NORTH-HOLLAND
MATHEMATICS STUDIES
185
Editor: Saul LUBKIN
Topological Algebras
V.K. BALACHANDRAN
NORTH-HOLLAND

TOPOLOGICAL ALGEBRAS

NORTH-HOLLAND MATHEMATICS STUDIES 185
(Continuation of the Notas de Matematica)
Editor: Saul LUBKIN
University of Rochester
New York, U.S.A.
2000
ELSEVIER
Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

TOPOLOGICAL ALGEBRAS
V.K. BALACHANDRAN
Ramanujan Institute for Advanced Study in Mathematics
Chennai, India
2000
ELSEVIER
Amsterdam - Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

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Printed in The Netherlands.

To my grand-children
Shobhana and Vivek
for delaying the completion
of the writing of the book

CONTENTS
Preface ix
Chapter 1: Algebraic Preliminaries
§ 1. Some Basic Concepts and Results 1
§ 2. Ideals and Radical 12
§ 3. Characters and Hypermaximal Ideals 22
§ 4. Extensions of Ideals 30
§ 5. Regular Representation and Primitive Ideal 36
§ 6. Real and Complex Algebras 43
§ 7. Spectrum and Quasi-spectrum 52
§ 8. Extended Spectrum and Extended Quasi-spectrum 61
§ 9. Strictly Real Algebras 67
Chapter 2: Topological Preliminaries
§ 1. Topological Groups and Linear Spaces 73
§ 2. Topological Algebras 84
§ 3. Completions of Topological Linear Spaces and
Topological Algebras 91
Chapter 3: Some Types of Topological Algebras
§ 1. Quarter-norms 100
§ 2. p -Seminorms 110
§3. Quarternormed Algebras; (F) Algebras 116
§4. p -Seminormed Algebras; p-Banach Algebras 131
§ 5. Bounded Linear Transformations on p-Seminormed
Linear Spaces 140
§ 6. Topological Algebras with Inverses 148
§ 7. Topological Zero Divisors 158
Chapter 4: Locally Pseudo-Convex Spaces and Algebras
§ 1. p -Convexity 174
§ 2. Locally Bounded Algebras 185
§ 3. Locally Pseudo-Convex Spaces 189
§ 4. Locally Pseudo-Convex Algebras 195
§ 5. Projective Limit Decomposition 201
§ 6. Metrizable Locally Pseudo-Convex Algebras 207
§ 7. Ample Algebras 213
§ 8. Topological Spectral Radius 216

Vlll
Chapter 5: Some Analysis
§ 1. Vector-valued Differentiability and Analyticity 222
§ 2. Exponential and Logarithmic Vector Functions 227
§ 3. Square Roots and Quasi-square Roots 234
§4. Complex Vector-valued Line Integrals and Cauchy's Theorems ...239
§ 5. Power Series Operations in Topological Algebras 253
Chapter 6: Spectral Analysis in Topological Algebras
§ 1. Spectral Properties 262
§ 2. The Resolvent Function 264
§ 3. The Pseudo-Resolvent Function 275
§ 4. Spectral Algebras 282
§ 5. Gelfand-Mazur and Other Similar Theorems 284
§ 6. Turpin's Theorem on Locally Convex Algebras 290
Chapter 7: Gelfand Representation Theory
§ 1. Ideals of Topological Algebras 296
§ 2. Gelfand Algebras 303
§ 3. The Gelfand Representation 311
§ 4. GB Algebras 324
§ 5. Holomorphic Functional Calculus for a Single Algebra Element .. .335
§ 6. Automorphisms and Derivations 344
Chapter 8: Commutative Topological Algebras
§ 1. Function Algebras 353
§ 2. Shilov Boundary 361
§ 3. Hull-Kernel Topology 370
§ 4. Completely Regular Algebras 377
§ 5. Holomorphic Functional Calculus for Several Commutative
Algebra Elements 388
§ 6. Shilov Idempotent Theorem 403
Chapter 9: Norm Uniqueness Theorems
§ 1. Norm-uniqueness Theorem of Gelfand 411
§ 2. Rickart Seperating Function 412
§ 3. Topological Modules 416
§ 4. Norm-Uniqueness Theorems for Non-commutative Algebras 419
Appendix 427
Type Chart 429
Bibliography 431
Index 435
List of Special Symbols 443
List of Special Abbreviations 445

