])
where the Todd genus of the complex manifold S is the cohomology class 1 € H°(S), the Chern character
ck(L) is equal to exp(ci) and its first Chern class c, is jj- ш where ш is the canonical symplectic form. Also
the Chern character cft(e) is p'ch@) = p"([M]) where [M] € H2k(M) is its fundamental class. We thus get:
Index(?le) = (exp (i- ш) р'([М]), [S]\ .
340
The class [M] is-represented by the differential form
Bxi)-
Ui Л ... Л yt"l dUk Л V,-' dVi Л ... Л Vt~'
so that its pull-back on 5 is translation invariant, and the computation of the right hand side of the above
formula takes place in the exterior algebra Л(Е ® ?"").
We shall now explain in some details the set-up of quantum field theory in a very simple case, and show that
the supercharge operator Q in supersymmetric theories yields a 0-summable K cycle. Thanks to the work
of constructive quantum field theory (cf. [J-L-W]) these results are valid in the interacting case, but for the
sake of simplicity we shall limit ourselves to the free theory. We shall show that even in the free case, the
infinite number of degrees of freedom of the theory in every (non empty) local region О implies that the К
theory K(Ul{O)) of the local observable algebra is infinite dimensional.
Let us first consider the simplest field theory, namely the free massive scalar field tp on the space-time:
X = S1 x H
with the Lorentzian metric.
The scalar field (p is a real valued function on Л' governed by the following Lagrangian:
where do = ¦§, is the time derivative (t € R) and d\ = jj is the spatial derivative, where the space is here
compact and one dimensional. The action functional is given by:
I(v) = / C(<p) dxdt= f L(t) dt
Thus, at the classical level, one is dealing with a mechanical system with infinitely many degrees of freedom,
whose configuration space С is the space of real-valued functions on Sl.
The Hamiltonian of this classical mechanical system is the functional on the cotangent space T*C given by.
+ (dp(x)J +m2<p2(x)) dx
where one uses the linear structure of С to identify T*C with С X С*, and where one views the field зг as
an element of С:
<p(x) ж(х) dx€C
As a classical mechanical system the above one is the same as a countable collection of uncoupled harmonic
oscillators. Indeed one can take as coordinates in С (resp. C") the Fourier components <pt = f <p(x) e~'kx dx
(resp. зг^), which are only subject to the reality condition:
P-k - Vt Vfc € Z (resp. jr_t = Wk).
Thus both spaces С and T'C are infinite products of finite dimensional spaces and the Hamiltonian Я is an
infinite sum:
Я = ^ - (т4 rk + {It2 + m2) ipt Jpt) .
341

The quantization of a single harmonic oscillator, say on a system with configuration space R, and Hamiltonian
^(p2+ui2q2), quantizes the values of the energy in replacing ~(p2+u2q2) by the operator j ( ("'*''5j) + u2q2 \
in the Hilbert space L2(H), whose spectrum is (up to a shift) the set {n Йы ; n € N}. The observable algebra
of the quantum system is generated by a single operator a* and its adjoint a obeying the commutation
relation:
[a, a'] = 1
and giving the Hamiltonian as H = hw a* a. These two equations completely describe the quantum system.
Its representation in L2(R), given by the equality:
is, up to unitary equivalence, its only irreducible representation. The (unique up to phase) normalized vector
0 such that ail = 0 is called the vacuum vector.
The reality condition <p_k = Tpk shows that for к > 0, the pair {—к, k) yields a pair of harmonic oscillators,
whose quantization yields a pair of creation operators a]., a"_t. The observable algebra of the quantized field
has the following presentation. It is generated by a?, at; к 6 Z with relations:
[ak,at] = 0 , [a»,a;] = 0 VJt ф t.
The Hamiltonian is given by the derivation corresponding to the formal sum:
Нь = / Ь.
о.\ at ;
The "vacuum representation" which corresponds to the vacuum state which is the infinite tensor product of
the vacuum states, is given by the Hilbert space:
and the tensor product representation of the algebra.
Equivalently one can describe this infinite tensor product as the Hilbert space L2(C, dft) where fi is a Gaussian
measure on C, all this being straightforward (cf. [Gl-J]).
The spectrum of the quantum field Hamiltonian Нь acting in ti is now:
Spectrum Нь = {? n^ huJk ; nk 6 N} .
Quantum field theory reconciles positivity of energy (i.e. Hb is a positive operator in Wj) together with
causality, which means that we have commutation at space-like separation for functions of the quantum field
<p(x,t) = e1'"* <p(x,0) e~itH\ with:
This commutation of <fi(x,t) with y(y,s) for (x,t) and (y, s) space-like in the space-time X = S1 X R is easy
to check directly. First, the operators ifi(f) = //(x) <p{x, 0) dx, f € C°° commute with each other, using
uit — w_fc. Next, with crt the automorphisms given by time evolution:
342
one computes [<r't (p(f),v>(<l)], for /, 0 € C~E'). яаЛ finds a scalar multiple of the identity, where the scalar
k(x,y,t) f(x)
where it satisfies the Klein Gordon equation:
and the initial conditions it(x,0) = 0, Jj i-(x,0) = SO.
It then follows from elementary properties of the wave equation that k(x,i) vanishes if (x,t) is space like
separated from @,0), and hence that the quantum field <p satisfies causality. All this is well known and we
refer to [Bo-S] for instance for more details. For our purposes we shall retain two easy, and well known,
corollaries.
Proposition 12. [Bo-S] a) Let 'hit, be the vacuum irprrsentaiiov. The quantum field y> is an operator valued
distribution.
b) Let О be a local region (i.e. a bounded open net) m X and U(O) be the von Neumann algebra in 'Нь
generated by the <p(f) with Support /Cf. Then when Oi and C?2 are space-like separated, one has
Fig.9. Space like regions Oi and O-± in space lime cylinder
The von Neumann algebras U[O) of local ohservahlt's are an essential part of the algebraic formulation of
quantum field theory ([КазЦ]). From our point of view in this book, the passage to a von Neumann algebra
343

only captures the measure theoretic aspect, and we shall now show that in the case of supersymmetric theories
we can refine the algebras U(O) to capture the finer topological and even metric aspects. We shall only treat
an example but the strategy works in general. The example we take is the free Wess Zumino model in 2
dimensions, with one complex scalar field <p of mass m and one spinor field ф, also of mass m. Thus the
Lagrangian is now
С =
Cs ; Сь = i (\д0 ч>? - |
- т2\ч>\2)
Cj — i ф у,,
- m ф ф , ф - ф* 70
where the Dirac matrices 7^ are say 70 = , 71 = I . , and a spinor field ф is given by a ijj
column matrix:
</> =
The bosonic part is quantized exactly as above, except that the field (p is now complex so that the reality
condition no longer holds and one needs twice as many creation operators, which are now denoted by a±
к 6 Z. They satisfy the commutation relations:
The time zero field <p(x) is given by:
<p(x) = Dx)-'/2 J2"(k)-1/2 (a-+(k)+a
where ш(&) = (t2 + m2I'3. The conjugate momentum is given by
and one has the canonical commutation relations:
= о
The quantum fields ip and jr are operator-valued distributions in the bosonic Hilbert space %ь of the vacuum
representation of the creation operators a^ffc), к 6 Z. In this representation the Hamiltonian Нь is given by
Hb = J :
where the Wick ordering: (cf. [Bo-S]) takes care of an irrelevant additive constant. Equivalently Tib is the
Hilbert space X2(C~oo(Sl),d/i) where d\x is the Gaussian measure, on the configuration space O~OO(S1) of
complex-valued distributions on S1, with covariance G:
G = -(d2/dx2+m2)-1/2.
Under this identification the field (p is represented by multiplication operators and тг(х) becomes — i 6/6(p(x)
(cf. [Gl-J] or [JLW]). Next let as describe the fermionic part of the model. It is worthwhile to first describe
the C"-algebra generated by the fermionic fields together with its natural dynamics generated by a derivation,
and then to find the ground state and use this vacuum state to construct the fermionic Fock space. The
C*-algebra generated by the fermionic field ф =
к € Z, which satisfy the following relations:
*'1 is generated by its Fourier components ф)(к), j = 1,2;
where {A,B} = AB + BA.
In other words we deal with the C*-algebra A associated to the canonical anticommutation relations in the
Hilbert space of L2 spinors on S1
The quantum fields г/>ь ф% are then given by
ф,(х) = E ф/(к) e~ik* \tx g S\
and they are A-valued distributions on Sl.
This specifies ф — Jm at time 0. Its time evolution is specified by the Hamiltonian which is given by the
derivation of A:
where Hj is the formal expression:
H{ = / @
7! дф - m
dx , ф = ф*
= E к(ф1(к)' ф,(к) - ф2(к)' ф2(к))-т(ф'1(к) ф2{к)+ф'2(к) ф2(к)).
The above derivation makes perfectly good sense, and defines a one-parameter group of automorphisms <т(
of A = CAR(L2(S1,S)). One has <rt — CAR(f/() where f/j is the one-parameter group generated by the
.. Г id —m
operator tf, = ^_m _.^
It is straightforward to check that the quantum field:
does satisfy causality, so that i/>(x,s) and ф(у, t) anticommute at space-like separated points.
The only difficulty in the quantization of the spinor field is the positivity of energy, since the order-one
operator H\ does have negative and positive eigenvalues. The answer is well known, and quite clearly stated
in C-algebaic terms:
Lemma 13. For any /3 6 [0,oo] there exists a unique KMSp state (pp on (A, <r(). It is a quasi free state.
For /3 = +oo the covariance of (p^ ts given by the sign F — H\\H\\~l of H\.
The representation of A associated to the ground state 1рх is the "Dirac sea" representation which can be
described as follows. The Hilbert space Hj is the antisymmetric Fock space over L2(S1, S) = L2(S')©?2(S1).
One lets ij-(fc) be the corresponding creation operators, for к € Z, the label of the natural orthonormal basis
of L2(Sl,S). The operators b'±{k) satisfy the canonical anticommutation relations, and are related to the
ф{(к) by the equalities which define the representation of A in 'Hj, namely:
344
345

И*) *!(*) - "(-*
where ш(к) = (A:2 + m2I/2 as above, and where
In this representation the one-parameter group <rt of automorphisms of Л has a positive generator, the
fermionic Hamiltonian:
H,=
! 71
: dx.
The Hilbert space of the full model is H = Чь ® "Hj, the tensor product of the bosonic and fermionic Hilbert
spaces. The quantum fields are given by the operator-valued distributions 4>(x) ® 1, 1® V>j(x), where tp is
the (complex) bosonic field and xpj are the two components of the fermionic field.
As above (proposition 12) we let U(O) be the von Neumann algebra of bosonic observables associated to any
local region О Q X. The full Hamiltonian of the non interacting theory is given by
Я = #4 ® 1 + 1 <g> Hh
and since the Wick ordering constants of the bosonic and fermionic parts cancel identically, one can write it
Я = J
- m) ф{х)) .
It is a positive operator in 7i, and admits a natural self-adjoint square root, the supercharge operator Q
given by ([JLW])
= -i= J_
(T'(x)-d<p(x)-imtp'(z)) dx+h.c.
where /i.e. means the adjoint operator.
On H = Hb®'Hj we let Г be the Z/2 grading given by liS^-l)"' where Nj is the Fermion number operator.
With the above notations it is straightforward to prove:
Proposition 14. a) The operator Q in H is selfadjoint, anticommutes with the Z/2 grading Г, and
Trace(e-" 1') < oo for any 0 > 0.
b) For any local region О С -X the subalgebra:
UL(O) = {TeU(O) ! [Q,T] is bounded}
is weakly dense m the von Neumann algebra Ui{O).
We refer to [JLW] for the proof of a). In fact by construction the pair {W,Q) is obtained as an infinite tensor
product ® fWt.Ot), where each Нь is a Z/2 graded Hilbert space with Z/2 grading Ft, and where the
infinite tensor product is relative to the unique (up to phase) normalized vector fit; Qt Qt = 0, Tt Qt = Qf
It is then an easy exercise to show that Q = ? Qt is selfadjoint in the Z/2 graded infinite tensor product,
provided each Qt is. Here each operator Qt, Jfc g 141 is of the form:
Qk=dVi+(d4,t)" in L2(Ek,Ah Ek)
where Et is the Euclidean real vector space Ek = C2 for k > 0 equipped with the non degenerate real
quadratic form:
346
ipk(z,u) = {2uit)-1 (k z1-kuu + m(zu+-zu)) V(i,u)?C'.
Also Q2 = Я = Нь ® 1 + 1 ® Hj, where the spectrum of Нь (resp. Hj) is the set of all finite sums:
E = E nt uit r>i € N (resp. nt 6 {0,1})
so that ехр(-/ЗЯ) is of trace class for any /3 > 0, and the number of eigenvalues of Я below a given E > 0
grows like exp (л \/E\ for E —> oo, with A a fixed constant.
b) To prove b) it is enough to show that for / € Cf(O) the following operators in H belong to Ul(O):
exp i(<p{f) + ?(/)*) . exp i(ir(/) + t(/)*),
and similarly for the Fermionic fields.
This follows from the boundedness of the commutators:
One has [Q, <p(f)] - ф_ I V>i(*) /(*) dx + -fe (f M*) 7(x) dx)''. which is bounded if / € L2. Similarly
[Q, "¦(/)] is bounded provided / and df belong to L2, so the result follows.
Corollary 15. Let О С X be a local region and A(O) be the pre С-algebra in U{O) generated by the
operators exp i(y>(/) +?>(/)'). exP »'('(/) + *(/)"). / e C? (°)- Then (W'<3' Г> « ° ^-summable К cycle
over A(O) for any local region О.
In fact A(O) QUL(O)as follows from the proof of 14 b). We can now state the main result of this section,
which shows that even though we are in a free theory, and however small a non empty local region О may
be, the К theory of the local algebra A(O) is non trivial, pairs non trivially with the supercharge operator
Q, and attests to the presence of infinitely many bosonic degrees of freedom localized in O.
Theorem 16. Let О С X be а поп empty local region. Then the index map:
Ind<j : K0(A(O)) — Z
ы non polynomial, and K0(A(O)) is of infinite rank.
To prove this we shall use lemma 10 and the construction of К theory classes given in this lemma. To prove
the result we can assume that О intersects 5' x {0} С X. Let then n € N and fh j = 1,... ,n be real valued
smooth functions on S1 with support in О П (S1 x {0}) and such that:
I
Let <pr be the real quantum field фг{х) = <p(x) + <fi'(x), and let irr be the conjugate momentum. Then the
following unitary operators belong to the algebra A(O):
Vj = exp i <Pr(fj)
The canonical commutation relations:
Vt =exp : T,(/t).
347

show that the operators Uj, Vfc generate a commutative subalgebra of A(O), and we can consider the element
of Kq(A(O)) given as in lemma 10 by:
By lemma 10, a; is a non trivial element of Kq(A(O)), since it pairs non trivially with the К cycle (ft, Q, Г):
Index Qx = 1.
We are applying lemma 10 in the case when E is infinite dimensional, but the same proof works. Here E = C
is the configuration space of the complex scalar field <p, and the quadratic form Ф is given by:
Ф(^) = - Re / (i(d<fi)<p + m <p2) dx.
By construction the above x € Ka(A(O)) pairs trivially with any cyclic cocycle on A(O) of dimension < 2n,
and pairs non trivially with Q, which is enough to show that the index map is non polynomial.
Note that the non polynomial index map of Q is given, using theorem 19 of section 8 by an entire cocycle
on A(O) namely the character c/t,(ft,Q,r) of the 9-summable К cycle (ft.Q.T) (Corollary 15).
By the results of [JLW] on constructive quantum field theory, proposition 14 still holds for the interacting
Wess-Zumino model in 2 dimensions.
We shall end this section by formulating two problems.
Problem 1. Prove theorem 16 for the interacting Wess-Zumino model.
Problem 2. Compute the entire cyclic cocycle c/t.(ft,Q,r).
348
Appendix A. Kasparov's Invariant theory
In this appendix we shall give without proof the basic results of Kasparov's bivariant theory, which is an
indispensable tool in order to construct К homology classes of C'-algebras. We shall follow [Kas 1 and [Sk 1
Let us first recall some notations from chapter II appendix A.
Given two C'-algebras A and B, an A — В C*-bimodule is defined as a C"-module ? over В together with
a * homomorphism from A to EndB(?). We say that a C*-algebra A is Z/2 graded if it is endowed with a
grading automorphism 7,76 Aut A, j2 — id. Any element x of A then has a unique decomposition as a
sum x = x0 + Xi with 7(x0) = xa,-r(xl) = -xi. The C'-subalgebra A1 = {x 6 A ; 7A) = x) is called the
even part of A, and the odd part is {x 6 A ; y(x) — —x}. The Z/2 grading is called trivial if A" = A, i.e. if
a = id. Similarly a C*-module ? over a Z/2 graded C"*-algebra В is called Z/2 graded if it is endowed with
a linear grading operator 7 : ? —* ? such that:
= 7@ 7(»)
€ ? , * € B.
When В is trivially graded one has 7 6 Е1к1д(?), and 7 is an involutive unitary of that C*-algebra. The
general case is quite useful since it allows to treat the even and odd cases in a unified manner.
Given a Z/2 graded C'-module ? over a Z/2 graded C*-algebra B, the C'-algebra EndB(?) is naturally Z/2
graded by the rule:
7(T) 7«) = 7(TZ) VT € EndBB) , f € ?¦
For А, В a pair of Z/2 graded C*-algebras, an А —В graded C-bimodule is given by a Z/2 graded C-module
? over В and a Z/2 graded * homomorphism from A to EndB(?).
In the Z/2 graded context, all commutators are graded commutators:
[z,y]=*y-(-l)dl!eidee» yx,
and more generally one uses the general rule that a transposition of two elements x, у introduces the sign
(_l)degxdegj (cf [Mi4]). As in appendix A of chapter 2 we shall denote by Endg(?) the two sided ideal
of compact endomorphisms of а С B-modu!e ?. AH the C*-modules will be assumed to be countably
generated (cf. chapter 2 appendix A).
Definition 1. [Kasi] Lei А, В be Z/2 graded C'-algebras. A Kasparov А — В bimodnle is a pair (?,F)
where ? is an A - В graded С -bimodule, and F 6 EndB(?), j(F) = -F, is such that for any a ? A one
has [F,a] € EndBB), a(F2 - 1) € EndB(?). a(F - F") € Ende(?).
One says that a Kasparov А — В bimodule is degenerate if the above 3 operators vanish for any а?Д i.e.
[F,a] = a(F--l) = a(F-F') = Q Va€A
The direct sum of Kasparov A — В bimodules is defined in a straightforward manner by:
Let B[0,1] = B®C[0, 1]. Then a Kasparov A-B[0,1] bimodule can be equivalent^ described as acontinuoui
field (?,, Ft), t e [0, 1], of Kasparov А-В bimodules, audit is called a homotopy between (?q,Fo) and (?i,Fi)
More specifically (?t,Ft), for t € [0, 1], is the Kasparov А-В bimodule obtained from (?, F) by compositioi
with the evaluation morphism:
p, : B[0, 1] - B.
349

In general, given two C-algebras Bt, B2 and a * liomomorphism p : Bx —> B2, one can associate to any
Kasparov A — B\ bimodule, the induced A — B2 Kasparov bimodule given by:
(?®Bl B2 ,
) = p.(?,F).
Similarly, given a * liomomorphism с : At —> A2, there is an obvious notion of restriction of any Kasparov
A2 — В bimodule to A\ — B.
Proposition 2. [Kasi] Let А, В be two С-algebras. The set KK(A,B) of homotopy classes of Kasparov
А— В bimodules endowed with the operation of direct sum is an abelian group.
Any degenerate Kasparov A — В bimodule is homotopic to the bimodule ? = {0} and represents О 6
The bifunctor KK(A, -S) is by construction homotopy invariant in both A and B.
Kasparov proved (cf. [Kasi]) that the equivalence relation of homotopy is in fact the same, modulo the
degenerate bimodules, as the apparently much more restrictive operatorial homotopy : given a (Z/2 graded)
A — В C*-bimodule ?, a homotopy (?, Ft)je[o,i) °f Kasparov A — В bimodules is called operatorial iff the
map t —> Fj is norm continuous.
Theorem 3. ([Kasi]) Let А, В be two С-algebras with A separable. Two Kasparov А- В bimodules (?j,Fj)
are homotopic iff up to addition of degenerate ones they are operator homotopic.
The key tool of KK theory is the Kasparov product ([Kasi]):
KK(A,B) x KK(B,C) -, KK(A,C)
which we shall now describe using the convenient notion of connexion introduced in [Co-S].
Definition 4. Let ?¦, be а В - С C'-bimodule, let Si be а С-module over B, and let ? = ?, ®B ?2,
Fi € Endc(?2). An element F 6 Endc(?) is called an F2 connexion on ?i iff for any { € ?\ the following
endomorphism is compact:
where f( = \® ^ € Endc(?i Ф ?), T( € Homc(?2, ?) being given by T((т)) = ( ® т; e ? , Vt; € ?2.
One should view F as a lift of F2, but the name of connexion is appropriate using examples coming from
pseudodifferential operators of order 0 on smooth manifolds and lifts of these operators to sections of vector
bundles.
The straightforward properties of connexions are listed as follows [Co-S].
Proposition 5. a) If [F?,b] = 0 V6 € B, then I <S> F2 makes sense and is an F2 connexion. Moreover
[T ® 1,1 © Fi] = 0 for any T € EndB(?,).
b) If [F2, b] 6 Elided) V4 € B, then there exists an Fi connexion for any (countably generated) C"-module
?1 over B.
c) The space of F? connexions is an affine space with associated vector space V = {T 6 Endcr(?) ; Tx and xT G
Endc(?) for any x € End^^) ® 1}.
d) // F is an F2 connexion then [F, x] € End?-(?) for any x 6 EndB(?i) ® 1.
e) //F is an F2 connexion and F' is an Ft connexion, then F + F', FF', F" are respectively F2 + F2, F2F2,
and F% connexions.
350
We can now give an implicit definition of the Kasparov product.
Definition 6. Let A,B,C be Z/2 graded C-algebras, (?u Fi) a Kasparov А- В bimodule, and (?,F2) a
Kasparov (B, C) bimodule. Let ? = ?i®b?2 viewed as an А — С C'-bimodule. The pair(?, F), F € Endc(?V
is called a Kasparov product of F, and F2 (and one writes F 6 Ft#F2) iff:
a) (?, F) is a Kasparov А— С bimodule (definition t).
b) F is an F2 connexion.
c) For any a € A, a[F, 1» 1, F]a' > 0 modulo End?.(?).
The motivation for a) and b) is clear. The reason for c) is the construction of the product [Kasi] using the
following formula
F = M(Fi ® 1) + N(l 1» F2)
which makes sense when ?\ is stabilized and where M and N are positive operators. The original construction
of Kasparov ([Kasi]) is slightly improved (cf. [Co-S]) as follows:
Theorem 7. Assume that A is separable. Let (?b i"\) be a Kasparov А — В bimodule and let (?2, F2) be a
Kasparov В — С bimodule.
a) There exists a Kasparov product (?, F), F G Fi#F?, and it is unique up to operatorial homotopy.
b) The map (?[,Fi) x (?2, F?) —* (?,F) passes to homoiopy classes, and defines a bilinear product noted
KK(A,B) x KK(B,C) -> KK(A,C).
The uniqueness of F is easy. Its existence is a corollary of the technical theorem ([Kasi]):
Theorem 8. ([Kasi]) Let В be a graded C*-algebra and ? a countably generated, graded C"-module over B.
Lei Ei, E2 be graded snbalgebras of EndB(?), and f С EndB(?) a graded vector subspace. Assume thai:
1) E\ has a countable approximate unit and contains EndB(?).
2) / and E2 are separable.
3) Ex ¦ E2 С EndB(?), [/,?,]C?,.
Then there exists M,N 6 Епс1в(?) such that
U > 0 , N > 0 , U +N = 1
MEi С EndB(?) , NE2 С EndB(?) , [f,M] С EndB(?).
The Kasparov product ®B enjoys the same general functorial properties as the composition of morphisms.
Each morphism p : A —- В of C*-algebras defines ir. an obvious way an element p € KK (A, B) (given by
the А — В C*-module ? — В with left action of A given by p, and operator F = 0).
There is a small notational difficulty since the composition of morphisms is written:
Hom(.4,B) x Hom(B,C)
(/>ь/>з) — Pi opu
vhile the natural notation for the Kasparov product is that of tensor products of bimodules, namely
351

К К (А, В) х КЩВ,С) -> КК(А,С)
which is also analogous to the contraction of tensors.
In particular (cf. [Kasi]) it is convenient to extend the Kasparov product to the following relative case:
For any C*-algebra A with countable approximate unit we denote by тл the operation of minimal tensor
product by A, i.e.
тА : KK{B,C) — KK{B ® А,С® A),
where the tensor products of C"*-algebras are minimal (or spatial ones) (cf. [Pe]). Given a Kasparov В — С
bimodule (?,F), its image by тА is given by (? ® A, F® 1), which is a Kasparov В ® A-C ® A bimodule.
Let then A\, Л2, 2?i, 52, D be C*-algebras, with A\, Ai separable, B, with countable approximate unit.
Definition 0. ([Kasi]) Let ц 6 KK(Ai,Bi ® D), x2 6 Л'Л'(?> ® Л2,В2). ГЛеп а?! ®д z2 е
A2, Si ® B2) к ife/inaf as
Then the associativity of the Kasparov product, i.e. the equality: ([Kas])
for i! 6 KK(A,B), i2 6 KK(B,O, x3 6 h'K(C,D) implies the stronger:
Theorem 10. ([Kasi]) JeMt, Л2, Л3, ?>i be separable, xx 6 /fff(ii,Bi®A), z2 Ё Л"А'(?>,®Л2, Bt®D2),
x3 6 KK(D2 ®A3, B3); then:
a i3 = r, ®D,
The associativity implies in particular that KK(B,B) is naturally a ring with unit for any separable C-
algebra B. In the form given in theorem 10 it is particularly convenient for the formulation of Poincare
duality ([Kas7] and [Co-S]). We shall come hack to this point in chapter 6.
Before we specialize A" A" to cases when A or В = С, we state two equivalent formulations of its cycles.
Definition 11. A Kasparov А- В bimodule (?, F) is normalized iff F = F' and F2 - 1.
One then has the following simple
Proposition 12. Let (?, F) be a Kasparov А — В bimodule. Then after addition of a degenerate bimodule
(?o,0), Ц is operator homotopic to a normalized Kasparov А — В bimodule.
Moreover since the construction is canonical, it shows that one does obtain the same Л'Л" theory if one
restricts to normalized Kasparov bimodules and liomotopies.
When В - С and A is unital and acts in a unital way in ? = "H. which is a Z/2 graded Hilbert space, one
can normalize (H, F) by simply adding to ~H the finite dimensional space KerF with opposite Z/2 grading
and 0 module structure over A. This removes the index obstruction to make F invertible.
Let Ci be the following Z/2 graded Clifford algebra over C, C\ = {A + ца ; А, ц e C} with a2 = 1 and Z/2
grading given by A + ца —• A — ца.
Proposition 13. Let A be a trivially graded С-algebra.
a) A normalized Kasparov A — С bimodule is exactly an even Fredholm module over A.
b) Let К = С with Z/2 grading 7 = _} \ and action of Ci given by A + ца — д VA,/i <= C.
Tften etiery normalized Kasparov Л ® Ci, С bimodule is of the form ["H ® К , F ® \ . ' I for a unique
orfd Fredholm module (H, F) over A.
The proof is straightforward. The next result due to [B-J] gives a useful construction of Kasparov bimodules
from unbounded operators in C*-module as well as a useful notion:
Definition 14. Let В be a C* -algebra, ? a C" -module over В then an unbounded endomorphism T of ?
is given by a pair of closed densely defined operators T and T" in ? commuting with the right action of В
such thai
1) <T?,!)) = <?,T*!/> V? 6 DomT , 1) 6 DornT*
2) 1 +T"T is surjective.
We refer to the thesis of S. Baaj [Baaj] for this notion introduced in [B-J].
Theorem 15. (cf. [B-J]) a) Let ? be an A — В С*-bimodule with Z/2 grading 7 and D an odd unbounded
selfadjoini endomorphism of ? such that
a) {a G A ; [D, a] bounded} is norm dense in A
/3) a{l + DmD)-1 € End°B(?) Va 6 A.
Then the following equality defines a Kasparov А — В bimodule:
F = D(\ + D"D)~'^2.
b) Every Kasparov A — В bimodule (?, F) is operator homotopic to one obtained from the construction a).
The proof follows using the equality:
to prove that [F, a] is compact for any 06 i.
One reason why the unbounded formulation of Kasparov bimodules is convenient is that it makes the external
product straightforward, and given simply by the formula:
?2,D-,) ={?l ©?2 , Di® 1
where the tensor product is a graded tensor product.
It is however more difficult to treat internal Kasparov product: KK(A,B) x KK(B,C) — KK(A,C) by
this method.
For our purpose it was crucial, in order to define the character of fl-summable Fredholm modules, to make
them unbounded, and theorem 4 of section 8 achieves this goal in a quantitative manner. By theorem 12
appendix В of chapter 2 (cf. [Co-H]) there exists a natural map:
KK(A,B)~. E{A,B)
for any C"-algebras .4,5. Moreover by [Sk3] this map is an isomorphism if Л is a nuclear (or even К nuclear
[S]) C*-algebra. Tile deformation associated to а К К class call be described very explicitely in terms of an
unbounded representations (?,D) (cf. theorem 15) of that class (cf. [Co-H] [Co-Mi]).
352
353

For general C*-algebras the two theories do not coincide in general ([Sk^]). Let us now briefly discuss the
G-equivariant case [Kass].
Let G be a locally compact (^-compact group. Then by a G-C"-algebra one means a C"-algebra A together
with a continuous action of G on A by * automorphisms, i.e. a group morphism G —» Aut A such that for
any a 6 A the map g —> g(a) is norm continuous. When A is also Z/2 graded the action of G is supposed to
commute with the Z/2 grading.
Definition 16. Let G be a locally сот-pact group and let А, В be Z/2 graded G-C-algebras. Then a G-
eqnivariant Kasparov А — В bimodule is a Kasparov A — B-bimodule together with an action of G on ?
{<?>?) G G x ? —> g? ? ?, preserving the Z/2 grading, and such that
a) g(a(b) =
Va
В , j 6G
с) a(gFg~l - F) 6 Endfl(?) Va 6 Л , j6G.
As above the homotopy classes of G-equivariant Kasparov A — В bimodules form a group, the G-equivariant
KK group:
KKa(A,B).
All the above results on the Kasparov product extend to the G-equivariant case ([Kas]), and the G-equivariant
theory is related to the crossed product C*-algebras A X, G, A Xmax G (reduced and maximal crossed
products) by a natural map valid both in the reduced and maximal case.
Theorem 17. ([Kass]) Both crossed product constructions extend to G-equivariant Kasparov bimodules and
yield natural maps, compatible with the Kasparov product:
KKG(A, B) — KK(AxG,B XG)
Г Г
KKG(A,B) — KK(A X G,B X G).
max max
Finally we shall end this appendix by the fundamental equality of G-equivariaut KK theory when G is a
semisimple Lie group ([Kas5]). Let H С G be a maximal compact subgroup, and assume to simplify that the
isotropy representation of Я in TC(X), X = G/H, lifts to Spin (or Spinc) (cf. [Kas5] for the general case).
Let then S be the G-equivariant Spinor bundle on X. One then has two natural G-equivariant Kasparov
bimodules for the G-C*-algebras Co(A') and С (with obvious G actions).
The Dirac element 6 € KKa(Co{X),C) is given by the (Z/2 graded) Hilbert space L?(X,S) of spinors, the
action of Cq(X) by multiplication, and of G by left translations. In unbounded form (cf. theorem 15) the
operator is D — px, the Dirac operator on Л'. '
The dual Dirac element 6 e A'A'G(C,Co(A')) is given by the (Z/2 graded) C*-module ? over C0(X) corre-
corresponding to the Hermitian vector bundle S with the obvious G-equivariant structure. In unbounded form
the operator, which depends upon the choice of a base point a € X, is given by:
(?{)(p) = «(p. «) ?(p) vt e ? . p e x,
where c{p,a) is the Clifford multiplication by the unique vector (p,"a) ? TP(X) whose image by the exponential
map at p is a:
Theorem 18. One has 6 ®c 6 = id e KKG(C0(X),C0{X)).
This result implies theorem 4 of section 7.
It is not true in genera] that ?®co(A") * = id Ё Л'Л'С(С,С) (cf. [Sk3]) but this holds for SO(n, 1) ([Kas4])
and SU(n, 1) [Ju-K]).
354
Appendix B. Real and complex interpolation of Banach spaces
A very useful method to prove inequalities is given by functorial constructions of interpolation spaces
F(B0,Bi) from apair (B0,Bi) of Banach spaces that are continuously embedded in alocally convex vector
space.
The first functorial construction is complex interpolation [C]. For any в € [0, 1] one defines a Banach space
В = [Bo, Si],
as the space of values /@), where / is a holomorphic function in the strip D={j6C, Rez € [0,1]}, with
values in Bo + Si (viewed as a Banach space for the norm ||г||в0+в, = Inf{||ro||a0 + ||xi||b,; x0 + x, = x}.
The function / is assumed continuous and bounded in the closed strip, with boundary values such that:
The norm on В is given by:
where
f(it)&B0,
l№ = ]
, vteR.
||/||o.i =SuP{||/(i0||Bo , ||/(i + «olk ;i6R}.
This construction is clearly functorial hi the interpolation couple (So, Si), which implies:
Theorem 1. ([C]) Let (B0,Bt) and(B'0,B'l) be two interpolation couples, and letT : Bo + B, -~> B'o +B[ be
a linear operator such that ТВ, С Sj and ||Tz||fl- < Cj||x||Bj, Vr e Bj, j = 0,1. Then, for any в 6 [0,1],
f
~ Cf
T defines a continuous linear map from [So, Si]» to [B'o, Bi]g of norm less than Cq ~ Cf.
One also has a reiteration theorem showing that, with Be = [S0,Bi]e and assuming that So П Bi is dense
in S»o D B«, one has:
[B,0,B,,],- = [Bo.B,], for fl = (l-e')
The simplest example of complex interpolation spaces are:
a) If (X,fi) is a measure space, then:
[L1, L°°], = Lp for 1 -0 = 1/p , pe [l,oo].
/3) If И is a Hilbert space, then:
[CiCH),K,], = C{-H), 1-0 = 1/p, p6[l,oo]
where C(H) is the Schatten ideal of compact operators T in И such that Тгасе(|Т|р) < сю, or equivalent^
that:
У < oo
where М„(Т) is the n-th eigenvalue of \T\ = (T*T)I/2.
In fact, these examples a) and 3) are special cases of interpolation between M, = L'(M,t) and M where
M is a semifinite von Neumann algebra with semifinite faithful normal trace r (cf. chapter 5).
Let us now pass to real interpolation theory ([Lio-P]). Let as above (So.Bi) be an interpolation couple. For
x Ё So + Si and t e R+, define:
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It is a continuous function of ( 6 R+, and by definition the real interpolation space [jBo.^i]«,p f°r
]0, 1[ , p € [l,oo], is the subspace of Xa + Xi determined by the finiteness of:
The Banach space norm on Ijp•') is obtained as:
\t» K(t,x)f dt/tj " = \\x\\Pi>.
For p = oo the above integral is replaced by the i°° norm of t \—<¦ t* K(t,x).
Using 1/p in place of p, we thus get an interpolation square. As above it is functorial in (Bo, Si), and one
has:
Theorem 2. [Lio-P] Let (Bo, Bx) and (B'a, B[) be two interpolation couples, and let T : Bo + Bi -* B'o + B\
be a linear operator such that ТВ, С B'j and ||Гх|Ц < C,||r||By, Vx e Bj, j = 0,1. Then for any
9 € ]0,1[ , p € [1, oo], T defines a continuous linear map from [Bo, Вг]е p to [B'o, B[]e p of norm less than
($-»C[.
What is quite remarkable in real interpolation theory is that the reiteration theorem applied to B«o,p0 and
^»i,pi gives a result independent of p0 and p\, thus strengthening for intermediate values of 9 the inequalities
proven for 9o,9i.
Let В be a Banach space such that:
Then the inclusion [5o,Bi]«,i С В holds iff for some С < oo,
INIa < сади,;' Nil,, Ух
The inclusion В С [Bo, 5i]«]Oo holds iff for some С < oo,
t' K(t,x)<C\\x\\B Vie B.
If both inclusions hold, one says that В is of class A'»Eo, Bi).
Theorem 3. [Lio-P] Let $0 < Oi, let B'o be of class h'eo(B0,Bi), and let B\ be of class Ke,(B0,Bi). Then
for any в' 6 ]0,1[, Рб [l,oo],
The combination of theorems 2 and 3 is a remarkably powerful tool to prove inequalities.
Let (Xt[i) be a measure space. Then the real interpolation spaces:
?<»•«>(*,/.) = [Л°°,Л1],,. , p=l/0e]l,oo[
are the Lorentz spaces, which are defined for any p, q 6 [l,oo] as follows. For any function / on X one lets:
M,(S) = M{« 6 -V ; |/(«)|>*} V*eR+
/*(<) = inf{« > о ; m/W < 0 v<er;.
Then / 6 ?(p'«' iff (f\i1/p Г@|' т)"' < °°-
For q = oo this means that t1^ /*(<) is bounded, or in other words that for some constant С < oo:
(i{«6J(; |/(u)| > s] <C s~p.
356
«¦"/•w?I
where /"(<) = i Jo' /•(*) ds Vt 6 R'+.
Similarly, let И be a Hilbert space and for p 6 ]1, oo[ , q€ [1, oo] let
?(?•«' = [AC,/:1]»,, , p= 1/9.
These define normed ideals of compact operators in "H, and:
т e ?(iM) iff
vhere М„(Т), for ngN.is the n-th eigenvalue of \T\ = (T"T)'/2.
357

Appendix C. Normed ideals of compact operators
In this appendix we recall the standard properties of symmetrically normed ideals of compact operators
([Goh-K]), and explain the relation between the Ma?aev ideal and the ideal Li(H) of section 6.
Let "H be a Hilbert space with countable basis and Tgta compact operator in fi. The characteristic values
д„(Т) are the eigenvalues of \T\ arranged in decreasing order and repeated according to their multiplicities.
One has:
Lemma 1. (cf. [Goh-K]) a) For any n e N, T € AC,
-1!! ; E n-dimensional subspace of H}.
b) Д„(Т) = dist(T, R,,) where Д„ is the set of operators of rank less than n and dist is the distance for the
operator norm.
Assertion a) is the minimax principle. For b) see [Goh-K] theorem 2.1. This lemma has many corollaries,
for instance:
ЫГ,)-
However the д„ for n ф 0 are not subadditive. The natural seminorms on К associated to the fin are given
by:
Lemma 2. (cf. [Goh-K]) a) For any Тек, n e N one has
an(T) = SupdlTBUi ; E n-dimensional svbspace of П)
where || ||i is the C1 norm: ||5||i = Trace(|S|) VS 6 C1.
b)
(cf. [Goh-K] lemma 4.2).
Definition 3. A symmetrically normed ideal is a two sided ideal J of ?(H) and a norm || ||j on J such
that:
a) \\ATB\\j < \\A\\ \\T\\j \\B\\ VA, В 6 ЦП) , T 6 J.
b) J is a Banach space for the norm || ||j.
Note that the inclusion J С AC is automatic. Also two symmetric norms on the same ideal J are necessarily
equivalent norms. "I
One usually requires the normalization:
||T||j = \\T\\ for any T of rank one.
358
It follows from the definition that for any T ? J
\\T\\J = II |T| \\j with \T\ = (
Moreover by unitary invariance of || ||j, the value of \\T\\j only depends upon the list of characteristic values
The restriction of || ||j to the ideal R of finite rank operators is thus determined by a functional Ф defined on
the cone Д of decreasing sequences (?n)n6N, of positive real numbers such that ?„ = 0 for n large enough.
The corresponding functionals are called symmetrically norming functions and are characterized by the
following properties: ([Goh-K])
о) Ф(?) > О for ( e Д, ? ф 0.
/3) Ф(Л?) = ЛФ(?) V? 6 Д, A > 0.
7) Щ + П)< *«) + Ф(ч) V?, 77 6 Д.
E) Let i, 77 6 Д be such that ?{;•<? Щ Vn 6 N. Then Ф(?) < Ф(т;).
о о
One can also require the normalization:
Theorem 4. [Goh-K] Let Ф be a functional on Д satisfying a) /3) -/) S). Then the following defines a symmet-
symmetrically normed ideal J^: Js, = {T eK ; Sup Ф(м^(Т))<оо} toAereMN'(T) = (/iO(T),/i1(T),...,MjV(T),0,0,...)«
Д for anj T 6 k. Moreover
||T||* = Sup Ф(м"(Т)).
N
Note that the sequence Ф(д^(Т)) is uon decreasing for any T. Conversely, for any symmetrically normed
ideal J there exists a unique symmetrically norming function Ф such that the restriction of j| ||j to the ideal
of finite rank operators is | ||ф:
\\T\\j = \\T\\* VT of finite rank.
This however does not imply that J — ^ф, and this nuance played a crucial role in section 3 of this chapter.
For any symmetrically norming function Ф, let ]\ be the closure in ,/ф of the ideal of finite rank operators.
Then one can assert that:
for any symmetrically normed ideal J.
It is however not true in general that Jj = ./ф. In section 3 we saw for instance that:
cir,«,) c C(p,oo) vP6[l,oo[.
In fact one can show (cf. [Goh-K] corollary 6.1) that J4 = J% iff ^ф is a separable Banach space.
Theorem 5. [Goh-K] Any separable symmetrically normed ideal J is of the form J% for a symmetrically
norming function Ф, uniquely determined by the equality || ||j = || ||ф.
For a given symmetrically norming function Ф one defines the dual norm Ф' by:
359

Г
ф'@ = Sup iY^& m ; г, е д , Ф(т») <
11=1
Then, provided /ф jt AC, one has the duality:
This duality is given by the pairing:
(A,B) =Тгасе(Л5) 6 С,
and use as essential ingredient the inequality: (cf. [Goh-K])
N n
As an example, let (жп) be a decreasing sequence of positive real numbers, жп —> 0 when n —> oo, and let
Ф»(?) = Sup
Then Фт is asymmetrically norming function whose dual is
Let us now list a number of symmetrically normed ideals with the corresponding norms.
\\П(г,,) =
ЧЯ
We assume here that p Ё ]1, oo[ , q? [l,oo]. For ? < oo one has ?<<*¦«> = jCjf"e), i.e. the finite rank operators
are dense in ?('¦*), but for q = oo this no longer the case, and one has the dualities:
For p, q Ё ]1, <x>[ one has the duality:
Forp = l,g = oo one lets
p' = 1 , 1/, + 1/»' =
and we denote the corresponding normed ideals by jC'1'00), jCj,1'00':
; <rN(T) = o(k>gN)}.
360
The dual space of C1^1'00' is the Ma?aev ideal
?(«¦¦') = i T 6 AC ; ^n-1Mn(
I i
The dual space of ?<-°°V is C^^K
oo
The natural norm on jC*00.1) is given by Y. N~2 "n(T), but it is clearly equivalent to the other norm given
l
Finally we let, as in section 8, ?i(W) be the symmetrically normed ideal:
Li(H) = {T 6 AC ; ^(T) = O((logn))}
with the symmetric norm:
||T||ii = Sup Li(N)-1 <rN(X).
N>1
One checks that Lio(H) = {T e K. ; /in(T) = o(logn)-1}. We shall show that the ideal ЩП) plays with
respect to the Mac,aev ideal ?(°°1' the same role as the ideal ?<P'°°' plays with respect to ?*'>.1' for p < oo
(cf. section 2). One has:
Lemma 6. Let f € C?°(R) be a smooth even function on R with compact support and let D be a selfadjoint
invertible unbounded operator in a Hilbert space "H. There exists a finite constant с = cj such that for any
e >0:
for any a 6 C('H) such thai [D,a] is bounded.
The proof is the same as that of lemma 2.13. As in that lemma one easily gets for any operator a,
\№eD),a]\\<e\\f% \\{D,a]\\-
Moreover if Supp / С [—к, к], the rank of f(eD) is majorized by the number of eigenvalues of |D|~' larger
than ek~i. There exists a universal constant A < oo such that
and it follows that:
1) < A(logn)-1 HO-'IU,- for n > 1,
log(rank(/(?D))) <?"! kX\\D-l]\Li.
The operator {f(eD),a} = T has rank N < 2 rank(/(e?>)) and norm < e\\f'\\i \\[D,a]\\. Thus its Mac,aev
norm
is at most (log N) ?||/'||i ||[Да]||, which by the above estimate concludes the proof.
361

Appendix D. The Chern character of deformations of algebras
The main theme of this chapter was the construction of the Chern character in К homology:
ch. : K'(A) — HC(A)
where we used Fredholm modules (Ti, F) as the basic cycles for К homology. We have seen in chapter 2
a variant of the Kasparov bivariant theory based on deformations of algebras. If we specialize E theory
to E homology; E(A, C), we obtain as our basic cycles the deformations of an algebra A to the algebra к
of compact operators in Hilbert space. (To be more careful one would have to stabilise both A and jfc and
replace them by SA, Sk but we shall ignore this point.) It is thus natural to extend the construction of the
Chern character eft,, done up to now in К homology terms, to the case of deformations.
This has been done in [Co-F-S] for closed * products, i.e. for deformations of the algebra of functions on
symplectic manifolds. We shall describe this construction in this appendix. It is thus restricted to the special
case A commutative, but should extend to the general case using the results of [Gerst].
We let W be a symplectic smooth manifold. We let ш be the closed symplectic 2-form on W, and let
п e C°°{W, A2T) be the 2-vector field on W dual to ш. The Poisson bracket on C°°(W) is given by:
P(f,9) = {/,?} = i(S)(dfAdg) V/,, 6 C°°(W).
By construction P is a Hochschild 2-cocycle:
where A = C™{W) and A is viewed as a bimodule over A. For each n we let Cn{A,A)> C"(A-4") be the
space of cochains continuous in the C°° topology.
Definition 1. A * product is an associative bilinear product on the space -
A, of the form:
of formal power series over
f*S= ?
for any
where CT €C7(A,A) for any r, and CQ{f,g)=^ fg, Cx - P.
Such * products exist on arbitrary symplectic manifolds ([WL]). We shall now consider a more restricted
class of deformations, the idea being that if the above formal deformation corresponds to a deformation of
A to an algebra of compact operators in Hilbert space, the canonical trace on jfc should yield a trace on the
deformed algebra given by:
By construction r(f) takes its values in C[[i/]], and we shall in fact only need the I first terms of its expansion:
Definition 2. [Co-F-S] A * product ts closed iff
Jcr(f,g)u,< = Jcr(g,f)u,1 Vr<e; f,g€C?(W).
This is equivalent to the requirement that
362
' , </,,) = j!gu>1,
0,/',...,/")= f C(fl /n)/° wl
Jw
n({)= Coefficient of i/' in //ui'
defines a trace on the algebra -4[[i/]].
If the condition of definition 2 holds for any value of r, we shall say that the * product is strongly closed.
Theorem 3. ([Om-M-Y]) On every symplectic manifold W there exists a closed » product.
The Moyal » product on W = R2' is defined by:
CV = (r!) Pr,
the r-th power of the bidifterential operator P. It corresponds to the Weyl pseudodifferential calculus and
is strongly closed.
Similarly for W — T" M the cotangent space of a Riemannian manifold, the * product given by the pseu-
pseudodifferential calculus with an asymptotic parameter (cf. [Wi]) is strongly closed.
Finally note that it is easy to give examples of non closed star products. Indeed using the symplectic measure
J ¦ u>1 we get a natural ,4-bimodule map from A to A":
and a corresponding map:
С eC"(A,A)->C€Cn{A,A') ,
Lemma 4. For a closed * product one has BCi = 0.
Here В : C2(A,A') —* Cl(A,A") is the В operator of cyclic cohomology (chapter 3 section 1).
Proof. One has (A, C2)(/, g) = JC2(f,gW, and AB0 C2 = 0 iff JC3(f,д)ш1 = /C2(j,flu.' Vf,g€A.
In fact one can see that, proceeding step by step from the knowledge of the С*, к < t, defining a star product
closed up to order t, the obstruction to the existence of Ct+i yielding a * product closed at order t + 1 is
given by:
(Ker4nKer5)/6(Ker5)= HC3(A).
We refer to [Co-F-S] for this point, as well as the discussion of equivalence classes of closed * products.
Let us now pass to the construction of the Chern character of closed * products, as an element of the cyclic
cohomology:
ch.((Ct)) e HC-(A).
Theorem 5. ([Co-F-S]) Let W be a symplectic manifold of dimension It, and let * = ? vr Cr be a closed *
product. Then the following equality defines the components of a 11 dimensional cyclic cocycle in the (b,B)
bicomplex on A = Cf°(W):
?>2*(/°,..., /") = n(f° * o(f,f-) *.. .*e(f-k-\ /")) vr e A , к = о, i,..., /,
where 9(/, g) = f * g — f g for any f,g 6 A, and n is as above the coefficient of i/ in f f wl € C[[i/]] for
f б ЛИ].
363
I