PREFACE
There are very few books devoted to general topological
algebras. This book is the outcome of an attempt to present a fairly
self-contained and systematic exposition of a number of basic
topics concerning such algebras. For the sake of completeness and to
increase the usefulness of the book as a reference source (for the
material treated) I have not hesitated in stating explicitly (and
proving) herein several corollaries and deductions emerging from
the main results. I hope to be able to follow this volume with
another treating algebras with involution and other topics.
In this book I have considered both complex and real algebras,
with and without unity. There is found in the literature on Ba-
nach algebras two types of real algebras: the strictly real algebras
and the formally real algebras. Both these types of algebras are
treated here in more general settings and it turns out that for such
strictly real algebras most of the results available for
corresponding complex algebras can be extended while such formally real
algebras share a few properties with the corresponding complex
algebras.
In the treatment of the spectrum I have made some
conventional changes. The usage of the term "spectrum of an element" is
limited to unital algebras. To take care of the non-unital case the
following procedure is adopted. If A is any algebra (non-unital
or not) it has its formal unitization A\ a unital algebra. The
spectrum with respect to A\ is denoted by a^ , and set for x
in A, a '(x) = °a(z) = aA\{x) ( the spectrum of x in Ai),
and call a \x) the quasi-spectrum of x . If A is itself unital the
spectrum <r(x) = <r^(x) make sense and we have the simple
relation a '(x) — a(x) |J {0} . It is to be noted that always 0 G a '(x)
so that a '(x) ^ 0. For a real algebra A we have, for x in A ,
besides a '(x),a(x) also the extended quasi-spectrum a '(x) and
extended spectrum &(x) (when A is unital) which are defined in
the following way. Every real algebra A has a complexification, a
complex algebra A. Set a '(x) = v'Ax) and a(x) — <r^(x) (note
that when A is unital A is also unital). The above changes or
extensions in the terminology for the spectrum I feel are ideologically
justified and technically useful.

X
The book consists of nine chapters. Chapter 1 is devoted to
algebraic preliminaries. Here I have included more material than
strictly needed in the following chapters; for instance it contains
an interesting result generalizing a well-known property of the
Heisenberg commutation equation but this is not used anywhere
in the book. The chapter can be profitably read independently for
the topics treated. For the definition of the circle operation in a
ring I have followed Perlis-Kaplansky rather than Hille-Jacobson;
thus the definition adopted here is at variance with the one
employed by Ricart or Bonsall-Duncan in their books (this has to be
bourne in mind when comparing results) but agrees with that in
Neumark's book.
Chapter 2 deals with some of the basic definition and results
concerning topological groups, topological linear spaces and
topological algebras. For dealing with continuity questions I have used
largely the approach via net convergence since I find this
particularly suitable. I have included in this chapter a construction for
the completion of a topological algebra using nets, imitating the
classical construction of Hausdorff for the completion of a
metric space. In connection with the construction I have isolated a
property which I have called "essentially bounded" applicable to
a net in a topological linear space. This property is weaker than
boundeness of the net ( as a set). In fact there are convergent nets
which are not bounded but every convergent - or even Cauchy -
net is essentially bounded.
In Chapter 3 I have considered some generalizations of the
norm: quarter-norm, (F) norm, ^-seminorm, ^-norm. These give
rise to different types of topological algebras: quarter-normed
algebras, (F) algebras, ^-seminormed algebras, p -normed algebras
and p -Banach algebras. Some properties of these algebras are
studied here. Based on special properties of quasi-inversion or
inversion, topologival algebras come under the following categories:
C algebras, Q algebras, I algebras, CQ algebras and CI algebras.
Various results concerning there categories are obtained. Finally
the chapter contains a large number of results pertaining to
topological zero divisors.
Chapter 4 is concerned with a generalization of the notion of
convexity called ^-convexity. This concept leads to some gener-