Thus the ?>2* satisfy b<p2t + Bfu+2 = 0 Vib = 0,..., I. They vanish by construction for jb > I.
They also vanish for 0 < 2ib < I, so that the relevant components are the ipu for t < 2Jb < 2?.
The component ?>и is always given by:
ел.
A simple computation in the 4-dimensional case, dim W = 4, i.e. I = 2, shows that
general, and is given by:
does not vanish i
in
= Jf°
This example shows that a closed * product does not in general come from a quantization, i.e. a deformation
of A to the algebra of compact operators in Hilbert space. Indeed a necessary condition for that to hold is
the integrality of the character:
(V, Л'о(Л)> С Z.
Since in the above example one can add to Ci an arbitrary cyclic 2-cocycle, the above integrality condition
does not hold in general.
As another example let M be a compact Riemannian manifold, W = T* M its cotangent space, and * the
* product associated to the pseudodifTerential calculus ([Wi]). It is a closed * product coming from the
deformation of C?°(W) to the algebra of compact operators given by the tangent groupoid of M (chapter 2).
In particuair it is integral, computes the Atiyah Singer index map, (chapter 2 section 5) and using invariant
theory ([At-B-P], [Gii]) one computes its character:
сЛ,(Сс°°(Т*М),*) = Td{T'M)
as the Todd genus of the symplectic manifold T*M, ([Co-F-S]) viewed as a de Rham current on T*M.
364
CHAPTER. V
OPERATOR ALGEBRAS
The theory of operator algebras, or von Neumann algebras (according to the terminology of J. Dieudonne),
is the noncommutative analogue of measure theory. By the same token, this theory is an essential ingredient
of the analysis of noncommutative spaces, which we have already described above. It is remarkable that
one can describe for such spaces the associated von Neumann algebra as well as the most general states and
normal weights on that algebra, in terms of random operators. This gives the general theory exposed below
an inexhaustible source of concrete and explicit examples, i» which the general theorems exercise their full
force.
In contrast with the commutative case, where the only interesting measure space, up to isomorphism, is
the Lebesgue space, the noncommutative case is far more complex. Nevertheless, we now have a complete
classification of the amenable von Neumann algebras, a class that is described by numerous equivalent
properties (cf. Sections 7 and 8).
In a certain sense, the theory of von Neumann algebras is a. chapter of linear algebra in infinite dimensions,
LUC UlSHH^nvJii испуги t* »u.» i т^ ~,...™...
on a Hilbert space S) that is closed for the weak topology, ami a C*-algebra or stellar algebra, which is a
norm-closed *-subalgebra of ?(#). Of course every von Neumann algebra is in particular a C*-algebra, but
it is not a very interesting C-algebra since it is not separable for the norm. (For vY compact, C(X) is
separable ъ> X is metrizable.)
The chapter is divided into eleven sections.
1) The papers of Murray and von Neumann.
2) Representations of C*-algebras.
3) The algebraic framework for noncommutative integration and the theory of weights.
4) The factors of Powers, Araki and Woods, and of Krieger.
5) The Radon-Nikodym theorem and factors of type 111*.
6) Noncommutative ergodic theory.
7) Amenable von Neumann algebras.
8) The flow of weights: mod(M).
9) The classification of amenable factors.
10) Subfactors of type Hi factors.
365

11) Hecke algebras, type HI factors and statistical theory of prime numbers.
Appendix A. Crossed products of von Neumann algebras.
Appendix B. Correspondences.
366
4
J
V.I. The papers of Murray and von Neumann
Let ft be a complex Hilbert space and let ?(Я) be the C-algebra of bounded operators from Я into ft. Thi
is a Banach algebra equipped with the norm
imi = sup
and with the involution T — T* defined by
Even if ft has countable dimension, the Banach space ?(Я) is not separable. There exists a unique closed
linear subspace of the dual C(f))' whose dual is C(ft); this is the space of linear forms on ?{?>) that may be
written
ЦТ) = Trace(pT) (VT 6 C(ft)),
where p is a trace-class operator, which means that \p\ = {/''p)'^2 satisfies
for every orthonormal basis (&) of ft. The norm of the linear form L is equal to Trace|/>|, and, equipped
with this norm, the space
?(Я). ={рб?(Я). Trace|/>| < oo}
is a Banach space, called the predual of the Banach space ?(J5).
When ft has countable dimension, the Banach space ?(Я), is separable; the duality between ?(ft)* and
C(ft) is exactly analogous to the duality between С (A) and (""(Л), wliere Л is a set. In particular, whenever
Я is infinite-dimensional, the space ?(Я). is not rellexive and the topology <r(C(ft), ?($)„)' is not the same
as the norm topology of ?(Я). Thus, it is much more restrictive for a subspace of ?(ft) to be closed for
<т(?(Я), ?(ft)*) than for the norm topology.
This distinction between the two topologies is essential for the sequel. Let M be a *-subalgebra of ?(ft)
containing the identity operator 1. The following conditions are equivalent:
1) Topological condition: M is cr(?(fl),?(fl).)-closed.
2) Algebraic condition: M is equal to the commutant (M'Y of its commutant M'. (The commutant of a
subset S of ?(ft) is denned by the equality S' = {T 6 ?(Я), TS = ST (VS 6 S)}.)
The equivalence between these two properties is von Neumann's double commutant theorem.
Definition 1. A von Neumann algebra on Я is a *-sitbaUjebra of C(ft) containing the identity operator
and satisfying the above equivalent conditions.
We cite some immediate consequences of the definition,
Let 5 С C(ft) be a subset such that S = S" = {T*, T e ,S}; then the commutant S' of 5 is a von Neumann
algebra.
Let M be a von Neumann algebra on Я and let Л/i = {T6 M, \\T\\ < 1} be the unit ball of M; then,
since Mi is <r(C{fi), C(ft).)-dose<i in the unit ball of the dualof C(ft),, it is compact for the topology
o(M], C(ft).). In particular, M is the dual of a Banach space Л/,, which is in fact the unique (closed) linear
subspace of M* whose dual is Л/ ([S3]).
Also called the vltraweak topology on C(ft) [Di?]-
367

One introduces, often with the name of weak topology, the topology on C(f>) arising from the duality with
the subspace of ?(ft). formed by the operators of finite rank, i.e., the topology characterized by
та-*т о сг„?,т»> —(Г?,ч) (уг.чея).
This topology is coarser than ir(?(ft),?(ft),), and, since the space of operators of finite rank.is norm-dense
in ?(•$)„, it coincides with <r(?(ft),?(ft),) on the bounded subsets of ?(.$). However, the term "weak
topology" is not a good one; what is in fact at hand is the topology of pointwise weak convergence.
The von Neumann algebras are the *-subalgebras of ?(ft) containing 1 that are closed for the topology of
pointwise weak convergence, since every commutant S', 5 С ?(Л)> has this property.
Examples of von Neumann algebras
1. Abelian von Neumann algebras. The description of this simple example will allow us to give the
abstract form of the spectral theorem and the Borel functional calculus. Let (X,B,ti) be a standard Borel
space equipped with a probability measure д, and let tt(L°° (X, fi)) be the algebra of operators from i'(X, fi)
to L2(X,n) defined by
Then M = ir(i°°) is a commutative von Neumann algebra, and in fact M = M'. The predual of M is the
space Ll(X,ti). Let x — n(x) be a Borel function from X into {1,2,.. .,00} and let {X,p) be the Borel
covering of X defined by
X = {(x,/) 6lxN, 1 < j < n{x)}, p(z,j) = x.
To the "multiplicity" function n one associates the representation ж„ of ?°°(X,/i) on ?2(Х,Д) (where Д is
given by J f(x,j)dji = j?, f{x,j)dfi), defined by
The image M = жпAая(Х,^)) is a commutative von Neumann algebra on the space f) = ?2(Х,Д), and if
n ф 1 then M' <f. M.
Definition 2. Let Mi be a von Neumann algebra on ftj (i — 1,2); me nay that M\ is spatially isomorphic
to Mi if there exists a unitary U : fti —> Л2 s"c* '*"' UMiV = Мт.
If Я is separable then every commutative von Neumann algebra on ft is spatially isomorphic to the space
жиA*сс(Х,р)) for a suitable space [Х,ц) and function 1».
Suppose then that T e ?(ft) is a normal operator GT* = T'T); let M be the von Neumann algebra
generated by T. One can take (X,B) to be the spectrum К С С of T, and the measure class fi given by the
spectral measure of T. The mapping that associates, to the restriction of a polynomial function y^a,;z*zJ
to K, the operator ^o.ijT'T'', may be extended to an isomorphism ir of i°°(ft.',/i) onto M with ж(г) = Т.
Thus, for every bounded Borel function / 011 Sp7\ f{T) has meaning and we have
(A,/, + A2/2)(T) =
A2/2(T),
(f о д)(Т) = f(g(T)).
The function 1 —• n(x) of A" into {1,..., 00} is unique modulo //. It expresses the multiplicity of the point x
in the spectrum K. The theorem on the bicommutant shows that every operator that doubly commutes with
T is a bounded Borel function of T. Finally, let N be a von Neumann algebra, not necessarily commutative,
and let T ? N; then T = Ti + iT-,, Tj = TJ; since Tt is normal and every I{Tj), f Borel, is also in TV, we
see that N is generated by the projections that it contains.
368
2. The commutant of a unitary representation. Let G be a group and let ir be a unitary representation
of G on a Hilbert space ft, (or, more generally, let A be a «-algebra and let зг be a nondegenerate *-
representation of A). The commutant
Я(х) = {ТбДЯ,), Тж(д) = *(,j)T (Vff e G)}
is by construction a von Neumann algebra.
The interest of Щж) lies in the following proposition:
Proposition 3. (a) Let E С Л 6e a closed subspace and P the corresponding projection; then
E reduces ж О Р Ё Я(ж).
(b) iet Bj, Ei be two closed subspaces reducing ж; then the reduced representations жЕ> are equivalent if and
only if
P, ~ P2 in Щж),
i.e., there is a U € Щж) such that U*U = Pu UV = P>.
(c) жЕ' is disjoint from жЕ'1 if and only if there exists a pivjcctton P in the center of Щж) such that PP, = P,
and(l-P)P2 = P2.
It is natural to say that the representation ж is isotypic if it does not have two disjoint subrepresentations.
This is equivalent to saying that Щж) is a factor in the following sense:
Definition 4. A factor M is a t>on Neumann algebra whose center reduces to the sealers С
Another immediate corollary of the proposition and of the Borel functional calculus is the following:
For ж to be itTedncible, it is necessary and sufficient Ihnt Щж) = С.
3. Finite-dimensional von Neumann algebras. Let. M be a finite-dimensional von Neumann algebra.
Forgetting the Hilbert space on which it is represented, regard i t as a semisimple algebra over the algebraically
closed field C. It is then the direct sum of a finite number of matrix algebras:
Here the sign '=' corresponds to the following definition:
Definition 5. Let Mi be a von Neumann algebra on the space f), (» = 1.2). We say that Mi is algebraically
isomorphic to M-> if there exists an algebra /ло»шг/)Ллч)» О of \[\ onto Aft such that &(x*) =¦ &{x)* (Vx G
Mi).
Let G be a group and let ж be a unitary representation of G on a finite-dimensional Hilbert space ft,- Then
Щя) is finite-dimensional and to the decomposition Л/ = ©J = 1 MU(C) there corresponds the decomposition
of ж into isotypic components тгц. of multiplicity nt.
4. The action of a discrete group on a manifold. Let V be a manifold and let Г be a discrete group
acting on V by diffeomorphisms. We make the following hypothesis:
(*) For every q G Г, п ф 1, the set {x G V, <jx = x} is negligible.
Let X be the set V/T and let p : V — Л' be the quotient mapping. Let Я„ = ^(p-'(o)) for all a e X. By
construction, fia has for orthonormal basis the et (x € V, p(x) = a).
Definition 6. A random operator .4 = (Aa)aex i* a family of bounded operators Aa 6 C(f)a) such that
the function
369

a(x,y) = (AHr)eT,e,) (x,ysV, p(x) =
ts measurable.
We write \\A\\ = esssup \\Aa\\ and call A bounded if \\A\\ < +00.
Proposition 7. Let M be the *-algebra of equivalence classes (modulo equality almost everywhere) of bounded
random operators, equipped with the operations
Then M is a von Neumann algebra (that is, a *-aigebra algebraically isomorphic to a von Neumann algebra
on a Hilberi space).
Moreover, the center of M may be identified with the commutative algebra L°°(X), where X is equipped with
the image of the Lebesgue class.
Note that Г acts ergodically on V iff ?°°(Л') = С.
Reduction theory '
Let Я be a separable Hilbert space and let 3 be t.he set of all factors M С C(f>). There exists a Borel
structure on 5 that makes it a standard Borel space. Let (Л', Б) be a standard Borel space, ц a probability
measure on (X,B), and t —> M(t) a Borel mapping of Л" into 5- Let M be the C"-algebra whose elements
x € M are the bounded Borel sections t — x(t) € M(t), identified if they are equal /j-almost everywhere,
equipped with the obvious operations and the norm
One shows that the C*-algebra M is a von Neumann algebra (i.e., is algebraically isomorphic to a von
Neumann algebra on a suitable space), for example by considering the action of M on the space
defined by
The construction of M is summarized by writing
M
= /
and M is said to be the direct integral of the family (Л/(/)),6л" with respect to the measure fi.
Theorem 8. Lei M be a von Neumann algebra on a separable IJilberi space. Then M is algebraically
isomorphic to a direct integral of factors
L
This theorem of von Neumann shows that the factors already contain the originality of all of the von Neumann
algebras: they suffice to reconstruct every von Neumann algebra as a "generalized direct sum" of factors.
1 Written by von Neumann in 1939, published in 1949 [N2].
370
Let G be a group and let x be a unitary representation of G on a separable Hilbert space. To the decompo-
decomposition of ft(x) as a direct integral of factors, there corresponds the decomposition of x as a direct integral of
isotypic representations.
Comparison of subrepresentations, comparison of projections and the relative dimension func-
function
Let G be a group and let x be a unitary representation of С on the Hilbert space f)T. Suppose x is isotypic;
it is then natural to expect, as in finite dimensions, that x is a multiple of an irreducible representation тЕ
that is a subrepresentation of x. In the correspondence between subrepresentations of x and projections
P 6 ft(x), it is easy to see that the irreducible representations correspond to the minimal projections of
Я(х):
x has an irreducible subrepresentation О the factor R(t) has a minimal projection о (Аеге exist a Hilberi
space J5| and an isomorphism ofC(S)t) onto R(t).
Every isotypic representation with an irreducible subrepresentation is a multiple of this subrepresentation.
For every factor M on Я having a minimal projection, f) can be factored as a tensor product Я = Я1 ® Й2
in such a way that
M = {T® 1, T operating on ft,}.
However, there exist groups G having an isotypic representation тг with no irreducible subrepresentation.
This penomenon does not occur if G is a real semisimple Lie group and x is continuous, or more simply if G
is compact and ж is continuous. However, it can occur for the regular representation of numerous discrete
groups: let Г be a denumerable discrete group and let Д be the union of the finite conjugacy classes of Г.
Then Д is a normal subgroup of Г; suppose Д = {1}. Let A' be the right regular representation of Г; then
Яд» = ^2(Г) is the Hilbert space having (cg)a^c as orthonormal basis, and one sets
One shows that the von Neumann algebra /?(A') is generated by the operators A(#) (g ё Г), and that the
vector €1 is cyclic and separating for Л(А'): R(X')e\ and /?(A')'fi are dense in f). The coordinates of Tc\,
where T ё Л(А')ПЯ(А')', are constant on the conjngacy classes of Г; since Д = {1}, it follows that Ts\ ё Cfi
and since ei is separating, T € C.
Thus, Я(А') is a factor and A' is isotypic. To see that Я(А') is not isomorphic to a factor C(f>), one makes
use of the following concept:
Definition 9. A trace r on a factor M is a positive linear form such that r(AB) = r(BA) for all А,В € M.
There exists a nonzero trace on ?(ft) only if f) is finite-dimensional, as one sees by verifying that if Й is
infinite-dimensional then every element of ?(#) is a sum of commutators. Now, one can define r on Д(А')
by
The property r(AB) = r(BA) may be verified for A and D of the form \(g) (g € Г) and may deduced
in general by bilinearity and continuity. Since т(\) = 1, a nonzero trace on the infinite-dimensional factor
Я(А') has been constructed.
Thus, there exist infinite-dimensional factors that have no minimal projection.
Let M be a factor. As in Proposition 3, translation of the concepts of equivalent representations and the
direct sum of representations into the language of projections yields the following concepts:
1) For a projection P, P € A/, one denotes by [P] the c(|iiivalence class of P for the relation Pi ~ Pi,
which holds if and only if there exists U € M with U'U = Pi, UU' = Pi-
371

2) For Pi and P2 such that PiP2 — 0, one denotes by [Pi] + [P,] the class of Л + Pi, it depends only on
those of P, and P2.
The hypothesis that Af is a factor implies that the set of classes of projections is simply ordered for the
relation [Pi] < [P2] defined by the condition that P{ < P2 for suitable representatives. This simply ordered
set has a partially defined law of composition that makes it possible to give meaning to an equality such as
[P] = ^[Q] with n and m positive integers.
Definition 10. A projection P € M is said to be finite 1/ Q ~ P and Q < P imply Q = P.
This property depends only on the class of P. In the language of representations, one would adopt the
following definition: a representation л- is finite if every subrepresentation it* equivalent to * is equal to тт.
If л- is not finite, then it contains an infinite number of subrepresentations equivalent to тг, and conversely.
Theorem 11. (Murray and von Neumann) Let M be a factor with separable prednal. There exists a mapping
D of the set of projections of M into R+ = [0,+co], unique up to a scalar factor A > 0, snch that:
(b) P,P2 = 0 => D(P, + P2) = D(Pi) + D(P2).
(c) P finite О D(P) < +00.
Moreover, up to a normalization, the range of D is one of the following subsets o/R+:
{1 ...,n}, in which case M is said to be of type 1„
[0,1]
[0,+CO]
{0,+co}
Hi
III.
One sees that, for M to have a minimal projection, it is necessary and sufficient that it be of type I. If Af is
of type In, n < со, then it is Л/„(С) and D(P) is the usual dimension of the subspace of C" onto which the
projection P € Afn(C) projects. If 11 = со then M = ?(#) with f) infinite-dimensional and separable, and
D(P) = dimension of the range of P.
What is remarkable in the II, case is the appearance of dimensions with arbitrary values in [0,1].
A factor Af is said to be finite if it contains no subfactor of type loo- This is equivalent to saying that Af is
in one of the cases I,, (n < со) or IIL. In particular, if M is infinite (not finite) then there exists no trace r
on Af such that r(l) = 1.
One of the remarkable results of the first papers of Murray and von Neumann is the converse:
Theorem 12. (Murray and von Neumann) If M is a factor of type Hi, then there exists a unique trace r
on M such that r(l) = 1.
Moreover, r € Af». One proof, due to F. J. Yeadon [Y], reduces this theorem to a powerful result of Ryll-
Nardzewski: in every о-(Л", A"*)-compact convex subset of a Banach space Л', there exists a point that is left
fixed by every affine isometry of the convex set. One applies this result by showing that if ip € Af» satisfies
II^H = ^A) = 1, then the closed convex hull A' of the orbit of ip under the action of the inner automorphisms
of Af (transported to M.) is a(M,,M(-compact.
One then obtains the existence of r € M., r(l) = 1, such that r(uxu*) = r(x) for и unitary in M and
i€ Af.
Algebraic isomorphism and spatial isomorphism
Let Afi and Af2 be two von Neumann algebras and let 0 he a «-algebra isomorphism of Mi onto Mi- Then
в is isometric (because Л/i and M2 are C*-algebras and \\T\\- = \\T"T\\ = spectral radius of T"T) and,
372
since the predual is unique, в is c(Mj, Af,-. (-continuous. If Mt acts on the Hilbert space Д- (i = 12)
the isomorphism в need not be spatial, for, even though they are isoinorphic, Afi and Af2 may have non-
isomorphic commutants. Let us fix Af: suppose M has a separable predual and let us try to describe all of
the isomorphisms * of Af onto a von Neumann subalgebra of C(H), where fi is a separable Hilbert space.
Making use of the reduction theory, we may further simplify matters by restricting attention to the case that
Af is a factor.
We are thus led to study, up to equivalence, the representations тг of Af on a Hilbert space Ят that are
continuous when Af has the topology o(M,M.) and Цв„) has the topology of the duality with ?(fiT),.
Since Af is a factor, the commutant Я(тг) = 3r(Af)' is also a factor and тг is isotypic. It follows, by forming
3Ti ф 7Г2, that two representations jrL and ttj are never disjoint and that every representation тг of Af is a
subrepresentation of an infinite representation that is fixed once and for all. This result may be extended
to the case that Af is no longer a factor. Every representation тг of M (continuous for a(Af, Af.) and
o(C(f),),C(?),),)) is a subrepresentation of a representation p that is properly infinite (in the sense that the
commutant R(p) contains a subfactor of type Ic) and faithful1 and which is chosen arbitrarily, for example
by starting with an isomorphism a of Af onto a von Neumann subalgebra of C(fi) and forming p = офоф..
Thus, once Af is known algebraically, there is no serious problem in determining all of the isomorphisms of
Af onto a von Neumann subalgebra of ?(Я) (see Section 9). The real problem is that of classifying the von
Neumann algebras up to algebraic isomorphism.
The first two examples of type II] factors, the hypnrfiitite factor and the property Г
Recall that if Г is a countable discrete group all of whose conjugacy classes are infinite and if A' is its right
regular representation, then the von Neumann algebra /?(A') is a factor of type Hi, denoted Я(Г).
In On rings of operators.IV [Mur-Ns], Murray and von Neumann showed that all of the factors Я(Г) are
isomorphic for Г locally finite (i.e., the union of an increasing directed family of finite subgroups), and that
if Г is the free group with two generators then one obtains a factor not isomorphic to them. Let N be a
factor of type Hi and r the unique trace (r(l) = 1) of N; one defines the Hilbert-Schmidt norm on N by
(This is the analogue of the norm ( J^ l°i;|2) " of a matrix («,;).)
This is a pre-Hilbert space norm on N and the corresponding metric is denoted d2.
The result of Murray and von Neumann is then as follows:
Theorem 13. (Murray and von Neumann [Mur-Na]) There exists, up to isomorphism, one and only one
factor of type Hi having separable predttal and such that:
(Vi], ...,!„€ Pf, Vc > 0) 3 finite-dimensional *-subalgcbra A' such that d2(xj,K) < ( (Vj).
We denote this unique factor by R, called, for obvious reasons, the hyperfinite factor.
In their paper, Murray and von Neumann show that every infinite-dimensional factor contains a copy of R.
For Г a locally finite, denumerable discrete group, one sees easily that Д(Г) satisfies the condition of the
theorem, hence is isomorphic to R. Choosing Г suitably, one sees moreover that R satisfies the following
condition, called the property Г:
(Vi'i, i'n € rt,V<; > 0) 3 unitary и of R such that
j=l n).
1 The term "faithful representation" is often used instead of "a representation тг whose kernel reduces
to 0" (jt(i) = 0 => x =0).
373

Murray and von Neumann then proved that if Г = Z is the free group with two generators, then the
factor Л(Г) does not satisfy Г. We shall return later on to this property, which was for Murray and von
Neumann only a technical tool. We quote these authors: "Certain algebraic invariants of factors in the case
Ui are formed, 1) and 2) in §4.6 and the property Г, of which the first two are probably of greater general
significance, but the last one so far has been put to greater practical use." In fact, the invariants 1) and 2)
that they mention are:
1) knowing whether N is anti-isomorphic to N (i.e-, isomorphic to the opposite algebra №, x -y=-yx for
<,9бП
2) the subgroup F(N) of R+ constructed as follows: let N = N®K, where К is a factor of type I»,. Then,
on N, the relative dimension function D has range R+, and if в 6 AutJV then there exists a unique
positive real number A = mod $ with D($(P)) = \D(P) (V projections P). The group F(N) = {mode,
$ 6 AutJV} is obviously an algebraic invariant of N.
In fact, an example of a type Hi factor not anti-isomorphic to itself was not obtained until tong after ([Coi]),
as well as the existence ([Co8]) of a type II( factor whose group F is distinct from R^. (the only calculable
examples always gave G = R+).
Finally, to conclude this review of the results of Murray and von Neumann, we mention that they had
succeeded in exhibiting a factor in the case III but they noted: "The purely infinite case, i.e. the case HI,
is the most refractory of all and we have, at least for the time being, scarcely any tools to investigate it."
(With the notations of Example 4, one can take V — Pi(R) and Г = PSLB, Z) acting by homographic
transformations. Then M is a factor of type III.)
374
V.2. Representations of C" -algebras
One of the key theorems in measure theory is the Riesz representation theorem. We state it only for X
compact and metrizable (.V nietrizable <=> С'(Л') separable).
Theorem 1. Let X be a compact metrizable space and lei L be a positive linear form on C(X), i.e.,
f 6 C(X), f(x) > 0 (Vz € X) => L(f) > 0.
Then there exists a unique positive measure ft on the tr-algebra В of Borel sets of X (i.e., the a-algebra
generated by the closed sets of X) such that
L(f) =
In particular, one can construct the llilbert space ia(A',B,/«) and the representation т of C(X) on h? by
multiplications. One knows in addition the von Neumann algebra generated by n(C(X)): it is precisely the
algebra of multiplications by the elements of L^fA'.B./i). The ст-atlditivity property of p is translated by
the equality
)
W/ / об/
where ip denotes the natural extension of L to L°°(X,B,[i) and where (eo)o6/ is any countable family of
pairwise orthogonal projections ea 6 iM(A',C,/i).
Now suppose that A is a noncommutalive O'-algebra with unit. The above concepts of positive element, pos-
positive linear form and ir-acklitivity have exact analogues, and this is I he point of departure for noncommutative
integration theory.
Positive elements in a C"-algebra
Let f) be a Hilbert space and let T ё C(f}). The following conditions are equivalent:
a) T = T' and Spect.rumT С [О.+те).
b) {ТС() >0for alUefl.
For a C*-algebra A with unit and for x ё A, the following properties are equivalent:
1) i = i* and Spectrum x С [0,+оо).
2) 3a € Л such that j.- = n"o.
3) 3a e A such that a" — a and x = a'.
4) ЗА > 0 such that ||j: - Al|| < A.
We then say that x is positive and we write x > 0. The condition 4) shows that the set of positive elements
is a closed convex cone in A, which is denoted Л+. If .4 = С(Л') then A+ = {/, f(x) > 0 (Vi € X)}.
Positive linear forms on a C*-algebra
Let Л be as above and let A' he its Bauach space dual. We say that L € A" is positive if L(x) > 0 (ix > 0).
We denote by A*+ the it(.4", ,4)-closed convex cone of positive linear forms on ,4. The analogue of the Riesz
representation theorem consists in the following construction (GelTand, Naimark, Segal).
Since L is positive, the condition 2) shows that
L(x'x) > 0 (Vz € .4).
375

It follows that the sesquilinear form {x^y)i = L(y"x) defines a pre-Hilbert space structure on A. Let fjL be
the separated Hilbert space completion and, for x 6 A, let ttl(z') be the left-multiplication operator defined
by
*l(x)v = xy for all У 6 A.
The inequality L(y"x"xy) < ||z||2?(y"y) (Vy 6 -4), which results from ||x||2 — x'x > 0, shows that T? defines
a representation of A on the Hilbert space #/,.
Just as, in the commutative case, the linear form L can be extended to the Borel functions / € JC°°(X,B,ii),
here the linear form L can be extended to a linear form on the von Neumann algebra tl(A)" generated by
t*l(A), and 'he extension, denoted by L, has the following <r-additivity property.
Definition 2. Lei M be a von Neumann algebra on the Hilberl space fi and let ф be a positive linear form
on M. We say thai ф is normal if
for every family (ea)ag/ of pairwise orthogonal projections.
Here ^2 ea denotes the smallest projection that majorizes all of the finite sums ]>2Г=1 еа;; it is an element
of M. One proves that, for ф to be normal, it is necessary and sufficient that it come from the predual Mm
of M.
Let us return to C*-algebras. Let A be such an algebra, with unity. The Hahn-Banach theorem, applied to
the convex cone A+ with nonempty interior, shows that the set
of states of Л is a nonempty convex set that separates the points of A. One is therefore assured of the
existence of "positive measures" and, by the Gel'fand-Naimark-Segal construction, of the existence of an
isometric representation of A as a C-subalgebra of ?(J5), ft a Ililbert space.
Moreover, S is (т(Л*,Л)-сотрас1, hence is the closed convex hull of its set of extremal points, the pare states,
which are characterized as follows:
ip is a pure state о the representation ttv is irreducible.
In fact, more generally, there is a bijective correspondence between the face of tp in the cone A*^ and the set
of positive elements of the von Neumann algebra Щп,?), given by
i> € A"+ is associated with у € ДGГ^)+ when V'(«) = {7V(a)l.yl}.
where 1 € f)v is the vector corresponding to the unit element of A.
Thus, every C-algebra A has sufficiently many irreducible representations. In the commutative case, the
irreducible representations of A coincide with the homomorphisms of A into C. In the noncommutative case,
the nature of the relation of equivalence between irreducible representatbns of A on a fixed Hilbert space
determines a privileged class of C*-algebras; the following theorem of J. Glimm is fundamental:
Theorem 3. (cf. [Die]) Let A be a separable C'-algebra. The following conditions on A are equivalent:
A) Every isotypic representation jr of A on a Ifilbert space ft, is a multiple of an irreducible represen-
representation.
B) For every irreducible representation it of A, the image л-(Л) contains the ideal ik(fi») of compact
operators on ft,.
376
C) Let ?) be a.separable Hilbert space and let Rep(A, ft) be the Borel space of irreducible representations
of A on fi; then the quotient by the relation of the equivalence of representations is eountably separated.
D) 1}т\ and tt2 are two irreducible representations of Л having the same kernel, then they are equivalent.
A C*-algebra satisfying the above equivalent conditions is said to be postliminal.
377

V.3. The algebraic framework for noncommutative integration aud the theory of weights
To be able to take into account positive measures that are not necessarily finite, it is necessary to introduce
the noncommutative analogue of infinite positive measures.
The initial data for noncommutative integration is thus a pair (M,tp) consisting of a von Neumann algebra
M and a weight ip on M in the following sense ([Com], [Pei], [Hi]):
Definition 1. A weight on a von Neumann algebra M is an additive, positively homogeneous mapping tp
of M+ intoR+ ~ [0,+00]. We say that:
a) ip is semifinite, iff, {x € Af+, <p(x) < 00} is cr(M, M.)-total.
b) ip is normal, iff, ^?(supxa) = supy>(xa) for every bounded, increasingly directed family of elements
ofM+.
The simplest example of an infinite weight is that of the usual trace for the bounded operators on a Hilbert
space Я- Setting M = ?(Й), for every T € M+ and every orthonormal basis (?„)„?/ of Я one has
^,) = sup {Traccvl, A of finite rank}.
0<Л<Т
ТгасеГ =
In fact, the first infinite weights studied were the traces in the following sense:
Definition 2. A weight on a von Neumann algebra Л/ is called a trace if it is invariant under the inner
automorphisms of M.
Thus, every weight on M which is canonically denned is a trace. The analogues of the concepts of convergence
almost everywhere and of the № -spaces, p € [l,oo], of the classical theory were obtained principally by
Dixmier [Dii] and Segal [Se,] for semifinite normal traces.
If ip is a trace, the set
CP = {x€M, P(|if)<co)
is a two-sided ideal of M, and ||z||p = ip(\x\'')llp defines a seminorm on Cp. The completion spaces V(M, ip)
have numerous properties that generalize the commutative case and the classical case of p'th-power summable
operators on a Hilbert space. In particular, if x > 0 and 4>(x) — 0 imply x = 0 (in which case ip is said to
be faithful), the predual M. of M may be identified with the space Ll(M,ip). Moreover, the intersection
L-(M, ip) П L°°(M, ip) is a Hilbert algebra:
Definition 3. A Hilbert algebra is я *-algcbra A equipped with a positive definite pre-Hilberi space inner
product such thai:
1) (*.y) = (v'.x')Dx,veA).
2) Tke representation of Л on A by left multiplications is hounded, involutive and nondegenerate.
The condition 1) defines a conjugate-linear isometry J of the Kilbert space completion Я of A onto itself.
Condition 2) makes it possible to speak of the left regular representation A of A on Я and therefore to
associate with it a von Neumann algebra X(A)" on f). Then:
a) The commutant of X(A) is generated by the algebra of right multiplications X(A) = JX(A)J.
b) The von Neumann algebra associated with the Ililbert algebra L-(M,ip) Л L°°(M, ip) may be identified
with M.
c) For every Hilbert algebra A, there exists a faithful semifinite normal trace r on the von Neumann
algebra X(A)" such that
r(X(y')X(X)) = (x,y) Ax,y€A),
and A is equivalent to the Hilbert algebra associated with r.
378
In general, a von Neumann algebra M does not have a faithful semifinite normal trace. For example, if M is
a factor, it has a faithful semifinite normal trace if and only if it is not of type III. A von Neumann algebra
M is said to be semifinite if it has a faithful semifinite normal trace (when M acts on a separable space, this
is equivalent to saying that M is a direct integral of factors none of which is of type HI).
For M semifinite, the supplementary tools that come out of the theory of Hilbert algebras make it possible
to prove results that are inaccessible in the general case, such as:
Theorem 4. Let Mi be a von Neumann algebra on Я{ (i = 1,2). Then the commutant (Af] ® A/2)' is
generated by M{ ® M^.
For semifinite von Neumann algebras M\ and Mi, this result is a consequence of the commutation property
a) for Hilbert algebras. Similarly, if G is a unimodular locally compact group and Ag is a Haar measure
on G, then the convolution algebra of continuous functions with compact support is a Hilbert algebra, and
the commutation result a) implies:
Theorem 5. The commutant Я(А') of the right regular if presentation A' of G on L (G, dg) is generated by
the left regular representation.
This theorem was proved, for not necessarily unimodular locally compact gronps, by J. Dixmier, who in-
introduced the concept of quasi-Hilbert algebra. Theorem 4 was proved, for not necessarily semifinite von
Neumann algebras, by M. Tomita in 1967. As a matter of fact, his theory of generalized Hilbert algebras is
the foundation for all of the noncommutative integration theory for weights that are not necessarily traces.
The Tomita theory owes much to M. Takesaki, who transformed the original, very difficult to decipher article
into an accessible text [T2].
The essentials of the Tomita-Takesaki theory can be summarized by the following definition and theorem:
Definition 6. A left Hilbert algebra is a *-algebra A, equipped with a posiiive definite pre-Hilberi space
inner product, such that:
1) The operator x —» x' is closable.
2) The representation of A on A by left multiplications is bounded, involutive and nondegenerate.
Thus, the only difference from a Hilbert algebra is that the closure S of the operator x —* x" can have
absolute value |S| ф 1. Let
Д = (adjoint of 5) о 5
be the square of the absolute value of 5. Then 5 = J\l/!, where J is an isometric involution, and the
fundamental result is as follows [T2]:
Theorem 7. Let A be a left Ililbert. algebra and let M be the von Neumann algebra generated by the left
regular representation of A. Then JMJ = M' and
Д"'Л/Д-" = M
for all t € R.
Moreover, just as the Hilbert algebras are associated with traces, the left Hilbert algebras are associated
with weights ([Com]):
Let ^bea left Hilbert algebra and let M be the associated von Neumann algebra. Then there exists a
faithful semifinite normal weight <p on M such that
379

Conversely, let M be a von Neumann algebra and let ip be a faithful semifinite normal weight on M. Then
Av = {x € M,
j:*) < oo},
equipped with the multiplication of Л/ and the inner product {x, y) = ip(y' x), is a left Hilbert algebra. The
associated von Neumann algebra may be identified with Л/ and the corresponding weight with ip.
Since every von Neumann algebra has a faithful semifinite normal weight (if M acts on a separable space then
in fact it has a faithful normal state), it follows in particular that every von Neumann algebra is isomorphjc
to the von Neumann algebra generated by the left regular representation of a left Hilbert algebra.
It is here that a remarkable discovery of Takesaki and Winniiik relates Tomita's theory—more precisely the
one-parameter group of automorphisms of the von Neumann algebra M denned by at(x) — Д"гД ""—to the
fundamental Kubo-Martin-Schwinger condition of quantum statistical mechanics which we already described
in 1.2. Of course the automorphism group a, is not unique; it depends on the Hilbert algebra A, that is, on
tlie faithful semifinite normal weight ip on M:
Definition 8. Let p be a faithful semifinite normal weight on a von Neumann algebra M. The group of
modular automorphisms is the one-parameter group (of ),eB of automorphisms of M associated with the
left Hilbert algebra Av.
One then has the following characterization.
Theorem 9. ([T2][Winn]) Let M be a von Neumann algebra and let ip be a faithful normal state on M.
Then the group (ct(°)i6b of modular aulomorphisms is 1he unique one-parameter group of automorphisms of
M that satisfies the К ubo-Mariin-S chiving er condition for /? = 1.
This characterization of {<rf) still holds for faithful scniifinite normal weights with the proper reformulation
of the KMS condition taking care of domain problems.
380
V.4, The factors of Powers, Araki and Woods, and of Ki-ieger
The noncommutative analogue of a probability space is a pair (M,<p), where M is a von Neumann algebra
and ip is a faithful normal state on Л/. The simplest example corresponds to M = Mn(C). Every state ip on
M may be written ip — Ti(p), where p is a positive matrix the sum of whose eigenvalues is 1:
<p(x) =
(Vz 6 Mn(C)).
One can therefore suppose that p is diagonal, with eigenvalue A, > 0 at the t'th row, and with A] > A2 >
... > А„ > 0. The list of eigenvalues of p is an invariant of ip called its eigenvalue list. The group of modular
automorphisms of ip is given by
o-f(x) = e"Hxe-ial, where Я = Log/;.
In particular, if ey denotes the canonical matrix unit, then
The system (Mtip) is analogous to a probability space having a finite number of points. To obtain examples
that are more interesting, the simplest procedure consists in carrying out the analogue of the construction
of infinite products of measures.
Thus, let (M,,,ipv)i,?N be a sequence of pairs (matrix algebra, faithful state); let A be the inductive limit of
the C*-algebras
,4„ = Mi ® M-.G) •••© Mv,
where Av С Au+\ by means of the mapping x —* x ® 1. On Д, which is a C*-algebra with unit, one defines
a state ip = (S*i/11 ^pv by the equality
One then considers the pair (M,ip) =(von Neumann algebra, normal state) associated with the pair (A,tp)
as in Section 3 of this chapter. If every <р„ is faithful then so is <p, and the group of modular automorphisms
of (M,ip) is given by
«zi ® •¦•® г„©1® •••) = <гГ(*1)®---®<гГ"(я„)® 1® •¦¦•
In fact, this construction of von Neumann algebras is due to von Neumann himself; however, one had to
wait until 1967 for it to reveal itself to be fundamental. For the thirty years that followed the birth of von
Neumann algebras, only three pairwise nonisomorphic factors of type III were known. In 1967 R. T. Powers,
who was trained as a physicist, succeeded in showing that if all of the pairs (М/,1|О„) are taken to be equal
to the pair (Mi(C),ip), where <p((aij)) = ояц + A — a)«22, then one obtains a family, with continuous
parameter a € @,1/2), of pairwise nonisomorphic factors of type Ш: Я\, A = a/(l - a) € @, 1) ([Pow,]).
In terms of the eigenvalue list (А„у )j = 1 Пи one has the following criterion:
1) M is type I if and only if
2) M is type II] if and only if
/2 - (Л^,I7-!2 < oo.
3) If А„, > * for some 6 > 0 for all v, then M is type III if and only if
381

Awinf {|(А„1/А„О - 1|2, С} = со
for some (and hence all) positive C.
After Powers' discovery, H. Araki and E. J. Woods undertook a classification, up to isomorphism, of the
factors that are Infinite tensor products of matrix algebras [Ar-W]. They showed in particular that, from the
eigenvalue list of the <р„, (А„у)> = 1„„, one can calculate the following two invariants:
"•oo(Af) = {A 6 @,1), M ® Дл isomorphic to M},
p(M) = {A 6 @,1), M ® Да isomorphic to R\}.
The computation of r^iM) is done as follows. For each finite subset / of the set of indices i/ 6 N one lets
X(I) be the product |~J {1,. .., ny} with probability measure A given by the product of the (А„^).
The asymptotic ratio of M is the set of all x €: [0,oo] such that there exists a sequence of mutually disjoint
finite index sets /„ С N, mutually disjoint subsets A",1,, A';; of Л'(/„) for each n such that a 6 K^ implies
A(o) / 0, and a bijection фп from A',1 to A'^ satisfying
and
lim max |* - А(^„(а))/А(а)| = 0.
n-*oo agA'i
They showed in addition that r^M) is a closed subgroup of Rl and that the equality roo(M) = A*
characterizes the Powers factor R\ among the infinite tensor products of matrix algebras. Moreover, by
studying p(M)^ E. J. Woods succeeded in showing the impossibility of classifying the factors by means of
real-valued Borel invariants.
On the other hand, W. Krieger had undertaken a systematic study of the factors associated with ergodic
theory. I shall begin by explaining his construction of a von Neumann algebra starting from an equivalence
relation with denumerable orbits on a standard Borel space (X,B) and a quasi-invariant measure ft. This
construction generalizes the first construction of Murray and von Neumann, which was itself inspired by
the crossed products of the theory of central simple algebras over a field. In its definitive form, it is due to
J. Feldman and C. Moore [Fel-M].
Thus, let (X,G, ji) be a standard borel probability space ami let 'R, С Л' х Л" be the graph, assumed to be
analytic, of an equivalence relation with denumerable orbits. One assumes that /i is quasi-in variant in the
sense that the saturation of a negligible borel set by 7\. is again negligible. One then considers the left Hilbert
algebra of bounded borel functions from 7v into С such that, for some n < oo, the set {j, f(i, j) ф 0} has
cardinality < n for all i. The inner product is that of L2G\.,/i), where
Г п- м- h
The convolution product is given by
(f*!l)h)=
where, for Yb 72 € И, one sets 7172 = 7 when 71 = (»,, ji), 72 = (»2,J2), Ji = '2 and 7 = (i\,ji).
The left Hilbert algebra A thus appears as a generalization of the algebra of square matrices. Since it has
a. unit element (the function / such that f(i,j) = 0 if i / j, and /A,1) = 1 for all i), it yields a pair
(Af, ip), where M is a von Neumann algebra and ip is a faithful normal state on M. We shall write simply
382
M = too('K, Д),. where H denotes the graph of the equivalence relation, equipped with the groupoid law
7172 = 7- It 's interesting from a heuristic point of view to make V. play (in noncommutative integration)
the role played (in classical integration) by the space X. The noncommutativUy is due to the existence
of a nontrivial groupoid law on 72 (the trivial law on the set X being x ¦ x = x for all x 6 X). Up to
isomorphism, the von Neumann algebra Ь°°(И,1л) depends only on the class of the measure /i. One then
makes the following definition:
Definition 1. Let (X,, S,-,/ii,7Zi) (i = 1,2) be equivalence relations with denumerabte orbits, as above; TZi is
said to be isomorphic to И2 if there exists a Borel bijecliov 0 of X\ onto X2 such that <>(/*]) is equivalent
to pi and, almost everywhere,
0(class of x) = class of &(x).
If T is a borel transformation of the standard Borel space (X,B) and if /i is a measure quasi-invariant
under T, then the orbits of T in X define an equivalence relation Ит', Ti is said to be weakly equivalent to
T-2 if Ht, is isomorphic to HT,.
Theorem 2. [Dy] Any two ergodic transformations with invariant measure are weakly equivalent.
Let T be such a transformation. The state <p (associated with the invariant measure) on the von Neumann
algebra L°° (Ит. f) is a trace, so that M = L°°(T!.t,p), which is a factor since TtT is ergodic, is of type Hi.
H. Dye showed, moreover, that it is the liyperfinite factor.
Around 1967, W. Krieger undertook a systematic study of the weak equivalence of the transformations
(X, B, f,T), where ц is quasi-invariant under T. He introduced two invariants ([Kri]):
r(T) = {A € [0,+co), V<r > 0, V.4 С .Y, ц(А) > 0,
ЗВ С A, ;i(B) > 0, and n € Z such that
•-A
< с (Чх€ В) and T"В С Л},
р(Т) = {А € R;, Зу ~ /1 with '^f^ € A1 (Vi € Л')},
and he showed that i- and p are not only invariants under weak equivalence, but that in fact r(T) coincides
with the Araki-Woods invariant
'"оо(Л-/) = {А, Л/0 R\ isomorphic to M],
where M = L°°(HT,n), and similarly that p(T) = p(M)-
This is actually a generalization of the results of Araki and Woods. For, let (Af,,, >Ри)„еы be a sequence
of pairs (matrix algebra, faithful state), and let (\uj)j = 1 „„ be the corresponding list of proper values.
Then the infinite tensor product von Neumann algebra ®^L, (Л/„,^„) may also be obtained by Krieger's
construction, from the following space and equivalence relation:
Л' = Д Xy, where Л'„ = {1 п„ } for each i/,
в is the cr-algebra generated by the product topology,
CO
/i — j I /*„, where /tvij) — ^v,j f°r j € Xu,
V. is the relation x = у О Xi = m for i sufficiently large.
38.4

The relation 72 is in fact equal to 72 x, where T is the transformation that generalizes the operation of adding
1 in the p-adic integers: to calculate Tx, one looks at the first coordinate г; of x that is not equal to the
maximum possible щ, one replaces it by ц + 1, and one replaces every preceding xj (j < i) by 1.
In fact there exist Krieger factors, i.e., of the form L°°GZt , /'), that are not infinite tensor products of matrix
algebras. The culminating point of Krieger's theory is the following theorem, proved around 1973 by means
of the invariants of factors that we shall discuss later on:
Theorem 3. (Krieger [Kr2]) Let (A',-,ft, fit,Tj) (i = 1,2) be ergodic transformations, where уц is quasi,
invariant and (A,-, ft) is a standard Borel space. Then, T\ is weakly equivalent to Ti (i.e., 72т, is isomorphic
to 727) if and only if the factors ?°° [TZt, , /*») are isomorphic.
This result clearly shows the need for recognizing whether an equivalence relation 72 with denumerable orbits
is of the form 72т for some Borel transformation T. In particular, when 72 arises from the action of a discrete
group Г, W. Krieger and I showed [Co-Kr] that if Г is solvable then 72 is of the form 7Ir- The case that
Г is an amenable group was solved by D. Ornstein and B. Weiss [Or-W] and the definitive answer to the
problem was obtained in [Co-F-W] by J. Feldman, B. Weiss and myself.
Theorem 4. [Co-F-W] К is of the form IZt if «nd only ifH is amenable in the sense of Zimmer [Zi]].
Zimmer's definition translates simply the amenability of the von Neumann algebra i°°G2, ц) (cf. Section 7),
so that the above two theorems imply ([Co-F-W]):
Corollary 5. Any two Cartan subalgebras of an amenable factor M = t°°G2,/i) are conjugate by an
automorphism of M.
Here, by Cartan subalgebra is meant a maximal abelian subalgebra A of M such that:
a) The normalizer of Л generates M (A is then said to be regular).
b) There exists a normal conditional expectation E of AI onto A.
Condition b) is automatic if A/ is of type Hi, thus one sees that:
The Ityperfinite factor R has. up to conjugation, only one regular maximal abelian subalgebra.
We note that, by a result of V Jones and myself, for nonamenable equivalence relations 72!, 72г, it is false
that the isomorphism ?°°G2i,/«i) «* ?""G22,/<2> implies the isomorphism of the relations 72] and 72г. Thus
one cannot hope to translate directly R. Zinimer's rigidity results (on equivalence relations arising from
ergodic actions of semisimple Lie groups [Zio]) into a rigidity theorem on the corresponding factors. The
problem of adapting his proof to the case of factors is a key open problem.
384
V.5. The Radon-Nikodyui theorem ami factors of type lllj
The Radoii-Nikodym theorem
The Tomita-Takesaki theory associates, to every faithful semifinite normal weight tp on a von Neumann
algebra M, a one-parameter group af of automorphisms of M, the group of modular automorphisms,
defined by
where A<p is the modular operator, the square of the absolute value of the involution x —<¦ x' regarded as an
unbounded operator in the space L?(M,tp), the completion of {x € M,tp(x'x) < oo} for the inner product
(x,y) = ip(y'x).
On the other hand, the theory of Araki and Woods associates to every factor M the two invariants r^ and p,
Гоо(М) = {А, М® Rx~M],
p(M) = {A, M® Rx~Rx).
Now, for the factors of Araki and Woods, a direct calculation based on their work yields the following
equalities:
1) >'оо(Л/) = p|SpectrumA,, (ip a faithful normal state он М).
2)
p(M) = {ехрBл-/Го), 3 faithful normal stnte tp on M such that <r?o = 1}.
These two equalities of course suggest the following definitions for an arbitrary factor M:
S(M) = P^SpectrumA,, {tp a faithful normal state on M).
T(M) = {possible periods of groups of modular automorphisms of M].
Clearly the first question consists in knowing whether the equalities r^ = S and p = ехрBэт/Х), valid
for the factors of Araki and Woods, remain true in general. A closely related question is the problem of
calculating the invariants S and T, The above definitions of these invariants show that in order to calculate
S and T one must pass in review all of the faithful normal states on M and calculate their groups of modular
automorphisms. Now in general, as is clear for the factors of Section 4 of this chapter, a factor is presented
with a priveleged state or weight for which the calculation of Av and of is easy. Thus the problem that
poses itself is to study the precise extent to which the group a^ depends on tp.
The complete answer to this problem in fact constitutes precisely the noncommutative version of the Radon-
Nikodym theorem.
Theorem 1. [C03] Let M be a von Neumann algebra, let tp be n faithful semifinite normal weight on M\
and letli be the unitary group ofM equipped with ihc topology (т(Л/,Л/.).
(a) For every faithful semifinite normal weight ф on Af there exists a unique continuous mapping o/R
info U such thai:
ф(х) = Ыи]хщ)I=-цг (Vr € M).
This is expressed by writing «, = (?></' : Dtp)t.
(b) Conversely, let t —» u, be a continuous mapping of Я into U such that
385

«.+,- = m«f (««<) (V/,/'€R).
Then there exists a unique faithful semifinite normal weight Ф on At such that (Оф : Dip) ~ u.
If M is commutative, M = Ьх(Х,В,ц), then ip and ф are positive measures on X that are equivalent to (t.
Therefore there exists a Radon-Nikodym derivative h : Л' — R+. The Л'1 (( e R) are in M = V(X, В,ц)
and h" = (Zty :Xty)t.
If Af is semifinite and r is a faithful semifinite normal trace on Af, then there exist positive operators
affiliated with M such that tp = т(рч>-), ф — r(py), and
The property о/(я) = u,af (x)u't (Vi € Af) shows that, although the group of modular automorphisms in
general varies with ip, its class modulo the inner automorphisms does not vary. We may then ask if it is not
altogether trivial. However, an easy argument based on the above theorem shows that
T(M) = {To, crjo is an inner automorphism}.
Moreover, a theorem of.). Dixmier and M. Takesaki [T2] shows that, assuming Л/, separable,
T(M) / R «- Л/ is not semifinite.
One then introduces the group OutJI/ = AutM/lnnA/ of classes of automorphisms of M modulo the inner
automorphisms and one associates to every von Neumann algebra M a canonical homomorphism of R
into OutAf:
6(t) = class of af (independent of the choice of ip).
In particular, T(M) = KerS is a subgroup of R. In fact, the range of 6 is even contained in the center of the
group OutM.
Moreover, one can calculate T{M) on the basis of a single faithful semifinite normal weight tp on Af, since
it suffices to determine the / for which af is an inner automorphism. For example, if Af is an Araki-Woods
factor M — ®"=1(M,,^»), the group of modular automorphisms for tp = ®^Li Vv 's known: it is
A simple calculation based on the eigenvalue list (A^j )J=i. „H of <pv then shows that
From this, one infers for example that Т(ДЛ) = {To, A'To = I}.
When M is a Krieger factor, or more generally when M = L°°Ai.,it) for some equivalence relation 7Z (see
Section 4; 1Z is not necessarily assumed to be of the form lip), one also verifies by an easy calculation that
T(M) = 27r/Lo(!/,GJ),
where p is Krieger's invariant, i.e., p(T\) is the set of all A > U for which there exists a v ~- [i such that the
Radon-Nikodym derivatives Ли(Sz)/dv[x) belong to Az for every Borel transformation S that preserves the
orbits of It. The latter calculation shows easily that the equality p(M) = ехрBтг/Т(М)) does not hold in
general.
386
Finally, we emphasize that Theorem 1, the analogue of the Radon-Nikodym theorem, makes it possible for
numerous von Neumann algebras, to describe explicitly 0// of the semifinite normal weights and their groups
of modular automorphisms.
Thus, let us take up again Example 4 of Section 1 and let us show how to represent every normal weight on
the von Neumann Af of random operators on X = V/T. Recall that a quadratic form on a Hilbert space Д
is a mapping q of ft into [ff, +00] such that:
a) ЧЫ + i) + ?(« - 4) = 2«@ + 2«(i,) (Vf, 4 6 Si).
b) Я(Ю = |Ap«(O (V€€fl,A€C).
It is assumed moreover that Dom? = {? € ft, q(?) < 00} is dense in ft and that q is lower semicontinuous.
Then there is a unique unbounded positive operator T such that q(?) — ||T''2f ||2 (V{ 6 ft).
Definition 2. A random form consists in giving, for all a € Л' and v 6 A"Ta(XI, a quadratic form ?„„
on f)Q, such that:
(lLa,A., = |A|w (VA€R).
B) The mapping x —» 4РAI„A)(?Г) of V into R is measvrablt for nil measurable sections и and ?.
Recall that p : V —- X is the canonical projection. One then sets / q = /v (fp(r)(ej,). Theorem 1 makes it
possible to prove:
Theorem 3. A) Let q be a random form and, for every pair (a,f), let Taitt be the positive unbounded
operator in fta associated with the quadratic form qOtU-
If f q = 1 then the following equality defines a normal stale tpq on M:
for every random operator A = {Aa)a?X¦
B) The mapping q —» tpq is a bijection between the random forms 4, /4 = 1, and the normal stales
on M.
C) tpq is faithful if and only 1/ the operator ТО:„ >s nonsingulitr for v / 0 and almost every a 6 X.
D) For every faithful normal stale <p= tpq on M, the group of modular automorphisms af is given by
(<rf(A))a = T%u АаТ-« for all A = (Ao) 6 M.
The factors of type ПЦ
The Radon-Nikodym theorem makes it possible to calculate the invariant S(Af) from a single faithful
semifinite normal weight ip on M. The centralizer Mv of tp is defined by the equality
Mv = {x € Л/, af(x) = x Cite R)}.
For every projection e ф 0, e € Mv, a faithful semifinite normal weight ipe on the reduced von Neumann
algebra eAfe =(г6 M, ex = xe = x] is defined by the equality
One then has the formula
eMe,x > 0).
S(M) = P) SpectrumA,,,,
1 With n = dim V, making use of the differential of the action of Г on V to trivialize the fiber T(V) along
each orbit о/Г.
387

where e varies over the nonzero projections of M^. Moreover, since e commutes with tp, the calculation of
of' (hence of the spectrum of AVt) is immediate:
af'(x) = af(x) (Vx 6 eMe).
Thus, the above formula permits calculating, for example, S(M) for M = L°°(T\.,fi) (section 4). One has
the equality
S{M) = ;•(•?),
where r is the invariant defined by Krieger as the set of essential values of the Radon-Nikodym derivatives.
Therefore, in general S(M) / rao(M). Moreover, there is a much more satisfying interpretation of S(M) as
the spectrum of the modular homomorphism 6.
Let us suppose that the predual M. is separable. Then, a one-parameter group (а(),ев of automorphisms of
M is of the form erf for some faithful semifinite normal weight tp on M if and only if the class ?(t*i) of a, in
OutAf is equal to 6(t) for all (. To put it another way, the set of all of the groups of modular automorphisms
for faithful semifinite normal weights on M is precisely the set of multiplicative Borel sections of 6.
For every faithful semifinite normal weight tp, the spectrum of
a^ in the following sense:
may be identified with the spectrum of
Spectrum av = {A e dual group of R, /(A) = 0 for
all / € i'(R) such that f f(t)afdt = 0}.
(The spectrum of it* can also be denned in terms of the supports of the distributions (a'*>(j:))A Fourier
transforms of the functions t —* af(z). One obtains in this way distributions with values in M, and
Spectrum it» is the closure of the union of the supports of the (<t*>(j:))a).)
We note also that in the above formula, H'+ is identified with the dual group of R by the equality (A,() = A"
(A e R+,( € R). The precise equality is then
R^. = Spectrum a43.
Therefore
Н'+П S(M)= Pi Spectrum»,
and this formula shows that when M is a factor, R + П S(M) is a subgroup of R^j.. Moreover, 0 € S(M) О
M is of type Ш, and one has the following cases:
HI0 S(M) = {0,1},
И1Л А е@, 1) : S(M) = AzU{0},
lib S(M) = [0,+oo).
In a certain sense, the above A expresses the distance between M and the semifinite factors; in fact, A is
related in a monotone and one-to-one fashion with a quantity that measures the obstruction to the existence
of a trace on M:
d(M) =diaiiieterAS/IiinM),
where .S denotes the metric space of normal states on M (with the metric d(tpi, tp2) = IIVi — Vail) ап^ where
innA/ acts on S by <p —* uipu" (n unitary in M). For M of type IHi, rl(M) = 0 ([Co-Sti]), thus one cannot
distinguish between two states of a factor of type HIi by means of a property that is closed and is invariant
under inner automorphisms.
388
Let us now give a heuristic interpretation of 5( Л/). We first, return to the origins of the terminology "modular
automorphisms". The first example of a left Ililbert algebra is the convolution algebra of continuous functions
with compact support on a locally compact group G. Let icj be a left Haar measure on G. The modulus of
the group is then the homomorphism Sa of G into R+ associated with the right action of G on dg. Moreover
the modular operator of the left Ililbert algebra is the operator of multiplication by the function 8a in the
space L2(G, dg), thus its spectrum is the closure of the range of 6G. One can then, always from a heuristic
point of view, interpret the invariant S(M) for a factor M as "the image of the modulus of AT'.
The factors of type ПЦ, A 6 @, 1), are characterized by the equality S(M) — A* U {0}; now, it is easy to see
that if G is a locally compact group and the image of the modulus Sg of G is A*, then G is the semidirect
product of the unimodular group Я = Ker5G by an automorphism a 6 Autff that multiplies every Haar
measure on Я by A. Conversely, every pair (Я, a), where Я is unimodular and a multiplies every Haar
measure on Я by A, yields by semidirect product a locally compact group G = H xa Z with Sq{G) = A*.
This analogy is confirmed by the following theorem:
Theorem 4. [C03] Let A € @,1).
(a) Let M be a factor of type ЦЦ. There exist a factor ff of type Hoo and a $ 6 AutJV multiplying every
trace of N by X (one then writes mod 0 = A) suck thai M is isomorphic to the crossed product of N by в.
(b) Let N be a factor of type Hoo and let в € AutiV with mod в = A. Then the crossed product of N by
$ is a factor of type ПЦ.
(c) Two pairs (Ni,$i) (i = 1,2) yield isomorphic factors if and only if there exists an isomorphism a of
N\onto N-2 such that the classes ofa&\a~l and &'±, modulo the inner automorphisms of N%, are the same.
Before studying the implications of this theorem for the problem of classifying factors, let us note some
important information concerning the general theory of factors of type ПЦ • The concept of trace is replaced
by the following:
Definition 5. A generalized trace ip on a factor M of type П1л, А ё @,1), is о faithful semifintie normal
weight tp such that SpA^ = S(M) and ip(l) = +00.
One proves [C03] the existence of generalized traces on M by studying the relations between the invariants
T(M) and S(M) and by showing that, except when S(M) = {0,1}, the invariant T(M) is determined by
S(M) (whereas, in the case Шо, Т(М) can be any ^enumerable subgroup, not necessarily closed, of R).
Moreover, the following uniqueness theorem holds: if tpi and ^2 are two generalized traces on M, then there
exists an tnner automorphism a such that <p? is proportional to <p\ о a.
The von Neumann algebra of type II», the N of the theorem, is none other than the centralizer Mv of a
generalized trace tp; its position in M is unique tip to inner automorphisms and it can be characterized as a
maximal semifinite subalgebra [C03].
Theorem 4 shows that the problem of classifying the factors of type Шл reduces to:
1) classifying the factors of type Hoc;
2) given a factor ff of type Пс, determining the conjugate classes of в in OutiV = AutJV/IimJV such that
mod $ = A.
These two problems are the principal motivations for Sections 6 and 7 below, problem 2) being subsumed
under noncommutative ergodic theory.
389

V.6. Noncommutative ergodic theory
Let (X, B, /*) be a standard Borel .space equipped with a probability measure /*, and let Г be a Borel
transformation of (X, B) that leaves /i invariant. Let M = L°°(X, S, /i) and let <p be the state associated
with ft. Then T determines an automorphism of M that preserves ip, by the equality
Conversely, every automorphism of M that preserves (p can be obtained in this way. Thus, classical ergodic
theory is, after translation, the same thing as the study, up to conjugacy, of the automorphisms of M that
fix p. In fact, one of the justifications for the theory is that all of the triples (X, B,fi) (hence all of the pairs
(M, ip)) with /*({:c}) = 0 for all x € X are isomorphic. Thus, to each different construction of such a triple
there will correspond a family of automorphisms of (X,B,/i) and the problem is to compare them. Similarly,
in the framework of noncommutative integration theory, there exist numerous different constructions of the
hyperfinite factor R, for example as the regular representation of a locally finite discrete group (Section 1),
as the infinite tensor product of the pairs (Л'/п(С),гп) with rn the trace normalized by rn(l) = 1, or again
by the theorem of H. Dye (Section 4). To each of these constructions, there correspond automorphisms of Я.
Indeed, R can also be constructed from the canonical anti-commutation relations on a real Hilbert space E,
thus obtaining an injective homomorphism of the orthogonal group of E into AutiJ. It follows that OntR
in fact contains every separable locally compact group.
Of course, one does not want to distinguish between two automorphisms of the form в and cr9o-~l , where
a € AutiJ. Let us adopt the following general definitions:
Definition 1. Lei M be a von Neumann algebra and let fli.fla € AuiAf be two automorphisms of M
(a) fli and #2 are said to be conjugate if there exists a a & AuiM such that fl2 = сгв^'1.
(b) 9\ and &i are said to be outer conjugate if there exists a o~ € AuiM such that в? = ав\<т~1
modulo InnM.
When M is commutative, the two definitions coincide because 1ппЛ/ = {1}. In the general case, we have
two problems: conjugacy and outer conjugacy. 1 shall begin by results that extend important results of
classical ergodic theory to the noncommntative case. I shall then discuss some specifically noncommutative
phenomena; the reader interested only in the automorphisms of R. can pass directly to Theorem 14 below.
Rokhlin's theorem
Let (X, B, /i) be a standard borel probability space and let T be a Borel transformation of (X,B) that
preserves /*. Then there exists an essentially unique partition of Л', Л* = Ui^i ^*< where each Xi is invariant
under T, and where:
1) For every i > 0, the restriction of T to Л\ is periodic of period * and
canl{7'Jx} = ; (VxSA'j).
2) For i = 0, the restriction of T to Л'о is aperiodic, i.e.,
cardfT^} = oo (Va.-e.Vo).
Rokhlin's theorem may be stated as follows. Let T be an aperiodic transformation of (X,B,/л). For every
e > 0 and n > 0, there exists a borel set E such that E,T(E) Tn~1(E) are pairwise disjoint and
fi(x\ 0 T'(E)) < с
Now let (N,t) be a pair (von Neumann algebra, faithful normal trace normalized by r(l) = 1) and let в be
an automorphism of N that preserves r. Then there exists a partition of unity in N, Y2T=o e> = '• wnere
each ej is a projection in the center of N that is invariant under 0, and where:
390
a) For every j > 0, the restriction of 9 to the reduced von Neumann algebta Nti satisfies the condition
that в> is inner and, for к < j and for every nonzero projection e < e;, there exists a nonzero projection
f <e such that
b) For j = 0, the restriction of в to the von Neumann algebra Nta is aperiodic: for every к > 0, every
nonzero projection e < ea and every e > 0, there exists a nonzero projection f < e with
As in the commutative case, this decomposition is unique. When ea — 1, в is said to be aperiodic.
Theorem 2. [C05] Let (N, r) be a pair (von Neumann algebra, normalized faithful normal trace)
be an aperiodic automorphism of N that preserves т. For every integer n > 0 and every e > 0, Me
a partition of unity ^,™ZQ Ej = 1 in N, where the Ej are projections, such that
\\9(Ej) - ?i+
Here, as in Section 1, we write ||i||3 = (r(|j|2))l/:! for all x e -ЛЛ
Entropy ([С02З], [Co-St2], [Co-N-T])
Let (Х,В,ц,Т) be as above and let V he a Borel partition of Л'. One calls entropy ofV relative to T a scalar
h(TV) that asymptotically counts 1/n times the logarithm of the number of elements of the partition
h(T,V) that asymptotically counts \j-
consisting of V,TV T"-lT:
h(T, V) = lim -
where, for a partition Q = (»y)je{i,....*}i
with tj(O = -tlogt (V<€[0,1]).
The entropy ofT is then defined to be the siipremum of the h(T,V). It is an invariant, calculable thanks
to the Kolmogoroff-Sinai theorem: /i(T) = h(T,V) for every partition V for which the T'V generate the
<r-algebra B. In particular, let us consider a Bernoulli shift: the translation by 1 in Пиег^"'/*»)> wnete
the A*,/ are all equal to {1 p] and all of the /!„ are the same measure j € {1, ...,p} —» \j. Then
МГ) = Е>=.1(^)-
Now, the Bernoulli shifts have an analogue in the noncommutative case; let us take the simplest that is
associated with au integer p. One considers the infinite tensor product (S)i,?z(Mi,t<Pv), where, for all v,
My — MP(C) is the algebra of all pxp matrices and ipv =the normalized trace. The shift Sp then yields
an automorphism of the pair (R, r), where R is the hyperfinite factor and г is its normalized trace. One
can then pose the following question: Are the Sr pairwise conjugate? This problem led me, together with
E. St0rmer, to the following generalization of entropy and of the Kolmogoroff-Sinai theorem, which enabled
us to distinguish the automorphisms Sp up to conjugacy.
These first results were limited to the nmmodular case, where the measurable space (X, B,/i) is replaced by a
von Neumann algebra M with a finite normal trace r, and where the automorphism в € AutAf preserves r.
The role of the finite partitions Q = (qj) of the space X is played by the finite-dimensional subalgebras
К С М, but at the very outset one encounters an entirely new difficulty: in the noncommutative case it is in
general false that the subalgebra of M generated by two finite-dimensional subalgebras Ki and K2 is again
finite-dimensional. (For example, the von Neumann algebra Я(Г) of the group Г = Z2 * Z3 = PSLB, Z) is
generated by two subalgebras K\ = C(Z2), Л'з = C(Za) of dimensions 2 and 3.) It is then impossible to
391

make use of the concept of composed partition V\ V Vi as in the commutative case. The key to resolving
this difficulty came from the noncommutative theory of in for mat ion, and in particular from the solution
by E, Lieb [Lie] of the Wigner-Yanase-Dyson conjecture on the concavity in the variable p € Af+ of the
functional
/я(р, T) = r([ff, Г] V~Pv7i)'. where p ? (OVtj.Te 'M.
An immediate corollary of E. Lieb's result is the convexity of the following functional:
S(p\,Pi) = 7-(
This enabled E. St0rmer and me to define a function H(K\,..., Л'„) € [0,oo), where the Kj are arbitrary
finite-dimensional subalgebras of Л/, that plays the role of #(A"i V Л'2 V ... V Л'„), that is, has the following
properties:
a) H(Nt,...,Nk) < Я(Яь...,Як) \tNj QPj for j = },...,k.
b) H(Nu...,Nt,Nt+u...,Np) < H(Nu...,Nt) + H(Nk+l,...,Np).
c) P,, ¦.., Р„ С P => Я(Я,,..., Р„, Я„+1,..., Р„,) < Я(Р, Р„+,,..., Рт).
d) For every family of minimal projections ea € N such that 22 е„ = 1, one has H(N) = 227HT(ea))-
e) If (N1 U JV2 U ... U JVjt)" is generated by pairwise commuting von Neumann subalgebras Pj С Nj, then
The functional Я is moreover continuous in the sense that it. satisfies the inequality
f) H(Nl,. .. ,Nk) < II(Pi,. ..,Pk) +Ylj II(Nj\Pj),
where the relative entropy functional H(N\P) satisfies
(Ve > 0, n € N) 3.5 > 0 such that:
dimJV — it ,
(v* e ЛММ1 < i) Зу е я э И* - y||2 <«.
(It is this stability of II(N\,..., Nt) under perturbation of the Nj that would not be satisfied by the functional
=> Я(ЛГ|Р) < <г.
It is then easy to define, for every automorphism 0 of the pair [At t r), the quantities
H(N,0)= _liiii i
tf@) = su()
N
Moreover, if M is hyperfinite then we liave the following analogue of the KolmogorotT-Sinai theorem:
Theorem 3. Lei Nk be an increasing sequence of finite-dimensional subalgebras of M, with {JN^ weakly
dense in M. Then
11@) = snp ir{Kt,0)
r
for every automorphtsm 0 of the pair {Al, T).
Using properties d) and e), one easily obtains:
Corollary 4. The shifts S,, of the hyperfinite factor Я are vairwise non-conjugate and H(Sn) = logn.
I refer the reader to [Co-Stj], [Pi-Po] for other entropy calculations. As in the classical case, the relative
entropy H(N\P) still has meaning for certain infinite-dimensional pairs (N,P) and plays an important role
in [Pi-Po].
392
I have deliberately omitted giving the explicit definition of the functional Я ([Co-St2]), since it only attains
its conceptual form with the solution of the general case (non-unimodular) given in [Coj3] and developed
in [Co-N-T], which I shall now describe.
Let M be a not necessarily finite von Neumann algebra. The functional S(pi,p2) defined above still has
meaning ([Аг]]). It is called the relative entropy. However, pi,pt are no longer positive elements of M
but are positive linear forms <p\,<p2 € M.. The convexity of this functional S(y?i,y>2) remains true in full
generality and has a simple proof ([Ко]], [Pu-W]).
The key concept that makes it possible to treat the general non-unimodular case is that of the entropy defect
of a completely positive mapping.
Definition 5. Let А, В be C"-algebras. A linear mapping T : A —> В is said to be completely positive
l/j for every n, ilie mapping 1 ® T ¦ Mn(A) — Mn(B) is positive.
I refer the reader to [Arv2] for further information. The principal interest of this concept is that, although
the category of C*-algebras and *-homomorphisms has relatively few morphisms, in any case not enough
to connect the general C*-algebras to the commutative C""-algebra4, the category of C"*-algebras and com~
pletely positive mappings is much more flexible, while not being too different from the former thanks to the
Stinespring-Kasparov theorem ([Arv^], [Kas^]). Thus, for example, if В is a commutative unital C*-algebra
and X = Sp(S) is the spectrum (a compact space) of B, then a completely positive mapping T, with
T(l) = 1, of a C""-algebra A into В is just a weakly continuous mapping of X into the space S(A) of states
of A, S(A) С A*. If A is a finite-dimensional C*-algebra and <p is a state on A, there corresponds to ip a list
(A;) of positive real numbers, called the eigenvalue list of <p, obtained by setting A; = tp(et), where the e,- are
minimal projections of A with sum 1, belonging to the centralizer A^ of <p in A. This list with multiplicity
does not depend on the choice of the e;, and 5Z Л; = 1 - One sets
Let A and В be finite-dimensional C'-algebras, with В commutative, let /i be a state on В (i.e., a probability
measure on X — Sp(S)) and let Г be a completely positive mapping of A into В such that T(l) = 1.
Definition 6. The entropy defect of T is defined to be the scalar
- f
Jx
(Here T*/i = fi о T and Г* is the image under T* of the pure state on В associated with x € X = Sp(S)
and S(-, ¦) is as above the relative entropy.)
One proves that s^T) > 0. This number measures the information loss due to the translation T of the
quantum system (A, tp), where <p = Г"> = /< о Т, into a classical system (X,/i). The typical case in which
this information loss s^T) is гего is the one in which T is the conditional expectation of A onto a maximal
abelian subalgebra В of the centralizer of ip.
Proposition 7. (a) Let Ti,T3 : А — В be completely positive, with fioTi = <p. Then, for every A € [0,1],
sw(ATi + A - A)T2) > ASwB-i)+ A - А)*„(Г2).
(b) If the state TJ is pure for every x 6 X = Sp(B), Йен
(c) IfT[ : Л, —> A is completely positive and nnital, tlien
393

(d) For every subalgebra
leiTB, = Е,оГ. Then
С В, lei Et be the conditional expectation of В onto B\ associated with /j, and
for every pair Bx,Bi of subalgebras of В (where B\ V B2 denotes the subalgebra generated by BT and B2).
It is property (d) that plays a determining role. Let M be a von Neumann algebra and let ip be a faithful
normal state on M. Proposition 7 makes it possible to define the functional H(Ni,..., Nt) in full generality,
where the Nt are finite-dimensional subalgebras of M. Since this functional depends on the choice of ipt it
will be denoted Hv. The formula that gives HV(NU .. ., Nt) is the same as that of [Co23] but is clarified
conceptually by the idea of entropy defect.
One defines HV(NU ..., Nk) to be the supremum of the quantities
J
where (Х,ц) is a finite probability space, В = С(Л'), T is a completely positive mapping of A into В such
that T'/i = ip, the Bj are subalgebras of B, and s,,G}) is the entropy defect of the mapping Tj - (EjoT)\Nj
of Nj into Sj.
In other words, one optimizes a commutative translation of the situation (M,<p, (Nj)) that one compares
with the commutative situation (B,/i,{Bj)). In the commutative case, the natural quantity is the quantity
of information or entropy S()i\ V Bj) of the partition generated by the Bj. However, because of the loss of
information in the translation, one must subtract the quantity ?; s,,(Tj), which yields the formula (*).
It is remarkable that in a great many cases one can effectively calculate the maximum Hv(Ni Nt) of
the quantity (*) over all possible commutative translations. This maximum Hv(Ni,..., Nt) is finite and
satisfies properties analogous to the properties a), b), c), d), e) listed above ([Co-N-T]). Of course it coincides
with the functional Я of [Co-St2] when ip is a trace.
In particular, if 9 e AutA/ is an automorphism that preserves <p, then the limit
HV{N,O) = lim ~Hv{N,O{N),...,0k(N))
exists and is finite for every finite-dimensional subalgebra Л' С M, and, with Hv(9) = supN HV(N,O), we
have the following analogue of Theorem 3:
Theorem 8. Let U be a von Neumann algebra, ip a faithful normal state on M, and 0 € kut(M) such that
cpoO = (p. Suppose that there exists an increasing sequence Nt С Nt+t of finite-dimensional subalgebras of
M whose union is weakly dense in M (M is then suiil to be hyperfinite) . Then
Hv@)= lim Hv(h\.,0).
t—oo
We thus have available, in the noncommutative case, the analogue of the Kolmogoroff-Sinai theory, and one
of the most interesting open questions is to use it in quant uni statistical mechanics in the same way that the
Kolniogoroff-Sinai entropy is used in the formalism of thermodynamics ([Ituei]).
The theory of entropy of the automorphisms 0 6 AutA/ of a von Neumann algebra that preserve a state
<p on M enables one to formulate the following variational problem. Let A be a (nuclear) C""-algebra and
let 9 € Aut/1 be an automorphism of A. For every self-adjoint element V = V" € A, by analogy with the
classical case [RueL] one defines, for fixe.d /3 e [0,+oc),
Pff(V) = sup (Hv(9)
394
where the supreinuni is taken over the compact convex set of states ip on A that are invariant under 0.
Suppose that the automorphism 9 is sufficiently asymptotically abelian for the following equality to define a
derivation 8 generating a one-parameter group <r, of automorphisms of A:
In
Moi
The general problem is then as follows:
Problem. Compare the 9-invanani states on A such that Hv(9) - 0tp(V) — Pp(V) with ihe siaies that
are /3-KMS for crt.
Let us now take up the discussion of the simplifications brought about by noncominutativity; we shall see,
for example, that all of the automorphisms Sp are outer conjugate.
Approximately iuner automorphisms
Let M be a von Neumann algebra and let M. be its predual. The action of AutM on M., equipped with
the norm topology, is equicontinuous; it follows that the topology of pointwise convergence in norm in M.
makes AutM a topological group.
In what follows, when I speak of AutA/ as a topological group, I shall always be referring to this topology.
To convince oneself that this is the right structure on AutA/, it suffices to observe that if M, is separable,
then the group AutA/ is Polish.
general, the group InnM С AutA/ is not closed; for example, for the hyperfinite factor Я, 1ппй = Aut&
we precisely, assuming M finite, in order that InnAf be closed in AutA/ it is necessary and sufficient that
M not satisfy the property Г of Section 1 ([Co2], [Sj]).
When M is a factor of type II,, the approximately inner automorphisms of M are characterized by the
following equivalence:
Theorem D. [Co4] Let N be a factor of type Hi wHh separable predual, acting on the Hilbert space Й =
L2(N,r), where т denotes the normalized trace of N. For an automorphism 9 € AutiV, the following
conditions are equivalent:
(a) 9 € Inn TV, i.e., 9 is approximately inner;
(b) ||?"=,e(aOMI = НЕГ=1а;М f°r <4 «>• e 'V andb> b" e N' (tAe commutant of N).
Condition (b) shows that there then exists an automorphism « of the C"-algebra C(N, N') generated by
N and N', such that cr(o) = 0(a) (Va € N) and «(ft) = ft (V6 € N').
Corollary 10. Let (JVi).=i.2 be factors of type \h with separable predual, and let (ft).=i,2 be automorphisms
of the Ni. Then
9\®9? approximately inner *> 9i and 92 approximately inner.
Centrally trivial automorphisms
Let N be a factor of type IIi with separable predual, г its normalized trace, and 9 an approximately inner
automorphism of JV, 9 € ШИ. Then there exists a sequence («t)t6N of unitaries of N such that, for
every x € JV,
9(x) = lim ukxul
it —CO
for the topology of L2(N,t): \\x - y||2 = (т(\х - y\2)Y'2-
This property is translated into an equality
395

9(x) = uxu' (Vx € N)
by introducing a von Neumann algebra containing ЛГ in the following way:
Definition 11. For every ultrafiHer ш € /?N\N let N" be the ultraproduct, Nw = the von Neumann algebra
(°°(N,N) divided by the ideal of sequences (xn)n€N such that Ит„_ш ||х„||2 = 0.
One proves that this ultraproduct is a finite von Neumann algebra ([Duf], [Ve]) even though, in general,
the above-mentioned ideal is not <r(?°°, ^(-closed. Moreover, N may be canonically embedded in the
ultraproduct Nw by associating to x € N the constant sequence (x)nlEN. The sequence of unitaries (ut)teN
defines a unitary u € N" and, of course, 9(x) = ttxu' for all x € N. This equality determines u uniquely,
modulo the unitary group of a von Neumann subalgebra of N" that plays a crucial role in the sequel:
Definition 12. Let N and w be as above; the asymptotic ceutralizer of N for ш is defined to be the
commutant Nw of N in N":
Nu = {;/ € JV", Vx = xy (Vx 6 N)}.
The construction of Nw is fuiictorial, so that, every automorphism в of N defines an automorphism 9Ш of JV^,.
As above, let 9 € InnTV and м g N" unitary be such that
9(x) = nxu' (Vx€N).
The question that arises is then the following: can и be chosen so that 9"(u) — н? One can multiply и
by a unitary v of Nuwithout changing the equality 9(x) = uxu' (x € N). Thus, setting w = u'ff"(u), the
problem is to find v € Nw with v'9w(v) = w. By construction, w is a unitary of Nu. Thus the problem
is to characterize the unitaries of Nw of the form v'9w(v), v g Nu. Rokhlin's theorem for noncommutative
ergodic theory yields a complete answer to this problem in the following form.
1) The partition of unity of the center of Nw associated with the automorphism 9Ш of Nw is formed by a
single ej =. ] and #? is outer for к < j and is equal to 1 for к = j.
2) For a unitary w 6 N^ to be of the form v'9w(v), v € Л'ш, it is necessary and sufficient that и>9ш(ш) ¦ ¦ >
<?-'(«>)= 1.
Moreover, the integer j depends only on 9 and not on the choice of ш € /3N\N. It is denoted by Ра(в) =the
asymptotic period of 9. This is the period of 9 modulo the normal subgroup CtN of AutTV consisting of the
automorphisms of N that are centrally trivial in the following sense:
0 g CtN if and only if 9Ш — I (for some ш € /3N\N, or eqnivalently for every ш € /?N\N).
Given this definition, let us return to the above problem. The problem now is to know whether iv9w(w) ¦ ¦ ¦ б?~Т(
1 when j = pa@) and w = u'0"(u). This reduces to asking whether (9")'(и) - и. Now, the period of 0" is
the same as that of 9, and the problem is to compare it with ;>„(<?); one has
9P' = 1, 9 € lunN => 3 unitaries (u,,)neN of N such that
в(и„) -n,-0asn- oo, 9(x) = Xvm^ и„хи'„ (Vx € N).
In particular, if pa = 0 then the condition is satisfied. This is the principal motivation for trying to determine
the group CtN in general. The following theorem may be deduced from [Co4].
Theorem 13. Lei N be a factor of type Hi with separable predual, acting on f) = L2(N,t), and let
9 6 Aut,V, U the unitary on L2(N, т) associated with 0 (the construction of lr is functorial). Let p = pa(9)
be the asymptottc period of 9 and let A e C, |A| = 1.
396
)= -J.
Then, in order that \r = 1, it is necessary and sufficient thai there exist an automorphism a\ of the C-
algebra generated by N, N' and U, such that
ax(U) = A/7, ax(A) = A (V.4 € C'(N, N').
The above theorem shows that 9\ ® 9i € CtGVi ® Ni) if and only if 9\ and 92 are centrally trivial. Another
interesting characterization of CtN, for N a factor of type II[ such that s(InniV) is noncommutative, where
e is the quotient mapping AutiV —» OutiV, is as follows ([Co5]):
e{CtN) is the commutant ofe(hmN) inOutN.
(It suffices to know e(C'tN) in order to know CtN, since InnTV С CtN always.)
If N is the hyperfinite factor Я, then CtR - InniJ.
The obstruction ~f(9)
Let M be a factor and let 9 e AutM. Let po{9) € N be the period of 9 modulo the inner automorphisms:
9> € InnM «¦ j € PoZ.
This is an invariant of 9 under outer conjugation. Suppose p0 ф 0 and let us try to find a 9' outer conjugate
to 9 such that 0'Po = 1. We have a honiomorphisni of Z/p0Z into OutM and the problem is to lift it into
AutM. Since the center of the unitary group U of Л/ is equal to the torus T = {г € С, |г| = 1}, the
obstruction associated with this problem is an element of H3(Z/p0Z,T), where the action of Z/p0Z on T is
trivial. This obstruction y(9) is in fact the po'th root of 1 in С characterized by the equality
и € U, S"°(x) = uxu' (Vx € Л/)
0(h) = ~/u.
The important point is then the existence of automorphisms 9, of factors like the hyperfinite factor, whose
obstruction -f(9) is ф 1. One can easily convince oneself of this existence by the following example, Let us
start with (A',B,n,(Ft)iee). where (X,B,pi) is a standard borel probability space and (fi)i€R is a (Borel)
one-parameter group of Borel transformations preserving the measure /i. Assume that each Ft, t ф 0, is
ergodic (for example one could take a Bernoulli (low [Sc]). Then the crossed product R of L°°(X,B,li) by
the automorphism associated with F, is the hyperfinite factor. The von Neumann algebra L°°(X,B,ii) is
contained in Я and the unitary U € R corresponding to F\ satisfies:
l)UfU' = foF1 (V/€?~(.Y,0,,«)).
2) Lco(X,B,n) and U generate R.
Since Ft, t € R, commutes with F\, it defines an automorphism 9t of R such that
9,(U) = U and 9t(f) = f°F, (f € L°°(X,B,p)).
Moreover, for each complex number A of absolute value 1, let <тл be the automorphism of R such that
and
= A/7.
By construction, 9 and <r commute with each other and 9t(x) = UxU~ (Vx € Я). Set a — 9i/prr^, where
p e N. Then a' = «кг.,,, so that if 7'' = I then <»''(x) = Ur.U' (Vx 6 R) and a(/7) = 7У. It follows that
pa(a) = p and 7@:) = 7.
A striking feature of this invariant j(9) is that it is a complex number 7 ф J in general. In particular, if one
lets 9 act, not on M but on M'', the complex conjugate factor, obtained by replacing Ax (A € C,x € M) by
Ax, one gets -/(9е) = %0). In fact, this is the first invariant that is sensitive to the Galois automorphism
г — ~ of С over R. This allowed me to construct a factor (of type III or of type Hi [C01]) not aiiti-isomorphic
to itself (see Section 1; the algebra Mc is always isomorphic to M" via the mapping x —> x').
397

The list of automorphisms of R up to outer conjugacy
For the hyperfinite factor Я, innR = AutR and CtR = hmR. In particular, pa(9) = pa(9) (V0 € AutH). We
thus have at our disposal two invariants of outer conjugacy, the integer po{9) and the po'th root of 1, y(9),
equal to 1 if po{9) = 0. On the other hand, we have observed the existence of an automorphism of Я having
a pair (P0j7) of invariants given a priori.
Theorem 14. [Co5] Let 9\ and 9? be two automorphisms of R. Then 9\ and 02 are outer conjugate if and
only if
l>o(9i) = Po(92), 7("i) = 7(*2).
Moreover, CtH0,i = тпЯод- From this, one deduces:
Theorem 16. [Co5] (a) Let fli and 9i be two automorphisms о/Яо,1- /" order thai 9\ be outer conjugate
io 9i, it is necessary and sufficient that
mod0i=mod02, Po(«i) = №(«2). 7(»i)=7№)-
(b) The following are the only relations between mod, p0 and 7:
modfl Ф 1 => po=O, 7 = 1;
Pa = 0 => 7 = 1.
For p0 = P Ф 0 and 7 g С, 7я = 1, there in fact exists an automorphism sj of R, unique up to conjugacy,
having as invariants Po(sp = P and f(sj) = 7, and whose period is the smallest possible compatible with
these conditions, that is, equal to p x (order of 7). In particular, all of the outer symmetries в € AutR of
R, 92 = 1, 9 ? 1ппЯ, are pairwise conjugate. The simplest realization of the symmetry s% consists in taking
the automorphism of R ® R that transforms x ® у into у ® x for all x,y € R. For p0 = 0, there exists, up to
outer conjugacy, a unique aperiodic automorphism 0 € AutR (i.e., with po(9) = 0). In particular, all of the
Bernoulli shifts Sp, although pairwise distinguished up to conjugacy by the entropy, are outer conjugate.
Corollary 15. The group Out/?, is a simple group with a denumerable number of conjugacy classes.
In fact, a result more general than the above theorem shows exactly the role played by the equalities
1ппЯ = AutR and CtR = InnA. One first shows, for every factor M with separable predual M», the
following equivalence.1
livaM/lnaM is nonabeliau *> Л/ is isomorphic to M ® R.
In this case, let 9 € innM. In order that 0 be outer conjugate to the automorphism 1 ® sjj of M ® Я for
suitable p and 7, it is necessary and sufficient that po@) = j'n(<?)- Moreover, for 9 € AutM to be outer
conjugate to 9 ® «J e Aut( M О R), it is necessary and sufficient that 7 divide the asymptotic period pa(9).
In particular, every automorphism 0 of M is outer conjugate 1.0 QO I. These results demonstrate the interest
of the invariant
\{M) = (Inn M П Ct .Af)/I"n M,
which made it possible ([Co]]) to show the existence of a factor of type Hi not anti-isomorphic to itself.
Among other developments on the subject, I mention the following. V. Jones [Jonej] has completely classified
(up to conjugacy) the actions of arbitrary finite groups on the factor R, by introducing invariants of a
cohomological nature generalizing y@) (more elaborate in the general case than in the cyclic case). Next,
A. Ocneanu [Oi] succeeded in classifying the outer actions of amenable discrete groups, up to outer conjugacy,
by using the techniques of tilings (of amenable groups) introduced by D. Omstein and B. Weiss [Or-W].
A counterexample due to V. Jones shows that one cannot hope to classify the actions of nonamenable groups.
Finally, V. Jones and T. Giordano [Gio-J] on the one hand, and E. Stermer [Stor] on the other, have shown
that R possesses, up to conjugacy, just one involutive anti-automorphism.
Let us now apply the above results to the Araki-Woods factor of type Поо-
Automorphisms of the Araki-Woods factor Я01 of type IIoo
The tensor product Rg 1 of the hyperfinite factor R by a factor of type 1^ is the unique Araki-Woods factor
of type IIoo ([Аг-W]). Recall that, for every automorphism 0 of a factor N of type Hoo? °ne writes mod 9 for
the unique A € R!j_ such that т о 0 = \r for every trace r on N. For N — Яо.ь
innRo.i = Kernel of mod = {9, mod@) = 1}.
398
Whereas the case mod 9 = 1 reduces to the case treated above, for every А ф 1 it follows from [Co-T] and
part (a) of the above theorem that all of the automorphisms 9 € Autflo i with mod в = A are conjugate (not
just outer conjugate). This is a remarkable phenomenon; in effect, for A an integer, one can describe the
nature of 9 precisely as a shift on an infinite tensor product of A x A matrix algebras. Thus, when mod в = А,
there exists a A x A matrix algebra in Яод. Let us denote it by Л" С Яод; it is such that:
1) the &(K) commute pairwise;
2) the 9'(K) generate the von Neumann algebra Яод, and this property remains true whenever 9 is
multiplied by an automorphism of modulus 1.
To summarize, we have arrived at the answer to problem 2) of the section on factors of type IIIa and we may
conclude that for each A € @,1) there is one and only one factor of type Шл whose associated factor of type
IIoo is Яод. One verifies directly that, for the Powers factor Яд, the associated factor of type lloo is Яод.
This shows the interest of the following subproblem of Problem 1) of the section on the Ilia's: characterize
the factor Яод among the factors of type IIco- One makes the following definition:
Definition 17. A von Neumann algebra M with separable predual is said io be hyperfinite if it is generated
by an increasing sequence of finite-dimensional subalgebrns.
(Equivalently ([El-W]) one can require that every finite subset of M be approximable by a finite-dimensional
subalgebra.)
It is immediate that /?o,i, and more generally every Araki-Woods or even every Krieger factor, is hyperfinite.
The above problem can then he reformulated in the following way:
Question 18. Is Яод the only hyperfinite factor of type IIoo?
This problem amounts to knowing whether the commutant of a factor with this approximation property also
has it. Around 1967, V. Ya. Golodets offered a proof of this; unfortunately, it contained an irreparable error.
However, in another article, the same author used his result to infer that a crossed product by an abelian
group does not affect the above approximation property. Although based on an unproven hypothesis, his
arguments [Gol] nevertheless show that if a factor M of type Шл is hyperfinite, then so is the associated
factor of type IIoo. This reinforced considerably the interest of the above question which will be solved in
section V.9.
To conclude this section, let us note that if the factor N of type IIoo is no longer assumed to be isomorphic
to Яод, for A € @, 1) there is in general an infinite number of conjugacy classes in OutAf of automorphisms
9 of modulus A ([CoJ, [Ph]). To each of these classes there will correspond a factor of type Шл, and the
corresponding factors will be pairwise nonisomorphic.
399