XI
alizations of locally convex spaces and algebras giving rise to;
locally ^-convex, locally pseudo-convex spaces and algebras; locally
sm. ^-convex, locally sm. pseudo-convex algebras; pseudo Frechet
algebras. After giving a basic treatment of these spaces and
algebras the projective limit decomposition, discovered by Michael, is
obtained for certain locally sm. pseudo-convex algebras. This
decomposition enables the extension to these algebras of some results
available for ^-Banach algebras. Also contained in this chapter is
a section on ample algebras and another on topological spectral
radius. The latter contains a proof of an important theorem, due
to Zelazko, for generating seminorm from ^-seminorm.
In chapter 5 some differential and integral analysis involving
vector valued functions is developed. The extensions to ^-Banach
algebras of the Banach algebra theorems of Nagumo (on the range
of the exponential function) and of Gleason-Kahane-Zelazko (on
characters) are obtained. There is a section devoted to square
roots and quasi-square roots which ends up with a useful result
regarding existence of idempotents. The final section concerns
power series operation in topological algebras and includes the
proof of a theorem of Mitjagin-Rolewvicz-Zelazko affirming local
submultiplicativity property for Frechet algebras on which all
entire functions can operate.
Chapter 6 is concerned with spectral analysis and applications.
Besides quasi-spectrum and resolvent function, I also consider the
pseudo-resolvent function which is a useful tool in the study of
algebras without unity. As applications of the spectral analysis
is obtained a number of Gelfand-Mazur type theorems which
include those due to Arens, Shilov, Zelazko. The last section of the
chapter is taken up with the proof of an interesting theorem due
to Turpin, affirming the local submultiplicativity property for all
commutative Frechet algebras which are Q algebras.
The Gelfand representation theory is the subject-matter of
chapter 7. For properly understanding and appreciating Gelfand's
results I have found it convenient to introduce two classes of
topological algebras called Gelfand algebras and spectrally Gelfand
algebras. Besides I have also introduced the class called GB
algebras i.e. algebras in which the spectral radius formulae of Gelfand
and Beurling (in a modified form) hold. The proof of the Beurling-

Xll
Gelfand-Zelazko theorem that every complex ^-Banach algebra is
a GB algebra is presented. Other topics considered in this
chapter are holomorphic functional calculus for a single element and,
automorphirms and derivations. The functional calculus in the
strong form has been developed for an element of a ^-Banach
algebra and using the projective limit decomposition theorem the
final result extended in the weak form to pseudo-Michael algebras.
The Singer-Wermer theorem on derivation has been obtained for
^-Banach algebras and an extended version for pseudo-Michael
algebras.
Chapter 8 deals with function algebras, Shilov boundary, hull-
kernel topology, completely regular algebras, holomorphic
functional calculus for several elements of a commutative ^-Banach
algebra and Shilov idempotent theorem. Finally, in chapter 9 an
exposition of the norm uniqueness theorems of Gelfand and
Johnson (extended to ^-Banach algebras) is given.
For writing this book I have drawn material and ideas from the
Banach algebra books of Neumark, Rickart, and Bonsall-Duncan,
from the memoir of Michael on locally convex algebra and
lecture notes of Zelazko on topological algebras. Besides, I have also
been influenced by the topological algebra book of Guichardet and
the treatment of Banach algebras in Rudin's book on functional
analysis. I wish to acknowledge here my indebtedness to these
authors.
I wish to thank P.S. Rema (a former Director) and S. Sri Bala
(the present Director) of the Ramanujan Institute for their
unstinted help in connection with the publication of the book. I
thank N. Vijayarangan (UGC project assistant) for his help in
proof reading.
I wish to record my thanks to G. Narayanan, (Assistant
Technical Officer (Computer)) of the Ramanujan Institute for his help
in the preparation of the well-executed laser print copy of the
book. Finally, I wish to express my appreciation to N.K. Mehra of
Narosa Publishing House for readily agreeing to publish the book
and for his understanding role in the production of the book.
V.K. Balachandran