V.7. Amenable von Neumann algebras
In this section 1 shall review the properties having to do with the approximation of a von Neumann algebra
M by finite-dimensional algebras. I shall show that in fact they all define the same class of von Neumann
algebras.
Approximation by finite-dimensional algebras
By definition, a von Neumann algebra M is hyperfinite if it is generated by an increasing sequence of finite-
dimensional subalgebras. An important reason for the interest in this class is the following result, due to
O. Marechal and based on the proof of Glimm's theorem.
Theorem 1. [Mar] Let A be a non-postliminal separable C'-algebra. Then, for every hyperfinite von Neu-
Neumann algebra M having no nonzero finite trace, there erisis a state <p € A* such that the von Neumann
algebra generated by тг^(А) (Section 2) is isomorphic to M.
Thus, a noncommutative integration theory that does not restrict itself to postliminal C*-algebras, that
is, to von Neumann algebras of type I, necessarily involves all of the hyperfinite von Neumann algebras.
Conversely, for the C'-algebra A that is the infinite tensor product of 2 x 2 matrix algebras, it is immediate
that all of the von Neumann algebras generated by A are hyperfinite.
The properties P of Schwartz, E of Hakeda and Tomiyaina, and injectivity
Until 1963, only two examples of nonisomorphic factors of type II were known (of course with separable
predual). Specifically, the property Г distinguished the hyperfinite factor R from the factor Z generated
by the regular representation of the free group with two generators. In 1963, J. T. Schwartz introduced a
property enabling him to distinguish R from Z(d R, both of which have the property Г. This property P of R
is based on the amenability of a locally Finite group. In fat!.. Schwartz showed that the following conditions
on a discrete group Г are equivalent:
1) Г is amenable, i.e., there exists я state Ф on ?°°(F) that, is invariant under translations.
2) M = Я(Г) acting on f) = B(Г) has the following property P: For every T € ?(M), there exists an
element of Л/ in the <т(?(Я),?(/)),)-closed convex hull of the uTu", и unitary in M'¦
Moreover, for every von Neumann algebra M on !Ts satisfying 2), he constructed a linear projection P of
norm 1 from C(fi) onto Л/, satisfying P(aTb) = aP(T)b (V«,6 € M, ГбОД)).
It is straightforward to verifiy that every liyperfinite von Neumann algebra satisfies Schwartz's property P.
A remarkable result of J. Tomiyama shows that, for every projection P of norm 1 of a von Neumann algebra
N onto a von Neumann subalgebra M, the following condition holds automatically [To]:
P(aTb) = aP(T)b (Vo, 6 € Л/, T 6 JV).
In [Hak-T], Hakeda and Tomiyama defined a property weaker in appearance than Schwartz's property P:
Definition 2. A von Neumann algebra hi on a Ililberi space Я is said 1o have property E if there exists
a projection of norm 1 from the Banach space C(f>) onto the Banack space M С C(f>).
For this reason, the von Neumann algebras satisfying
property E are also called injective.
Property E is not descriptive: it says, a priori, very little about, the von Neumann algebra M, but it does
have remarkable stability properties:
1) If M is an injective von Neumann algebra act.iug on a llilhert space jj, then the comnmtant M' of M
is injective.
400
2) If (Ma)agl is a decreasing directed family of injective von Neumann algebras, then Г)„€/ Ма is injective.
3) If (М„)а€/ is an increasing directed family of injective von Neumann algebras, then the closure of
Uap/ hfa is also injective.
4) Let M be a von Neumann algebra with separable predual and let M = fx M(t)di*(t) be a disintegration
of M into factors M(t); then M is injective «> M(i) is injective for almost all t € X.
5) Let M be a von Neumann algebra, ./V a von Neumann subalgebra, and G a subgroup of the normalizer
of N in M. Suppose that N and 5 generate M, N is injective and Q is amenable as a discrete group;
then Л/ is injective.
6) Let M be an injective von Neumann algebra and Г an amenable discrete group acting on M by auto-
automorphisms; then
is injective.
Properties 2) and 3) show that if fy is a separable Hilbert space, then the monotone class generated by
the von Neumann algebras of type I contains only injective von Neumann algebras. Property 4) essentially
reduces the problem of classifying these algebras to the case of injective factors. Property 5) shows that
every amenable group of Militaries generates an injective von Neumann algebra. Finally, it is easy to see
from property 6) that, for a factor M of type \\\\, A € @, 1), to be injective, it is necessary and sufficient
that the associated factor of type И,» be injective.
Among the most important injective von Neumann algebras, we cite:
a) The crossed product of an abelian von Neumann algebra by an amenable locally compact group.
b) The commutant von Neumann algebra of any continuous unitary representation of a connected locally
compact group.
c) The von Neumann algebra generated by any representation of a nuclear C*-algebra (see below for the
definition).
Semidiscrete von Neumann algebras
Let M be a factor of type I on a Hilbert space fy. It gives a decomposition of Я as a tensor product
f) = f)l ® fj2 in such a way that M = ?(У),) © 1 and M' = 1 ® C(fJ). One then recovers C(f)) as the
tensor product of M and M'. In one of Murray and von Neumann's first articles, they showed that for every
factor M on У), the homomorphism ;/ of the algebraic tensor product
M © M' = { J]a; ® bt, a; € M, bt € M'\
into C{f>), defined by
is injective and has <т(?()э), ?()э).)-dense range.
In [E-L], E. Effros and C. Lance succeeded in pushing the analysis much further by studying i) from a metric
point of view. Let A (resp. B) be a C'-algebra with unity acting on a Hilbert space f)A (resp. )Эв). Consider
the norm on the algebraic tensor product A © В that arises from its action on f}A ® #g. This norm on
A © В makes the completion a C'-algebra, is independent of the choice of the (faithful) representations of A
on ?)a and В on f)g, and is characterized by a highly useful theorem of M. Takesaki as the smallest norm
on A © В for which the completion is a C'-algebra. It is denoted || )|min and one writes A ®min В for the
completion C*-algebra ([Ti]). The C'-algebra A is said to be nuclear if || ||min is the only рге-C* norm on
A © B, for every В.
Effros and Lance succeeded in characterizing the factors A/ for which the above mapping tj is isometric, by
means of a property that is a strengthening of the metric approximation property for the predual M* of M.
401

The predual Mm is not only an ordered space (for the cone A/,+ ) but is matricially ordered, in the sense
that the tensor product vector space M. ® -V/,,(C) is ordered for every n, as the predual of M® Afn(C).
A completely positive mapping X of M, into M. is by definition a linear mapping such that X® 1л/,, is
positive for every n. The result of Effros and Lance [E-L] is then as follows:
Theorem 3. Let M be a factor operating on a Hilbert space $). In order that т/ : M ®mm M1 —»¦ C(ft) be
isometric, it is necessary and sufficient that the identity mapping on M. be tke pointwise limit in norm of
completely positive mappings of finite rank.
One then defines a semi-discrete von Neumann algebra by the above approximation property of its pred-
predual M». We note that when the factor M is such that neither M nor M' is of type loo or II^, and f) is
separable, a corollary of Takesaki's theorem on the min-norni enables one to prove:
Corollary 4- A factor M acting on a separable Hilbert space $), such 1hat neither M nor M' is of type I^
or Hooj Is sevii-discrete if and O7ily if the C"-algebra C(Af, Л/') on $) generated by M and M' is simple
(i.e., has no nontrivial two-sided ideal).
This corollary is very important when it is brought together with the following characterization of the type II]
factors that do not have the property Г:
Theorem 5. [Co4] Let M be a [actor of type II] with separable predual, acting on L2(M, т) = Д, and let
C'(M, M') be the C'-algebra generated by M and Af. Then:
M does not have property Г <=> С (Af, M') contains the ideal
k(f)) of compact operators.
An example of a type IIi factor for which C'(M,M') contains the ideal k(f)) was given by C. Akemann and
P. Ostrand in [Ak-O]. Thus, combined with the corollary, the theorem shows that every semi-discrete factor
of type II] has the property Г. In fact, Effros and Lance proved in their paper [E-L] that every Araki-Woods
factor is semi-discrete. They also proved the implication
semi-discrete
injective.
The relations between the various properties pertaining to approximation by finite-dimensional algebras can
be summarized in the form of a diagram:
/
hyperfinite
Schwartz's property /' semi-discrete
Ilakeda-Tomiyama property E
— inject-ivity
Happily, the situation is in fact remarkably simple:
Theorem 6. [Co4] Let f) be a separable Hilbert space. For a von Neumann algebra acting on Д, the above
four properties are equivalent.
I defer to the next section the description of the corollaries of this theorem relative to the classification
problem. Let us begin witli a problem of terminology: we surely have at our disposal the right class of
von Neumann algebras for the noncomniutative theory of integration. Indeed, by O. Marechal's theorem,
this theory should cover at least the hyperfinite case, and, by the results of Effros and Lance, the case of
"property E" suffices to account for all of the von Neumann algebras associated with nuclear C"*-algebras.
In [Co4], the term "injective von Neumann algebras" was adopted to denote the above class. Among the
advantages of this choice there is above all the simplicity of the definition by means of the property E.
402
However, this terminology has the disadvantage that it is suggestive neither of the fact that it is concerned
with an approximation property, nor of the analogy with the amenability of discrete groups. The solution
therefore seems to be to choose the term "amenable von Neumann algebra", which is fortunately justified
by the equivalence between the above four properties, and a fifth:
Definition 7. A von Neumann algebra M is said to be amenable if, for every normal dual Banach bimodule
X over M, the derivations of M with coefficients in X are all inner.
I refer the reader to the papers of Johnson, Kadison and Ringrose, who established the foundations of the
cohomology of von Neumann algebras with coefficients in Banach bimodules ([Jo,], [Jo2]).
Having accepted the term amenable to denote our class of von Neumann algebras, we then have the following
corollary, an easy consequence of [E-L] and [C04]:
Corollary 8. Let A be a separable C'-algebra. Then, A is nuclear «¦ for every state p on A, the von Neu-
Neumann algebra vv(A)" generated by irv(A) is amenable.
403

V.8. The flow of weights: mod(M)
In this section we shall describe in detail the main invariant of type III factors, the flow of weights. This
invariant emerged from the solution [C03] of the problem of existence of hyperfinite factors not isomorphic '
to infinite tensor products of type I factors and the discrete decomposition of factors of type IIIo (section a\
below). In its final form it is due to M. Takesaki [T3].
a) The discrete decomposition of factors of type IIIo
We have seen above (section 5 theorem 4) that any factor M of type Шл cai* be uniquely decomposed as
the crossed product M = N xe Z of a factor N of type II^ by an automorphism 0 € Aut(N), mod@) = A.
We shall now describe an analogous decomposition for arbitrary factors of type IIIo ([C03]). There are two
nuances with the Шл case. The first is that N will no longer be a factor but a semifinite type lloo von
Neumann algebra. The second is that in order to state the uniqueness part of the decomposition we need
to introduce the ergodic theoretic notion of induced automorphism 9e where 9 € Aut N and e € Z(N) is an
idempotent in the center of N.
Proposition 1. Let Z he a commutative von Neumann algebra with no minimal projection, and 9 € AutZ
be an ergodic automorphism. Then for any поп zero projection e ? Z there exists a unique sequence of
projections е„ € Z such that:
e = ^en , 9n(en)<e , e9k'(en) = 0 for к = 1 n - 1.
1
CO
Moreover one has e =?^ 0'l(en).
1
We refer for instance to [Str] for the simple proof. For brievity a commutative von Neumann algebra is called
diffuse if it does not contain any non zero minimal projection.
Definition 2. Let N be a von Neumann algebra with diffuse center Z{N) and $ ? Aut(AT) be ergodic on the
center of N. Then for any projection e ? Z(N) we let 0e be the automorphism of the reduced von Neumann
algebra Ne determined by:
M*) = ?«..(««) VxeTV,
where (en) is the sequence of projections en ? Z{N) of proposition 1.
We say that ве is the induced automorphism by 0 on Ne. Next let (TV, r) be a von Neumann algebra with
semifinite faithful normal trace т. We shall say that an automorphism 9 € Aut(TV) is a contraction iff there
exists A < 1 such that
A) toO < А т.
We can now state the existence and uniqueness of the discrete decomposition of arbitrary factors of type
Theorem 3. ([Co3]) Let hf be a factor of type III0.
1) There exist a type ll^, iwn Neumann algebra N and a contraction 0 € Aut TV such that the following
isomorphism holds:
M = TVx» Z
2) Let (Nj,$j) be as in \J with Nj X$3 Z isomorphic to M then there exist non. zero projections ey
such that the induced automorphisms 0jej of Nej are conjugate.
404
This theorem has, due to the discreteness of the group Z involved in the crossed product, the same analytical
power as the structure theorem 5.4 for factors of type Шл-
The uniqueness statement 2) was originally formulated in terms of outer conjugacy [Co3] but improved to
conjugacy in [Co-T].
0) Continuous decomposition of type III factors
The above theorem 3 together with 5.4 leaves untouched the understanding of the structure of factors of type
IIIj. This structure was elucidated by M. Takesaki [T3] as a corollary of the following general continuous
decomposition of type III von Neumann algebras.
Theorem 4. ([T3]) Let M be a type III von Neumann algebra.
1) There exists a type Hoo von Neumann algebra N with semifinite normal trace т and
of automorphisms @\)\?tr , в\ € Aut ЛГ with то 9\ = At VA € R+, such that the
a one parameter group
following isomorphism
holds:
м = Nxe r;.
2) The above decomposition is unique up to conjugacy.
The uniqueness part was originally formulated in tenns of outer conjugacy but improved to conjugacy in
[Co-T]. The great virtue of this theorem is its generality and the simplicity of its proof which is a direct
corollary of the biduality theorem for crossed products (theorem 4 of appendix A) and of the Radon Nikodym
theorem 5.1. The von Neumann algebra N is the crossed product:
B)
N = MX,* R
of M by an arbitrary modular automorphism group <r* and by 5.1 is independent of the choice of ip. The
action on N of the Pontrjagin dual group R^ of R yields the one parameter group @л)лея- °f automorphisms
of TV.
Definition 5. The flow of weights W(M) is the restriction of the action of (вх)хе«'+ io Ле center Z(N).
Stated like this, it is not clear that the flow of weights is defined functorially. We shall give below in section
-f) a functorial definition which will justify the name as well. The invariants S(M) and T(M) of a factor are
easy to reformulate in terms of the flow of weights, one has:
C)
S(M) П R; = {A € R; ; Ox = id}
D) T(M)= {T€R , 3u€Z, ифО , 9Л(«) = A'T« VA g R^}.
It is also straightforward that:
E) M is a factor о Ox is ergodic on Z.
Thus one gets in particular the following analogue of theorem 5.4 for factors of type III].
Corollary 6. [T3] Let M be a factor of type III,.
1) There exists a factor of type IITC, TV and a one parameter group of automorphisms в\ € Aut N; mod 9\ =
A VA € R^ such thai M - TV x9 R;.
405

2) The above decomposition is unique up to conjugacy.
With more work one can also derive theorem 5.4 and 3 above from the general theorem 4.
7) Functorial definition of the flow of weights
Let M be a von Neumann algebra and tp a semifinite normal weight on M. The support e = s(tp) is the
unique projection e € M such that: or) <p/Me is faithful, /?) tp(x) = tp{exe) Vx € M+.
Definition 7. Two semifinite normal weights tp, ф are equivalent iff there exists a partial isometry u,
u*u = s(tp), u«" = s(i>) such that:
ф(х) = <p(u*xu) Wx€M+.
For type III factors the equivalence of projections yields trivial results and the equivalence of weights is the
correct substitute. We shall say that a weight tp on a properly infinite von Neumann algebra M is of infinite
multiplicity iff tp is equivalent to the tensor product tp ® Trace of ip by the canonical semifinite faithful
normal trace on the type loo factor. There is a slight ambiguity in the definition since one has to give an
isomorphism M ~ M ® F where F is the type loo factor, but since Aut F = JntF one can just use any
isomorphism F ~ F ® F.
We can now state the analogue of the comparison of projections of Murray and von Neumann.
Theorem 8. [Co-T] Let M be a properly infinite von Neumann algebra (for instance a type III factor).
1) Let H be the set of equivalence classes of semifinite normal ivetghts of infinite multiplicity endowed with
the operation:
(Class p)V (Class 1/') = Cl
where ip © ф = ip + ф if s(<p) s(^) = 0. Then S if the boolean algebra of a-finite projections in a unique
commutative von Neumann algebra Z.
2) The action ofR'+ by multiplication:
AfClass ip) = Class(Ap) VA € R+
determines a one parameter grovp of automorphisms of Z whose continuous part ifi canonically isomorphic
to the flow of weights of definition 5.
To understand the theorem one has to remember that a commutative von Neumann algebra Z is uniquely
determined by the boolean algebra (t(Z) of projections e ? Z with the operation:
In 1) one obtains a very large commutative von Neumann algebra Z whose tr-finite projections e € Z exactly
classify the semifinite normal weights of infinite multiplicity on M. For each A€HJ. the formula 2) defines
an automorphism B\ of Z. The continuous part of this action of R^. on Z is given by the following von
Neumann subalgebra:
F) W = {x € Z ; A —- 0x(x) is strongly continuous}.
One has moreover a very simple characterization of those (p for which class(y>) € W.
Proposition 9. ([Co-T]) Let ip be a semifinite normal weight of infinite multiplicity. Then class (tp) € W
iff the modular automorphism grovp erf of ip is integrable, i.e. iff {x e M ; /^ af(x'x)dt 6 M] is weakly
dense in M.
406
There exists moreover a largest integrable weight, the dominant weight, unique up to equivalence, which was
already used in [C03] in the proof of the converse of the Radon Nikodym theorem. In the type IIIi case this
weight plays the same role as the generalized trace of section 5. It is characterized, among semifinite normal
weights of infinite multiplicity, by the invariance: tp ~ \<p VA € R^..
The construction of the flow of weights given in theorem 8 is obviously functorial. An idempotent e € W is
just an equivalence class of weight and the action of R^. is just multiplication of ip by A € R^..
This allows in particular to extend to the type III case the definition of the module mod(a) of automorphisms
a ? Aut(M).'
Definition 10. Let M be a type III factor, then fora € Aut(M) the module mod(a) is the automorphism
of the flow of weights determined by the equality:
mod(a) (class tp) = class(poa).
This allows ([Co-T]) to obtain the analogue in the type III case of the classification of automorphisms of
section 6.
It also suggests the notation mod for the functor from von Neumann algebras to flows given by theorem 8
2).
E) Virtual groups and the flow of weights as modular spectrum
The theory of induced representations of locally compact groups led G. Mackey ([M3]) to a natural gene-
generalization of the notion of closed subgroup which has great heuristic value and was further developed by A.
Ramsay [Ra] and R. Zimmer [Zii]. The non transitive impriinitivity systems suggest the following definition:
Definition 11. Lei G be a locally compact abelian group. Then a virtual subgroup of G is given by an
ergodic action a of G on a commutative Von Neumann algebra A.
Given an ordinary closed subgroup Я of G one lets A — L°°(G/H) and a be the action of G by left
translations. The group H is then recovered as the isotropy group. The virtue of this notion is that many
concepts extend from the special case of closed subgroups to arbitrary virtual subgroups.
For instance if Hi,H2 are two closed subgroups of G then the closure Я = Н1Н2 of Я1Я2 is obtained as
the isotropy group for the action 01 0 I of G on the following commutative von Neumann algebra:
G)
A =
A2f = {: € A, ® A2 ; av(y) ® a2(g l)(z) = z Vg € G]
where Aj = L°°(G/Hi) on which G acts by left translations 04.
Now obviously the construction G) works for virtual subgroups as well and the special case of closed sub-
subgroups gives its heuristic meaning.
Now the invariant 5(Л/) for factors of type III is a closed subgroup of R^. (section 5), but there is no general
formula for S{Mi ® Mo) in terms of S{M\) and 5(Л/о). Indeed one can construct factors of type lllo whose
tensor product is of arbitrary type Шл ¦ This difficulty is completely removed if one uses the above idea of
Mackey. Indeed the flow of weights mod(M) of a type III factor is by construction an ergodic action of R^.
on a commutative von Neumann algebra and hence:
(8)
mod(M) is a virtual subgroup of
Moreover the relation C) between S(M) and mod(M) shows that except in the IIIo case, mod(Af) is an
ordinary closed subgroup of R^. and is equal to S(M).
407

Thanks to G) we can define the closure of the product of two virtual subgroups and one has:
Theorem 12. [Co-T] Let Mt, M2 be two type III factors then
mod(Mi ® Mi) = (mod Mi) пкх)(Мг).
In other words the flow of weights of M\ ® M2 is obtained by formula G) from the flow of weights mod(Mj).
Thus, unlike the invariant S, the invariant mod is compatible with tensor products.
As a next example of the suggestive power of the idea of virtual subgroups, let us note that for general
locally compact groups G with closed subgroup Я one has:
(9) Я is of finite covolume in G О the action of G on A = L°°(G/H) has a finite invariant measure.
(We mean of course in the measure class of the Haar measure.)
This suggests to say that a virtual subgroup has "finite covolume1' when the corresponding action has a finite
invariant measure. This notion plays an important role in ft. Zimmer's work ([Zi2]).
In our context it allows us to restate the main result of section III .6 as follows:
Let (V, F) be a codimension 1-foliation with non vanishing Godbillon Vey class then the virtual subgroup
mod(M), M - W*(V,F), is of finite covolume in R^.
As a final example let us mention the computation of the (low of weights mod(M) for factors associated to
ergodic equivalence relations 71 with countable orbits (cf. section 4) on a measure space (X, /i). Let us, as in
section 4 endow 7i, the graph of the equivalence relation, with the obvious groupoid law and with the measure
[i which plays the role of a left Haar measure on this groupoid. Let us then consider the homomorphism
b : 7i —* R^. given by the lack of right invariance of the left Haar measure Д, i.e. by:
(This is well defined a.e. modulo /*.)
The range 6G?) of this homoniorphisni is well defined as a virtual subgroup of R^J. (cf. [Ram]) and one has:
Proposition 13. [Kra] [Saua] Let 72. be an ergodic equivalence, relation, M = L°°G?,/() be the associated
factor (def. 4.1). Then the flow of weights of M is given by:
We have already seen this in a different guise when we described in chapter I the flow of weights mod(M) for
the von Neumann algebra W{ V, F) of a foliation. The range of a homomorphism 6 from a measured groupoid
7S. to an abelian locally compact group G is defined in general from the action of G on the commutative von
Neumann algebra of functions / € L°°(TZ x G) which are invariant under the action of 7S. on 7S. x G given by:
408
V.9. The classification of amenable factors
In this section we shall give the complete classification of amenable von Neumann algebras. All von Neumann
algebras here are assumed to have separable predual. By the reduction theory, any amenable von Neumann
algebra M can be written as a direct integral, M = / M(t) d/i(t) of amenable factors. We shall now give
the complete list, type by type of amenable factors. (We ignore the trivial type I case.)
Factors of type Hi
As a corollary of theorem 1.13 and 7.6 one has:
Theorem 1. ([Co4]) Any amenable factor of type Hi is isomorphic to the Murray-von Neumann hyperfinite
factor R.
I refer to [Co4] [H4] [P01] for simplifications of the original proof.
Now any von Neumann subalgebra M of R is automatically finite (it has a finite faithful normal trace)
and amenable (using the canonical normal conditional expectation E : R —» M given by the orthogonal
projection on M С L2(R)). Thus one gets:
Corollary 2. [C04] Let M be a von Neumann subalgebra of R then M is isomorphic to a direct sum of
tensor products М„(С) ®С„, R® Co, where the Cj are commutative von Neumann algebras.
In particular all subfaciors of R are either finite dimensional or isomorphic to R.
It is straightforward that a discrete group Г is amenable iff the von Neumann algebra Я(Г) of the left regular
representation of Г is amenable. This is a necessary coherence of the notation. One can then analyse R(V)
as follows:
Corollary 3. [C04] Let Г be an amenable discrete group and R(V) the von Neumann algebra generated by
its left regular representation 111 ?2(Г). Then Я(Г) is a direct sum of tensor products Mn(C) ®С„, R®Ca
where the Cj are commutative von Neumann algebras.
When all non trivial conjugacy classes of Г are infinite one has Л(Г) ~ R. As we shall see in appendix В
the analogy between the notion of amenability for discrete groups and for type Hi factors goes quite far (cf.
theorem 21). We shall now describe the analogue of the invariant means of discrete amenable groups.
Theorem 4. [C04] Let N be a factor in о (separable) Hilbcrf spaceTi. Then, unless N is finite dimensional,
N is tsomorphic to R <=> there exists a state Ф on C("H)
such that Ф(хТ) = Ф(Тх) Vxe N , T e C(H).
Such astate cannot be normal. By definition a hypertrace is a state Ф on C(H) satisfying the above condition.
Even though such states are not normal they can be obtained by nice formulas in some examples. For instance
if we let D be the analogue of the Dirac operator on the non commutative 2-torus T,, в ? Q and consider
Gi,D) as a Л'-cycle over the C*-algebra .4», the following equality defines a hypertrace Фи, on the weak
closure R = A'f of Ae in H:
A)
Ф„(Г) =
VT € C(~H)
where Tr^, is the Dixmier trace for a given choice of a; (chapter IV). One can also use a result of G. Moko-
bodski ([Me]) allowing to make <S>W universally measurable and commuting with integrals: Фш (/ Tad/i(a)) —
/Фи(То)^(«)-
In the case of discrete groups Г, amenability is equivalent to a condition involving finite subsets, the Folner
condition:
409

i, ...,}„ ?Г, Ve > 0 , 3 finite non empty subset. F of Г such that.
(where If is the characteristic function of F and ]| ||2 is the norm in ^2(Г).)
In a very similar way the existence of a hypertrace on a factor N in Ц is characterized by the following
condition:
Vzi,..., х„ 6 N , Ve > 0 , 3 finite dimensional projection P on И such that
||*i P-Pxi\\HS<e\\P\\HS
(where ||T||hs = (Тгасе(Г*Г))'/2 is the Hilbert Schmidt norm of operators).
Factors of type II^
Any factor M of type IIoo is the tensor product N © F of a factor of type Hi by the factor F of type loo-
Moreover if M is amenable so is N, thus theorem 1 implies:
Theorem 5. [C04] There exists up to isomorphism only one amenable factor of type 11^ namely йод =
R®F.
The notation R0,i comes from the Araki-Woods notation for the only TTPFI of type П„о. We thus obtain
the solution of 6.18:
Corollary 6. [C04] Any hyperfinite factor of type 11^ is tsomorphic to йц.
Note that the proof of this corollary is very indirect, hyperfiniteness is only used through the apparently
much weaker amenability which passes to the Hi factor N if M = N ® F. There is no direct proof of the
hyperfiniteness of N.
We have already given ill chapter I examples of foliations (V,F) such as the Kronecker foliation, whose
associated von Neumann algebra W(V,F) is the hyperfinite factor of type Hoc,-
As another large class of occurences of this factor one has:
Corollary 7. [C04] Let О be a connected separable locally compact group and let A be ike left regular
representation of G in L~(G). Then R(\c) is a direct integral of factors which are either of type I or
isomorphic to Rot\.
This result follows from theorem 5 and a result of Di.xmier and Pukanzky showing that no factor of type HI
occurs in the decomposition of /?(A<j).
Factors of type НЦ, A 6 ]0,1[
Let, M be a factor of type Шл, A 6 ]0,1[, and let M = N Xe Z be the discrete decomposition of M
(theorem 5.4). Then if M is amenable so is N as one sees using the natural normal conditional expectation
E(llanUn) — ao 6 N from M to N. Thus since ;V is a factor of type II^o it is by theorem 5 isomorphic to
йо,1 and 0 6 Aut(fio.i) satisfies mod(fl) = A. By the results of noncommutative ergodic theory (theorem
6.16) one knows that в is unique up to outer conjugacy, which thus implies:
Theorem 8. [C04] Lei A ? ]0,1[. There exists up to isomorphism only one amenable factor of type Шл, the
Powers factor R\.
410
We refer to section 4 for the description of Ял as an ITPFI. As above the analogue of theorem 8 holds with
amenable replaced by hyperfinite: Ял is the only hyperfinite factor of type Шл.
These factors Ял describe the measure theory of the simplest fractals, namely the self similar Cantor sets
of chapter IV section 3 example 23. Let us first recall that given a pair (Ay?) of а С -algebra A and a
state »oni there is a canonically associated von Neumann algebra: the weak closure A" of A in the GNS
representation (cf. section 2). Let А" С [0,1] be the self similar Cantor set of 4.3.23, and let C(K) act by
multiplication operators in ?2(D) as in 4.3.21 where D С К is the denumerable set of endpoints of К. А
selfsimilahty <r of К is the restriction to Domain(<r) = К П J (where J is a closed interval) of an affine
transformation x — <r(x) = ax + Ь of R which preserves K, i.e.
C) <т(х) е К Vl 6 Domain (<r).
By construction a preserves D and thus can be represented in ?2(D) as a partial isometry «„ such that:
и„(еь) = 0 if b $ Domain (a)
D)
taUb) = ?»(|.) if 4 6 Domain (a)
where (еь)ь€О is the canonical orthonormal basis of C2(D).
We let A be the C*-algebra in К = P{D) generated by C(K) and the selfsimilarities u9. It can be described
in many equivalent ways as in chapter II.2. Let then p be the HausdorfT dimension of К and let f> be the
positive linear form on the C*-algebra A given by:
E)
(a) = Tru(o|dxn Va 6 A
with the notations of 4.3.23. Recall that the restriction of <p to C(A') is (up to a constant) the HausdorfT
measure Ap on К.
Proposition 9. The factor A" associated to the pair {A,ip) is the hyperfinite factor Rfp, of type III,,,, where
p is the similarity ratio of K-
With the notations of 4.3.23 one has ff = ^y.
The analogue of the construction of section 11, with the global field Q replaced by the field of rational
fractions over a finite field F, yields the factors R(,,->) for any /3 6 ]0,1].
Finally we already gave in chapter I examples of foliations (V, F) with W(V, F) = Ял for arbitrary values of
a e]o,i[.
Factors of type IIIo
The analysis of amenable factors of type IIIo combines the discrete decomposition (theorem 8.3) with
Krieger's theorem (theorem 4.3). Of course any Krieger's factor is hyperfinite (and amenable), conversely
one has:
Theorem 10. [C04] Any amenable (or hyperfinite) factor of type III0 is a Krieger's factor.
Moreover such factors are in one to one correspondence with ergodic flows by the remarkable result of W.
Krieger:
Theorem 11. [Kr2] 1) Two Krieger factors Л/ьЛ/з
isomorphic:
«re isomorphic iff their flows of weights are
411

~ mod(A/2)-
2) For any ergodic non transitive flow there exists a (unique) Krieger factor M with this flow as flow of
weights.
Thus, in other words, the natural extension of the invariant S as a virtual subgroup of R^. is a complete
invariant.
Before this result the discrete decomposition was already used in [C03] together with an ergodic theory result
of W. Krieger, to exhibit a factor M of type IIIo which is hyperfinite but is not isomorphic to any ITPFI
(Araki-Woods factor). By now we have the following characterization of ITPFI among hyperfinite factors:
Theorem 12. [Co-W] Let M be a type III hyperfinite factor. Then M is isomorphic to an ITPFI iff its flow
of weights mod(AZ) is approximately transitive.
We need to define approximate transitivity for an action a of a locally compact group G on a commutative
von Neumann algebra Z:
Definition 13. The action a of G on Z is approximately iransitive iff for any normal states [t\ /in on
Z and ? > 0 there exists a normal state /i on Z such thai the distance, in norm, of pj to the convex hull of
G /i is less than € for any j = 1, ..., n.
One can show ([Co-W]) that any measure preserving flow with non zero entropy is not approximately
transitive. It follows that the factor of type IIIo, hyperfinite with this flow as flow of weights cannot be an
ITPFI.
Corollary 14. ТЛеге exists 1ype IIIo hyperfinite non ITPFI factors.
Let us stress finally that theorem 11 shows the equivalence of two classification problems, and this in a
functorial manner. It had been shown by E.J. Woods, using the theory of ITPFI that the set of isomorphism
classes of ITPFI is not countahly separated, and thus that such factors cannot be classified by a countable
family of real valued invariants. We have seen in chapter I how to construct foliations (V, F) such that
W(V,F) is hyperfinite of type III0.
Factors of type IIIj
Let M be a factor of type III] and let Л/ = N >He R+ be its continuous decomposition (theorem 8.6). Then
Л^ is a factor of type 11^ and if Л/ is amenable so is Л^, so that by theorem 5, the factor Л^ is isomorphic to
Ko.i- We do not however have a direct proof of the analogue of the non commutative ergodic theory result
(theorem 6.16) for flows. The difficulty is that, flows are incompatible with the use of ultraproducts. Making
use of discrete crossed products instead, to reduce from IIIi to III), A 6 ]0, 1[, I showed (cf. [Co]) that the
uniqueness (up to isomorphism) of amenable factors of type II Ii would follow provided one could prove the
following:
"Let M be a factor of type IIIi. Then for any pair of normal states <p ф ф on M there exists a bounded
sequence (x,,) in M such that ||[y>, г„]|| —<¦ 0 and ||[</>, z,,]|| -/ 0 when n —<¦ 00."
This property is easy to prove for the Araki-Woods factor of type IIIi, R<x>- In 1983 V. Haagerup was
able to prove the above property for arbitrary amenable factors of type IIIi, making use of the continuous
decomposition and of the relative comniutant theorem ([Co-T]). This result concludes the classification of
amenable (or equivalently hyperfinite) factors:
Theorem 15. [H5] There exists up to isomorphism only one amenable factor of type IIIi, the factor Я00 of
Araki and Woods.
Here are some examples of occurences of this factor.
412
1) We have seen already in chapter I that the Anosov foliation (K, F) of the unit sphere bundle V of a
Riemann surface of genus > 1, gives the factor R^ = W( V, F).
2) The von Neumann algebra W@) generated by bosonic quantum fields with support in a local region 0,
as in chapter IV section 9 proposition 12, are all isomorphic to Roo- This result, in the free field case, is due
to H. Araki. The explicit computation of the modular automorphism group for the vacuum state restricted
to U(O) is only done in some very special cases and remains a challenging open question.
3) We have seen that the factors Ях of type Шл describe the measure theory of the self similar Cantor sets.
Similarly the factors R,» describes the measure theory of the quasi Fuchsian circles of chapter IV theorem
3.17. One lets A be the C*-algebra crossed product of the algebra of continuous functions on the quasi circle
by the action of the quasi Fuchsian group Г. The positive linear form у is given by the formula (with ТГь,
the Dixmier trace):
yj(a)= Tru{a\dZ\f) Va ? A.
The obtained factor is of type IIIi and isomorphic to Ra>.
4) We shall see in section 11 that the notion of module in basic number theory [Wei] is intimately related
to our functor mod, and that the factor R^ appears naturally from the statistic of prime numbers.
413

V.10. Subfactors of type IIj factors
V. Jones first extended the classification (theorem 6.14) of actions of finite cyclic groups on the hyperfinite
factor R to arbitrary finite groups С [Jonei]. There is in this situation a Galois-type correspondence between
G and the fixed point von Neumann algebra Ra = {x 6 R ; gx = x Vj 6 G}. V. Jones then went on and
investigated arbitrary subfactors N of R satisfying the following finiieness condition:
Definition 1. Lei N С M be a subfactor of a type II] factor M iken N is of finite index iff ike commutant
of N in L2(M) is finite (i.e. of type IIJ.
Here L2(M) is the Hilbert space completion of M for the inner product (x,у) = т(х'у) where г is the unique
tracial state on M. It is a bimodule over M with the action by left and right multiplication (cf. appendix
В for the description of the trivial correspondence in general). By a result of Pimsner and Popa [Pi-Po] JV
has finite index in M iff M viewed as a left N module is finite and projective.
The Murray and von Neumann dimension function (theorem 1.11) yields for any Hi factor N and any normal
representation i of N in a Hilbert space %, a real number dim^r(W) which satisfies the following conditions.
1) dimw(?, t) 6 [0,+co], dimw(Z2(JV)) = 1.
2) dini^(A, "¦) = diniAr(J5', »*) if and only if the representations т and ir1 are equivalent.
3) dimw(©r(a...*..)) = ?Га'т(я„,=г»).
4) If € 6 k(N)' is a projection and ire is the restriction of я to the space ей, then
5) dim^r(A,!r) •dim^r.ffl) = 1, where N' denotes the commutant of *(N).
The index of a subfactor /V is defined as follows:
Definition 2. Let N С M be a subfactor of a type IIi factor M then tke index [M : N] of N in M is the
multiplicity
dmiN{L2(M)).
This index satisfies the following easy properties:
Proposition 3. (a) Lei N be a subfacior of M of finite index. For every representation (Й, i) of M of
finite multiplicity, the restriction irN of я to N has finite multiplicity, and
A\mN(t),tN) = [M : N] • dimA,(a, jt).
(b) Let N and M be us in (a) and lei P be a subfactor of N of finite index. Then P is a subfactor of M of
finite index, and
[M : P] = [M : N][N : P].
(c) IfN,M,f),T are as in (a), then the commutani !г(Л/)' is я subfacior of ir(N)' of finite index, and
[x(N)' : t(A/)'] = [M : N].
Property (a) is immediate; (b) follows from (a), and (c) follows from property 5) of the dimension function
diniM-
The above example RG С Я of fixed points of finite group actions, as well as the inclusion;
Я(Г,)СЯ(Г2)
414
of the type Hi factors associated to discrete groups I"i С Г2 with infinite conjugacy classes, only yield
integral values for the index.
One obtains noil integral values for the index in a rather trivial manner as follows:
Let M be a factor of type IIi and e 6 M a projection. We denote by Me the factor Mt = {x 6 M, xe =
ex = ж}. In order that Me be isomorphic to Mi_e, it is necessary and sufficient that the positive real number
A0/(l - Ao), where Ao = TrM(e), belong to the group F(M) (Section 1). Assuming this to be the case, let в
be an isomorphism в : Me —> M,_e and let
N = {х + в(х), xe Me).
By construction N is a subfactor of A/, and a straightforward calculation of dimjv(L2(A/)) shows that
[M : JV] - — + г-.
Aq I — Aq
The group F(R) of the hyperfinite factor R is equal to R+. Thus, the above construction yields, for every
real number a > 4, the existence of a subfactor JV of Я with [R : JV] = a.
All this shows that the set 2 С [1, oo[ of values of index of subfactors satisfies:
{l,2,3}U[4,+oo[ С Е.
The first great result of V. Jones is the following:
Theorem 4. [Jone2] Lei S be the subset o/R+ consisting of the vahes of the index [M : N], where M and
N are factors of type IIb JV с A/. Then
S = {4 cos2 -; n e N, n > 3} U [4, +oo).
We shall briefly describe the basic construction of V. Jones. It extends to this situation the idea of iterated
crossed products which was already crucial in the construction of the automorphisms sj of theorem 6.14.
Here we do not have any group G but only the inclusion JV С M of its fixed point algebra JV in M. First
the analogue of the inclusion of M in its crossed product by G is the inclusion of finite factors:
A) А/ С JN'J , JV' = commutant of JV in L-(M)
where J is the canonical antilinear isometric involution in L2(M),
B) Jx = x' Vie L-(M).
One has M' = JMJ (cf. section 3) so that the inclusion A) holds. Moreover since the inclusion A) of type
II, factor is of the same type as the original inclusion JV с М (it also has the same index by 3 c)), the
operation
C) N С M -* M С M\ = JN'J
can be iterated. It yields an inductive system Mi of type lit factors such that:
D) Mk С Мь+1 Vfe , [Mt+i : Mb] = [M : N] V*
and one can endow the inductive limit Л/те with its unique tracial state which we denote by r.
415

The crucial fact now is that in the inclusion A) one has a canonical projection t\ € J N' J = M\ given by:
E)
is the orthogonal projection of L2(M) on the closed subspace N = {xl ; i 6 N}~.
By construction J et J = e^ because N" = N. Moreover et is the probabilistic conditional expectation of
M on N and in particular is an iV-bimodule map from M to N so that ([To]):
F) ej 6 N' , е, 6 JWJ.
Using the bicommutant theorem one then checks (cf. [Joneo])
G) Mi is generated by M and e\.
Now when the construction A) is iterated we get a sequence of projections en 6 М„ and one checks (cf.
[Jone2]) that they satisfy the following relations.
Lemma 5. Let r be the unique iracial state on Mm. The (eo) form a sequence of projections em = e?, =
ejj, 6 Mo such that:
a) ei ej = ej e; Vi, j 6 N , \i-j\ > 1.
/?) em + i em em + i = [M : N]'1 em+1 , em em+1 em = [Jtf : iV] em V?n > 1.
7) r(r em+i) = [M : iV] r(x) for any element x of the algebra generated by e\,... ,em.
The relations a) /3) imply that the algebra Am generated by elt..., em is a finite dimensional C-algebra.
The proof of theorem 4 ([Joneo]) is based on the analysis of the AF C"-algebra inductive limit of the Am's.
The obtained C'-algebra A(t) only depends upon t = [M : N]~l and is non trivial only for i~l 6 S with 2
as in theorem 4. The first example of non-integral index corresponds to n = 5; the index is then the square
of the golden number '+У* = 2cos j, and the AF C'-algebra generated by the projections e; is canonically
isomorphic to the C-algebra associated with the parameter space for the Penrose tilings (cf. Chapter 2).
I refer the reader to [Goo] for more detailed information. By analyzing the relative commutant of N in М„,
V. Jones also introduces an invariant consisting of a Dynkin diagram which is finer than the index [M : N],
and is canonically associated with a subfactor N С М.
The classification of the subfactors of index 4 cos2 J of the hyperfinite factor was carried out by A. Ocneanu
and S. Рора ([Ог], [Р02]). Finally, V. Jones' discovery of new polynomial invariants for knots [Лопез], based
on his analysis of the subfactors, has had a remarkable impact 011 low-dimensional topology. However, it lies
beyond the scope of this book, and we shall just explain below the relation with the Hecke algebras.
/3) Positive Markov traces on Hecke algebras
We shall explain in this section the link between the relations 5 a) /3) and the Hecke algebras of biinvariant
functions on SLn(Fg) where F, is the finite field with q elements. We need a general definition which will
be used again in section 11.
Definition 6. Lei Го С Г be a subgroup of a discrete group Г such thai the left action of T on Г/Го has
only finite orbits. Then the Hecke algebra й(Г,Г0) is the convolution algebra 0/Г0 biinvariant functions on
Г with finite support m Г0\Г/Г0.
More explicitely the convolution / * /' of two such functions is given by:
(8)
(f*f')(y) =
Го\Г
416
When Г is finite, one can view ft(r, Го) as the subalgebra of Го biinvariant elements in the group ring СГ.
Let then q be a power of prime and F, the finite field with q elements. Let Г = Stn(F,) and Го С Г be the
Borel subgroup given by upper triangular matrices. The Bruhat decomposition Г = U Го to Го where
Sn is the permutation group gives a natural basis (Тш)ш€5, for А„(д) = А(Г, Го), where
(9) tw(g) = 1 if Я 6 Го ш Го , tw(g) = 0 if я ( Го u> Го.
Moreover one checks, using ij = <„, where oy is the transposition (i, i + 1),
Proposition 7. The а1яеЬга fin(q) = А(Г, Го) is generated by the (,-, i = 1, ...,n - 1 and admits the
following presentation: ¦
a) (tj + l)(ij - 9) = 0 , i=l,...,n-l
b)<; tj =tj U Vi,j , \i-j\ > 1
c) U+i U ti+i =titi+,ti Vi = 1,..., n - 2.
For any integer n > 1 and complex number q 6 C, let t),,(q) be the algebra over С with generators tit
i = 1,..., n - 1 and the presentation a) b) c). We also allow the value n = 00 and Лто(?) is the inductive
limit of the ftn(q), pnm where />„,„(<>,„) = tj,m with obvious notations.
For q = 1, f)n(q) is the group ring C5n of the symmetric group. For q ф 0, q" ф 1, f>n(q) is isomorphic to
CSn but for q" = 1, q Ф 1, or q = 0 the algebra f)n{q) is not semisimple.
For q ф -1 let e,- = 75т 0 then the ejt j > 1 generate «„(<?) and the presentation а) Ь) с) becomes, with
< B + + ?-1)-1
A0)
a') ej = ej Vj > 1
4') eiej=ejei if |i - j| > 1
e, ei + 1 - t e;+i = e; ei+1 e, -t et Vi
The relations a) /3) of lemma 5 are stronger since while a') b') are unchanged c7) is replaced by the stronger
A1) c") ej + i ev ei+, -t ei+i = e{ ej+1 ei - t e{ = 0.
The V Jones construction thus gives:
Theorem 8. [Jone4] Lei q 6 [l,oo[U {e"^m ; m = 3,4,...} and t = B + q + q-1)'1. Let ?„(?) be the
above n/je4ra with the unique involution * such that e'j = ej Vj > 1.
1) There exists a unique trace r on f)^q) such that r(l) = 1 and that for any n r(x en+i) = i r(x) Vs 6
я„(?);
2) The trace r is positive (rU'z) > 0 Vj: 6 ««,(</)) and the weak closure о/Лсо(?) in the GNS represen-
representation is the hyperfinite factor R.
3) The von Neumann snbalgebra of ftx(q)" generated by the ej, j > 1, is a subfactor N С Я, [Я : N] = *"'¦
There are two refinements, due to A. Ocneanu Q. First consider for q + e 6 R the involutive algebra
(f> (q) *) as defined in theorem 8. Then this involutive algebra admits non trivial Hilbert space represen-
representations 'iff q 6 [l,oo[ U {e'2*/"' ; m = 3,4,...}. Next, with q as above and z 6 С there exists a unique
normalized trace ф,_г on !ix(q) satisfying the following Markov property:
A2)
) = z Ф(х)
417

Moreover with q as in theorem 8 the trace ф,л on the involutive algebra Яю(,),», is positive iff z € f0 ,1 4
and the pair (q, z) satisfies one of the following conditions (П): J Ж
q> i,
A3)
? = e2"/-and3*eZ, \k\ < m - 1 , к # 0 with z = (^+ ~}f^ gt) B + a + ,-¦)'¦ = f.
As in theorem 8 one obtains for each allowed value of (q,z) a subfactor of the hyperfinite factor Д, whose
index has been computed by H. Wenzl [Wen].
V.ll. Hecke algebras, type III factors and statistical theory of prime numbers
In this section we shall discuss an example of a quantum statistical mechanical system, arising from the theory
of prime numbers, which exhibits a phase transition with spontaneous symmetry breaking (cf. [Bos-C]) The
original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).
Let us first resume our discussion of quantum statistical mechanics of chapter I section 2. Thus a quantum
statistical system is given by:
a) The C* algebra of observables: A.
/3) The time evolution (<rt),^tt, which is a one parameter group of automorphisms of A.
An equilibrium or KMS' state at inverse temperature /3 is a state y> on A which fulfills the KMS^ condition,
A.2) and V.3, i.e. for any 1,}бД there exists a bounded holomorphic function, (continuous on the closed
strip), Fx<y(z), 0 < Imz < /3 such that:
F^i) = <p{x <r,(y)) V<€R
418
Fx,,(t + г/3) = f{tr,(y)x) Vi € R.
In the simplest case where A = Afjv(C) is the algebra of N x N matrices, any one parameter group of
automorphisms (tfiJieB of A is of the form:
<rt{x) = eil" x e-"H 4xlZA,ien
for some selfadjoint element Я = Я" 6 A. Then for any /3 6 [0,oo[, one has a unique KMS^ state for <r(,
and it is given by the formula:
, ч Tracefe-"" x) w
^{X)= Traced») V*€A
Note here that Я is only defined up to an additive constant by &t, so that the normalization factor:
Тгасе(е~'ш), cannot be recovered from cr,. However the following formula holds:
Log Traced-*"') =Sup (S(<p) - /V(#))
where y> varies over all states on A and S(<p) is the entropy of the state:
S(<p) = — Trace(pLog/>) for ip = Trace(/>•).
In a slightly more involved situation, that, of systems without interaction it is still true that for any /3 € [0, oo[
there exists a unique KMS^ state. More precisely one has the following immediate:
Proposition 1, Let A = ® А„ be an infinite tensor product of matrix algebras А„ = Мц„(С), and
(Л = ® <j", a product time evolution. Then for any J3 > 0, there exists a unique KMS^j state fp for (A, <rt),
and one kas ipp =<8> lpp,v, where pp,* *s the unique KMSjs state for{Av^o'")-
For interesting systems with interaction, one expects in general that for large temperature, i.e. small /3, the
disorder will be predominent so that there will exist only one KMS^j state. For small enough temperature
some order should set in and allow for the existence of various thermodynamics! phases, i.e. of various KMSp
states. It is a very important general fact of the С algebraic formulation of quantum statistical mechanics
that for a given /3 every KMS^j state decomposes uniquely as a statistical superposition of ег<гете KMS^j
states'.
419

Proposition 2. ([Br-R] [И]) Lei (A,<rt) be а С-dynamical system and C e [0,oo[. Then the
states is a compact convex ckoquei simplex.
space o
For a careful discussion of the link between extreme KMS^ states and thermodynamical phases we refer the
reader to [H].
As a simple (classical) example illustrating the coexistence of phases at small temperature one can think of
the phase diagram for water and vapor (fig.l) or better for the ferromagnet (fig.2). In the latter example when
the temperature T is larger than the critical temperature Tc, of the order of 103K\ the disorder dominates,
while for T <Tc the individual magnets tend to align with each other, which in the classical 3 dimensional
set-up yields a set of thermodynamical homogeneous phases parametrized by the 2 dimensional sphere of
directions in 3 space.
This example serves to illustrate the phenomenon of spontaneous symmetry breaking: The group SOC) of
rotations in R3 is a symmetry group of the dynamical system one starts with, and for large T, T > Tc, the
equilibrium state is unique and hence invariant under rotation. For small T however, T < Tc, the group
5OC) acts non trivially on the set of thermodynamical phases and the choice of an equilibrium state breaks
the symmetry.
The C*-algebraic formulation of this is straightforward. One has a (compact) group G of automorphisms of
the C-algebra A which commutes with the time evolution:
ag 6 Aut A , Vj € G , ag<r, = <rtag WgR.
Such a group obviously acts on the compact convex space of KMS^j states and hence on its extreme points.
We shall now describe (the precise motivation will be explained below) a C"-dynamical system intimately
related to the distribution of prime numbers and exhibiting the above behaviour of spontaneous symmetry
breaking.
The C"-algebra ,4 is a Heche algebra, which contains the algebra of usual Hecke operators of number theory
[Seri], i.e. those related to Hecke correspondences for lattices in С [Seri]. The latter algebra is commutative
and is essentially the algebra of composition of double cosets:
у 6 GIB,Z)\GIB,Q)/G?B,Z).
Recall that given a discrete group Г and a subgroup Го which is almost normal:
"The orbits of Го acting on the left on Г/Го are finite"
one defines the Hecke algebra "Н(Г, Го) as the convolution algebra of (C-valued for our purposes) functions
with finite support on Г0\Г/Г0. More specifically given two such functions /, /' 6 И(Г, Го), their convolution
is
Го/Г
In this formula/ and /' are viewed as Го biinvariant functions on Г with finite support in Го\Г/Го-
To complete И to a C""-algebra we just, close it in norm in the following regular representation of ti in
<*(Г0\Г) (cf. [Bi]).
Proposition 3. Let Го С Г be an almost normal subgroup of the discrete group Г. Then the following
defines an (involutive) representation А о/Т<(Г,Го) in ^2(Г0\Г).'
ro\r , v/ e H.
Го\Г
420
One checks that A(/) is bounded for any / 6 Ti. The involution on ti such that:
is given by the following equality:
€ Г0\Г/Г0.
Thus we let A be the C*-algebra norm closure of И(Г,Г0) in ?2(Го\Г). A good notation for it, compatible
with the discrete group case (chapter II section 4) is:
Й=Сг'(Г,Го).
Let us now define the one parameter group of automorphisms <r, 6 Aut A We first need to introduce a
notation. Since each Го orbit on Г/Го is finite we shall let, for у € Г
L(y) = cardinality of Г0GГ0) in Г/Го
ЯG) = cardinality of (Го7)Го in Г0\Г.
Thus by construction L(y) 6 1ST, R{y) 6 1ST, R(y) = Цу~у), L and Я are both Го biivariant functions.
Proposition 4. Lei Ta С Г be ян almost noniial subgmtip of the discrete group Г. There exists a unique
one parameter group of automorphisms <r, 6 Aut(C;(T, Го)) such thai
(<г.(Л)G)=
vT€r0\r/r0.
In fact <r( is the restriction of the modular automorphism group <rf for the state onM = A(W)" given by
the unit vector corresponding to the coset Го 6 Г0\Г.
Let us now consider the Песке algebra H for the groups:
Г = PQ , Го = Pi
where P is the group of 2 x 2 matrices P = j L I ; aa'1 = a~la = 1 j.
One checks that Pi is almost normal in Pq.
We shall now describe the phase transition with spontaneous symmetry breaking for the dynamical system
corresponding to Г = Pq , To — Pi-
Let us denote by V/з the following function on the group Q/Z. Given n = ? € Q/Z, with a, 6 € Z, with о
relatively prime to b > 0, one lets 4 = Пр*р be the prime factor decomposition of Ь and one sets:
</,„(,,) = Пр-*-"A -/-')A -p-1).
Theorem 5. ([Bos-C]) Let {A,tr,) be the C" -dynamical system associated to the almost normal subgroup Pi
Q. Then:
a) For 0 < /? < 1 there exists a unique KMSp state fP. Its restriction to C*(Q/Z) С С*(Г, Го) (through
the inclusion Q/Z С Г0\Г/Г0 of the umpotent subgroup j [ * " j j С Р) is given by the above function of
positive type фр on Q/Z. Each <p» ¦> « /«с'<т $Ше a»d ilte associated factor is ike hyperfintte factor of type
Ilh : Ясс
421

b) For C > 1 ike KMS^j states form a simplex whose extreme points <ppx are parametrized by imbeddings x
of the field К = Qaj (the field of roots of unity) К -i- С and whose restrictions to C*(Q/Z) are given by the
following formula:
These states are type /TO factor states.
The normalization factor is the inverse of the Riemann Q function evaluated at /3.
In other words the critical temperature here is Tc = 1 and at low temperature (/? > 1) the phases of the
system are parametrized by all possible embeddings of К = ЦаЬ in the field of complex numbers.
As we shall see below the Galois group G = [K : Q] does act naturally as automorphisms G С АиЦС"(Г, Го»
commuting with the time evolution, and the spontaneous symmetry breaking occurs for 0 > 1.
We shall now explain how the above C"-dynamical system is related to the distribution of prime numbers
and to class field theory.
/3) Bosonic second quantization and prime numbers as a subset of R
It is a saying of E. Nelson that first quantization is a mystery while second quantization is a functor. In the
bosonic case this functor S from the category of Hilbert spaces to itself, assigns to every Hilbert space %
the new Hilbert space S7i given by:
sn = ©~=os" n
where SnTi is the nth symmetric power of 7i endowed with the following inner product:
¦>*.)=
Given an operator T in % (more generally T : Wi —> W2), the operator ST in SH is given by:
, •••{„) =
• (Т{„) V& € П.
Even if T is bounded, ST is not bounded in general but if T is selfadjoint (unbounded) so is ST. Thus we
shall work with such operators. One has the following formula:
Trace( ST) =
1
det(l -T)
which makes good sense if || T ||< 1 and T 6 С1(П). The problem we shall now consider is the following: to
give a simple characterization of selfadjoint operators T in И whose spectrum is the subset J"cR formed
of all prime numbers, each with multiplicity one:
V = {2,3,5,7,11,13,17,...} CR.
The corresponding problem for the set N = {0,1,2,3,...} of natural numbers, or N* = ISI\{0}, is easier,
and was solved by Dirac's paper [Dir] which inaugurated quantum field theory. In that case the solution is
simply that there exists an operator a such that:
aa" — a" a — 1 , a" a = T.
(For 1ST one requires that aa" be equal to T.)
Let us now state the result for the subset V С R:
422
Lemma 6. [Bos-C] Let T be a selfadjoint operator in a Hilbert space ti then (counting multiplicities):
Spectrum T = V <=> Spectrum ST = 1ST.
Proof. Let us first assume that Spectrum ST = N". Then, as quite generaly SpecT С SpecST, (using the
inclusion П С S7{) we see that ? = Spec(T) С N*. Let us show that PCS. Indeed let p e P not be in
2. Then because ? С N* one has p ? E" for any n (with E" = {*i*2 • • *n,*j 6 ?}). This shows that
p $. Spec(ST) = US" whence a contradiction. Thus PCS. If к € Ъ\Р then astgP" for some n > 1 this
would mean that к is not a simple eigenvalue for ST. Thus V = E. The converse is obvious from Euclid's
unique factorization theorem, but we shall fix the corresponding notations: we let 7ii = {?(!>) be the Hilbert
space with basis (ер)р$т>> and we identify it to the one particle subspace of SWi = ^2(IST), the Hilbert space
of square integrable sequences of complex numbers, with canonical basis the ?„, n € ISI". We shall denote by
T the operator:
and by ST the corresponding operator:
ST :
— е2(Ы') ; (ST)e,, = пе„ Vn e N*.
Also we shall let H = log(ST). It is the generator of a one parameter unitary group Ut = exp(HH) = Г",
whose role is made clear by the following special case of formula (*) which is the Euler product formula for
the Riemann ? function:
For Res > 1; <(*) = Trace (ST)' = ^.т-у
Now the meaning of lemma 6 is that the subset V С R has a neat definition provided one is ready to use the
formalism of bosonic quantum field theory. That formalism includes the algebra of creation and annihilation
operators, respectively a*(f) and a(jj), for f,i) 6 И, given by:
It also includes the time evolution, in Heisenberg's picture, given by:
(r,(x) = Ut x I'i = e"" x e'ilH V* 6 R.
In our case the corresponding C-algebra in SH = C2(N") and time evolution are given by the following:
Proposition 7. 1) For every peV let цр be the isomciry in t2(№) given by the polar decomposition of the
creation operator associated to the unit vector ep 6 П. The С-algebra C*(ISI") generated by the fip 's is the
same as that generated by Ike isometnes:
Mn , /in €t =?in V»eN' is 1ST.
2) This С-algebra is the infinite tensor product:
C*(N*)= ® rp
per
where each rp is the C"-algebra generated by цр and is ihe Toeplitz C'-algebra.
3) The equality <r,(x) = e"" * e~itH, 4x € С"(гЧГ), t € R where И = log(ST), defines a one parameter
group of automorphisms ofC"(N') and <r, = ® <r,:P; <rtp{fip) - pli fip Vi €R-
per
423

We recall that the Toeplitz C-algebra г is the C"-algebra with presentation u; u'u = 1. If u is any non-
unitary isometry in a (separable) Hilbert space the smallest C*-algebra containing u is isomorphic to r. This
C-algebra is nuclear so that the finite tensor products ®р<„тр are unambigously defined. Their inductive
limit, ® тр is the C*-algebra С"AЧГ).
The C-dynamical system thus obtained is not very interesting because it is without interaction G.3).
Nevertheless the corresponding unique KMS^j states will be useful later and are given by the following
corollary of proposition 1 and of the Araki-Woods classification of ITPFI (section 4).
Proposition 8. [Bos-C] a) For every /3 > 0, there exists a unique KMS^ state on (C*(N*),<T|). It is the
infinite tensor product:
where f0lP is the unique KMS^ state, on the Toepliiz algebra, for <rtiP. Its eigenvalue list is:
{(l-p-")p-"" ; neN}.
b) For /3 > 1, the state fp is of type 1^ and ts given by:
W(x) = <(/?)-' Tracefe-"" x) Vx 6 С (NT).
c) For ji = 1, the state ipp is a factor state of type IIIi given by:
w(z) = Trace^fe"" x) Vz € C(N')
where Trace^, is the Dixmier trace.
d) ForO < /3 < 1, ip/i is a factor state oftype III\ and the associated factor is the factorH^ of Araki-Woods.
Statement d) for /3 = 1 is due to B. Blackadar ([Bl2]). We refer to chapter 4 for the definition of the Dixmier
trace, whose general properties make it clear that the equality c) defines a KMSi state.
y) Products of trees and the non commutative Hcckc algebra
In this section we shall relate the C"-dynamical system C""(ISI"), o-,, of section /3) with basic number theory
notions [We] and get. to the Песке dynamical system of theorem 5.
Let P be the ax + Ь group, i.e. the group of triangular 2x2 matrices of the form , with a invertible.
We view it as an algebraic group, i.e. as a functor A —> PA from abelian rings to groups. It plays an
important role in the elementary classification of locally compact (commutative and non discrete) fields (cf.
[We]). Indeed given such a field Л' then the group G — PA- is a locally compact group, and as such it has
a module: 6 : G —• H^, obtained by the lack of invariance of a left Haar measure dg on G under right
translations:
(Or equivalently
d(glc) = S(k)dg Vi € G.
= S(g)~1dg as measures on G.)
This module 6 : Рк —> R + is 1 on the additive group and its restriction to the multiplicative group (extended
by 0 on K\K') yields a proper continuous multiplicative map:
mod a- : A" —> R +
such that the open sets {k € A';modA-(i) < e} form a basis of neighborhoods of 0 (cf. [We]). The image of
S is a closed subgroup of R^_ and except for the case of the archimedian fields R or C, this closed subgroup
is discrete equal to A* for some A 6 ]0,1[ whose inverse q = A is called the module of K. The function
424
modjf (ж - у) = d(z,y) is then a ultrametric distance giving back the topology of К ([We]). Given x € К
the integer v(x) such that гткх)к(г) = q~v(-x^ is called the valuation of*.
Proposition 9. (cf. [We]) Let К be a non-discrete commutative locally compact field, К ф R or C. Then
there exists a prime p such thai modjf (p) < 1. Call R, Rx and P the subsets of К respectively given by
Я = {x € A' I modA-(z) < 1} , Я* = {x € К | modA-(z) = 1},
J = {x<Z К |modjf(x) < 1}.
Then К is ultrametric; Я is the unique maximal compact subring of K; Rx is the group of invertible elements
of R; J is the unique maximal ideal of R, and there is ж € J such thai J = хЯ = Ях. Moreover, the residual
field k — R/J is a finite field of characteristic p; if q is the number of its elements, the image of A'x in HJ
under modjc is the subgroup o/Hj generated by q; and modjc(т) = q~l.
As a basic example the field Qp of p-adic numbers is defined (given any prime number p), as the completion
of the field Q of rational numbers for the distance function:
<Kx,y) = \x-y\v
where for x 6 Q, x = p" f (with ?), a, b integers and я, 4 relatively prime to p) one sets:
The maximal subring R of A' = Q,, is the ring Z,, of p-adic integers and the residual field k = R/J is the
finite field Fp.
One obtains in this way, together with the inclusion Q С H, all the inclusions Q С A" of the field of rational
numbers as a dense subfield of a local field A'. Such inclusions, (or rather in general, equivalence classes of
completions) are called places and to distinguish the real place QcH from the others, the latter ones are
called finite places.
Putting together the inclusions of Q in its completions <}„ = A' parametrized by the places of Q one gets a
single inclusion of Q in the locally compact commutative ring of adeles which is the restricted product of the
fields Qv. More specifically this ring is the product Я х Л where the ring Л of finite adeles is obtained as
follows:
a) Elements x of A are arbitrary families (?,,); xp 6 Qp such that xp 6 Zp for all p but a finite number.
b) (x + y)p = xp + yp; (xy)P = xpyp define the addition and product in A.
c) Finally A has a unique topology of locally compact ring such that the subring 7t = IIZp is open and closed
and inherits its compact product topology.
We shall now relate the C"-dynaniical system (C*(IST), <r,) of section /3) with the locally compact ring A oi
finite adeles.
We just need to recall that given a non-unimodular locally compact group G one has a natural one parametei
group of automorphisms a, of С (G) given by the following formula valid say on ?'(G):
(*) ЫЛ)(з) = На)" f(u) vff e G , * e R.
(This group is the modular automorphism group of the Plancherel weight on C'(G).)
Proposition 10. Lei A be the ring of finite adeles ouerQ, and "R. its maximal (open) compact subrtng. Le
G be the locally compact group G - PA, and e 6 C'{G) the characteristic function of the open and compac
subgroup Рк С Pa- Then:
425

1) One has e = e" = e2, and the reduced C'-algebra C"(G)e = {x € C(G) ; ex = xe = x] is canonically f
isomorphic to the С-algebra C'(N") of section /})¦
2) One has o-t(e) = e Vi g R, and the restriction of a, to the reduced C* -algebra C*(ISP) is the time evolution
of section 0). S:f
«ft
We think of the characteristic function of Р-ц as an element of Lx(G,dg) С C"(G), with dg the unique left ^
Haar measure which gives measure 1 to Р-ц. The group G is solvable and hence amenable so that there is no '¦
distinction between C*(G) and the reduced C*-algebra C'(G). The proof of proposition 10 is not difficult 1
because it reduces immediately to a local statement: That if A' = Qp and R С К is the maximal compact Й
subring, the C*-dynamical system (C*(Pa-),it,) given by (¦), reduced by the projection ep = characteristic Л
function of Pr, is isomorphic to the Toeplitz C-algebra rp, with the time evolution atp of proposition 7 3).
We then take for pp 6 C"(P^p)tp the isometry given by the following L1 function:
I =
1 if & € Я , val(a) = 1, and is equal to 0 otherwise.
The C-dynamical system (С"(Ра),<т{) of proposition 10 is without interaction, exactly as is (С*(гЧГ), <т()
(proposition 8), and there is an exact analogue of proposition 8, which states the existence and uniqueness
(up to scale) of KMS^j weights on the above system. One needs to use weights because one is dealing with
non-unital C*-algebras. At the technical side such weights have to be semicontinuous and semi fin He (for
the norm topology) (cf. [Com]). It is however instructive to work out the explicit formula for those KMS^
weights. Using the natural isomorphism of the Pontrjagin dual A of the additive group A with itself, one
gets an isomorphism:
C-(PA)=Ca(A)xA-
where the multiplicative group A* acts by homotheties in the locally compact, space A. The KMS^j weight
on C'(Pa)< °~t is then the dual weight of the following measure \ip on A:
Ja-
Here d'j is the Нааг measure on the multiplicative group A*, j —> \j\ is the module, and the formula makes
sense as such for /? > 1, and by analytic continuation for 0 < /? < 1 ([We]).
It is clear that to obtain a C*-dynamical system with interaction we need to use not only the locally compact
ring A but also the fundamental inclusion:
QC.4.
We shall use the corresponding inclusion Pq С Pa ~ G in the action of Pa on the C*-module ? = C*(G)e
over C'(N') given by the isomorphism of proposition 10: C(G)e = С"AЧГ). Indeed, given any C"*-algebra
В and (self adjoint) projection e 6 B, the space ? = Be = {x 6 В ; xe = x) is in a natural way a (right)
C*-module over the reduced C"-algebra Bt = {x g В ; ex — xe — x). One thus lets:
(«,ч)=Гче в, , ч?,г,е?= Be
taef. v( e? , ae вс.
This C*-module has moreover a natural left B-module structure, given by F, ?) —> b? 6 ?, V6 € B, { € S.
In our case ? — C* (G)e is a space of functions on G which are invariant by right multiplications by elements
of Ptz С G, or in other words it is я space of functions on the homogeneous space:
426
This space Л is by construction the restricted product of the spaces:
Д, = PoJPi,
relative to the base point given by Pjp.
Proposition 11. The homogeneous space Ap = Pq,/Pz, over the group P9r с GiB, Qp) is naturally
isomorphic to the (set of vertices of the) tree o/5iB,Qp). The group Pqr acts by isometries of A and
preserves a point at oo.
Let us recall (cf. [Ser2]) that the tree of S?B, A"), where К is a local field, is defined in terms of equivalence
classes of lattices in a two dimensional vector space V over K. With the notations of proposition 9, a lattice
L с V is a sub Д-module of V which is of finite type and generates V as a vector space. The multiplicative
group A" operates on the set of lattices by (L, x) —* xL for л 6 К", and one lets T be the set of orbits of
this action of K*. Given a lattice L С V and a class Л' € T there exists a unique representative V € A'
such that V С L and L' <(_ i? (with ж given by proposition 9). Then L/V — TJ./ir"V. and the integer n,
which only depends upon the classes of L and ?', defines a distance d on T, by the equality:
rf(class of L , class of L') — n.
Using the set of pairs with mutual distance equal to 1 to define a 1 dimensional simplicial complex, one gets
a tree, the tree of SLB, A'), and the above distance is the length of the unique path joining two elements of
this tree (cf. [Ser2]). The group GL(V) of automorphisms of the vector space V acts on the set of lattices
by
(L,g)-.gL Vff 6 GL(V),
and since this action commutes with that of A" it gives an action, by isometries, of GL(V) on the tree T.
Let us identify V with A'2, GL(V) with GIB, A"), and consider PK as a subgroup of GLB,K) : Рк -
l\ ; a e A'* , Ь e A' >. Let Lo be the lattice R2 С A'2. Then one checks that PK acts transitively
on T and that the stabilizer of the class of Lo is Pr. We thus get a canonical identification T = Pk/Pr-
Taking A" = Qp yields the conclusion.
Proposition 12. 1) The homogeneous space G/Р-ц = Д is canonically isomorphic to the restricted product
of the trees Tp wiik base point *, and the action of G on A is simplicial.
2) Me subgroup Pjj С Pa = G ads transitively on A, and Hie isoiropy subgroup of the base point * is Pi.
We can thus identify Д with Pq/Pi, and we shall now get the Песке algebra of theorem 5 from the commutant
of the action of Pq in the space of functions on Д. We need for that purpose to obtain Hilbert spaces first,
and the construction will follow from the following general lemma applied to the C*-module ? = C"(G)e
over C*(ISP) and the time evolution <r, of C"(ISP).
Lemma 13. Let С be a unital C" -algebra, ? а С-module over С, (<Ti)ier я one-parameter group of
automorphisms ofC, /3 6 ]0,oo[, ipp a KMSp state on C, and"Hvt the Hilbert space oftkeGNS construction
for fp.
a) Let Tip be (he completion of ? for the. inner product given by:
Then the action of the endomorphisms, Endc(?), on ? extends by continuity toTip.
b) There exists a unique representation p of C" (the opposite algebra ofC) in tip such that for any l.&'Hp
and a ? С in the domain of 0"_^/2 one has:
427

This representation commutes witk ike left action of Endc(?).
The Hilbert space Tip is the tensor product:
Tip = ? ®c Hip,,
so that the first assertion follows (Chapter II appendix A). The second assertion also follows, using Ti^p as
a left Hilbert algebra.
We apply this lemma with С = C*(ISP),? = C'(G)e, and <r, € Aut С given by the time evolution (proposition
7 3)) of C* (ISI*). As ? is a space of functions on Д, so is each of the Tip, and for each a € Д we let sa be the
characteristic function of {а} С Д. The vectors ea, a € Д, are of unit length in each Tip and always span a
dense subspace of Tip- For /3=1 they give an orthonormal basis, so that Hi = ^2(Д). In general, the inner
product is uniquely determined by the following function of positive type on Pq:
*= base point of Д
and the computation of фр yields the following:
where Ь — Y[pkp is the prime factor decomposition of the denominator b > 0 of the irreducible fraction n = j.
The coramutant of Pq in 'Hp is given by a right action p0 of the Hecke algebra A = C*(PQtPi) of theorem
5 as follows:
Proposition 14. Let 13 e ]0,oo[.
1) The following equality defines a faithful unitary representation of A0 = C* (Pq, Pz)° in Tip;
*= ? )
2) For eachp, рр(А®) generates ike commuiani of Pq in Tip.
Thus the Hecke algebra A = C'(Pq, Pz) appears uniquely as the norm closure, in each Tip, of the algebra of
operators in Tip which commute with Pjj and preserve the linear span of the ea, a € Д. The latter algebra is
normalized by any g € Рд = G, and it follows that the group С = Pa/Pq = -4*/Q* acts by automorphisms
on A. This action is independent of the real parameter ji and will be denoted by:
jgC -*вя 6 Aut Л.
The compact group С is the idele class group ([We]). By construction the fixed point algebra Ac = {a €
A ; Og(a) = a Vj ? C} is isomorphic to C*(ISP). It is this action of С on A which is the symmetry group
of the dynamical system (A,<rt) of theorem 5, with spontaneous symmetry breaking for 0 = 1.
428
Appendix A. Crossed products of von Neumann algebras
Let M be a von Neumann algebra, G a locally compact group and a : G —> Aut M a continuous action of G
on M. The group Aut M is endowed with the topology of poiiitwise norm convergence in the predual M, of
M. Thus for each f>€*f, the map g —. a9(tp) is norm continuous, let dg be a left Haar measure on G and
Л be the left regular representation of G in the Hilbert space L2(G),
A)
The action a of G on M is encoded by the homomorphism a : Jtf —• M ® t°°(G) given by:
B) 5(x) = (a;'(*)bec 6 ?°°(G, M) = A/ ® L°°(C)
which satisfies the following equivariance condition:
C)
Vy € G , г 6 W.
Definition 1. The crossed product MxaG is the von Neumann subalgebra of M ® C(L2(G)) generated by
a(M) and 1 ® A(G).
The equality 3 shows that- the crossed product contains as a weakly dense subalgebra the finite sums'.
S5(x9)A(jr) х„еМ.
When the group G is discrete any element of the crossed product can be uniquely written as a strongly
convergent sum:
S5(i,)J(j)
where the xg are determined as matrix elements in the natural basis of C2(G) (cf. [Di]). This is no longer the
case when the group G is not discrete and a good description of the generic element of Mx>aG is lacking
in general.
Definition 2. Let a, a' be two actions of G on M. Then a and a' are outer equivalent iff there exists a
strongly continuous map g — ия from G to the unitary group U(M) such that:
!) M»iS3 = "si а»Д"л) Vffi,tf2 S G
2) a',(x) = ug ag(x) ug Vz 6 M , g 6 G.
The crossed products Л/ Xa G and M Xo' G are then canonically isomorphic using the unitary и € M ®
Lm(G), и = (Hj-Ojec, which satisfies:
« 5{x) ч* = a'(x) Vz € A/
D)
Чя) u Afj,)-1 =«o(«,)
When G is abelian let G be the Pontrjagin dual of G and let (g,g') for g 6 G, g' € G be the canonical
pairing with values in T = {: GC , |-| = 1}.
429

Appendix B: Correspondences
Proposition 3. [T3] Lei G be an abelian locally compact group, or an action ofG on a von Neumann algebra
M. The following equality defines a canonical action a of the Pontrjagin dual G of G on the crossed product
M Xia G:
Sg,(y) = A
where (V Z)(9) = {9,9') «Ы
У A
Vy 6 M xia G , g' 6 5
Щ
By construction a fixes pointwise the von Neumann suhalgebra M {= a(M)) of M'xCrG and multiplies
1® A(j) by {9,9') 6T.
The central result of the theory of crossed products of von Neumann algebras is the following;
Theorem 4. [T3] Let G be an abelian locally compact group and a an action of G on a von Neumann
algebra M. There exists a canonical isomorphism 0 of the double crossed product (Л/ >ae G) X~- G with
M®C(L^(G)) which transforms the double dual action a into an action outer equivalent to the action a® I
ofG on M®C(L2(G)).
This theorem has since then been extended to non abelian groups and Hopf von Neumann algebras (cf. [Str] '4
[En-S]). "i.
•H
We shall describe in this appendix a third natural notion of morphism between von Neumann algebras. We
already met the following two notions:
1) p : M —*¦ N is a normal involutive algebra homomorphism.
2) T : M —> N is a completely positive normal linear map.
While 1) is the most obvious notion of morphism and basic concepts were introduced for automorphisms
(section 6), the notion 2) plays a key role in section 7 and in the entropy theory (section 6). In the
commutative case: M = Ц»(Х,цх), N = ?°°(У,^у) the notion 1) corresponds to a non singular map
V> : У —» X, while the notion 2) corresponds to a non singular map у —> py from У to positive measures on
X.
We shall introduce a third notion of morphism, intimately related to 1) and 2):
Definition 1. Let M,N be von Neumann algebras. Л correspondence from M to N is a Hubert space 7i
which is an N — M bimodule.
Thus more explicitely we are given commuting normal t representations irjv of N and vMv of M° in Л where
M° is the opposite von Neumann algebra. To make notations lighter we shall write whenever no confusion
can arise:
To justify the terminology (one could simply call'
mutative case.
an N — M bimodule) we shall first consider the com-
comLet then (X,fix), (Y, Py) be standard measure spaces, M - L°°(Jf,^A-) and N = L°°(Y}fiy). Then the
most general correspondence "H between M and Af is given by a measure class /i on X x Y with projections
Prx(^)> Pry(^) absolutely continuous with respect to /*x> /'v < and an integer valued ^-measurable Junction
n(s,t) ((s,t) 6 X x У). The Hilbert space V. is equal to / Я(а,,) d\i{s,t) where #(s,t) is a Hilbert space of
dimension n(s,t) E {0,1 oo} while the structure of bimodule is given by:
430
B)
(»«/)(*.') = 9@/(*)«(».') 4feM,geN,teH.
In general the measure /i is not absolutely continuous with respect to px * (iy, this measure represents the
graph of the correspondence, while the function n represents the multiplicity of the correspondence.
If in the above example we take (X,fix) = (У.I*y) and p equal to the image of fix — t*Y on the diagonal
Д = {(ж,ж),a: e X) while n(s,s) = 1 Vs 6 X, we get the identity as a correspondence from M to
N = M. The Hilbert space H is equal to L2(X, px) and the bimodule structure corresponds to the standard
representation of M.
We shall now descrihe with special care the identity correspondence ?2(M) for an arbitrary von Neumann
algebra M.
a) Half densities and the identity correspondence
Let M be a von Neumann algebra, M,+ the positive cone of the predual of M. Recall that any <p 6 M+ has
a well defined support e = s(<p), e = e*, e2 = e, e 6 M, and that the modular automorphism group af is a
one parameter group of automorphisms of the reduced von Neumann algebra Me = {x & M; ex — xe — x].
We shall introduce the notation:
C)
trf(x) = <pil x ip-H Vx 6 Mc , V( 6 R
431

and we shall still use it for imaginary values of t and elements of M, for which <rf(z) exists by analytic
continuation. Consider now a monomial of the form:
where a,- 6 M, <р{ 6 M+ and z,- 6 C, Re z,- > 0.
We shall give to such a monomial the degree ? г, 6 С. This degree plays the same role as the degree of
densities in differential geometry.
When the degree of S is equal to 1 we define the integral or expectation value F) of S using the following
rules:
D)
,,..., г„)
for ф faithful, ay 6 A/, ? z* = 1, where the function -F is (cf. [Ar]) the unique holomorphic bounded function
in the tube {(z,) 6 C";Re z, > 0,?z; =1}, with boundary values given for ib<2, • •-,<n-i 6 R by
E) f (»ti, «2, • ¦ ¦, ««-1,1 - i E *,-) = V(ai < (аг) <+.Л°з) • • ¦)¦
In order to pass from Ho a monomial of the form D) one uses the following rule:
F)
<рг = (D<p :
where (Dip : DrpJ is the value at z of the Radon-Nikodym derivative (section 5). Note that this operator
is bounded if ip < rj/ and Re г < 1/2. It follows from [C03] and [Ho] that the above rules are consistent
and that the expectation value (S) is well denned, independently of the choice of ф, say with i>>T, ipi, and
Re z{ < j which can always be assumed. The outcome is that one can manipulate the above monomials, or
linear combinations of such monomials of the same degree a, exactly as densities of degree a in differential
geometry. The commutation rules are given by C) and F).
The expectation value F) satisfies:
G)
(at) = Fa) Va 6 M.
In particular a 1 density defines a normal linear form on M given by a 6 M —> (аи) 6 С.
There is a natural involution 6 —* 6*:
(8)
(<Ч ip\' a2 ^f,2 .. .а„ <p'n- an+i)" = a' + 1 <р*„" a'n...ip\' a\
which replaces the degree a by a and is antilinear. Let us consider the sesquilinear form on | densities given
by
(9) {«i.fc) = («i«;).
One shows that this sesquilinear form is positive and moreover that:
A0) {Si 62) = F2 61) V«!,«5 J densities.
It follows immediately that the Hilbert space L2(M) thus obtained is a normal bimodule for the action:
432
A1)
, SeL2(M).
Moreover one can show that any element of L7(M) admits a canonical left polar decomposition of the form:
A2) 6 = u<f>il'i
where и is a partial isometry with initial support u*u = s(v)> V 6 M.+ (cf. [Co] [Ho]).
The canonical involution J of ?2(M), given by.
A3)
JS = 6' V6 6 L2(M)
is isometric (by A0)) and it obviously exchanges the left and right actions of M on L2(M).
Finally L2(M) is endowed with a natural positive cone L2(M)+ whose elements are the i densities of the
form y1/2, <p € M+ (cf. [Co] for a characterisation of the self dual cones in Hilbert space obtained in this
manner).
Definition 2. Lei M be a von Neumann algebra. The identity correspondence between M and M is the
canonical bimodule L~(M) of 1/2 densities 071 M.
To describe this bimodule, one may also use an auxiliary faithful (semi finite normal) weight v. Then the
Hilbert space L2(M, is) (completion of {x 6 M,i/(x"x) < 00} with the obvious prehilbert space structure) is
naturally equipped with:
A normal * representation iru of M (by left multiplication).
An isometric antilinear involution Ju such that:
Jv л"„(А/) 3V = irv(M)' (comnmtant of irv(M)).
Then the equality ]Г„(х°) = Л, Ti/(x)* Jv defines a normal * representation of M" in L2(M, v) which hence
becomes an M — M bimodule.
The Hilbert space ?2(A/,i/) comes equipped also with a natural self dual cone L2(M,u) +, whose elements
are in bisection with the positive cone Л/.+ of the predual of M by:
^ € L2(M,v)+ — wf,f € Л/+ ([Ho] Lemma 2.10)
where ui(i( (x) = (т„(х) ?,?) Vi € M.
It follows immediately that there exists a unique unitary equivalence U of the bimodule L2(M,v) with L2(M)
such that:
A4)
vr-
Since L2(M,v) is the completion of Donii/2(''') = {x € M ; v(x*x) < 00}, we let )}„ be the canonical map
from Dom1/3(i') to L2(M,u). Then the isometry U allows to extend the notation y>1/2, 9 6 A/+ to weights
A5)
Vi 6 Dom,/2(i/).
433

/?) Correspondences and * homoinorphisms
Let M, N be von Neumann algebras. Let p be a normal * homomorphism of M in N. We do not assume
that p(l) — 1, then />A) = e is a projection, and the Hilbert space L2(p) = {? € L2(N) , ?e = f} is an
N — M bimodule with:
A6)
, x e M.
Proposition 3. Assume that JV is properly infinite.
a) Every correspondence H(*) between M and JV is equivalent to an L2(p).
b) The interiwinning operators from L2(pt) to L2(p?) are the elements у o
Рг(х) У = У Pi(x) Vх € M.
JV px(l) such that:
Proof, a) As JV is properly infinite, the representation xN of JV in H is subequivalent to the standard
representation of JV in L2(N). Thus we can assume that H = L2(N) e, where e is a projection, e g JV,
and that irN(y) ? = y(, Vy g JV , ? € L2(N) e. The comniutant of irw(JV) is then the algebra of right
multiplications in Z2( JV)e by elements of eJVe, so жд/о determines a normal * homomorphism p, p(l) = e of
M in JVe, such that:
*M'(x°) « = « />(*) VieM, V«€i2(/V)e.
b) With the obvious notations, the intertwiiming operators from ir^, to irj, correspond to the elements of
/J2(l) JV />i(l), and the intertwinning condition with respect to the action of M is exactly
у Pi(x) =/»(x) у Vx€M.
Q.E.D.
Corollary 4. Let в1,в2 € AutJV then ?2@,) is equivalent to L2(92) iff e{9\) = e(9i) in Out JV =
Aut JV/lnn JV.
If JV is not properly infinite, proposition 3 does not hold (in general there need not be any non zero *
homomorphism of M in JV while there is always the coarse correspondence between M and JV). This
however will not create any difficulty since, letting F be the factor of type I,» of all bounded operators in
B(N), the von Neumann algebra N = N <g> F is properly infinite and replacing JV by JV does not affect the
correspondences from M to JV. (Let И be a correspondence from M to JV, then И <g> B is in an obvious way
a correspondence from M to JV. Conversely, let e = 1 ® en g JV, where (eij)ij^n is the canonical system of
matrix units in F, then if 'H is a correspondence from M to JV the subspace e 7{ is a correspondence from
M to JVe = JV.)
Let 7{ be a correspondence from M to JV and Mi, Ni be von Neumann subalgebras of A/, JV. It is clear that
by restriction of the bimodule structure of И we obtain a correspondence from Mi to JVi. This operation
of restriction does not look so natural from the point of view 1), so even though 1) and 3) are equivalent
(proposition 3) it is important to keep both of them.
We shall now describe several examples of correspondences for which the associated homomorphism p is less
natural.
Examples 5. a) In the commutative case, M = i^f.Y, /ja'), JV = ?°°(У, ру) we can consider the coarse
correspondence which associates to each x € Л' an arbitrary point of Y. This means that we take the
measure fi — fix x ^y and the multiplicity function n(s,t) = 1 V(s,t) 6 X x Y.
( *) To avoid useless complications we shall assuir
that both M and N have separable predual, and that ft is separable.
434
This construction extends immediately to an arbitrary pair of von Neumann algebras M and JV. The coarse
correspondence is then the obvious bimodule of Hilbert Schmidt operators from L2(M) to L2(N):
A7)
T&C2{L\M),L2(N))
y{T)x = yTx Vy € JV , x € M , T € C2.
Equivalents one can use L2(M) ® L2(N) with:
yd i) Vx € M , у g JV.
b) Let Г be a countable group acting freely by non singular transformations of the measure space (X, fix),
then the restriction to L°°(X,fix) С M = L°°(X, fix) ХГ (the crossed product by Г) of the identity
correspondence of M is the graph in Л' x X, with its natural measure class, of the equivalence relation x ~ у
iff Bg бГ,р = у.
c) Let M — N >ч„ G be the crossed product of a von Neumann algebra JV by an action a of the locally
compact group G (cf. appendix A). Then every unitary representation ir of G in a Hilbert space ff, defines
canonically a correspondence from A/ to M as follows:
П, ~ L2(M) ® H, , the right action irwo(x°) , x g M is
A8)
the left action тгд/ of M is given, using M = N ><ia G, by:
, >? € я,
j) Coefficients of correspondences and completely positive maps
A correspondence from the von Neumann algebras M to JV is by definition a representation ir of the C-
algebra JV®max M° which is 6inorma/i.e. whose restrictions to both M° and JV are normal representations.
The coefficients of such representations, i.e. the functionals:
" € JV Omax M° -
€ С
€ m (
are exactly the iinormel states ([E-L]). We refer to [E-L] for the definition of the C*-algebra JV ®ы„ М°
corresponding to such states. We shall now relate these coefficients with completely positive normal maps
from M to JV. We let v be a faithful semifmite normal weight on JV and use the following:
Proposition 6. a) let П be a normal left- N-module. Then D(H,v) = {( € 4 ; 3c < oo , \\y?\\ <
с "(y'y)l/2 Vy € Dom,/2(i')} is dense m H-
b) The equality 1„(<р) = i/'2 <p i/~'l- , \/<p € Face(i/) =: {ip € M+ , <p < Xu for some A > 0} defines
a completely positive linear map Iv from the face of is in JV, to the domain of v in M. Its inverse x —*
i/1'5 x f1'2 Vx g Dom(i/) is also completely positive.
The proof is straightforward (cf. [Coi 11] [E-L]).
Definition 7. With N,7i,u as m a), a vector ?€H is called v-bounied iff t, € D{H,i>).
435

We can now state the precise relation between coefficients of correspondences and completely positive normal
maps.
Proposition 8. Let M, TV be von Neumann algebras, v a faithful semifinite normal weight on N.
1) Let 7i be a correspondence from M to TV, and ? a i/-bounded vector (definition 7). Then there exists a
unique completely positive map P from M to N such that for any x 6 M, у 6 TV, one has:
2) Let P be a completely positive normal map from M to N such that i/(P(l)) < oo, Then there exists a
unique pair (W^) where H. is a correspondence from M to N, ^6W and:
a) N?M is dense in H
j3) ? is a u-bounded vector and for any x 6 M and у € N, i/(y*y) < oo one has
Proof. Both statements are immediate from proposition 7 and the GNS construction for binomial states.
Corollary 9- // N is properly infinite and P : M —- N is a completely positive normal map, there exists a
normal * homomorphism p : M —+ N and a partial isomctry v 6 N, v*v < p(l), vv* = Support P(l) with
P{x) = P(lI'2 v p(x) »• ЯAI/2
It follows from proposition 3.
We shall introduce the following notation for the coefficients:
Notation 10. Let M, N, и and'H be as in proposition 8 1). Then for any u-bounded vectors ?i)?a 6 "H we
let (?i,?2)i> ^e ^l€ unique normal map P from M to N such that:
(y 6
P(x) v1'
V* € M , у 6 N.
For any a € M we have
(j-") = (?i,?2)v @1) and (^1,^2 a)i/ A) = ((.хЛъ)» (xa") for any x g M.
Lemma 11. a) Let be N be such that t — e-,"F) 6 N extends analytically from t € R to Imt 6 [0, i]. Then
for any $1 € D(H,v) one has b ?, € D(H,v) and (b b,C)r A) = ^jflKfi,?!), (x) Ух 6 M (and also
(fb*6M*) = (ei,fcM*) W/sW)' Vx€M).
b) Let V1 be another weight on N, with v1 > \v for some A > 0, then D{H, i>) С D(H, iS), the Radon-Nikodym
derivative (Dv1 : Dv)t € R extends analytically from i € R to Imt € [-5,0] and with b = (Di/ : Dv)_t/2
one has for any ?i,?> € DGi, v), x € M:
= *'(«!,6J,
Composition of correspondences
In the previous sections we related correspondences with the two classical notions 1) and 2) of morphisms
for von Neumann algebras. While for 1) and 2) the composition of morphisms is the obvious one, defining in
a canonical manner the composition of correspondences requires more care and.will be dealt with in detail
below to avoid any confusion.
Let Л'/i, N, Mo be three von Neumann algebras, "Hi a correspondence from Mi to Af and 7^2 a correspondence
from TV to M?. Thus И, is in particular a left. /V-nloclule and V.2 a right TV-module and we shall construct
canonically a tensor product:
436
A9)
П = «2 ®JV «1
which will be naturally a correspondence from Mi to Mi- The subtle point is that, while H will be defined
as any tensor product as the linear span of simple tensors satisfying simple algebraic relations, the basic
tensors generating И are not just of the form ?> ® ft, (j 6 W, but rather
B0)
ft ® «'ХП Ь , fj 6 W; , " 6 ЛГ+, 1/ faithful.
In fact it is quite useful to also allow v to be a semifinite faithful normal weight but this will introduce no
difficulty and will not change the tensor product.
Here we have three variables ft, ft, v and thus besides the obvious bilinearity of B0) in ft, ft, and the
simplification by TV:
B1) ft x®v-4- ft =ft® i/-1'^!/1'2 * "~'/2)ft . V* 6 TVnDom otf/J
(which uses of course our notation 3), we shall have the following relation which eliminates the choice of p:
B2)
ft
2 ft = ft
V2 «/-I/2)ft,
whenever one has и < t/ so that i/1'2 i/'2 = (Dv : Dv')^^ 6 TV. It is clear that relation B2) allows,
using 1/ = v\ + vi for Vj 6 TV,+ , to compare simple tensors B0) for different faithful Vj 6 TV+. The case of
weights will not be more difficult.
To proceed carefully we shall first of all fix v and construct a Hilbert space generated by the simple tensors
B0) and satisfying the relation B1). We shall then use B2) to eliminate the choice of v.
Let us define a sesquilinear form on the algebraic tensor product:
B3)
, v)
of tii by the dense subspace of (/-bounded vectors in Hi (definition 7) by the following equality:
B4)
where <pj € N+ are given by:
B5)
B6) <рг{У) = (ft У,П2)
We have used as a notation, ft ® i/~1/2 ft instead of ft ® ft.
Proposition 12. a) The equality B4) defines a positive sesquilinear form and the relation B1) holds in the
associated Hilbert space: "H^ ®v "Hi-
437

b) Let v < j/ be faithful semifinite normal weights on N then with b = (Dv : Dv')_l/2 the following map
у
defines an isomeiry Hi ®ui Hi
"Hi ®„ Hi:
A) =b®
, i/',).
Proof, a) The positivity follows from the complete positivity of the map /„ = i/'2 • i/~'/2 from JV» to N
(cf. proposition 6). To prove B1) one checks using B4) that for x 6 Dom <r^_if1 one has:
for any ?2,
b) follows from the equality Iv(<p) = b /„/(yN* proven in lemma 10 b) above. This shows that V is an
isometry and the faithfulness of v shows that it has dense range.
Theorem 13. 1) Let N be a von Neumann algebra and Hi a (normal) left module, Hi a (normal) right
module over N. There is a canonical Htlbert space H-= Hi ®nHi generated by the ?2 ® с'2 ?i/ {2 € Hi,
v € ЛГ + , faithful, and ?x € D(H,vx) and satisfying the relations B1), B2) and B4).
2) Let Hi (resp. Hi) be a correspondence from A/i to N (resp. from N to Mi) and H = Hi ®л/ Hi as in
1). Then the following equality defines a correspondence between Mt and M?:
Proof. 1) follows from proposition 12.
2) It is enough to check that if both Xj are uuitaries the formula 2) defines an isometry in 7i which is easy
using B4).
Remark 14. Using proposition 12 we see that ?2 <g> i//2 ^i is still well defined as an element of % when v
is a faithful semifinite normal weight and ?i ? D{%, u).
Definition 15. The correspondence "H.2®N %i from M\ to M% is the composition of the correspondences
We can easily compute the coefficients of the composition:
Proposition 16. Let'Hit'Hii'H = %'>®n'H\ as above: Let и be a faithful semifinite normal weight on N and
let i/2 be a faithful semifimte normal weight on Mi- Then for any ?i, r)i 6 DCHi, v) and &, rfe € D{'H2, t/2)
one has:
In other words the coefficients of the composition are obtained by composing the coefficients of the cor-
correspondences.
The proof of proposition 16 is immediate using B4).
Let us now relate the composition of correspondences with the composition of * homomorphisms. Thus we
let p\ : Mi —*¦ N and p-> : N —- Mi be * homomorphisms and L^(pi) the associated correspondences.
Proposition 17. The correspondence ?2(/?з) <3>n L2{p\) ls canonically equivalent to L2{p2 ° pi)-
438
It follows from a slightly more general fact: Let Hi be any correspondence from N to Af2. Then consider
the following M% — Mi bimodule:
B7) H = Hi /n(l) , фъвхЧК = *2 ? /ч(*,) V^j 6 Mj.
One then has:
Lemma 18. The above bimodule "Hi />i(l) is canonically equivalent with "Hi ®jv -^2(pi)-
Proof: One checks that the following formula gives the desired equivalence, recalling the construction
(section a) of L2(N). as half densities, so that i/'2 ? € N for any ^-bounded element of L2(N) viewed as
a left JV-module.
vF ® v-ir> ti) = b{"~1'2 6) V& € % , «i e ?>(Wi,i/).
Using the above propositions 4 and 17 one checks that the composition of correspondences is associative but
this can be of course done directly as in theorem 13 with a specific equivalence. We shall end this section
with the basic notion of contragredient of a correspondence.
Definition 19. Let H be a correspondence from M to N then its contragredient *H is the correspondence
from N to M given by:
x4 у = (у* ? x')- vseH , хем , ye n
with H the conjugate Hilbert space ofH.
We have used the canonical antilinear isometry ? —> ? from H to "K.
Example. Let p : M —> N be a * isomorphism, then one checks that L2(p) — i2(p~').
The meaning of L-{p) for » homomorphism is far less obvious.
Theorem 20. Let Hi be a correspondence from Mi to N, Hi a correspondence from N to M? then one has
a canonical equivalence of correspondences from Л/т to Mi:
(H2 ®jv Hi)' =
The meaning of the theorem is that in the
'/2 ?
Proof. Let v be a faithful normal semifinite weight on N. The meaning of the theorem is that in the
construction of Hi ®nV.2 of theorem 13 we could equally have used expressions of the form: {2 i>~'/2 ®?i>
{2 € D(H, 1/), ? 1 € Hi and defined the inner product using the following analogue of B4):
B8) F v-in®(i , >?2 v
with ipj e N. given by B5) B6).
One diecks indeed that the following map defines the desired unitary equivalence:
?¦) Correspondences, hyperflniteness and property T
In this section we specialize to factors N of type Hi and exploit the following analogy between correspon-
correspondences from N to N and representations of discrete groups. First, since such correspondences are exactly
439

binormal representations of the C*-algebra N ®max № all classical notions of representation theory are
available such as:
Unitary equivalence, Irreducibility, Intertwinning operators ...
We also have a natural topology on any set of equivalence classes of correspondences from N to N using, as
in representation theory (cf. [Di3]), the coefficients to define it.
Finally by section 6) we have the analogue of the tensor product of representations, but of course there is a
fundamental difference since the product %i ®N 7i2 is no longer commutative (and OutJV is not commutative
in general).
The identity correspondence is a unit for this product and is the analogue of the trivial representation of
discrete groups. The coarse correspondence from N to N (cf. example 5 a)) has the same absorbing property
for the product as the regular representation of discrete groups and is the analogue of the latter in general.
With this small dictionary at hand we can now translate two key notions of the representation theory of
discrete groups: amenability and property T of Kazhdan and see what they give:
Recall that a (discrete) group Г is amenable iff the trivial representation is weakly contained in the regular
representation. Here we have:
Theorem 21. Lei N be a factor of type lip Then the identity correspondence is weakly contained in the
coarse one iff N is hyperjinite.
Proof. First recall (example 5 c)) that if Я(Г) = N is the von Neumann algebra of the left regular
representation of a discrete group Г we have a natural functor from representations of Г to correspondences
from N to N. Since this functor is obviously continuous it is enough to write R = Я(Г) for some amenable
discrete group Г with infinite conjugacy classes to check the property for the hyperfinite factor. Conversely if
the identity correspondence of N is weakly contained in the coarse one it is immediate that N is semidiscrete
using theorem 7.3. Thus the conclusion follows from theorem 7.6.
Next, recall that a (discrete) group Г has Kaahdan's property T iff the trivial representation is isolated.
Theorem 22. Lei Г be a discrete gronp with infinite conjugacy classes. Then Г has property T iff the
identity correspondence of the Hi factor Й(Г) is isolated.
Using the above functor it is trivial that if Г does not have property T then the identity correspondence of
-R(F) is not isolated. Conversely, if Г has property T one shows that the identity correspondence is isolated
using the following functor from correspondences (from Й(Г) to Я(Г)) to representations of Г:
To the /J(r)-bimodule % one assigns the unitary representation it of Г in И given by:
»(»)« = j«9-' v9er, f en.
One checks easily that x contains the trivial representation of Г iff "H contains the identity correspondence.
This gives the desired result.
The great interest of the "property 7™ factors of theorem 23 is that their outer automorphism group Out(JV)
is countable as well as their fundamental group F(N), ([Cos]) leading to the first example F(N) ф R+.
The factors with property T enjoy remarkable rigidity properties. For example, there does not exist a
nontrivial sequence Nn С Nn + l of subfactors with UNn dense in N. The main problem is as follows:
Problem 1. Show that ifTi and 12 are not isomorj>hic, then /2A*1) and
are not isomorphic.
Of course, here Г\ and Г 2 are rigid in the sense defined above (for amenable groups, the situation if radically
opposite; see section 9). The only results known are 1) that if I\ is rigid m the above sense and if Г 2 is
discrete in S?B,R), then there does not exist any hoinomorphism of fi(Ti) into Я(Гг), 2) that one can
distinguish countably many discrete subgroups Г of Sp(n,R) by their associated factors Й(Г) ([H8])
Problem 2. Determine the fundamental group of Й(Г) for Г rigid.
440
CHAPTER VI
THE METRIC ASPECT OF NONCOMMUTATIVE GEOMETRY
In the present chapter, we shall begin a systematic investigation of the metric properties of noil commutative
spaces. We shall first show that the basic data of the geometry of Gauss and Riemann can be reformulated
in operator-theoretic terms. This will be done using an elliptic differential operator of order one, the Dirac
operator. We shall see how this self-adjoint, operator D acting in the Hilbert space # (of L2 spinors),
together with the representation in f) of the *-algebra of functions on the manifold, readily yields the metric
of geodesic distance:
''(P.'/) = infimum of the length of paths 7 from p to q. (*)
The geometric spaces of Gauss and Riemann are defined as manifolds in which the metric is given by the
formula (*) and the length of a path is computed as the integral of the square root of a quadratic form in
the differential of the path:
length of 7
= f
(**)
Fig. 1. Riemanuiaii distance
441

These geometric spaces form a relevant class of metric spaces, inasmuch as:
1) They are genera] enough to include numerous examples. (Vom non-Euclidean geometries, surfaces em-
embedded in R3, to spacelike hyporsurfaces in general relativity.
2) They are special enough to deserve the name of geometry since, being determined by local data, all of
the tools of differential and integral calculus may be brought to bear on them.
In our context, it will be rather clear from the outset that our basic data
3. Product of continuum by discrete and the symmetry breaking mechanism.
4. The commutant and Poincare duality.
5. The standard U{\) x SUB) x SU{Z) model.
consisting of a Hilbert space #, an involutive algebra A of operators in # and a self-adjoint unbounded
operator D in fy, will satisfy satisfy 1). In fact, we can already give the following list of new geometric spaces
covered by such data, in addition to Riemannian manifolds:
a) Discrete spaces.
Q) Spaces of non-integral Hausdorff dimension.
y) Group rings of discrete subgroups of Lie groups.
S) Configuration space in supersymmetric quantum field theory.
We already discussed j3) in section 4 of chapter 4 and -)) 6) in section 8 of chapter 4.
Thus, our main task will be to show that the new class of spaces still deserves the name of geometry. For
this, we shall replace the tools of the differential and integral calculus by the operator-theoretic tools of
the quantized calculus developped in chapter IV. This will allow us to understand the notion of dimension
of a space and to develop the analogue, of the Yang-Mills union in the case of dimension 4, in which we
shall prove a central result: the inequality bet-ween the second ('lieni number of a /\"-theory class and the
minimum of the Yang-Mills action.
The notion of manifold itself will be reached only after an understanding of Poincare duality, i.e., expressing
that the Л'-homology cycle defined by the Fredholm representation (#, O) yields the fundamental class of the
space under consideration. The obtained notion of manifold is directly inspired by the work of D. Sullivan
[SU7] who discovered the basic role played by the A'-houiology fundamental class of a manifold.
The main example of a space to which all of these considerations will be applied is the Euclidean space-time
of pbysics, i.e., space-time but with imaginary time. What we shall give is a geometric interpretation of the
now experimentally confirmed low-energy effective model of particle physics, namely the Glashow-Weinberg-
Salam standard model. This model is a gauge theory model with group ?7A) x SU{2) x SU(S) and a pair of
complex Iliggs fields so as to provide masses by the symmetry-breaking mechanism. We interpret this model
geometrically as a pure gauge theory, like electrodynamics, bm. on a more elaborate space-time E' = E x F,
the product of ordinary Euclidean space-time by a finite space F. The geometry of this finite space is
specified by a pair (#,?)) as above, where f) is finite-dimensional and the self-adjoint operator D encodes
the nine fermion masses and the four Kobayashi-Maskawa mixing parameters of the standard model.
The values of the hypercharges do not have to be fitted artificially to their experimental values but come
out right from a simple unimodularity condition on the space ?".
Our analysis is limited to the classical context and does not at the moment address the questions related to
renormalization, such as the existence of relations between coupling constants or the naturalness problem.
Nevertheless, our more geometric and conceptual interpretation of the standard model gives a clear indication
that particle physics is not so much a long list of elementary particles as it is the revelation of the fine
geometric structure of space-time.
The content of this chapter is organized as follows:
1. Riemannian geometry and the Dirac operator.
2. Positivity in Ilochschild cohomology and inequalities for the Yang Mills action.
ЛУ1
443

VI.l. Riemaunian manifolds and the Dirac operator
Let M be a compact Riemaunian spin manifold and let D = d\t be the corresponding Dirac operator
(cf. [Gii]). Thus, D is an unbounded self-adjoint operator acting in the Hilbert space Й of L2 spinors on the
manifold M.
We shall give four formulas below that show how to reconstruct the mefric space (M,d), where d is the
geodesic distance, the volume measure dv on M, the space of gauge potentials, and, finally, the Yang-Mills
action functional, from the purely operator-theoretic data
(Aft, D),
where D is the Dirac operator in the Hilbert. space !) and weie A is the abelian von Neumann algebra of
multiplication by bounded measurable functions on Л/.
Thus, A is an abelian von Neumann algebra on S), and knowing the pair (Jrj, A) yields essentially no in-
information (cf. Chapter 5) except for the multiplicity, which is here the constant 2^dl^, where d = dimM.
Similarly, the mere knowledge of the operator D in f) is equivalent to giving its list of eigenvalues (An)neN,
An € R, and is an impractical point of departure for reconstructing Л/. The growth of these eigenvalues, i.e.,
the behavior of |An| as vi —. oo, is again governed by the dimension (t of M, namely, |An| ~ Сп1/Л as n —• со.
What is relevant is the triple (A ft, D). Elements of A oilier than the constants do not commute with D, and
the boundedness of the commutator [D, a] already implies I lie following regularity condition on a measurable
function a:
Lemma 1. If a is a hounded measurable. Junction on M. Ihr n I/it densely defined operator [D, a] is bounded
if and only if a is almost everywhere equal to a Lipscliil: function f, \f(p) — f(q)\ < Cd(p,q) (Vp, q 6 M).
Here, d is the geodesic distance in Л/. The operator [D.a] should be viewed in effect as a quadratic form
(,4 "(<¦(, D,t)-{/)(, a',,),
which is well-defined for f,7) in the domain of D; its hoiiudedness signifies an inequality of the form
\{a(,nV) - (Di,a-,i)\ < c\\?\\\\ii\\ (V?,4 6 dom?>).
The proof of the lemma follows immediately from [IV].
Now, every Lipschit.z function on Л/ is continuous and the algebra о I" Lipschitz functions is norm-dense in
the algebra of continuous functions on M; it follows thai the (""-algebra C(M) of continuous functions on
M is identical to the norm closure A in C(t>) of A = {a ? A; [D.a] is bounded}.
By Gelfand's theorem (Chapter 2), we can recover the compact topological space M as the spectrum of A.
Thus, a point p of Ы is a «-homomorphism p : Л — С,
p(,,h) = i>(a)p{b) (Vn, he Л).
Any such lioniomorphism p is given by evaluation of it al i> for some point ;> € M,
p(e) = aii,) € C.
All of this is still qualitative,- we now come to the first interesting formula, giving us a natural distance
function, which turns out in this case to be the geodesic distance:
Formula 1. For any pair of points p,q 6 ,V/, the geodesic distance between them is given by the formula
d(P,'/) = sup{|«(/>)-nG)l; «€ A ||[D,«]|| < 1).
'Ill
The proof is straightforward, but it is relevant to go through it to see what is involved. The operator [?>,а],
which (Lemma 1) is bounded if and only if a is Lipschitz, is given by the Clifford multiplication t'7(da) by
the gradient da of a. This gradient is ([Fe]) a bounded measurable section of the cotangent bundle T"M
of M, and we have
||[?>,a]|| = esssup||da|| = the Lipschitz norm of a.
It follows at once that the right-hand side of Formula 1 is less than or equal to the geodesic distance d(p,q).
However, fixing the point p and considering the function r;(r/) = <l(q,p), one checks that a is Lipschitz with
constant 1, so that ||[I>,a]|| < 1, which yields the desired equality. Note that Formula 1 is in essence dual to
the original formula
d(p,q) = infimum of the length of paths 7 from p to q, (*)
in the sense that, instead of involving arcs, namely copies of R inside the manifold M, it involves functions a,
that is, mappings from M to R (or to C).
This is an essential point for us since in the case of discrete spaces or of noncommutative spaces X, there are
no interesting arcs in X but there are plenty of functions, namely, the elements a € A of the defining algebra.
We note at once that the right-hand side of Formula 1 is meaningful in that general context and it defines
a metric on the space of states of the C"-algebra A, the norm closure of A = {a € A [D, a] is bounded}:
Finally, we also note that, although both Formula 1 and the formula (*) give the same result for Riemannian
manifolds, they are of quite different nature if we try to use them in actual measurements of distances. The
formula (*) uses the idealized notion of a path, and quantum mechanics teaches us that there is nothing like
'the path followed by a particle'. Thus, for measurements of very small distances, it is more natural to use
wave functions and Formula 1.
We have now recovered from our original data {A,Si,D) tin; metric space (M,d), where d is the geodesic
distance. However, we still need tools of Riemannian geometry that, are not implied by the metric structure,
the first obvious example being the measure given by the volume form
jm /d|1'
where, in local coordinates i',j,i,, we have
This brings us to our second formula, which is nothing more tlmn a restatement of H. Weyl's theorem about
the asymptotic behavior of elliptic differential operators ([Gu], [Gii]). It does, however, involve a new tool,
the Dixmier trace Тг„ (cf. chapter IV.2), which, unlike asymptotic expansions, makes sense in full generality
in our context and is the correct operator-theoretic substitute for integration:
Formula 2. For every f <= A, we have /„ fdv = c(d) Тг„(/|О|"<'), when d = dim M, c(d) =
(By convention we let D~' be equal t.o 0 on the finite dimensional snbspace KerD.)
Let us refer to chapter IV.2 for the detailed definition and properties of the Dixmier trace Тгш. We can
interpret the right-hand side of the equality as the limit, of the sequence
Ф15

where the A;- are the eigenvalues of the compact operator JD~d, or equivalently, as the residue, at the point
s = 1, of the function
(,"(*) = Trace(/XT''") (lies > 1).
The crucial fact for us is that the Dixmier trace makes sense independently of the context of pseudo-
differential operators and that all properties of the integral jAl /do, such as positivity, finiteness, covariance
etc., become obvious corollaries of the general properties of t,he Dixmier trace:
A) Positivity: Tr^T) > 0 if T is a positive operator.
B) Finiteness: Tr^T) < со if the eigenvalues of |T| satisfy ?" /(„(T) = O(log TV).
C) Covariance: Tra(UTW) = Тгш(Т) for every unitary U.
D) Vanishing: Ттш(Т) - 0 if T is of trace class.
Property D is the counterpart of locality in our framework,- it. shows that the Dixmier trace of an operator
is unaffected by a finite-rank perturbation, and allows many identities to hold as we shall see in Section 2.
Now, getting the integral of functions, i.e., the Riemaiininu volume form, is a good indication but quite far
from the full story. In particular, many distinct Rieiuauniiiii metrics yield the same volume form. Since our
aim is to investigate physical space-time at the level of elementary particle physics, we shall now make a
deliberate choice: instead of focusing on the intrinsic Riemannian curvature, which would drive us toward
general relativity, we shall concentrate on the measurement (using (fj,D)) of the curvature of connections
on vector bundles, and on the Yang-Mills functional, which takes us to the theory of matter fields. This line
is of course easier since it does not involve derivatives of the (/„„.
Let us then clearly state our aim: it is to recover the Yang-Mills functional on connections on vector bundles,
making use of only the following data:
Definition 2. A A'-eycle (f),D), over an algebra A icilh inrolniion *, consists of a representation of A
on a Hilbert space fy-iogcther with an uvlwunded self-adjoin I operator D with compact resolvent, such that
[Dt a] is bounded for every a 6 A.
If the eigenvalues \n of \D\ are of the order of n1^ as n —¦ oo, we say that the A'-cycle is (d,oo)-summable
(cf. Section 2 of chapter 4). On the algebra of functions on a compact Riemannian spin manifold, the Dirac
operator determines a A'-cycle that is (rf, oo)-summable, where (/ = dim Л/. Finer regularity of functions, such
as infinite differentiability, is easily expressed using the domain of powers of the derivation 6, 6(o) — [\D\, я].
We shall not he too specific about the choice of regularity; our discussion applies to any degree of regularity
higher than Lipschitz.
The value of the following construction is that it will also apply when the *-algebra A is noncommutative, or
when D is no longer the Dirac. operator (cf. Section 3). The reader can have in niiud either the Riemannian
case or the slightly more involved case in which the algebra .4 is the «-algebra of matrices of functions on a
Riemannian manifold, provided one bears in mind that, the notion of exterior product no longer makes sense
over such an algebra.
We shall begin with the notion of connection on the trivial bundle, i.e., the case of 'electrodynamics', and we
shall first define vector potentials and the Yang-Mills action in that case. We shall then treat the general case
of arbitrary hermitian bundles, i.e., in algebraic terms, of arbitrary hcrmitian, finitely generated projective
modules over A.
We wish to define jb-forms over A as operators in Jo of the form
where the a?i are elements of A represented as operators on fl. This idea arises because, although the
operator D fails to be invariant under the representation on Й of the unitary group U of A,
446
U = {и € А; и'и = гш" = 1},
the following equality shows that the failure of invariance is governed by a 1-form in the above sense-
ш„ = u[D,u*], that is,
Note that ш„ is self-adjoint as an operator in Я. Thus it is natural to make the following definition:
Definition 3. A vector potential V is a self-adjoint element of the space of \-forms 53 "otA а{]> where
4 € Л.
One can immediately check that in the basic example of the Dirac operator on a spin Riemannian manifold,
a vector potential in the above sense is exactly a 1-form v on the manifold M and that this form is imaginary,
the corresponding operator in the space of spinors being given by the Clifford multiplication:
The action of the unitary group // on vector potentials is such that it replaces the operator D + V by
u(D + V)u', thus it is given by the algebraic formula
yu(V) = u[D.u"]+ uVu' (u €//).
We now need only define the curvature or field strength 0 for a vector potential, and use the analogue of the
above Formula 2 to integrate the square of 9: the formula
YM(V') = Tru.(fl2|?»|-1)
should give us the Yang-Mills action.
The formula for в should be of the form 0 = dV' + V2; the only difficulty is in defining properly the 'differential'
dV of a vector potential, as an operator in ft.
Let us examine what happens; the naive formula is:
If V = ^»ip,«{] then ,1V = Y}
Before we point out what the difficulty is, let ns check that if we replace V by yu(V), where
yu(V) = «[?», ri'] + Y, "<'i[D,a{]u-,
then the curvature is transformed covariautly:
ju{V)- = u(iH' + V-)u'.
As this computation is instructive, we shall cany it out in d<-iail. First, in order to write ju(V) in the same
form as V, we use the equality
[D,a\]u- = [D,a\u"} - rrj[D,u'].
Thus, 7«(V) = u[Au'] + ?««o[Aa>']-?IinneJi[A""]. and we have
dj,.(V) = [D, »][?>, «*] + ]Г[?>, иа{,][О, a\ „'} - ^[D, ucPoa[][DУ].
We now claim that the following operators in f) are indeed equal:
a) dTu(V) + T«(VJ,
0) u(dV + V2)u\
447

For, the operator a) is equal to
(V) + (u[?J,u*] + uVu'J
= d7u(V) + u[?J,u*]«[?J, u*] + u[D, u'Jul'w" + uVu*u[D,
= d7u(V)-[D,«][D,u']-[D, »]Vu" + uV[D, u"] + uV2
= ^[D, uaj][?J,o{u*] - ^[D,«4a{][D, «*] - [D, u] W
, u
where the last equality follows from
, u*] = udVu',
The difficulty that we overlooked is the following: the sanir vector potential V might be written in several
ways as V = 53 ao[?*> ai]> so "lat "le definition of dV as
»aJ0[D,a{][D, и"] = «V[D, «•].
is ambiguous.
To understand the nature of the problem, let us introduce sonic algebraic notation. We let il'A be the
universal differential graded algebra over A. It is by definition ucjtial to A in degree 0 and is generated by
symbols do (« 6 A) of degree 1 with the following prcsenuiliou:
a) d(o6) = (doN + nd6 (Vo,4 € A),
/3) dl = 0.
One can check that п'A is isomorphic as an .4-bimodnle to the kernel Ker(m) of the multiplication mapping
m : A ® A —¦ A, the isomorphism being given by the mapping
53о,- C3> 6i € Ker(m) — ^о>A6; € П'Л
The involution * of A extends uniquely to an involution on Q' with the rule
7) (do)' = -do".
The differential d on п'А is defined unambiguously by
d{a°dal . ..do") = doM«l . . .do" (Va> € A),
and it satisfies the relations
E) d2u = 0 (Vw € ПМ),
?) d(u!U2) = (dui)ui2 + (-1)д"'ш^1ш2 (Vuj € WA).
Proposition 4. A) T/ie following equality defines a t-npnsciilalion л oftke universal algebra U"(A) on f):
!r(a°d«1da") = o°[?J o1] [»«"] (V
B) Ie( Jo = Kerjr С П* ''e </ie gmdeil Iwo-sulcd ideal of 0' jincn 6j( /q'' = {ш е П4, ?г(ш) = 0}; (uen
J = Jo + dJo w » graded differential Iwo-sulcd ideal ofu'{A).
The first statement is obvious; let us discuss the second. By construction, Jo is a two-sided ideal but it is
not in general a differential ideal, i.e., if w 6 &(A) and ?r(t*j) = 0, one does not in general have 7r(dw) = 0.
4'I8
This is exactly the reason why the above definition of ?[?),аУ[?>,а'1] as the differential i
ambiguous.
Let us show however that J = Jo + dJo is still a two-sided ideal. Since d2 = 0 it is obvious that J is then a
differential ideal. Let ш 6 J^' be a homogeneous element of J; then ш is of the form ш = u/\ + dwj, where
u, g Jo nfi', ш-г € Jo nfl'-l. Let ш' € пк' and let us show that uiui' € J<'+''). We have
However, the first term belongs to Jo П f)'+t' and w-ja/ 6 Jo П П4"*'.
Using B) of Proposition 4, we can now introduce the graded differential algebra
Let us first investigate ?1%, Q\) and Од.
We have /flQ° = Joflfl0 = {0} provided that we assume, as we shall, that A is a subaigebra of ?(i^) Thus
п%=А.
Next, J nu1 - Jo ПИ1 +d(Jonn°) = Jonn\ thus П]г, is the quotient of Я1 by the kernel of -я, and it is
thus exactly the ,4-biniodule tt(SI) of operators ш of the form
Finally, J Пп2 = Jo П П2 + d( Jo П fI) and the representation ?r gives an isomorphism
More precisely, this means that we can view an element ш of Qj, as a class of elements p of the form
p = ? a,[0. «]][?>,«;] D 6 Л)
modulo the sub-bimoclule of elements of the form
Po = 53[d, 6°][o, 6j] nj e >t. 53 6i'[°. *Л = °)-
It is now clear that since we work modulo this siibspace jr(d(Ju nfi1)), the question of ambiguity in the
definition of du for ш 6 "'(П') no longer arises.
The equality (*) makes sense for all fc.
and allows us to define the following inner product on Qj,: for each it let JJj. be the Ililbert space completion
of jr(fi') with the inner product
(Ti,T-,)t = Тг,ДГ,-Г,|ОГ') {Щ € *(П*)).
Let P be the orthogonal projection of фд. onto the orthogonal complement of the subspace Tr{d{JoC\Qk~1)).
By construction, the inner product (Ри,,^) = (Л],^п) for uij 6 т(Й*) depends only on their class
in Пд. We denote by Л* the Ililbert space completion of Пд for (.his inner product; it is, of course, equal
toPflfe.
Proposition 5. A) The actions of Л on Ak by lefl and right muliiplicaiion define commuting unitary
representations of Л on Ak.
B) The functional YM(V) = {dV + V~,dV -t- V~) is рояШге* qiiartic and invariant under gauge transfor-
transformations,

y,,(V) = udtT + uVu" (Уи в U(А)).
C) The functional I(a) = Тг„@э|О|~"*), 0 = ;г(с1а¦ + a1) is positive, quartic and gauge invariant on {a g
П'(Л), а = а*}.
D) One has YM(V) = inf {/(<*); тг(а) = V].
Let us say a few words about the easy proof. First the left and right actions of A on #t are unitary. The
unitary of the right action of A follows from the equality Ttw(Ta\D\~4) = Tiu,(aT\D\-lt) VT € C(H),
a € A. Since ?r(d( Ja ПП')) С 1г(П4) is asub-bimodule of <r(S2') it. follows that/" is a bimodule morphism:
P(a{b) = aP{t)b (V«,4 € A € € Sik).
Thus A) follows. As for B), one merely notes that, by the above calculation, with dV now unambiguous,
9 ~ dV + V2 is covariant under gauge transformations, whence the result. For C), one again uses the above
calculation to show that dor + a~ transforms covariantly under gauge transformations.
Finally, D) follows from the property of the orthogonal projection P: as an element of Л2, dV + V2 is equal
to P(b(da + or)) for any n with гг(а) = V, and since the ambiguity in jr(dn) is exactly 7r(d(,7o Л П1)) one
gets D).
Stated in simpler terms, the meaning of Proposition 5 is I.hat the ambiguity that we met above in the
definition of the operator curvature 0 = til' 4- V~ can be ignored by taking the infimum
YM(l') = infTU<?-|D|-')
over all possibilities for в — dV + K", dV =
nevertheless quartic by B).
,а^] bring ambiguous. The action obtained is
We shall now check that in the case of Rieniannian nramfoUls with the Dirac /C-cycle, the graded differential
algebra Qp is canonically isomorphic to the de Ilham alytbm of ordinary forms on M with their canonical
pre-Ililberi space structure- The whole point is that Proposition 5 gives us these concepts in far greater
generality and the formula in D) will allow extending to this generality, in the case d = 4, the inequality
between the t.opological action and the Yang-Mills action YM (cf. Section 2). (We refer the skeptical reader
to the examples of Section 3.)
We now specialize to the Riemaiinian case, where A is the algebra of functions (with some regularity) on the
compact spin manifold A/, and D = дм is the Dirac operator in the flilbert space L~(M,S) of spinors. We
let С be the bundle over Л/ whose fiber at. cacli p 6 M is the complexified Clifford algebra Cliffc(Tp (M)) of
the cotangent space at p 6 M. Any hounded measurable section p of С defines a bounded operator f(p) on
Я = L2(M,S). For any /°,...,/" € A one has
^/"d/1 .. .d/") = i-'-H/M/1 d/- ¦ ... ¦ d/"),
where the usual differential d/ is regarded as a section of C, and ¦ denotes the product in C.
For each p 6 M, the Clifford algebra C,, has a Z/2 grading given by the parity of the number of terms ?j,
?j € T*(M) in a product ?i ¦ ?•> ¦ ... ¦ ?n, and a nitration, where1 Cj, is the subspace spanned by products
of n < к elements of T*{M). The associated graded algebra is canonically isomorphic to the complexified
exterior algebra Дс (Tp*(M)) and we shall write <Tt . CUI — AcO'p) f°r 'he quotient mapping.
Using the canonical inner product on С given by the trace in the spinor representation, one can also identify
Дс with the orthogonal complement of Otk~1} in C^'1; eqnivalently, if we let Ck be the subspace of C^of
elements of the same parity as fc, then д? = С1' в Ск~~.
The differential algebra Q'D is determined by the following lemma:
¦150
Lemma 6. Let (fl, D) be the Dirac K-cycle on the algebra A of functions on M and let к g N. Then, a
pairTi, T2 of operators in S) is of the form Ti = я-(ш), Т, = n(du) for some ш 6 uk(A) if and only if there
exist sections pi, p-> of Ck and Ct+1 sucA that
Tj = i(Pi) (} = 1,2), id<7k(Pl) = <rt+x(n).
Here o-i(p1) is an ordinary fc-form on M and d is the ordinary differential. Note that for к > d = dimM
one has o-t(p) = 0. The proof is straightforward.
We can now easily determine the graded differential algebra П~о. First, let us identify я-(П') with the space
of sections of C*; Lemma б then shows that
jr(d(Jonnt-1)) = Kerffj...
(If pis a section of C* with trt(p) — 0 then the pair of sections px = 0, p? = p of C* and C* fulfills
the condition of Lemma 6, so that p — ir(dtj) for some ш, т(ш) = 0.) Thus <rt is an isomorphism Яд 2
sections of Лс(^*)' wh'0^ again by Lemma 6, commutes with the differential. We can then state:
Formula 3. The mapping a°dal ... da" —> !na°d,_al ...dta" for a? € A extends to a canonical isomorphism
of the differential graded algebra il"D with the de Rltam algebra of differential forms on M. Under this
isomorphism, the inner product on Q.kD is itie Riejnanviav inner product of k-forms:
Jm
Л *u/.
The last equality follows from the computation of the Dixmier trace for the operator in fi — L?{M,S)
associated with a section p of the bundle С of Clifford algebras (cf. Section 3):
TUHDr') = I trncc(p(,-))cli.(p).
Jm
As an immediate corollary of Formula 3, we have
YM(V)= /"||dr||2dt>.
Let us now extend the definition of the action YM to connections on arbitrary liermitian vector bundles.
First of all, we need to express in algebraic terms — i.e., using only the involutive algebra A — the concept
of hermitian vector bundle over M. A vector bundle E is entirely characterized by the right Л-module E
of sections of E (having the same regularity as the elements of .4); the local triviality of E and the finite-
dimensionality of its fibers translate algebraically into the statement that ? is a direct summand of a free
module AN for some finite N, or, in fancier terms, that ? is a finitely generated projective module over A.
The hermitian structure on E, that is, the inner product {?, i;)p on each fiber Ep, permits constructing a
sesquilinear mapping
( , ) :? xf. — A,
given by {?,T))(p) = {Z(p),'l(l>))p- The mapping ( , ) satisfies the following conditions:
1) (?а,п4) = а'((,т))Ь (V?,n € ?, a,b e A),
2) ((,()> 0 ОЛ;6Л,
3) ? is self-dual for ( , ).
Thus, the liermitian vector bundles over M correspond bijectively to the hermitian, finitely generated pro-
projective modules over A in the following sense:
451

Definition 7. Let A be a *-algebra until unity and hi ? be a finitely generated projective module over A.
A hermitian structure on ? is given by a .scsqnilinear mapping ( , ) : ? x ? —* A satisfying the above
conditions 1), 2) and 3).
We shall use this concept only in the case where .4 is a subalgehra stable under the holomorphic functional
calculus in a C*-aIgebra, in which case all reasonable notions of positivity coincide in A.
In this case, all hermitian structures on a given finitely generated projective module ? over A are isomorphic
to each other and are thus obtained as follows: one writes ? as a direct summand ? = eAN of a free module
?0 ^= AN, where the idempotent e 6 Mj^(A) is self-adjoint, and one then restricts to ? the hermitian
structure on .4" given by
The algebra Endj»(?) of endomorphisms of a hermitian, finitely generated projective module ? has a natural
involution, given by
With this involution, End^(i') is isomorphic to the reduced *-nlgebra еМц{А)е.
As above, we now let (Д,О) be a /\'-cyde over A, and ft), I he ^4-himodule of operators in Л of the form
V = ?>i[AM. °.-'6> € A
Definition 8. Let ? he a hermitian, finitely (jencnilul ptojeilmc module over A. A connection on ? is
given by a linear mapping V : ? — ? О л ^1> sncli that
+ (, О ila (V? E ?. a E A).
A connection V is compatible (with Hie metric) if and only if
<?¦ v>,) - (Щ,r,) = dtf,,,) (V?,ч е <?)¦
The last equality lias a clear moaning in Q)}. In the computations, one should remember that (do)* = —da*
(Vae.4), and if Vf = ? fr о c*, w; € 0]j, then (V?.i,) = E-7 №.'/>¦
Such connections always exist; for S expressed as r.A1** n.s ;iI>ovl\ one may take V as follows:
= eij, whore щ = d^.
Two compatible connections V and V on ? differ by an clement Г 6 Иотд(?,? ®д Dj,).
As in Proposition 5, we shall now give two equivalent definitions of the action functional YM(V) on the
affine space C{?) of compatible connections.
The group?/(?) of unitary automorphisms of ?, U(?) = {» € У,нАд{?); «и* = "*•' = 1}, acts by conjugation
-yu(V) = uVu" on the space C(?). To dvfii»? the curvature 0 of a connection V, one first extends V to a
unique linear mapping Vfroni ? to ?, ? = ? 'Г'л OJ,, such tlial
e?, ue n'D),
and one diecks that this mapping satisfies
for every homogeneous rj (z ? and и 6 Qq. It then follows that 0 = V2 is an endomorphism of the right
Q*}-module ?; it is determined by its restriction to ?, agciiu deuotod 0:
в е 11атл(?,?®лП2о).
Next, using the inner product on Qf, and the hermitian structure on ?, one gets a natural inner product on
Using this, we make the following definition.
Definition 9. YM(V) = {в, в).
By construction, this action is gauge invariant, positive and quartic. It is moreover obvious from the above
Formula 3 in the case of. the Dirac Л'-cycle on a Riemannian spin manifold that one has:
Formula 4. Let M be a Riemannian spin manifold with Us Dirac K-cycle (fi,D). Then, ihe notion of
connection (Def. 8) is the usual one, and
= J
where в is the usual curvature o/V.
Thus, we recover in this case the usual Yang-Mills action. For computational purposes, and also to see the
curvature as an operator in fj, we shall now mention the, easy adaptation of Proposition 5, D) to the general
case.
First of all, any compatible connection in the sense of Definition 8 is the composition with ж of a universal
compatible connection, i.e., a linear mapping fulfilling precisely the conditions of Definition 8:
V :?— ^©дП1.
To see the surjectivity of the mapping ?r : CC(?) —» C(?), where CC{?) is the space of universal compatible
connections, it is enough to check that the special 'Grassinannian' connection Vo is of this form and that
7Г is asurjection of Q' onto ?г(П1). Next (cf. [ ]), a universal compatible connection extends uniquely as a
linear mapping
V : ? ®a П* — ? ®A П*
such that V is equal to V on ? (g> 1 and such that
for every homogeneous ?/ in ? ®д П* and u» € П*.
The curvature в = V2 is then an endomorphism of the induced module ? = ? ®a П* over П*, and ж(в)
makes sense as a bounded operator in the llilhert space ? QA >J, as does the following operator Dv-'
the analogue of the action / of Proposition 5, A) is then given by
One then proves in the same way that, for a given compatible connection V 6 C(?),
); tt(Vi) = V}.
453

VI.2. Positivity in Hocliscliild cohomology and inequalities for the Yang-Mills action
The residue theorem of section IV.2 is a basic result that, allows one to express the Ilochscliild cohomology
class of an (n, oo)-summable Л'-cycle by means of the Dixmier trace of suitable products of commutators.
However, the Dixmier trace enjoys the fundamental property of being positive:
тг„(Г)>о (чт е с^-^Нъ), т>0).
This shows that the Hochschild class of an (n,oo)-summable Л'-cycle (Д, D,f) in fact lias a positive repre-
representative, given by the following positive Hocliscliild cocycle:
фи(а° a") = Tru,((l + 7)a°[?J,«1]. .. [D, «"]?)-") (Va° a" € A).
(*)
By definition (cf. [Co-G]) a Hochschild cocycle 0 on a *-algehrn A is positive if it has even dimension n = 2m
and the following equality defines a positive sesquilinear form on the vector space ,4®(m+0:
(o° ® o1 ® ... ® o", 6° ® 61 ® . .. © 6'") = 0F"Vi",nl,... , a"', 6"",..., 61*)
for any a', Ы eA.
In general the positive Hochschikl cocycles form a convex rout1
Z'l(A.A') С Z"(A.A')
in tlie vector space Z'J of Hochschikl cocydes on A.
To familiarize ourselves with the notion of positivity, we shall consider the example where A is the *-algebra
of smooth functions on a compact, manifold M (and of course only consider continuous multilinear forms
on M).
For n = 0, the space Z" = Z"(A,A") is the space of 0-dimen.sional currents on M, and Z\ is the cone of
positive measures in the usual sense.
For n = 2, a Hochschild cocycle class С is characterized by a 2-dimensional de Rham current С which is
directly obtained by antisymmetrization from any cocycle ^ 6 C:
(C, /"d/1 Л d/2) = -2(v{f, /',/-) - 9(/n, /-, /')) (V/' 6 Л).
In the class С there is a unique element <lv that is skcw-syi cliic in the last two variables; it is given by
Фс(Л/\/'')= <с,/М/' л,|/-) (V/J eA).
The mapping С —- Ф^ is the natural cross-section that, one usually uses to identify tie Rham currents with
flochschild cocycles rather than Hochschild cohomology classes. It- docs, however, have one bad feature: the
cocycle АФ<7, A € С, is never positive. (Otherwise the equality ?/(<?) — (C,</d/Ad/) would define a positive
measure; but since Lj ~ —Lf one gets Lf — 0 and С = 0.) This shows that if one sticks to this canonical
representative, the notion of positivity remains hidden. As we shall see, if we let M be a 2-dimensional
oriented compact manifold and take for the class С the fumlanipntiil class [M] of M (a well-defined de Rham
current of dimension 2), the positive representatives <p 6 Z\. ^ ('^7r*)~i С °^ *"^'ls class will correspond to
conformal structiires (j on M. More precisely, since Z+ С\{'1ъ>)~1 ('¦ is convex, the correspondence will be
established between conformal structures on M and the extreme points of Z^.DBt7-)~1 С (cf. fig.2).
Thus, let g be a con formal structure on Л/ or equivalent I y, since1 M is oriented, a complex structure. Then,
to the Lelong notion of positive current ([Le]) there corresponds i.he positivity in our sense of the following
Hochschild 2-cocycle:
where д and д are inherited from the complex structure.
45-1
An immediate check shows that the antisymmetrization of <ря is B*»)"' [A/], where i = \/^T:
/°d/'Ad/2.
The positivity of <ps corresponds to the Dirichlet Hilbert space structure on the space of forms of type A,0).
It is also clear that the mapping g —» ipg is an injection, since one can read off from ips what it means for a
function / to be holomorphic in a given small open set U С М.
Indeed, from the positive inner product on A® A associated with tp3 one reconstructs the Л-bimodule of L2
forme of type A,0) as well as the complex differentiation d:
<Pg
С
Fig.2. Conformal structures and extreme points of Z\ П С
Each ip3 is an extreme point of the convex set Z\ П C, and as for the converse, the exposed points of this
convex set can be determined as follows: for any element of the dual cone (Z^.)* of Z+, of the form
d
G= Y. д„ЛхЦс1х"Г etf(A),
where g^v is a positive element of Md(A), one can show, assuming a suitable condition of nondegeneracy,
that the linear form
attains its minimum at a unique point in Z\ П B?ri)-' C, and that this point is equal to ips, where g is the
conformal structure on M associated with the classical Riemannian metric:
455

n cyclic
, qres the notion of fundamental class in cycl
cohomology. It can be taken as a starting point for developing complex geometry in the n on commutati
case.
Finally, let us note that the cocycle tps can be expressed in terms of the Dirac operator D for g on M:
where the Z/2 grading 7 of the spinor bundle is provided by the orientation of M and the choice of spin
structure is irrelevant.
This shows that, for d = 2, the formula (*) for a positive representative of the Hochschild class of a B,00)-
summable /f-cycle is the relevant construction of positive llochschild cocycles.
We shall now concentrate on the case d = 4. Let us first state an easy proposition for the case of Riemannian
spin manifolds of dimension 4:
Proposition 1. Let M be a 4-dimensional compact Riemaviiian spin manifold and let D be the corresponding
Dirac operator in ike Hilberi space Sj = L"{M,S) with Z/2 grading 7. Then the Hochschild class of the
following positive 4-Hochschild cocycle on A = C^iM) is Ihc fundamental class of M:
°, ¦¦¦./") = 8*2Ti-" (A + 7)/"[D. /']¦•• [A f]D-4).
The cocycle i/v, is independent ofw and of the spin structure ои Л/; it depends only on the conformal structure
of M and is equal to
In the last expression, the trace tr is the natural trace on the Clifford algebra Cx at every point x 6 M, the
differentials if' are regarded as sections of tlie Clifford algebra bundle, and dn is the Riemannian volume
element on M.
We shall now consider the general case of a D,oo)-suinniablc /\-cycle (Si,D,f) on a *-algebra>l and show,
using the positivity oi-фш, how to extend the usual inequality <t(EJ + c?(E) < YM(V) between Chern
classes of vector bundles E and the value of the Yang-Mills action on arbitrary compatible connections V
on E.
In Proposition 5 of Section 1, we gave a general definition of the Yang-Mills action, of the form
YM(V) = Tcu,(x@fD-d),
where (a", 00) is the degree of summability of (fl,D). Now, the curvature has homogeneity degree 2 in D,
that is, if we scale D and replace it by AD, then ж(в) gets replaced by A2?r@). Of course YM(V) then
gets replaced by A4~~ YM(V), and this shows that we can hope for an inequality relating it to topological
invariants of finitely generated projective modules only when d = 4, which we shall assume from now on.
We shall now give an equivalent definition of the action YM(V) of Section 1, which will make the desired
inequality fairly obvious. Let W(A) be the universal differential graded algebra over A, reduced by the
condition dl = 0, and with the involution (da)* = —d(o') (Vn ? A).
Definition 2. Lei ? be a hermitian, finitely generated pwjeclive module over A. Then, a universal
compatible connection on ? is a linear mapping V of ? to ? ©^ Cl] such that:
a) V(?a) = (V?)a +? Q da (V? € S, a e A),
b) <?, V^> - (Щ, n) = d((, n) (V?, 4 € ?).
Such connections always exist and it is also clear that in the context of Section 1, since UD is a quotient of
Q(A), any connection V relative to Qn comes from a universal compatible connection through the compo-
composition
where p : il1
P-V:? —>
= Q,XD is the quotient mapping.
Let CC(?) be the space of universal compatible connections on ?, This space is in general quite large; for
instance, for ? = A it is the self-adjoint part of il'(A), which, if we take for example A to be the algebra of
(smooth) functions on a manifold M, is a vector space of functions of two variables f(x, y) A, у 6 M) such
that
a) f{x,x) =0 (Vi€M),
0) /(*,») =Ж^) (Vz,y6M).
Of course, as in Section 1, the group U(?) of unitary endoinorphism of ? acts by gauge transformations
7U(V) = ti(Vu') on the space of universal compatible connections and the curvature в makes sense, as an
element of Ногпд (?, ? ®л Q'(A)), and is covariant under gauge transformations. The mapping V -+ p • V
is covariant and the two curvatures are related by 0p.v = p{0v).
We shall now assume that the A"-cycle (Я,О) is ечеи (i.e., we are given a Z/2 grading 7) and D,co)-
summable. We now describe two traces on Q(A)- The first is positive and yields the action YM(V) of
Section 1; the second is closed and yields topological invariants of finitely generated projective modules ?.
Lemma 3. If (Я, D, 7) is a D,<x)-summable K-cycle over A, then the following equalities define traces on
Ueven(A):
A) r(a°da' ...da4) = TTu(a0[D,al]... [D,o4]D-4) (V«> € A);
B) <S(a°da'...da1) = Tru,Ga°[D, a']... [?J,a4]?J-4) (Vn' 6Л).
In fact, both are traces on W(A) but Ф is a Z/2 graded trace, i.e.,
Let ? be a hermitian, finitely generated projective module over A, and extend г to a unique trace f on the
endomorphisms of the induced module ? 0a &'{Л)\ then
YM(p • V) = г((?") ).
Since the operator A — 7) is positive and commutes with !г(ПеУеп(^4)), we get the general inequality
т(в-) > Ф@2) (VV 6 CC(?)).
It remains to understand the topological meaning of the term Ф@2). This will follow from the pairing
between A'-theory and cyclic cohomology as expressed in terms of connections and curvature (cf. Chapter 2)
but requires the following additional hypothesis. We know that if В is the natural boundary operator
В :
IIC:\A),
then ВФ = 0, simply because, by the residue formula (chapter IV.2), the Hochschild class of Ф is (assuming
H'(A,A') to be Hausdorff) the same as the Hochschild class of a cyclic cocycle, the character of (Я, D,y).
However, we shall need the following hypothesis:
Hypothesis 4. ВФ = 0 as a cochain.
We shall of course have to check the hypothesis in the relevant examples, but we can already note that in
the case of the Dirac operator on a Riemannian manifold Л/ (of dimension d = 4), we have
450
457

which is a cyclic cocycle and therefore satisfies Hypothesis 4.
Note also that in general since В0Ф is already cyclic,
the condition ВФ = 0 means in fact that В0Ф = 0, i.e., that Ф is a cyclic cocycle. The corresponding cyclic
cohomology class
[Ф] € IIC*(A)
is not, in general, the same as the cyclic coliomology class ch. (Й, D, y) of the Chern character of the Л"-сус1е,
but it differs from it by lower-dimensional classes, i.e., by S(HC'-(A)).
We may now conclude:
Theorem 5. Let A be a *-algebra, ($,D,7) a D,oo)-summalilc K-cycle over A, and Фш the Hochschild
cocycle
Фа,(а° а4) = Tru,(Ta°[D, a1]... [D, n4]D-4).
Assume thai ВФ„ = 0. Then, for any hermitiau, finitely дсиспШ protective module ? over A, we have
<[?],**) <ym(V)
The left-hand side is the pairing between Л'-theory and cyclic cohomology. In the case of the Dirac operator
on a compact spin manifold M, the cocycle Ф„ is (8л-2) times the usual Yang-Mills action. Thus, the
above inequality is the usual one: ci(EJ + c2(E) < YM(V) for any compatible connection V on a hermitian
vector bundle E on M.
458
VI.3. Product of continuum by discrete and tlie symmetry breaking mecbaiiism
We have shown how to extend, to our context of finitely sunmiable A'-cycles (Я, D) over an algebra A, the
concepts of gauge potentials and Yang-Mills action, as woll as tlie way in which this action is related to a
topological action in the case of dimension <l. In this section we shall give several examples of computations
of this action. We first recall briefly its definition and use the opportunity to add to it a fermionic part.
We are given a «-algebra A and a (rf, oo)-summable A'-cycle (Д,О) over A. This gives us a representation
on fi of the universal differential algebra Q?A'.
*(aaAa> ... da*) = a°[D, a1]... [D, a"] (W 6 A),
which defines a quotient differential graded algebra
' ПЬ(Л) = U'{A)/J, J = Jo + AJo, J^k) = П* П Кегтг.
A compatible connection V on a hermitian, finitely generated projective module ? over A is given by a linear
mapping
V : ? - ? Qa Ob
that satisfies the Leibniz rule and is compatible with the inner product. The affine space C(?) of such
connections is acted on by the unitary group U(S) of the «-algebra of endomorphisms End^(?). This action
transforms covariantly the curvature 0 = V2 of such connections, and
is a gauge invariant quartic, positive action on C(?) (if. Section I)-
In the case of the trivial module ? = A (with the right action of .4 on itself), a vector potential is a self-adjoint
element V of Q}D, and the following expression is also gauge invariant:
(Ф,(D + ж(\'))ф) (Ф е ъ, v e n'D),
where the unitary group U = U(?) = U(A) acts on fi by restriction of the action of A, whereas it acts on
vector potentials by gauge transformations:
7u{V) = udoo + «I'm' (u eu, v e 0],)-
This is the fermionic action that we want to add to the action YM(V); it extends to the case of arbitrary
hermitian, finitely generated projective modules ? over A. by means of the next lemma.
Lemma 1. Let A, ?, (>5, D) be as above. Then:
A) The tensor product ? ®д >J is a Ililbert space wilb inner pivduct given by
B) For any compatible connection V, the following equality defines « self-adjoint operator Drt in the above
Hilberi space:
Thus, the fermionic action is now given by
(<l;Dv</') {Ф€?&a Я, V € C{?)),
and one checks that it is invariant under gauge transformations by elements of li(?).
The total action is
459

?(V, ф) = AYM(V) + (</., Dv</'>,
where A is a coupling constant.
We shall compute it in three cases: a) the discrete case of a 2-point space; b) the product case of a 4-
dimensional manifold by case a); and c) the irrational rotation algebra Ag. In order to see what the relevant
concepts are in the O-dimensional case a), we first need to discuss product spaces briefly. We are given two
triples
(Ai,Sit,Dt), (Л2,$2,А>)
and we assume that one of them is etien, i.e., that we are given a Z/2 grading, say 71 on Sit. The product
is then given by the triple (A,Sj,D), where
A =
>2, D = D, <S I + 71 ® D2.
This corresponds to the external product of К -cycles. There is an ohvious notion of external product of
hermitian finitely generated projective modules over the Aj.
Next, the formnla D2 = D\ ® 1 + 1©D|, which follows from th« anticommutatton of D\ with 71, shows that
dimensions add up, that is, if D, is (pj,oo)-sutnmable then D is [pt + p2, oo)-summable; moreover, once the
limiting procedure Lim^ is fixed, one can show that if one of the two terms is convergent then
Тг„((Т, ®T2)|?J|-') =Tru,(T1|?J,|-'1')TrIU(T2|D,|-'") (V7} € ?(Я,-)).
All of this is true provided that pj > 1, but in the case we are interested in (Example b)) we have p\ ~ 4,
pi = 0. The corresponding formula turns out to be
where Trace is the ordinary trace. To understand how this occurs, one can use the following general equality,
assuming that |D|~' б ?('>'~':
= Tru,(T|D|-'>)
for all T€ ?(J5).
Thus, the O-dimensional analogue of the action YM(V) is just given by Тгасе(тг@J).
2. Example a). The space we are dealing with has Iwo points a and b. Thus, the algebra A is just the
direct sum С ф С of two copies of С An element / ? A is given by two complex numbers f(a)tf(b) € C.
Let (Я,О,7) be a O-dimensional A'-cycle over A; tlion Jj is finile-dimenswnal and the representation of A
in ft corresponds to a decompoiti f ft di f ft
in ft corresponds to a decomposition of ft as a direct sum fi —
'«-[ТА
If we write D as a 2 x 2 matrix in this decomposition,
, with the action of A given by
D =
we can ignore the diagonal elements since they commute exactly with the action of A. We shall thus take
D to be of the form
D-
0 Д,,
Dba 0
4E0
where Dbo = D;t and Ot<1 is a linear mapping from $„ to fit. We shall denote this linear mapping by M
and take for 7 the 2/2 grading given by the matrix = 7. We thus have
Let us first compute the metric on the space X — {a, b], given by Formula 1 of Section 1. Given / ё A we
have
M
1 i/() 11 — Г
0 j'[ 0 /(*)JJ~l-.V<(/F)-/(a))
Thus, the norm of this commutator is \f(b) — /(o)|A, where A is the largest eigenvalue ||.M|| of M. Therefore
rf(a, 6) = sup{|/(a) -
Let us now determine the space of gauge potentials, the curvature and the action in two cases.
case a): ? = A (i.e., the trivial bundle over .V)
First, the space Q1(A) of universal 1-forms over A is given by the kernel of the multiplication m : A® A —» A,
m(f ® g) = fg. These are functions on Л' х Л' that vanish on the diagonal. Thus, Q1(A) is a 2-dimensional
space; if e 6 A is the idenipotent e(n) = 1, eF) = 0, this space has as basis
ede, A -e)de,
so that every element of П'(Л) is of the form Acde+ /i(l-e)d( 1-e). The differential d : A -» Q'(A) is the
finite difference
d/ = (A/)ede - (Д/)A - e)d(l - с), Д/ = f(a) - f(b);
it is a derivation with values in the bimoclulc fiM.4) which foils to be commutative: /w ф ш/ for w € П1,
/ел
Also, if Л1 / 0 then the representation jr : п'(А) — ?(Й) is in.jo.ct.ive on ill[A), so that П'(Л) = Пд
We have
A vector potential is given by a self-adjoint element of Яд, i.e., hy a single complex number Ф, with
Since V = —0ede -Ь ФA — e)de, its curvature is
0 = dV + V- = -Idedc - «Silccle + («Ecde - ФA - c)deK,
and, using the equalities ede(l - e) = ede, e(de)e = 0, A - e)dc( 1 - e) = 0, we have
0 = -(Ф + Ф)Aес1е - (ФФ)Ле(к.
Under the representation jr, we have 7r(de) = J and ?r(dede) = - —MM' \
yields the following formula for the Yang-Mills action:
'IG1

V
YM(V) = 2(|Ф + 1|2 - 1JТгасе((ЛГЛ<J),
where Ф is an arbitrary complex number. The action of the gauge group U = GA) x [/A) on the space of
vector potentials, i.e., on Ф, is given by
7„A/) = udu" +uVu";
for и = uoe + «ьA - e), this gives
7u(V) =(uoe + ut(l-e))(uade-u6de)
+ (uae + ut(l - e))( -ФеЛе + ФA -е)с1е)(пае +7ГЬA - ej)
— иащФе<&е + щпаФ{\ — e)de,
which, on the variable 1 -Ь Ф, just means multiplication by иьИа.
Thus, in this very simple case our action YM(K) reproduces the usual situation of broken symmetries (Fig.3);
it has a non-unique minimum, |Ф + 1[ = 1, which is acted upon nontrivially by the gauge group.
^|^Ц1ДДУ
Fig.3. The potential YM(V)
The fermionic action is in this case given by
(^ (D +t(K)H),
where the operator D + ?r(l/) is equal to
[О М'] Г 0 Ф.М'1 Г 0 A + Ф)АГ]
[М 0 j [ФМ 0 ]~[A+Ф)Х 0 J'
which is a term of Yukawa type coupling the fields A + Ф) anil ф.
462
1
1
1
;
case /3): Let us take for ? the nontrivial bundle over .V = {«,6} with fibers of dimensions no and nb,
respectively, over a and b. This bundle is nontrivial if and only if пл ф пк; we shall consider the simplest
case na = 2, nb = 1. The finitely generated projective module ? of sections is of the form
? = SA\
where the idempotent / 6 M2(.4) is given by the formula
, [A,1) 0 1 [1 0]
1 ~\ 0 A,0)J - [O ej
in terms of the notations of a).
To the idempotent / there corresponds a particular compatible connection on 8, given by Vo? = /d? with
obvious notations. An arbitrary compatible connection on 8 has the form
V« = Vo? + pi,
where p = p' is a self-adjoint element of Л/2(Л},(.4)) such that fp = pf = p. If we write p as a matrix,
1 />">t p'>'>
I/ -I / --J
these conditions read as follows:
ffti = Pi\, <tf>Ti = Pn — №.'<¦• P\'ie = Pi2i
thus we get
P\\ =-Фlede + Фl(l-e)de, fbi - Ф-jcde, pi2 = p'n, Pi7=0,
where Ф,, Ф2 are arbitrary complex numbers.
The curvature в is given by
0 = fdfdf + fdpf + p-
- [° 0 1 + [ d' (JPl2)el ,\PlU>U +Р12Й21 PllPl2]
[0 ededcj [edp2l 0 j [ рцрп P21P12I
An easy calculation gives the action YM(V) in terms of the variables Ф|,Фз:
YM(V)= (l + 2A — (|Ф, + 1|2 + |Ф22)J)тг((>СЛ1J).
It is by construction invariant unde> the gauge group U(l) x UA). What we learn in example /3) rather
than a) is that the choice of vacuum corresponds to a choice; of connection minimizing the action, and in
case /3) there is really no preferred choice of Vo, the point 0 of the space of vector potentials (case a)) having
no intrinsic meaning. In fact, the space of connections realizing the minimum of the action YM is a 3-sphere
whose elements have the following meaning. Let Ea (resp. Ei) be the fiber of our hermitian bundle over the
point a (resp. 6) of Л*; then dilute = 2, dmi-Efc = 1. As we saw above, the differential d : Л —> Q'(^) is the
finite difference. One way to extend it to the bundle E is to use an isometry u : Еъ —* Еа and the formula
All minimal connections V are of the form
Щ = (Д{)„ © ede + (Д?Ь © A - e)d( 1-е).
463

Since the minimum of YM(V) is > 0 we also see that, the bundle E is not flat; it does not admit any
compatible connection with vanishing curvature. Since the dimension of the space X is 0, the action YM
has of course no topological meaning. However, we shall now return to the 4-dimensional case and work out
the case of the product space in detail.
3. Example b). D-dimensional Riemannian manifold M) x B-point space X).
Let us fix the notations: M is a compact Riemannian spin 4-manifold, Ai the algebra of functions on M
and (JS[, Di,75) the Dirac Л'-cycle on Ai; let Ai, /Ь> Di '>e as in Example a) above, that is, Ai = СфС,
Л2 is the direct sum Л2 = &1,а © &1,ь, and Da is given by the matrix
0 M*'
[M 0
j and D = Di ® 1 + 75 0 D2-
The algebra A is commutative; it is the algebraof complex-valued functions on the space У = Mx X, which
is the union of two copies of the manifold M: Y = Ma U Мь-
Let us first compute the metric on 5' associated with the A'-cycle (#,D):
Let A - Ai ®Л2, Я =
Г 0 M-]
'¦=[м oj
p{
/ел
To the decomposition Y = Ma U Мь there corresponds a decomposi tion of A as Aa
is a pair (fa, h) of functions on M. Also, to the decomposition of $2 as
, so that every f GA
2 г.я*вг,6,
there corresponds a decomposition J5 = #„ Ф ^b* i'l which the action of / = (fa, fit) € -4 is diagonal:
In this decomposition the operator D become
дм О I
where дм is the Dirac operator on Л/ and 75 is the Z/2 grading of its spinor bundle.
This gives us the following formula for the 'differential' of a function f & A:
The differential [D, f] thus contains three parts:
a) the usual differential d/a of the restriction of / to the copy Ma of M;
j}) the usual differential d/j of the restriction of / to the copy Мь of M;
7) the finite difference Д/ = f(pa) — f(pb), where р„ and ]ц, are the points of Ma, Мь above a gi
p of M.
given point
The norm of the operator [D,f] can he computed easily: if Л is the norm of M, i.e., the largest eigenvalue
of \M\- (М'МУ'-, then
llfD fill - ess sun Г "/^^
where ||d/o(p)|| is the length of the gradient of /„ at p € /!/„.
4C4
We thus obtain, easily:
Proposition 4. A) ТЛе restriction of ihe metric d ом Д/„ и Мь to each copy (Ma or Mb) of M is the
Riemannian geodesic distance of M.
B) For each point p = р„ ofMa, the distance d[pa,Mb) « equal to A and is attained at ihe unique point pb.
Now recall that, given a metric space (Y, d) and two subsets V'i, Y2 of Y, their Ilausdorff distance d(Y\,Yi)
is given by
d(Yi, У2) = sup{d(i, y2), x € У\; rf(j,У,), i € У2}.
Thus, the metric d on Ma U Мь = Y is clearly related to the following definition of the Gromov distance
between two metric spaces (A'i,di) and (Л'?,^):
Definition 5. ([Gro,]) Let (A'i,d|) (i«rf (Л'з.Л) к dm) metric spaces. Then, given ( > 0, iue Gromov
distance 6(Xi,Ar2) »^> smaller than e if and only if there crisis a metric d on X = Ari U A'2 such thai
a) d\Xj = djt and b) d(XuX->) < <¦¦
Thus, we see that if we try to take different Riemannian iiK'trics <ja, уь on the two copies М„,Мь of M, say
by letting
A, 01
then Proposition 4 fails unless the Gromov distance between //,, and tji is less than 1 = I/A, A = \\M\\.
Let us now pass to the computation of the „4-biniodn!e Q[D of 1-fonns over the space У. The above com-
computation of [D,f] = *{df) for / e A shows that an element t» of the A-bimodule ulD = x(Ql) is given
by:
a) a usual differential form ша on Ma;
/3) a usual differential form ui on Mf,
7) a pair of complex-valued functions й„, 6ь on M.
The corresponding operator in fi is given by
\ii(u>a)Ql «„75©.VP] _
[ his ® M п(шь) О I J
the bimodule structure over A is given, wil.h obvious notations, by
(f<.,fbHua,u>l,,K,h) = (/«-„,/,,ЗД„/„^,/Л).
(W..W4. #„,*»)(/„./t) = (/»-•«,/t-t,/**»,/««*)¦
465

The involution * is given by (ша,шь, 6а,6ь)" = (— "'a. — "*. *6, ^Л-
/=(/«./»)
Fig.4. Differential of a function on a double-space
The terms 6a,h correspond to the bimodule of finite differences on passing from one copy Ma to the other
copy Мь of M. Note that even though A is commutative, this bimodule is not commutative; for, if it
were commutative then the finite difference would fail to be a derivation. With the above notations, the
differential / 6 A —> Jr(d/) reads as follows:
/ = (/a, h) - (d/a,dA,/4 - /a,/a " h) € &„¦
When we project on M, the bimodule Q\, can be viewed as a 10-dimensional bundle over M, given by two
copies of the complexified cotangent bundle, and a trivial 2-dimensional bundle:
however, one must keep in mind the nontrivial bimodule structure in the last two terms.
As in the case of the Dirac operator on Riemaimian manifolds (Section 1, Lemma 6), let us compute the
pairs of operators of the form -a{p) = Tb w(dp) = T2 for p e П1(А). Given p = ? fjdg, 6 QX(A), with
/j, ijj в A, we have
where u, = ?/jad<7ja, шь = YUjtd9jb and
<*>» = 53 -W*4 ~ &«)• *4 = 53 />*(*'« ~ *')•
We have jr(dp) = X) T(d/j)T(d3:j), which gives the 2 x 2 matrix
1
MM'\ '
466
where fo = ?d/>«d9ja a" f* = Z)d/j'd»jb are sections of the Clifford algebra bundle C2 over M, whereas
Using the equalities
4a
we can rewrite ;ja and ^6 as follows:
As in the Riemannian case (Lemma 6 of Section I), I lie sections ?,,{), of C2 are arbitrary except for
<T2((a) = dua and (Ti{?ii) = d^6- This shows that, the subspaa; 7r(d( Jn nil1)) of w(il2) is the space of 2 X 2
matrices of operators of the form
T =
0
0 7(sM0lJ'
where <%and 6 are sections of C°, i.e., are jnst arbitrary scalar-valued functions on M, so that Tfe) = ?,,
A general element of л"(П") is a 2 x 2 matrix of operators of t.lie form
т _ f-7(<*«) © 1 + /i, 3 M'M 75'Т(Л
-7(<н)©1 + Л(,®ЛШ*] '
where аа,аь are arbitrary sections of C2, /io, 'it are arbitrary functions on M and 0а,0ь are arbitrary sections
of C1 (i.e., 1-forms). We thus get:
Lemma 6. Assume that M'M is not a scalar multiple of Ihc identity matrix. Then, an element of№D is
given by:
1) a pair of ordinary '2-forms а„,аь on A/;
2) a pair of ordinary 1-forms 0а,13ь on M\
3) о pair of scalar functions ha,hi, on M.
The hypothesis M'M / Aid is important since otherwise the functions ha,kb are eliminated by )r(d(JonfJ')).
Using the above computation of ir(dp) we can, moreover, compute (<ra, <ть, А>, A, fta, Ль); for the differential
dui of an elemental = (aio,an, 6a,St) of UlD, we get:
1) <ra = daia. <гъ = doij;
2) 0a — Olb — O»a — d^a, 0b — ^a — Olfc — dfo;
3) /la = *а + *Ь, 'И = 6a + 6t.
Thus, we see that the differential doi € flj, involves tile differential terms duia, don, dio, dib as well as the
finite-difference terms a»a —an, 6a -hSb, but in the combinations such as on —wa — d6a imposed by d(d/) = 0.
Next, let us compute the product ljoj' € fip of two elements a» = (a»e, шь, 6a, 6ь), a1' = (шв,ш^, ?а,^ь) °^ ^Ь>
we get:
1) <r* =шаЛа1а, <т-ь
2) 0a = «aO.J - iao.0
467

3) ha=taft,ht=StS'a.
The next step is to determine the inner product on the space Q2D of 2-forms given in Section 1. By definition,
we take the orthogonal complement of ir(A{Ja Л Q1)) in ir(Q2) equipped with the inner product {Ti,T^) =
*). An easy calculation then gives:
Lemma 7. Let X(M'M) be the orthogonal projection (for the Hilbert Schmidt scalar product) of the matrix
Л4*Л4 onto the scalar matrices Aid. Then, the square norm of an element ((Tat&b, 0at0b, ha,h^) ^
given by
) /
Jm
tt((M'M-X(M'M)J) x / (|
v ' Jm
where Na = dimij,,, JVj = dimiij.
We are now ready to compute the action YM(V). We shall take the hermitian bundle on Y — Ma UJVffc that
has complex fiber C2 on the copy Ma of M, and trivial with one-tiimensional fiber С on the copy Мъ of M.
In other words, we consider the product of the example of Section 1 by the example a), /3) on the space X.
From the above description of Qjj, we see that if V is a compatible connection on E then it is given by a
triple:
a) a usual connection Va on the restriction of E to Л/д;
/3) a usual connection Vj on the restriction of E to Л/j;
7) a section и on M of the bundle \\om(Eb,Ea) of linear mappings from the fiber Еъ,р to the fiber Eap.
Both a) and /3) have the obvious meaning, while 7) prescribos the value of the finite-difference operation on
sections ? of E. At the point pai this finite difference is
(д«)(р«) =
whereas at the point pt it is given by
) - «,.? ы e ?„. = с2,
Of course, the choice of и is given by a pair Ф1, Ф2 of complex scalar fields on M, namely the two components
of u(l) for the basis of C2 (cf. Example a), /3)).
The gauge group U = End^i^) is the group of unitary endoinorpliisms of the bundle E over Y = Ma U Мъ,
or, equivalently, the group
U = Мар(Л/,С/A) X UB)).
Its actions on the U('2) connection Vn and on the (^A) coniux'tion Vi are the obvious ones, and the action
on a field u e Hom(?|, ?„) is given by compasition.
Let us be more explicit in the description of V as a linear mapping from the space ? of sections of E to
? ®A &b ¦ An element of ? <3)a ^d is given by:
1) a usual differential form w = (ша,и;ъ) on Y = Л7а U Л1ъ, with coefficients in ?:
2) a section 6 = (Sa,6b) on V = М„ U Л/j of the bundle E.
The mapping V is then given by
for any section ( = ({„,{4) of E.
4C8
Since the restriction of E to Д/а is trivial with fiber C2, we may as well describe Va by a 2 X 2 matrix
1" i2 of 1-forms on Л/ that is skew-adjoint. Similarlv, Vj is Riven by a single skew-adjoint 1-form
Ы21 W22j
[ш!]], and и by a pair of complex fields A -г-уьУг) — "(i)-
With these notations, the connection V is given by the equality
where ? = /Л2, / 6 M2(^l) being the idempotent / = у Ч , с = @,1) 6 Л, and where /J e M2(f2},) is
the 2x2 matrix whose entries are the following elements of Qj^:
Pu =
or, equivalently,
= (ш?._,, 0,0,0),
Ni -&H 0 oj-Lw oj• L о о JJ
The curvature ^ is then the following element o
which is easily computed using the above compulation of
<l : nj, - nj,, Л : «J, x п), - Пь.
As we saw in Lemma 6, elements of Q^ have a different iai degree and a finite-difference degree (or, /?) adding
up to 2. Let us thus begin with terms in 0 of bi-degree B,0). To compute them we just use the formulas 1)
following Lemma 6:
<та — Аша, &ъ ~ &иъ; (та = ^;а Л л'а, <Тй = ый Л о/j.
We thus see that the component (?<201 of bi-degrce B,0) is the following 2x2 matrix of 2-forms on M»UMj:
(?!,2-0> = do/1 +ы" ЛШ°, 0[--'" = du;' +Ш' ЛШ"
Next, we look at the component (?(']) of bi-degree A,1) and use the formulas 2):
0a = ub — ua — с1й„ fia = й„ш}, — шой^
/Зь -Ша—ШЬ - d6b A = 6jw'a - ШЬ6'Ь
Thus, ff*1'1* is the following 2x2 matrix of 1-forms on Ma U Л/j:
»
1 О
о о
,, oj [ о oj L-Si лА Ы oJ
-dwi -(
},)(^i + l)-w«!S52 0
1) - (w!2 - wf, )V2 0
Similarly, we have
'169

„A.1) _ [1 0
о oj - [ о о
] fwfi wf,l _ [w{i 0] [Vi V2]
J Ki W22J L 0 Oj L 0 0 J
IPs I
0 0 _
'о ""' о
Finally, we have to compute the component 0(°i2); we use the formulas 3):
(2 - "ll)
We then have
fi 4>г\ , \ч>\ О] fvi V2I
0 0 J + [?-, Oj [ 0 0 J
?->(l+9i)]
0 J'
@,2, _fl 01 Г1 Oj [?,+
"» - [0 oj + [0 oj [ *,
sl f
J [
_ [1 + Vi +
vi Vsl flPi "
0 0 J [V3 0
fl 0
0 I 0 0
Thus,
where each Ij is the integral over M of a Lagrangian density given by the following formulas.
First,
/2 : |dw" +ш' Ли°рЛл + IcL/pJVj,
where iVa = dimi5a, iV^ = dimiifi and the norms are the square norms for the curvatures of the connections
V and V*, respectively.
Next,
: 2
V
ir(M'M),
where V is the covariant difTerentiation of a pair of scalar fields, given by
Finally,
470
where Ax is the orthogonal projection in the Hilbert-Schmidt space of matrices onto the orthogonal com-
complement of tbe scalar multiples of the identity. These terms are obtained, with the right coefficients, from
the computation of tbe Hilbert space norm on Q^ in Lemma 7.
The fermionic action is even easier to compute. We have
where ф 6 ? <8u Я, *(ф = ф, is given by a pair of left-handed sections of 5 ® f)a denoted by \ ,\ and
right-handed section of 5® Я„ denoted by ф . Both Jo and J\ are given by Lagrangian densities:
Ja:
Ji :
+h.c.
We can now make the point concerning this example b): modulo a few nuances that we shall deal with, the
five terms of our action
Io + h + h + Jo + ii
are the five terms of the Glashow-Weinberg-Salam unification of electromagnetic and weak forces for N
generations of leptons (where N = Na = Nb is the dimension of fta, f)b).
Let us describe, for instance from [J,E], and using conventional notations of physics, what are the five
constituents of the G.W.S. Lagrangian, which we write directly in the Euclidean (i.e., imaginary time)
framework. For each constituent, we give the corresponding fields and Lagrangian:
where:
1) Co- The pure gauge boson part is just
where Gp«a = d^Wua — д„\У„„ +.'/fa4<r^iil'/»o and F;1I/ = 0,,В„ — д^Вц are the field strength tensors
of an SU{2) gauge field W^a and a UA) gauge field B,,.
2) Cj\ The fermion kinetic term lias the form
>.7y iv,lo + »V yb<.) А
У^в^) /я],
where the fc (resp. /я) are the left-handed (resp. right-handed) fermion fields, which for leptons and
for each generation are given by a pair, i.e., an isodoublet, of left-handed spinors (such as I ), and
a singlet (ея), i.e., a right-handed spinor.
We shall return later to the hypercharges 1't, Yr, which for leptons are given by Yi = —1, Уя = —2.
3) ?ф: The kinetic terms for the fliggs fields are
'J 2 "" 2
~l('
where Ф = «' is an SUA) doublet of complex scalar fields Ф1 and Фо with hypercheurge Уф = 1.
471

4) Су. The Yukawa coupling of Higgs fields with fermions is
• Л)],
where #^/' is a general coupling matrix in the space of different families.
5) Cv'- The Higgs self-interaction is the potential
1
Cv = /Г(Ф+Ф)- 1л(Ф+ФJ,
where A > 0 and ft1 > 0 are scalars.
All of these terms are deeply rooted in both experimental and theoretical physics, but we postpone an
evocation of this point to the complete model invoking quarks and strong interactions as well. For the moment
we shall establish a dictionary, or change-of-variables, between our action and the Glashow-Weinberg-Salam
action.
The first obvious nuance between the two actions is that our action invokes a U(l) and a UB) gauge field
while the G.W.S. action involves a U(l) and an SUB) gauge field (however, cf. [Wein] for an interesting
perspective on this point). We shall thus, in an artificial manner, reduce our theory to V(l) X SUB) by
imposing on the connections V* = d + ы" and V* = d +w' the following condition:
Ad hoc condition: tr(wa) = ш ¦
Let us now spell out the dictionary.
Noncommutative geometry
Classical field theory
chiral fcrmion /
differential components of
connection и", и>ь
finite-difference component
of connection A+4"), <S*
h
h
Io
Jx
pure gauge b
Higgs field Ф
Ca
C*
Cy
Cf
Cy
It is moreover straightforward, using the above 'ad hoc condition', to work out the change of variables from
our fields ip,u,8 to the fields /, W, В,Ф which give the equality
<T2YM(V) + (
= C(f, W, В,Ф),
where the right-hand side is a special case of the G.W.S. Lagrangian, with a few constraints. These relations
are meaningless for three reasons. The first is that the model incorporates neither the quarks nor the strong
interaction, the second is the artificial nature of the 'ad hoc condition', and the thud and most important is
that, due to renormalization, the coupling constants such as rj,;/', /t. А, Hjj> that appear in the G.W.S. model
are all functions of an effective energy Л to which, for the inoinoui, we can give no preferred value.
In Section 5 we shall remove the first two defects by explaining how to incorporate quarks and strong
interactions, as well as how the hypercharges of physics occur, from a conceptual point of view.
472
8. Example c): The noncommutative torus.
In this example we shall treat the noncommutative torus (cf. Chapter 4 section 6) from a metric point of view,
and show how the classical solutions of the extrema of the Yang-Mills action modulo gauge transformations
yield an ordinary torus.
Thus, let us begin with the following «-algebra Ae, where 0 6 [0,1] is an irrational number and A = e(9) =
exp 2Tifl:
<S(Z2) being the vector space of sequences (en,m)n,m?Z that decay faster than the inverse of any polynomial
in (n,m). The product m Ae is specified by the relation
VU = \UV, A = e@) =exp2irifl,
and the involution by U' = U~l, V = V~l.
In chapter 3 we computed the cyclic cohomology of this algebra; thus, HC(Ae) is 1-dimensional and is
generated by the unique trace r0 of Ae,
whereas beyond Stq 6 HC~ the cyclic colionioiogy ПС'2(Аи) i« generated by the cyclic 2-cocycle
Г?{а ,а,СГ) = 27Г1 ?_, l»i»'-»>"'i ) "n<,,moan,,m,an,,m,-
The above considerations involve only the smooth algebra A»; we shall now fix the metric. To this end we
take the element G of Qt{Ae) denned by
G= {(\U)(dU)' +(c\V){i\V)',
and we solve the following extremum problem (which, in the commutative case, gave us the conformal
structure on a Riemann surface):
Lemma 9. On the intersection of the cyclic cohomology class rj + (Кегб) П (KerB) with the positive cone
Z\ in Hochschild cohomology, the functional G defined by
v, V)
attains its minimum at a unique point кр~> given by
We shall now use the noncommutative analogue of a conformal structure, i.e., the positive cocycle y>2i
together with the trace To, to construct the analogue of the Dirac operator for Ae, that is, we shall obtain a
B,oo)-summable A'-cycle (f),D) on An- The llilbert space )j is the direct sum f) = Яо Ф Яг of the Hilbert
space ij0 = L2(A), r0) of the G.N.S. construction of r0, and a llilbert space Я, of forms of type A,0) on the
noncommutative torus which is obtained canouically from 9-^ as follows:
Proposition 10. Let A lie a *-algehra and ip? e 2%{A,A~) 11 positive Hochschild 2-cocycle on A. Lei Я, be
the Hilbert space completion oJQ}(A) equipped with the inner product
473

Then, the actions of A on fix by left and right multiplications are unitary.
They are automatically bounded if A is apre C-algebra.
Thus, Я, is a bimodule over A (i.e., a correspondence in the sense of Chapter 5) and the differential
d : A —> п*(А) gives a derivation which, for reasons that will become clear, we shall denote by д : A —> Я,.
In our specific example, the computation is straightforward and gives the following formulas for Я1 and d:
f)x = L2(Ae,ro) as an ./4»-bmiodule;
д : A —> Я1 is given by d(a)n,m = ('* + »n)<it>,m.
One can immediately check the following:
Proposition 11. Lei A = Ae ad on the left on both Яо = L^(Ae,i\>) and Я] (associated with tpi by
Proposition 10). Then, the operator
in Я — Яо Ф ^i defines tt B,oo)-summable K-cycle over An-
The Z/2 grading у is given by the matrix 7 = .
We have thus arrived at the desired quantization of the cocycle i?2. Indeed, a calculation yields the formula
Тги,(A + 7)«°[Д«1][А«2№-2) =V2(aa,a\a2). (.)
The point is that if we had started not with rj but with a multiple Агг, where A > 0, this would have
replaced ip2 in Lemma 9 by А^ь and would thus have modified (lie inner product о/Я], multiplying it by A.
This is equivalent to the replacement of д by A'/2<9 keeping the inner product of Я1 fixed, in view of the
equality
(a°0a\da-) = <S2iuo,a',a2').
Thus, this replacement of ть by Ar2 replaces D by \4-D and clearly leaves unchanged the left-hand side
of formula (*), and shows that there is a unique normalization of ip? for which it holds. In fact, this
normalization is dictated by the integrality of the pairing of ^2 with Л'о(^4«):
The same independence of the choice of trace r0 applies to (lie homogeneous formula (*), but of course the
value of the trace
06 Ae — Ti-^fn/J)
does depend on the scaling of D.
Note, finally, that we have found a completely canonical procedure for constructing the A'-cycle (fy,D,7)
over Ae from the fundamental class in cyclic cohomology, i.e., the choice of orientation, and the formal
positive element
G = dU(dU)' + dV(dV)' e O'iiAe).
This possibility of going backwards from cyclic cohoniology 10 A'-hoinology is at the moment limited to the
2-dimensional situation.
474
We shall now use the B, oo)-summable Л'-cycle (ft, D, 7) over An and compute the action YM(V) on connec-
connections V on hermitian finitely generated projective modules over As- Our first task will be to determine the
differential graded algebra Q[)(Ae). We thus need the analogue of Lemma 6 of Section 1. By construction,
the Hilbert space f) is the direct sum Я = Jj0 ф Я, of two copies of the left regular representation A of At
in L2(Ae, t"o), and we have:
Lemma 12. A) If k > 1 then ^@*) С ?(Я) is the Ae-bimodnle of all 2 x 2 matrices with entries in
and of degree (-1)* for the Z/2 grading defined by 7 = . . .
B) Lei Oij 6 Ae and consider the elements
Then there exists an ш G
with a = ж(ы), j) = ir(iluj) if and only г/022 — *n = #'«12 — da^x.
This is not difficult to verify, first, the image under ir of the subspace {aU~xdU +bV~1dV; a, b 6 Л»} of
Q1 (As) already gives all elements of the form
0 A(ol2)
A(e21) 0
with o,j 6 As- This shows that ir(Q') has the desired form, and so does т(П*) since it contains the t'th
power of jr(ft').
To verify B), one uses the commutation дд" = д'д as well as ihe equalities
where w 6 Q' (Ae) is given by
ы = -a(U-ldU - ^U--d(ir2)) (вбЛ,)-
It is straightforward to compute the Dixmirr trace Tru,(TZ)~-'), where T = (A(o,-,)) is a 2 X 2 matrix of
elements A(ejj) (o,j 6 Ae), sincere (and hence Л/з(Л«)) has a unique normalized trace r0. Thus we get:
With the notation of Section 1, Proposition 5, we thus see that, as an ./4e-bimodule, the Hilbert space
completion Я2 of ir(Q2) for the inner product given by the Dixmier trace may be identified with two copies
of L2(Ae, rg) (each viewed as an *4t-bimodule).
The projection P is the projection on the orthogonal complement of the elements ir(dw) (ы 6 П1, л-(ы) = 0).
Thus P is the projection on the orthogonal complement of the pail's (йц,<ь2) with a^j G Ae, an = a??.
This shows that we can identify the .Дв-bimodule Q% with L2(A#,ro). Let us compute the differentials
d : Ae -> ulD, d : CllD -> П^, and the product Qj> x CllD — u'-D.
Proposition 13. As an As-bimodnle with involution, TrfQ1) is isomorphic to the direct sum of two copies
of Ae. With this identification and the above isomorphism of Q~D with L2(>l#,ro), we have:
B) de = (<5ie, i52e), where the Sj e Der(>le) are the unique derivations such that 0 = i5i+i62, d* = —i
C)d(o,,O2) = 2i(M'»i)-«i('»2));
D) (ai,a2)(ii,&2) = "iii - «2*2-
-17Г)

The proof is straightforward. This proposition shows thai, llie notion of compatible connection V 6 C(?)
on a hermitian, finitely generated projective module ? over Ae, as defined in Section 1, is identical with the
notion of compatible connection that we introduced with M. RJeHel in [Co-R] and which depended on the
natural parallelizable structure of Ae . Moreover, up to normalization, the two actions YM—the first as
defined in Section 1, the second as defined in [Co-R]—do coincide. We can thus exploit the results of [Co-R]
and state:
Theorem 14. Let ? be an arbitrary hermitian, finitely generated projective module over A$ and lei d be
the largest integer such that ? = Ad is a multiple of a finilely generated projective module Л. Then the
modulus space of the equivalence classes under U(?) of the compatible connections V on ? that minimize the
action YM(V), is homeomorphic to (T2)rf/?d, the quotient of the dyth power of a 2-torus by the action of
the symmetric group Ej.
This shows that even though we started with the irrational rotation algebra, a fairly irrational or singular
datum, the Yang-Mills problem takes us back to a fairly regular situation.
We should again stress that the only data used in setting up the Yang-Mills problem are 1) the *-algebra>l(?,
and 2) the formal 'metric'
(№&U* +dV'dV* G Q^_{Ae).
We shall now give a thorough description of all the finitely generated projective modules over Ae and of the
actual connections that minimize the action YM.
Let us recall from chapter 3 theorem 3.1/1 that tho isomorphism classes of finitely generated projective
modules over An are parametrized by the interseciion of I lie lattice Z~ with the half-space
0},
i.e., by ([Pi-V,]),
The mapping that associates with a finitely generated projective module ? the positive real number p — Bq
is the pairing with the trace r0 that is the generator of JlC°{Ae). It thus plays the role of the dimension
and indeed it is the Murray-von Neumann dimension over l,h<; Hi factor R that is the weak closure of Ae
on the Hilbert space ft (of the Л'-cycle (f), ?>)).
The mapping that assigns to ? the integer '/ e Z plays the role of the first Chern class and is given by the
pairing with the fundamental class г2 е HC'-(Ae) in cyclic cohoinology. It will be relevant to compare it
below with the action YM.
Thus, let (p, fl) 6 Z2, fl > 0, be a pair of relatively prime integers (p = 0 being allowed). We remind (chapter
3.3) that a finitely generated projective module ? over Ae is obtained as follows. Let <S(R) be the usual
Schwartz space of complex-valued functions on the real line and define two operators Vi, V2 on S(R) by
= ФШ") (s 6 R, « 6
where t = p/q — в and e(.s) = exp2ir«s (Vs 6 R). Of course,
V2Vl=c(c)\\V-,.
Next, let Л' be a finite-dimensional Hilbei-t space and let. w\. «¦-_, he unitary operators on К such that
We make ? into a right _4«-module as follows:
(U = (Ц® «<!)?, iVr
476
By theorem 14 of chapter 3 section 3 the above modules, together with the free modules A%, give us the
complete list of finitely generated projective modules ? over Аи. Since, given such an ?, all of the hermitian
structures on ? are pairwise isomorphic, we need only describe, one of them on .S(R) <8> K.
We view ? as the space S(R, A') of Л'-valued Schwartz functions on R and we define the hermitian metric,
i.e., the ./^-valued inner product, by the formula
where we have identified the elements a of As with sequences /(in, 11) of rapid decay on Z2:
It is not difficult to check that this inner product defines a heniiilian structure on ?.
(All of the above constructions have a clear geometric origin if one remembers that the algebra Ae corresponds
to the non commutative space of leaves of the irrational Kionecker foliation of the 2-torus (cf. chapter 2).
Then the closed geodesies of the 2-torus are transverse 1,0 lhe foliation and yield the above description of
hermitian finitely generated projective modules. However, we do not want to perturb our canonical route
from Л» and the metric dUAU' +dVdV* to the Yang-Mills problem under discussion.)
We are now ready to describe the connections V that minimize l.he action YM. By Proposition 13, B),
giving a connection V on a finitely generated projective module ? is equivalent to giving the two covahant
differentials V,- (j = 1,2) such that
) (V{ e ?,« 6 AeJ = 1,2)
(Vj is a linear mapping from ? to ?). Also, by Proposition 13, l-he curvature is, under the identification of
uj, with L2{Ae, n>), equal to the eiidoiuorpliisin V,V-j - VjV, of ?. We shall thus say (as in [Co-R]) that
the curvature is constant if the endomorphism ViV? — V2V] of S is a scalar multiple of the identity.
Let ? be the herniitian, finitely generated projective module S(R, I\) as above; we define a connection V on
? by
(VseR),
where e = p/q — 0 as above. Straightforward calculations show that this is indeed a connection and that it
is compatible with the hermitian metric defined above.
Moreover, one checks that the curvature 0 = V1V0 — V?V! is constant and is given by
д 2™'-i
0 = id.
All of this remains true, with the same value of the const nut curvature, for the connections V obtained
from the formulas
= 2iri(s/f )f(») + 2iri<r1€(.i),
where <7i and 02 are commuting self-adjoint operators in Я thai, commute with tui and w?. We then have:
Theorem 15. [Co-R] A) /1// of the above connections V an compatible, have constant curvature —2wi/e
and minimize the action YM(V).
B) Every compatible connection V that minimizes the action YM 4ns constant curvature — 2xi/« and is
gauge equivalent under U(?) to a connection of the form V17.
To conclude this section we note that with ? as above, both the dimension (i.e., the Murray-von Neumann
dimension p — 9q) and the curvature constant
477

are irrational numbers, but their product is an mteger, which gives the pairing between ? viewed as an
element of K0(A») with the cyclic coliomology fundamental class rj of At (cf. chapter 3).
476
VI.4. The notion of manifold in noncommiitativc goometi-y
Let X be a quantum space and A the corresponding «-algebra. We saw above that the giving of a /if-cycle
(f),D) over .4 yields a metric с!((р,ф) on the state space of Л, permits constructing a differential graded
algebra Q"D, and, in the (J, oo)- summable case, recovers integration in terms of the Dixmier trace. The
nontriviality of the A'-homology class, i.e., of the stable honiotopy class, of the К-cycle (f), D) played a
crucial role in the residue theorem (chapter IV.2) in ensuring the nonvanishing of the Dixmier trace on the
li-dimensional forms Qjy.
In this section we shall first expound classical results of the theory of manifolds and their characteristic
classes, in particular those of D. Sullivan, which exhibit the central role played by A'-homology. We shall
then explain how to formulate Poincare duality in A'-honiology in the non commutative context. This
will allow us to get closer to the concept of a manifold in noncomniutative geometry, and we shall see
how ordinary manifolds, the noncommutative tori and, later, Euclidean space-time as determined by the
U(l) x SUB) x 5?/C) standard model, fit in with this algebraic notion of manifold.
a) The classical notion of manifold
A J-dimensional closed topological manifold Л" is a compact space locally homeomorphic to open sets in
Euclidean space of dimension d. Such local homeomoi phisms are called charts. If two charts overlap in the
manifold one obtains an overlap honieomorphisin between open subsets of Euclidean space. A smooth (resp.
PL...) structure on Л* is given by a covering by charts so that, all overlap homeomorphisms are smooth
(resp. PL ...). By definition a PL homeomorphisni is simply a homeomorphism which is piecewise affine.
Smooth manifolds can be triangulated and the resulting PL structure up to equivalence is uniquely deter-
determined by the original smooth structure. We can thus write:
A)
Smooth
PI. => Top.
The above three notions of smooth, PL and Topological manifolds are compared using the respective notions
of tangent bundles. A smooth manifold Л" possesses a tangent bundle TX which is a real vector bundle over
X. The stable isomorphism class of XA in the real A'-theory of A' is classified by the homotopy class of a
map:
B)
A' — HO,
Similarly a PL (resp. Top) manifold possesses a tangent bundle bin, it is no longer a vector bundle but
rather a suitable neighborhood of the diagonal in A' x X for which the projection (x,y) —> x on X defines a
PL (resp, Top) bundle. Such bundles are stably classified by the honiotopy class of a natural map:
C) A' — В PL
The implication A) yields natural maps:
(resp. В 'Гор).
D)
ВО — В PL — В Top
and the nuance between the three above kinds of manifolds is governed by the ability to lift up to honiotopy
the classifying maps C) for the tangent bundles. (In dimension -I this statement has to be made unstably
to go from Top to PL.) It follows for instance that cvny PL manifold of dimension d < 7 possesses a
compatible smooth structure. Also for d > 5, a topological manifold Xd admits a PL structure iff a single
topological obstruction 6 6 #4(A',Z/2) vanishes.
For d = 4 one has Smooth = PL but topological manifokls only sometimes possess smooth structure (and
when they do they are not unique up to equivalence) as follows from the works of Donaldson and Freedman.
479

The КО orientation of a manifold.
Any finite simplicial complex can be embedded in Euclidean space and has the homotopy type of a manifold
with boundary. The homotopy types of these manifolds with boundary is thus rather arbitrary. For closed
manifolds this is no longer true and we shall now discuss this point.
Let X be a closed oriented manifold. Then the orientation class fix 6 Hn(X,Z) = Z defines a natural
isomorphism:
E)
об Н' X -af
e //„_; Л'
which is called the Poincare duality isomorphism. This continues to hold for any space Y homotopic to X
since homology and cohomology are invariant under homotopy.
Conversely let X be a finite simplicial complex which satisfies Poincare duality E) for a suitable class fix,
then X is called a Poincare complex. If one assumes that -V is simply connected (wl(X) — {e}), then [Mi-S]
there exists, a unique up to fibre homotopy equivalence, spherical fibration E ^* X over X (the fibers
p~ 'D), 4 6 Л' have the homotopy type of a sphere) which plays the role of the stable tangent bundle when X
is homotopy equivalent to a manifold. Moreover, in the simply connected case and with d = dimX > 5, the
problem of finding a PL manifold in the homotopy type of .V is the same as that of promoting the Spivak
normal bundle to a PL bundle. There are, in general, obstructions for doing that, but a key result of D.
Sullivan [ICM, Nice 1970] asserts that after tensoring the relevant abelian obstruction groups by Z [5], a PL
bundle is the same thing as a spherical fibration together with а КО orientation. This shows first that the
characteristic feature of the homotopy type of a PL manifold is to possess а КО orientation
F) uxer<O.(X)
which defines a Poincare duality isomorphism in real К theory, after tensoring by Z[l/2]:
G)
KO-(X)u-> — «ni/A- 6 A"O.(-V)i/2-
Moreover it was shown that this element i>x € А'О„(Л*) describes all the invariants of the PL manifolds in a
given homotopy type, provided the latter is simply connected and all relevant abelian obstruction groups are
tensored by X [\]. Among these invariants are the rational Poutrjagin classes of the manifold. For smooth
manifolds they are the Pontrjagin classes of the tangent vector bundle, but in general they are obtained from
the Chern character of the КО orientation class их- These classes continue to make sense for topological
manifolds and are honieomorphism invariants thanks to the work of S. Novikov.
We can thus assert that, in the simply connected case, a closed manifold is in a rather deep sense more or
less the same thing as a homotopy type X satisfying Poincare duality in ordinary homology together with
a preferred element их 6 KO.(X) which induces Poincare duality in A'O" theory tensored by Z [1/2]. In
the non simply connected case one has to take in account the eqnivariance with respect to the fundamental
group wt(X) = Г acting on the universal cover Л".
/3) Bivariant K-theory and Poincare duality
The main property of (the stable homotopy class of) the fnndt4inental class in Л'-homology is that it gives
a Poincare duality that exchanges Л'-homology with A'-lheory. In the noncommutative case this is a little
more involved than in the commutative case, so we shall fii-st review ([Kas7], [Co-S]) how Poincare duality
is formulated in bivariant A'-theory.
Kasparov's bivariant Л'-theory ([Kasi]) is a hifunctor l\ К from the category C*-Alg of C-algebras (with
morphisms given by algebra *-homoniorpliisms) 1.0 the category of abelian groups. The abelian group
associated with a pair (A, B) of C"-algebras is denoted h'h'(A,Ii); it is covariant in В and contravariant
480
in A, so that, for instance, to a morphism p : Ai — ,<U there corresponds a mapping p' : KK(A2,B) —>
KK(Ai,B). This bifunctor has the following properties.
1) For A = С, КK(C,B) is naturally isomorphic to the Л'-theory group K0(B).
2) For В = С, КК(А,С) is the Л'-homology of A; in particular, to every Л'-cycle on a *-algebra A there
corresponds an element of К К (А, С), where A is the norm closure of A in C(f)).
3) К К (A, B) is homotopy invariant. This means that the morphisms p" and p. of abelian groups associated
with a morphism p of C*-algebras depend only on the homotopy class of p.
4) A bilinear, associative intersection product is defined, given C*-algebras A\, Ач, B^t B2 and D:
KK(AUB,® D)®D KK(D®A2,B2) —• KK(A, ® A2,Bi ® B2).
We refer to Chapter 4 Appendix A for the properties of the intersection product. One way to remember
these properties is to think of the elements of KK(A,B) as (homotopy classes of) generalized morphisms
from A to B, the intersection product then being composition. For any C'-algebra the intersection product
provides the abelian group KK(A,A) with a ring structure, whose unit element will be denoted 1л-
Now let A and В be C*-algebras and let us assume thai we have two elements
ae KK(Af)BX), lie KK(C,A®B)
(*)
such that
0<?A a - In 6
0Ou<> = 1,1 e К К (A, A).
It then follows from the general properties of the intersection product that there are canonical isomorphisms
A'.(.4) = KK(C,A) 2 Л'А'(В,С) = K"{B),
h'.(B) = KK(C, В) =? К Л(.!,С) = К'(А)
that exchange the Л'-theory of .4 with the Л-homology of 13.
More explicitly, the mapping from K«(A) to K'(B) is given by Hie intersection product with a:
x e A'.(.-i) = л'л'(СЛ) — *o,i " e кк(в,с) = к'(в).
The inverse mapping from K'(B) = K'K(B,C) to А"„(Л) is given by the intersection product with /3:
у 6 K-{B) = A7v(B,C) — /3Go .7 6 A'A'(C, A) = K.(A).
More generally, for any pair of C*-algebras С and D, we have canonical isomorphisms
KK(C, А О D) S KK(C 1 j B, D),
h'K(C, В 0 D) 2 KK{Cc-j A, D),
which show that the above pair a,)i establishes a duality between A and В with arbitrary coefficients.
In ([Kas7],[Co-S]) an example of this duality was worked out with A = C(V), В = C0{TV), where V is a
compact manifold, A is the C-algebra of continuous fund ions 011 I', and TV is the total space of the tangent
bundle of V. The differentiable structure of V then provides, through the pseudo-differential calculus, the
desired elements a 6 KK(A ® B, C), /? 6 А'Л'(С,.4 & В) fulfilling the above condition (*).
The Thorn isomorphism for vector bundles ([KaSj]) provides a natural Л'Л'-equivalence (i.e., an isomorphism
in the category of C*-algebras with КI\(A, B) as the morphtsms from A to B)
C0(TV) йСг,
-181

where CV is the C'-algebra of continuous sections of the bundle over V of Clifford algebras Cp = Cliff c (TP(V)).
We thus get, using the Л'Л'-equivalence, a natural duality between A = C(V) and В = Cv¦ We shall now
describe in greater detail the corresponding elements
a 6 KK(C(V)®Cv,C)), 0e KK{C,C(V)®CV).
Since, as a rule, Jf-homology is always more difficult than Л'-theory, we shall concentrate on the description
of a. The description of/3 is much simpler.
We shall describe a as a very specific A'-cycle on C( V)®Cv ¦ we let ft be the Hilbert space of square-integrable
differential forms on V, where V is equipped with a Rieniannian metric g:
We let D — d + d* be the self-adjoint operator in ?) given by the sum of the exterior differential d with its
adjoint d*. The action of C(V) on f) is the obvious one, by multiplication. For the action of Cv we have
the following:
Lemma 1. 1J(?),D) is the K-cycle over С{V) given above, Ihen the comtnuiani on f) of the algebra
generated by C(V) and the [D, f] (/ 6 C(V), \\[D,f]\\ < oo) is cavonically isomorphic to the algebra of
bounded measurable sections of the bundle С of Clifford ahjehma.
Indeed, the commutant of C'(V') on ft is the algebra of bounded measurable sections of the bundle End(A)
of endomorphisms of Дс T*, so that it is enough i.o compute lor each /> ? V the conimutant of the algebra
generated by tlie operators 7(?), ? ? T*(V), where
However, 7 defines a representation of the Clifford Algebra C,, on the Hilbert space /\CT? with 1 6 Дс Т" as
cyclic and separating vector, so that its commutant is also given by a canonical representation of Cp, given
explicitly by the formula
7p(O>/ = (-lf"(( Л ./-.>;/) (V,/ 6 /\cT;(V)).
This Л'-cycle (ft, D) over C(V) © Cv defines the fundamental class of V in /v'-homology, a 6 KK(C(V) ®
CV,C)), and yields for instance the construction of the Dirac operator with coefficients in a Clifford bundle
([Gii]) as the natural mapping
K.(Cv)— K'(C( К))-
The Л'-theory class/? e А"Л'(С, C(V)®CV) is easier to describe; it is just the family, parametrized by p 6 V,
of Bott elements /3,, 6 Л".(CV) obtained from the Bott periodicity applied to a small disk centered at p 6 V.
In general, Cv is not Morita equivalent to C(V). Giving a Spin0 structure on V determines such a Morita
equivalence and thus permits replacing « and /3 by equivalent, elements
a e к к (C( v) 0 C( I-'), c), ;ie к к (С, с( v)®c{ v)).
This time or is given by the Dirac Л'-cycle on V and Ihe two representations of C(V) on ft are identical, thus
yielding the diagonal representation of C(V) ® C(V) on f>. This is a very special feature of the commutative
case: if Л is abelian then every Л-module is in a trivial way an .1-bimodiile, since one then has the diagonal
homomorphism A® A — A. In general, as we saw above, the fundamental class in Jv'-homology involves an
algebra A, its Poincare dual В and {A, B)-bimodules.
We shall now review the construction in Section 3, Example c) of the A'-cycle (fi,D,y) on the «-algebra Ae
and discover that the ^4«-moclule ft is in fact in a canonical way an .4»-bimodule, thus yielding the desired
fundamental class for the noncommutative torus:
48-2
[<J),D,7)] =«? Л'А'(.-1. 0 АвХ),
where Ae, the norm closure of Ae, is the irrational rotation С-algebra.
Recall how we constructed ?): we have Й = %&?),, where f>0 is the Hilbert space of the G.N.S. construction
relative to the unique normalized trace r0, and f)t is given by Proposition 10 (section 3). On До we have
natural actions of Ae on both sides (left and right) on the G.N.S. representation relative to a trace. On ^
Proposition 10 of section 3 shows that we also have a natural bimodule structure on Ae- We thus see that Si
is naturally an Ae ® A%-moda\e and it remains to verify that the operator Z) = ¦ still a Л'-cycle
for Ae®A°e- In fact, we have the following analogue of Lemma 1:
Lemma 2. (a) For any a, ft 6 Ae the commutator [D, a ® 4°] is bounded.
(b) The commuiani on Я of ihe algebra generated by the left action of Ae and ihe commutators [D,a]
(a 6 Ae) is ihe weak closure R = {Aae)" of ihe right action of Ae on f).
The assertion (a) is clear. To see (b), just use Lemma 12, which shows that (with the notations of that
lemma) the «-algebra generated by Ae and the [D, a] {a e Ae) is the left action of M^(Ae) on f).
This is a good point at which to comment a little on noncoiumutative measure theory in this example; the
von Neumann algebra {A°)" is the hyperfinite factor R of type II] (Chapter 5), and the Dixmier trace gives
the following formula for a hypertrace on R:
V{T) = TX(rZT2) (VT 6 C(S}))
(cf. Chapter 5).
The Z/2 grading 7 = L . on Й = #0 Ф #, then turns (ft, D,j) into an element a 6 KK(Ae ® Ae,C).
We leave it to the reader to determine a Л'-theory class /3 S Ka(At & At) = А'Л'(С, Ae ® Ae) such that the
pair («,/3) satisfies the rules (¦) of Poincare dualilty.
The lesson that we want to draw from this section is that ihe A'-liomology fundamental class of a quantum
space is given by a bimodule, not just a module, (ft,D).
We should of course stress that, we are interested in the actual Л'-cycle (fj, D) over A ® В and not only in
its stable homotopy class.
7) Poincare duality and cyclic coliomology
The above discussion involved Poincare duality in A'-theory; we shall now turn to cohomology. We showed in
Section 1 how to associate to a Л'-cycle (ft, D) on a «-algebra A, a differential graded algebra U'D which, in
the case of the Dirac Л'-cycle on a Riemannian manifold, gives the de Rliam differential algebra of ordinary
forms. Now, U"D makes sense in general and, as does every differential graded algebra, it has a cohomology
ring H'(U"D) which, in general, will fail to be graded-commut.ative. We shall now explain how in the finitely
summable case, using the Dixmier trace, one gets natural mappings
A) П*,(^)-.Я"-''(В,В*),
B) Hk(umD(A)) — PHCd~k(B),
that relate a pair А, В of Poiucarednal algebras, where }Г(В,В~) and Ii'(B) are, respectively, the Hochschild
and periodic cyclic cohomologies of the algebra B.
Note that the cyclic cohomology H'(B) of an algebra Б does not in general have anatural ring structure. In
fact even in the commutative case, say if В is the algebra of Lipschitz functions on a simplicial complex X,
the cyclic cohomology H'(B) of В is the homology of Л\ which, unless Л" is a manifold, has no natural ring
structure.
483

Thus, let (ft, D, 7) be an even (d, oo)-summable A'-cycle over A <S> B. In order to construct these mappings
we shall need the following conditions, which, thanks to the above Lemmas 1 and 2, are fulfilled in the
examples.
The order 1 condition on D: [[D, a], b] = 0 (Ve 6 A, b 6 B)
Note that since [a,b] = 0 (Ve 6 A,b 6 B), this condition is symmetric in A and B. Note also that given
the Л"-сус1е (Я, D) on A, there is a largest algebra В fulfilling the above condition, namely, if M is the
von Neumann algebra commutant of A U {[D, a], a ? A] then
В = {x e M; [D,x] is bounded}.
The second condition is the one we already met in Theorem 5 of Section 2; namely, we want to assume that
ВФ„ = 0, where Ф„ is the Hochschild cocycle on A ® Б given by
Тг„G1°[Д x1]... [D, z"]?>-") = Фы (x°,..., xn).
The closedness condition: Тг„(т[Л, x1}... [D, x"]D—) = 0 (Ых* e A ® B)
This condition is fulfilled and easy to check in the examples of Riemannian manifolds with the Dirac К -cycle
or in the C(V) ® Cv example discussed above. Let us verify it. in the case of the noncommutative torus,
A = Ag, В = At. We have to show that
а1 0b'][D,a-Q b2]D~'J) = 0 (W,b> 6 Ae),
the action of a ® b on f) being given by the 2x2 matrix
where Л (resp. A') is the left (resp. right) regular representation of Ae on L?(Ae,r0) (cf. Lemma 12 of
Section 3). Since D = „, , we have
0 \(да)У(Ь) + \(а)\>(дЪ)]
в-<,)А'F) + А@)А'(Э-*) 0 J
just using the derivation rule for both д = 6\ + tS2 and 0* = —#i + i6?. Next, for any 2x2 matrix
Xij = ^iaij)^'{bij) with <iij,bij € Лв, one verifies that, by the uniqueness of the trace on Ae,
TM[.Y,j ]?>-") = то{аи)то(Ьи) + го((»22)го(&з2)-
Now, [D, a1 ® b^D, a2 ® b2} = [Л'у], with
Xn = {Xida^X'ib1) + Xla^X'idb'^iXld'a^X'ib-) + X(a2)X'(d'b2)),
Л'22 = (Xid'a^X'ib') + Xia^X'id'b^JiXida^X'ib2) + X(a2)X'(db2)).
Thus, Tr^fTjD.e1 ® bl][D,a2®lr]D-2) is the sum of 8 terms
T0(dald'a2)T0(b'b2) + т^да^а2) та{Ь'д"Ь2)
+то(а'д'а2Iь{(ОЬ1)Ь3) + та(а\г)т„(И^ д'Ь2)
-То{д'а1да2)т1)(Ь1Ь1) - Мд'и1 а')^!I сП2)
Dne sees that these terms add up to 0, thanks to the following relations:
a) ro(dald*a2) = rold'a'da2) (using transposition aiul the commutation of д and d"). Similarly, T0(9b1d'b2) -¦
ibid'b^b2).
484
0) ro((aa')o2) = -ra(alda2) (transposition), T0(bld'b2) = -^((a'i1)*2).
We are now ready to construct the mappings A and B.
Lemma 3. Lei (f),D,-j) be a (d,co)-sutnmab!e (A,B)-bimodule satisfying the order 1 condition. Then:
A) For every к < d and a 6 пк(А), the following formula defines a Hochschild cocycle Ca 6 Zd-k(B,B'):
Ca(b°,...,bd-'<) = TTa,(jir(a)bolD,b1]...[D,bd-k}\D\-d) (W 6 B).
B) If the closedness condition is satisfied, then Ca depends only on the class of a in ^^(.4), a^d we have
Here Bo is the cyclic cohomology operator
(Bo?>)F°, ...,&«-') = v( i.»°. ¦•-,*«"') + (-DV(»°. ¦-.,»«-', l).
To verify A), simply note that ir(«) helongs to the coniniutant of i>; this is sufficient because when computing
the coboundary
one gets only two terms
Тг„Gл-(аN°[Л, 41]... [D, bd-k]b'l-k+i /_>"'')
-Тг„G1г(аL'|-1 + 14"[Д41]...[Д4''-*]1Г''),
and these terms cancel due to the commutation of hd~k+1 with D~d modulo trace class operators.
To verify B), let us consider the differential graded algebra QD{A ® B). First, we have [da, 4] = 0 (Va g A,
b 6 B) and similarly [d4, a] = 0, using the order 1 condition. Next, for a 6 QdD(A ® B), the value of
ТГш (T"(aa)D~d), where oo 6 Qd{AOB) is any representative of a, is well-defined. Indeed, it depends only
on Tr(cto), and if ao G d./0~l then it vanishes by the closediiess condition. We shall denote this value by J a,
and we note that it gives a closed, graded trace on the differential algebra Qd(A ® B).
Using the natural homoniorphism QD(A) —• Qd(A&B), the quotient of ЩА) —> ЩА8В), we see that Ca
only depends on the class a 6 Q*j(.4). By construction, the Hochschild cocycle CaD0,. .. ,4''"*) vanishes if
апУ 4^ > J > 1. is zero. Thus,
(BaCa)(ba bd~k~l)= /„A4° ilfc1 ...db"-"-1
Pi-oposition 4. ?e* (Jj, Д7) 4c an (A,B)-bnnodvle which is (rf,iDo)-SM?nmo4/e and satisfies both the order 1
and the closedness condition. Then:
A) ForO < к <d, the mapping a — С„ is well-defined fmm Q^iA) 1o the Hochschild cocycles Zd~k(B,B').
B) The image under С ofKerd С QkD(A) is contained 111 ZCd~k(B), i.e., Ca is a cyclic cocycle if da - 0
(m П*+'(-4))-
C) The image under С of Imd С Пко{А) is contained in ImB, when В : Hd-t+'(B,B') -> HCd~k(B) is
the cyclic cohomology operation.
D) С defines a mapping from H'(UD(A)) to periodic cyclic cohomology Ii"{B) that is compatible with the
natural filtrations.
485

The first assertion follows from Lemma 3. To prove B), just note that by Lemma 3, if da = 0 then B0Ca = 0
but since Ca is a Hochschild cocycle it then follows that Ca is also a cyclic cocycle. To get C), note that if
a = d/3 then first of all, da = 0, so that Ca is a cyclic cocycle by B). Then, by B) of Lemma 3, we have
Ca = (-l)*~'BoC73 and, since C» is cyclic, ACO = (d - L- + l)Ca, where A is cyclic antisymmetrization;
thus Ca = G_'i*+"i' BCp belongs to the range of B.
Thus, С is a well-defined mapping from #*(По(<*)) to HCd~k(B)/lmB, and the assertion D) follows.
6) Bivector potentials on an (А, ?)-biniodule (fj,D, j)
Let А, Б be л pair of f-algebras and let (#, D, j) be an even A"-cycle on A ® Б" that satisfies the order 1
condition:
[[D,a},b]=0
, b 6 B°).
(*)
Since (f), D, 7) is a Л'-cycle over A ® B° = C, the entire construction of Section 1 applies, in particular the
concepts of a vector potential V and its Yang-Mills action YM(V) when (S$,D,y) is (d,oo)-summable. We
only want to remark here that if one considers the group G = Uj. x И& which is the product of the unitary
group of A by that of B, then there is a natural affine subspace of the space Уд^до of vector potentials for
A ® S°, which satisfies the following conditions:
a) V is invariant under the affine action of G ~Ua x Uq.
0) For every К 6 V the operator D+V also satisfies the order 1 condition (*).
Proposition 5. IfV = V^ +VBo is the snbspace o/V^qb» of .turns of vector potentials relative to A and B°,
then V satisfies the above conditions n) and /1).
The action of the unitary group of AqB" on vector polonl inl.s is determined (cf. Section 1) by the equality
g(D + V)g* — D + ~f3(V). Let us specialize it to elements ;/ = mi e Ua x Ua and verify a). Let V =
V" + Vb E Уд + VB . Then
uv(D + V" + Vb)i"u" = uiiDb'ii" + uV°u' +vVbv',
since, by the order 1 condition, every element V of V (resp. K'of V') commutes with Б (resp. A). Next,
uvDv'u' = u{D+v[D, v"]) = P+ «[/?,»¦] + v[D,v'],
using again the order 1 condition. Thus we get
which shows that the action of the gauge group is the product action. One checks 0) similarly.
-186
VI.5. The standard U(\) x SUB) x SUC) model
In this section, we shall start from the main point of the computation of our Yang-Mills functional in
Example b) of Section 3 (referred to briefly as Example 3b)), i.e., in the case of the product of a continuum by
a discrete 2-point space. The point is that we recovered the Glashow-Weinberg—Salam model for leptons with
the five different pieces of its Lagrangian, from this simple modification of the D-dimensional) continuum.
The question we shall answer in the present section is the following: Can one, by a similar procedure,
incorporate the quarks as well as strong interactions?
Before embarking on this problem, some preparation is required to explain better what our aim is. First,
there is at present A993) no question that the standard model of electro-weak and strong interactions is
a remarkably successful phenomenological model of particle physics. Since I did not take any part in its
elaboration, I shall refrain from a survey of the experimental roots of this model or of the long history of
its elaboration. I refer the reader to the beautiful book of A. Pais [P] or to the more technical papers [Ell].
This seems an absolute prerequisite for a mathematician reader of the present section, who might otherwise
underestimate the depth of the physical roots of the model.
Next, by the work of t'Hooft this model is renormalizable ([Bo-S], [F-S]), a necessary requirement for applying
the only known perturbative recipe for quantizing the theory. It nevertheless has problems, such as the
naturality problem [Ell], which make specialists doubt that, it is really of fundamental significance, thus
leading them to look for alternative routes, grand unification, technicolor. These alternate routes all share
a common feature: they deny any fundamental significance to the lliggs boson.
Our contribution does not throw any new light on the above theoretical problems of the standard model, since
it is limited to the classical level. However, it specifies very precisely which modification of the continuum, in
fact its replacement by aproduct with ^finite space, entails that the Lagrangian of quantum electrodynamics
becomes the Lagrangian of the standard model. As we shall see, the geometry of the finite set will be, as
advocated above, specified by its Dirac operator, and this will be an operator in a finite-dimensional Hilbert
space encoding both the masses of the elementary particles and the Kobayashi-Maskawa mixing parameters.
Once the structure of this finite space F is given, we merely apply our general action to the space (continuum) x
F to get the standard model action. In many ways, our contribution should be regarded as an interpreta-
interpretation, of ageometric nature, of all the intricacies of the most accurate phenomenological model of high-energy
physics; if it makes the model more intelligible to a mathematical audience, then our purpose will in some
small measure be achieved. It does undoubtedly confirm that high-energy (i.e., small-distance) physics is in
fact unveiling the fine structure of space-time. Finally, it gives a status to the Higgs boson as just another
gauge field, but corresponding to a finite difference rather than a differential.
a) The dictionary
We shall first need to refine our dictionary between the language of nonconimutative geometry and that of
particle physics.
In the basic data of noncommntativp geometry, namely (.4, .t>, D), the llilhert space f) and the operator D
have a straightforward translation (cf. Example 3b)):
/S = the Hilbert space of Euclidean fermions.
D = the inverse of the Euclidean propagator of fermions.
The algebra A is related to the gauge group of the second kind, but the relation deserves to be spelled out
at the mathematical level. It is given by
*-algebra A —¦ unitary group U(A).
We need to make some comments on this functor U from «-algebras to groups.
First of all, a similar functor, namely A -~ GL(A), which replaces an algebra A by its group of invertible
elements, plays a fundamental role in algebraic Л'-theory ([Mil]).
487

щ
Next, when the above functor U is applied to Mn(A), the algebra of n x n matrices over A, and one retains
the corresponding inclusion of U{n) = W(A/n(C)) С G = И(М„(Л)), one recovers uniquely the algebra A
from the pair of groups U(n) С G, provided that n > 1 . Indeed, at the Lie algebra level one knows the
inclusion of the complexified Lie algebra of U(n), i.e., the Lie algebra of matrices, in the complexified Lie
algebra of U(Mn(A)), i.e., again the Lie algebra М„(А) with bracket [A,B] = AB - В A. For simplicity let
n = 2, let e,j be the usual matrix units in A/2(C), and let %H = ец — e2a = I . Then, the equation
[H, x] = x characterizes the subspace of M?(A) of elements of the form ft „ , a 6 A.
The algebra structure on the vector space {л; [Н, x] — x] is then uniquely recovered from the equality
[[z,e2i],eu],y] = z
for x =
0
1 ГО ab
]-*=[o о
This shows that the replacement of an algebra by its associated groups loses very little information, at least
in the nonabelian case. Giving the algebra A from which (lie group G = U(A) comes, singles out a very
narrow class of representations of G. Indeed, there is a natural mapping
Res
Rtp(A) —- Rep(G)
which associates, to every unitary representation it of A on a llilbert space Д,, its restriction to the unitary
group U{A). This mapping is by construction compatible with direct sums, but—and this is an essential
point—while group representations p\,pi € RepG can be tensorcd, yielding a representation p = p\ ®/>2 of
G on Syx ® J52i there is no corresponding operation for representations of an (involutive) algebra.
In particular, the above mapping Res : Rep(>t) — Rep{U(A)) is in general far from snrjective. Just take
the simplest example, A = Л/„(С) so that G = U(n), and the range of Res consists of the multiples of the
fundamental representation ip of 0(n) in Cn. In other words, to assume that a representation p of G comes
from a representation of A by restriction is a very restrictive condition.
Now, the group G does have a clear significance as the group of local gauge transformations and we shall
take it as a characteristic of its representation in the one-particle space of Euclidean fermions that this
representation (of U(A)) comes by restriction from a representation of A.
Notice also that it is only through the group G and (lie corresponding algebra A that Euclidean space-time
enters the picture, since in quantum field theory the space-time points merely play the role of indices or
labels except in their occurrence in the construction of the group of local gauge transformations. Thus, the
last piece of the dictionary is:
¦-algebra A —* H(A) = Group of local gauge transformations
on Euclidean space-time.
All of this works perfectly well for the Glasliow-Weinberg-Salam model for leptons, as shown in Example 3b),
and we shall see how the incorporation of quarks and strong interactions will require the setup of Potncare
duality and himodules of Section 4. We first need to recall the detailed description of the standard model.
C) The standard model
Just as for the Glashow-Weinberg-Salam model for leptons, the Lagrangian of the standard model contains
five different terms,
? = ?g + ?/ + ?„ + ?i- + ?v,
which we now describe together with the field content of the theory.
1) The pure gauge bosou part
where G,ra is the field strength tensor of an 51/B) gauge field W^, F^ is the field strength tensor of a
U{1) gauge field B^, and Н^ь is the field strength tensor of an 5?/C) gauge field !/„». This last gauge field,
the glnon field, is the carrier of the strong force; the gauge group SU{3) is the color group, and is thus the
essential new ingredient. The respective coupling constants for the fields W, B, V will be denoted 9, g', g",
consistent with the previous notation.
2) The fermion kinetic term ?j
To the leptonic terms •
g'^
one adds the following similar terms involving the quarks:
^'
«У ^
For each of the three generations of quarks
two right-handed SU('2) singlets (such as
• \b
one has a left-handed isodoublet (such as
), and each quark field appears in 3 colors so that for instance
there are three ur fields: u^, w^j, u^. ЛИ of these quark fields are thus in the fundamental representation 3
of Sf/C).
The hypercharges Y/,, Yr are identical for different generations and are given by the following table:
e,;i,r vtvltvT u,c,l d,s,b
YL -1 -1 1/3 1/3
Yn -2 't/3 -2/3
These numbers are not explained but are set by hand so as to get the correct electromagnetic charges Qem
from the formulas
2Qem = YL + 2I3, 2Qom = Ул,
where 7з is the 3rd generator of the weak isospin group SU{2).
3) The kinetic terms for the Higgs fields
where tp = ^s an SUB) doublet of complex scalar fields with hypercharge Y^ = 1. This term is exactly
th i h GWS dl f l
488
the same as in the G.W.S. model for leptons.
4) The Yukawa coupling of Higgs fields with ferinions
'189

A- = -
¦ Л)].
where Hjji is a general coupling matrix in the space of different fermions, about which we must now be
more explicit. First, there is no Hj/< ф 0 between leptons and quarks, so that ?y is a sum of a leptonic and
a quark part. Since there is no right-handed neutrino in this model, the leptonic part can always be put into
the form
?y
,lepton = ~Gt(Lt -<p)eR-
- GT(Lr <p)tr + h.c,
where Le is the isodoublet e> and similarly for the other generations. The coupling constants Ge, Gp,
GT provide the lepton masses through the Higgs vacuum contribution.
The quark Yukawa coupling is more complicated owing to new terms which provide the masses of the up
particles, and to the mixing angles. We have three new terms. The first is of the form
GLuRf>,
(*)
where the isodoublet L — \ is obtained from a left-handed up quark and a mixing qt of left-handed
L (IL J
down quarks (taken from the three families), the two others h.-we a similar structure but with the up quark
replaced by the charm and top quarks respectively. Also, ф needs to have the same isospin but opposite
hypercharge to the Higgs doublet y? and is given by
= J<p', J =
0 1
-1 0
(**)
We refer to [Ell] for more details on this point, to which we shall return later.
5) The Higgs self-interaction
Су =ц2ч
has exactly the same form as in the previous case.
Thus, we see that there are essentially three novel features of the complete standard model with respect to
the leptonic case:
A) The new gauge symmetry: color, with gluons responsible for the strong interaction.
B) The new values 3, 5, ^ of the hypercharge for quarks.
C) The new Yukawa coupling terms (*).
We shall now briefly explain how these new features motivate л corresponding modification of our Exam-
Example 3b), which led us above to the G.W.S. model for leptous. First, our model will still be a product of an
ordinary Euclidean continuum by a finite space.
In Example 3b), for the algebra A of functions on the finite spare, we took the algebra С„ ф С». But since we
then considered a bundle on {a, b] with fiber C2 over a and С over b, we could have in an equivalent fashion
taken А = Л^(С)фС and then dealt with vector potentials, instead of connections on vector bundles. Let us
see how C) leads to replacing A = А/о(С)ФС by A = НфС, where H is the Hamilton algebra of quaternions.
The point is simply that the equation (**) which relates tp and tp is the same as the unitary equivalence 2 ~* 5
of the fundamental representation 2 of SUB) with the complex-conjugate or contragradient representation,
i.e., we have
0 e uB),
su(-2).
490
Let us simply remark that x € Л/2(С), JxJ'1 = x defines an algebra, the quaternion algebra H.
Next, let us see how A) leads us to the formalism of bimodules and Poincare duality of Section 4. Indeed,
let us look at. any isodoublet of the form "L of left-handed quarks. It appears in 3 colors,
"I ui 
which makes it clear that the corresponding representation of SU{2) x 5?/C) is the external tensor product
2si/B) ® 3si/C) of their fundamental representations. It is easy to convince oneself that even if one neglects
the nuance between U(n) and SU(n) in general, there is no way to obtain such groups and representations
from a single algebra and its unitary group. The solution that we found, namely to take (>t,B)-bimodules,
with В=Сф Мз(С) (and A = С ф Н as above) is in fact already suggested by the following picture in the
paper [Ell] of J. Ellis, very close to that of the diagonal Д С А' х X in Poincare duality:
SUB)
X
•r : "R>aS
G
Ш
¦a
colour
Fig.5. The apparent "generation" or "family" structure of fundamental fermions. The horizontal axis
corresponds to 5?/C) colour properties, the vertical axis to SUB) x ?/A) representation contents
We just refine it by taking algebras—С ф H for the y-axis, С ф Мз(С) for the x'-axis—instead of groups,
which allows us to better account for the leptons (by the С of С ф Л/з(С)).
Finally, we shall get a conceptual understanding of the numbers B) from a general unimodularity condition
that makes sense in noncommutative geometry, but we need not anticipate on that point.
We are now ready to describe in detail the geometric structure of the finite space F which, once crossed
by R4, gives the standard model.
491

7) Geometric structure of the finite space F
This structure is given by an (.4,B)-module (S),D,j), where A is the «-algebra СфН while В is the «-
algebra С Ф Л/з(С). Unlike В, the algebra A is only an algebra over R. The «-representations jt of A on
a finite-dimensional Hilbert space are characterized (up to unitary equivalence) by three multiplicities: n+,
n_, m, where Й. = C"+ ф Cn- Ф C3m; if a = (\,q) € A = С фН, then jr(a) is the block diagonal matrix
jr(a) =
N j*? U ®idmj ,
where the quaternion q is qr = a + /?j with a,/? € С С H. The representation of the complex «-algebra В on
Я gives a decomposition
in which В acts by jtF) = b0 Ф A ® 61) for Ь — (bt,,b\) S С ф М3(С), thus the commuting representation
of Л is given by a pair jr0, щ of representations on S)o and !i\- The (.4,B)-bimodule Й is thus completely
described by the six multiplicities: (n°+,n°_,m0) for To and ("V,"!."»1) for it\. We shall take these to be
of the form
(where N will eventually be the number of generations N = 3). We shall take the Z/2 grading 7 in Й to be
given, as in Example 3b), by the element 7 = A, —1) of the center of A. Finally, we shall take for D the
most general self-adjoint operator in # that anticoramntes with 7 (Dj = —yD) and commutes with С ® B,
where С С Л is the diagonal subalgebra {(A, A), A S C}. (As we shall see, D encodes both the masses of the
fermions and the Kobayashi-Maskawa mixing parameters.) It follows that the action of A and the operator
D in S)o (resp. S)t) have the following general form (with </ = n +/3j S H):
0 -
(/
f
0
0
0
'/) =
0
7
0
0
f
0
uo
0
0
a
-H
0
a
о
-/? «J
o-
0
0
Ot.
Do =
Md
. 0
0
Me
u 0
0
M,
0
0
0
Л-Л,
0
0
0
Щ
0
0
where A/e, A/u, Af^ are arbitrary N x N complex matrices.
Since tto is a degenerate case (Mu = 0) of тг), we just restrict, to ж, in order to determine Q'D(A).
A straightforward computation gives т\(^2 ajda'j) with <ij,a'j e Л, a, = (>4,4j), 4j — <4 +Pjl, 4'j = a'j
we have
0 A'
where X, Y are the matrices
л- = I I, у
-ЩЩ MX
vith
492
It follows that fi{j(.4) = H ф Н with the .4-bimodule structure given by
(VA ?C,,e H),
and the differential d being again the finite difference:
(just set $i = yjj + tp23> 42 — <Pi + ^У with the above y;'s).
Finally, the involution * on ?2b(>t) is given by
The space J/ of vector potentials is thus naturally isomorphic to H, and a similar computation shows that
Q?D(A) = H ф Н with the .4-biinodiile structure
') = (Л1ЛЛ',№'/) (VA.A'eC, ?,<?'€ H);
j, q'2) = (
the product Q'D x SJJ, —> SJJ, is given by
(tfi
and the differential d : QlD -* Qj, by
d(»i.'ft) = ('/1 +'/=,91 +'b)-
Thus, the curvature в of a vector potential V — G,17*) is
9 = dV + V2 = G +q'+ qq',4 + 7" +,,",,) = (|1 + ,|' - 1)A, 1),
where q —> |^| denotes the norm of quaternions.
We thus see that the action YM( V) = Тгасе(тг@)-) (we are in the 0-dimensional case) is the same symmetry-
breaking quartic potential for a pair of complex numbers as in Example 3b).
The detailed expression for the Hilbert space norm on Q^>(>1) — НфН is given, for w = G1,72), Qj — aj +/?jj,
by
where
A, = Тгасе(|Л/е|") +ЗТгасе(|Л/,1Г1 + |Л/„|4),
M, = eTracedM^I'IMMp),
А, = iTracedA/el4 +S(\Md\* + |Л/„|4 + 2|JWd|2|Mu|2)).
Finally, we shall investigate what freedom we have in the choice of the self-adjoint operators DOt Dx in J5o,
#i in the above example. Two pairs (Jr)-,.Dj) ami (У)'-,?)') give identical results if there exist unitaries
t/j : J5j —*¦ S)'j such that:
a) UjDjU- = Dj (j= 1,2),
493

p) Ujirj(a)U; = тг<(о) (Vo e A, j = 1,2).
Making use of this freedom, we can assume that Da is diagonal in J5o and has positive eigenvalues e\, 62,
e3. Thus, the situation for Do is described by these 3 positive numbers.
For #), a general element of the commutant of *\(A) is of the form
rv, о о on
0 V2 0 0
О О И, 0
LOO 0 V'3J
where the Vj are unitary operators when U\ is unitary.
Conjugating D\ by Vi replaces Л/j and Mu, respectively, by
М'л = Vi MdV3", M'u = V-, Mu V3.
We thus see that we can assume that both Mu and Mj are positive matrices and that one of them, say Mu,
is diagonal.
The invariants are thus the eigenvalues of Mu and Afj, i.e., a total of 6 positive numbers, and the pair
of maximal abelian subalgebras generated by Mu and Л/,/. Since any pair Ai, Ai of maximal abelian
subalgebras of A/3(C) are conjugate by a unitary IV, IKyliiV = Ai, which is given modulo the unitary
groups U(Aj), there remain 4 parameters with which to specify W so that WMjW" is also diagonal. Such
a W corresponds to the Kobayashi-Maskawa mixing matrix in the standard model.
6) Geometric structure of the standard model
We shall show in this section how the standard model is obtained from the product geometry of the usual
4-dimensional continuum by the above finite geometry F. Thus, we let M be a 4-dimensional spin manifold
and (Ь2,|9д/,75) its Dirac К -cycle. The product geometry is, according to the general rule for forming
products, described by the algebras
А =Г(И)8(СфН), В=Г(«H(СфМ3(С)).
The Ililbert space Я = L2(A/,SH flF, where f)F is described in 7) above, i.e., S$F = 30®(Si®C3). There
is a corresponding decomposition И = Яо Ф [ffi 0 С3), with corresponding representations ttj of A on Hj.
Then D = дм ® 1 + 75 0 DF, where DF is as above. This give.s a decomposition D = Dq ф (D\ ® 1), where,
according to 7), we take Me, Mu and M4 to be positive matrices:
Do =
¦ дм 0 1
75 0 Me
0
¦ дм ® 1
0
7s 0 Md
0
75 0 M,
дм® 1
0
0
дм® l
0
Тя ® М„
дм
7s
а,
о ¦
0
0i.
© мй
0
/0 1
0
0
75® М
0
дм® 1
We shall first restrict attention to the algebra A, the case of В being easier. Note that A — C~(Af, С) ф
C°°(A/,H), so that every a g A is given by a pair (/,</) consisting of a C-valued function / on M and an
H-valued function q on M.
494
Let us first compute S}{>(.4). Given p = ^a.da', ё ?2'(Л), with a,,a', € A, we have a, = (/.,«,),
a', = (/,',??), where /„, }', are complex-valued functions on M and g,, jj are H-v
the form
l-valued functions on M, of
Then
q, =«.
® Mu
^75 ® H. -
*.(/>) =
where A = Ylf'df, is a C-valued 1-form on Л/, and Wi + W'jj = IV = ? ?»d?i is an H-valued 1-form on M
(cf. [At4]).
Also, tpj, (p'- axe complex-valued functions on M given by the same formulas as above for the finite geometry,
namely,
This means that the pair A7,17') of H-valued functions, given by 4 = <p\ + ipzi, q' = v\ + V2J1 satisfies
where, Аа'я — (q's — Ад,А'я — i/^) is the fmite-dilTorenee operation, wliile the >t-bimodule structure on the
space of H ф H-valued functions (gi ,172) is given by
(/,<1){<Н,<12) = (/Ч1,'1Ч2), (</ь'Ы(/,</) = (>li'l,4-*f) for all (f,q) € A.
This shows that ?2j,(>t) is the direct sum of two subspaces:
where:
?}д , the subspace of elements of differential type, is the space of pairs (A, IV) consisting of a C-valued
1-form A on M and an H-valued 1-form W on M\
!){,''', the subspace of elements of finite-difference type, is the space of pairs D1,42) of H-valued functions
on A/, with the above >t-bimodule structure.
The geometric picture is that of two copies Mr and Me of Л/, with C-valued functions on Mr, and H-valued
functions on Ml More generally, the differential forms on Mr are C-valued, whereas on Ml they are H-
valued, exactly as in Atiyah's book [At.,]. Of course the finite difference mixes both sides, so they are not
independent.
Given an element a = (f,q) of A, the element da of QlD has a differential component (d/,d4), given by the
C-valued 1-form d/ and the H-valued 1-form dq; and a finite-difference component D — /, / — 4).
The involution * on Q'D is given by
so that a vector potential V is givon by:
495

a) an ordinary ?/A) vector potential on M;
/?) an SU{2) vector potential on M (cf. [At.,]);
7) a pair q — a + /?j of complex scalar fields on M.
The next step is to compute Q"q(A) as well as the product
U'o x QlD -. SJj,
and the differential d : П}, —> SJ|,.
We shall first state the result. It holds provided a certain uomlcgeneracy condition is satisfied, namely, that
the following matrices are not scalar multiples of the identity matrix:
Ml or \г(М\ + М„2), Mj or M? or Ml.
The result then is that an element of 0,2D(A) has:
1) a component of type B,0) given by a pair (F,G) consisting of a C-valued 2-form F and an H-valued
2-form G on M;
2) a component of type A,1) given by a pair (шьыо) of quaternionic 1 -forms wi, w2 on M;
3) a component of type @,2) given by a pair G1, 72) of qiiaternionic functions on M.
Moreover, with the obvious notations, the following formulas hokl:
d{(A,W),(quqi))
= {(dA,d\V),(drn
((A,W),(qi,q3) ¦ ({A',W')A4\,4i))
W-A),(q,
+ 72)}.
they show that we are dealing with the graded tensor product of i.he differential algebra of A/(the de Rham
algebra) by the differential algebra Qp of the finite space F.
In order to compute the Ililbert space norm on Qp{A), wo have to explicitly write down the class in
t(u2{A))/v(J) associated with an element ((F, G), (i.'i,u?|, G1,1ft)) of U'JD(A). We shall first write what
happens for лч; the case of тг0 is a degenerate case obtained by taking М„ = 0. The subspace v(J) only
interferes with the elements of degree 0 in the Clifford algebra, and any element of jr(.7) is given by 5 complex-
valued functions a, /?, 7, Y, Z on M, whose representation in fy^ is given by:
r® 1 0 0 0
0 <*@1 0 0
0 0 Tn Ti2
0 0 T2i T22
In the Hilbert space Йо, the same element is represented by
a® 1 О О
о г;, г;,
о ц, ти
wi th
T -
\ М*№-
The elements (F,G) and (wbw2) of degree B,0) and A,1) have canonical representatives given by the
following expressions, where wt = at + /?jj and ak, /3k are complex-valued 1-forms on M:
tor Sju
о
о
'7(ЛO5 © М«
'7(O2O5GA/U
О j7(oi
i27(G) 0 1
•7(«]O5 ©
-»7(?iOs
О
О
for й,,
0
0
in Йо.
О О
The component G1, 72) of degree @,2), 7, = n; +/?,j, has the following representative modulo
MdMu 0
a, M,T
0
0
Л/2
0
0
72
0-
0
The noudegeiieracy condition ensures that the various terms «11th as «iA/J do not disappear modulo *(J)-
It is now straightforward, as in Section 3, to compute the action. One gets the five terms of the GA) x 5GB)
part of the standard model, with the new Yukawa coupling terms C). But before we discuss the coefficients
with which they arise we need to show how to reduce the gauge group U(A) x U{S) of the theory to the
global gauge group U(\) x SU('i) x SU(Z) and obtain the intricate table of hypercharges of the standard
model. Note that we did not make the straightforward calculation of vector potentials and action for the
algebra B, which yield a pure gauge action with group U(\) x IJC).
497

e) Unimodularity condition and hypercharges
We shall now see how, from a general condition of unimodularity valid in tile general context of noncommu-
tative geometry, one obtains the intricate table of hypercharges of elementary particles:
e, д, т
YL -1
YR -2
-1
u,c,t d,s,b
1/3 1/3
4/3 -2/3
(Note that since we are dealing with the Lie algebra of U(l), this table is only determined up to a common
scale.)
To obtain these values and at the same time obtain the global gauge group U(l) x SUB) x SGC), we shall
simply replace the local gauge group U(A) x U(B) by its unnnodular subgroup SU relative to A.
In our context, the notion of determinant of a unitary makes sense provided that a trace т is given. More
precisely by [Harp-S], given a C-algebra С and a self-adjoint trace r on С (i.e., т(х") = т(х) for all x € C),
one obtains the phase of the determinant of a unitary и as follows:
1 f'
Phaser(«) = ir-. I T(u(t)'u(t)-l)dt,
^7гг Jo
where u(() is a smooth path of unitaries joining u to 1. Thus, this phase is only denned in the connected
component U0(C) of the identity, and it is ambiguous, by the image (т, Л'о(С)) of K0(C) under the trace r,
which is a countable subgroup of R.
The condition Phaser(n) = 0 is well-defined and gives a normal subgroup of U(C). We let STU(C) be the
connected component of its identity element.
Now let A and В be *-algebras and let ($y, D) be a (d, oo)-sunimahle bimodule over A, B. We shall apply
the above considerations to the C'*-algebra С on J5 generated by A and В and to the family of traces r on
С given by the self-adjoint elements p — p" of the center of A:
т„(х) = 1iw(pxD'd) (V* € C).
We thus get a normal subgroup Sa(C) of the unitary group of С by intersecting all of the Sr, т = Tf as
above. Since U(A) x U(B) is a subgroup of W(C), its intersection with Sa(C) gives a normal subgroup
S(A, ?>) of U(A) x U(B). We shall see what S(A,B) is, in simple examples, but our main point now is:
Theorem 1. Let [A,B,'H,D) be ihe product geometi-y of a 4-dimensional Riemannian spin manifold by the
finite geometry F. Then, the group S(A, B) is equal to Мар(Л/, {'A) x SUB)xSUC)) and its representation
in "H is, for ihe f/(l) factor, given by ihe above table of hypeivharges.
By construction, A = C°°(M)®Af, Б =
it follows by a straightforward argument that
= L-(M,S)® bF, D- дм ® I + 75® DF, and
S{A, в) = Map(M, S{AF, Sf)),
where S(Af, Bf) is defined as above, but using the ordinary trace in the finite-dimensional space fiF instead
of the Dixmier trace. Thus, we need only compute the group S(Af, Bf) over a point-, and its representation
in Sip- Now, every self-adjoint element p of the center Z(Af) is of the form p — Xie + A2(l - e), where the
Xj are real numbers, e = @, 1)ёСфН and 1 — e = A,0). It follows easily that
S(Af,8f)= (U(Af)
(SU(eS)F)
498
In other words, the unimodularity condition means that the action is unimodular on both ef>F and A — e)f>F.
Let, then, U be an element of U(Af) x U(Bf)- It is given by a quadruple
и - ((A,«),(«,«)); x eu(i),4 e SUB),u eU(i),v eu(z).
We have ?iF = ЯоФ^!® С3) and, with the notations of example b) of section 3, the action of V on S)F is
given by
This operator restricts to both eS)F, and A — e)S)F, and we have to compute the determinants of these
restrictions. With JV generations, we get
det(Ue) = u2N x (det(t))JJV, det(y,_e) = (Аи)" х (dett)JJV,
hence the unimodularity condition means independently of N that
A = и, det v = u~'. (*)
It follows that S(Af,Bf) = U(l) x Si'B) x SUC). Let us compute the table of hypercharges, say, for
example, by taking u as generator, as it is represented. Since A = u, for S)o we get
Аи 0 ] _ IV 0 ]
0 qu\ ~ [ 0 qu\ '
which corresponds to hypercharge 2 for ец, дл, тд and 1 for ei, щ,. Г/, and ve, v^, vT. For J5j we get
v = «ой'3 with vo € S?/C) and u1'3 a cube root of и, so that
tt,(A, q)u
-1/3 _
Г Аи" О
0 A"lu-
0 0
0-
which corresponds to hypercharge | for dft, sr, Ьц; -?¦ for иц, сд, ir; and —j for the left-handed quarks.
An overall sign is of course irrelevant (change и to u~~l), so we get the desired table.
Remarks. 1) Let A be the algebra of functions on a 'i-dimensional spin manifold M, let (J5, D) be the sum
of N copies of Ike Dirac K-cycle (L-,dn/) on M, and IS the commutant of AU[D,A] on Si. Then, the group
S(A,B) is (Ле group Map(M,SU(N)) of local gauge transformations associated with the global gauge group
SU(N).
2) To the above reduction of the gauge group there corresponds a similar reduction of bivector potentials V,
which in the case of Theorem 1 means that V is iraceless in ihe spaces eS)f and A — e)Sip, and in ihe case
of Remark 1 yields the corresponding SU(N) pure gauge theory.
We are now ready to compare our theory with the standard model of electro-weak and strong interactions.
We first note that the bimodule ($), D) over A, & of Theorem 1 is not irreducible, i.e., the commutant of the
algebra generated by А, В and D does not reduce to C. With generic Me, Mu, Md, one can check that this
commutant is С so that (S),D) splits as a direct sum of 4 irreducible bimodules, three for the three lepton
generations, i.e., for ?H, and one for j^ which is irreducible.
Theorem 2. Let (A,B,S),D) be the product geometry of a ^-dimensional spin Riemannian manifold by
the finite geometry F. Let S(A,B) be the reduced gauge group and V the corresponding space of bivector
potentials. Then, the following action gives the standard model with its 18 free parameters:
499

where Ад, Ав belong to the commutanl of the bimodule
(ф, Dvifi),
, Ob are the respective curvatures.
The proof is straightforward, given Theorem 1 and the above computations.
Let us now note that, at the classical level which we have not left so far, there is a natural 17-dimensional
subspace given by the following restriction:
Ад, Ав belong to the algebra generated by A and B.
The corresponding classical relation can only have a heuristic value. In the limit of large top mass, i.e.,
neglecting the other fermion masses, one gets the following:
ifA.4>0;
2)
m,
Here x is the ratio between the two eigenvalues of the operator Ад, so that the value x = 1 is the most
natural, leading to m, = '2ni\y. None of these relations is a physical prediction but if they were nearly
satisfied by the physical values of in, and m// it would show that the change of parametrization in the
standard model given by the above theorem is qualitatively a good one at the classical level.
In order to transform our interpretation of the standard model into a predictive theory it is important to
solve the following problems:
1) Find a non trivial finite quantum group of symmetries of the finite space F.
2) Determine the structure of the Clifford algebra of the finite space F, given by the linear map from the
space H ф H of 1-forms into the algebra of enclomorpliisms of Tip which to a 1-form Sa,- dbi associates
500
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