CHAPTER I
ALGEBRAIC PRELIMINARIES
§ 1. Some Basic Concepts and Results
1.1.1. Recall that in a ring R an element e; (respy. > er) is
called a left (respy. right) unity if e\ a — a (respy. aer = a) for
all a in R. If e is both a 1. (=left) unity and a r. (=right) unity
then e is called a unity. If R has a unity, R is called unital. We
always assume that the unity e^O.
The meaning of a positive power am of an element a is clear.
Also, in a ring R with unity e, we define for any element a,
a0 = e. This is well-defined since unity is unique (see 1.1.2).
1.1.2. LEMMA. If R has a l.unity e\ and a r. unity er then
necessarily e\ = er — e (say) and e is a unity of R which is
moreover unique.
PROOF, e; = (ei)er = e;(er) = er. The uniqueness of e is
clear.
1.1.3. Let R be unital with unity e. An element a^ (respy.
a~ ) is called a I. (respy. r.) inverse or l.i. (respy. r.i.) of an element
a if af a — e (respy. aa~l = e). An element a-1 is called an
inverse of a if it is both a l.i. and a r.i. If the inverse a-1 of a
exits we call a invertible or regular.
1.1.4. LEMMA, (a) If a has a l.i. a;~ and a r.i. a"1 then
al = a~ = a {say) and a is the unique inverse of a. In
particular, the inverse of an element, whenever it exists, is unique.
(b) The invertible elements of a unital ring R form a group
G{ = Gi(R) under multiplication.
(c) If a e Gi then -a e G,- and (-a)-1 = -a-1.
PROOF, (a) of1 = a;_1e = a^^aa;1) = (a^1a)a;1 = ea~l =
a"1.
respy. = respectively.

2
Algebraic Preliminaries
(b) It suffices to observe that if a, 6 are invertible then b~1a~1
is the inverse of ab and a is the inverse of a-1.
(c) This is an immediate consequence of the identity xy =
(-x)(-y) (x,yeR).
1.1.5. LEMMA. (Kaplansky). In a unital ring R (with unity
e) if an element a has a unique I. (or r.) inverse then a is
invertible.
PROOF. Suppose that a has a unique 1.inverse 6, so that
ba — e. Then
(ab — e + b)a = aba — a + ba = a — a + e = e.
By uniqueness of 1. inverse, ab — e + b — b or ab — e. Therefore,
by 1.1.4 (a), a is ivertible.
1.1.6. Let F be a field and A an (associative, i.e., linear
associative) algebra over F. Given a subset S of A, there exists
a smallest subalgebra A(S) containing S, called the subalgebra
of A generated by S; A(S) is the intersection of all subalgebras
of A containing S. Explicitly, A(S) is the set of all finite sums of
the form Yi^k^k, where Ajt G F and (each) x^ a finite product
of elements from 5 (clearly this set is a subalgebra A(S)
containing 5 and every subalgebra containing 5 contains A(5)). If
A has a unity e then we have also the smallest subalgebra Ai(5)
containing 5 and e. We have
Ai(S) = Fe + A(S) = {Xe + x:XeF,xe A(S)}.
If A is only a unital ring then we have the analogous subrings
A(S) and Ai(5). Here Ai(5) = Zc + A(5).
A subalgebra (respy. subring) Aq of a unital algebra (re-
spy, ring) A is called a subunital algebra (respy. subunital ring) of
A if the unity e(of A) G AQ; then A0 is automatically unital with
e as its unity. Note, however, that a subalgebra (respy. subring) of
A can be unital without being subunital (i.e. Aq can have a unity
eo 7^ e). For example in the algebra A of 2 x 2 diagonal matrices
over a field F the subalgebra Aq comprising diagonal matrices
with second entry 0 is unital but not subunital (e = diag (1,1),

§ 1. Some Basic Concepts and Results
3
e0 = diag (1,0)).
1.1.7. Let A be an algebra (or a ring). For a,b G A we
write a <-> 6 if ab = 6a and say that a and 6 commute. For two
commuting elements a,b we have the Binomial Theorem.
(« +6)" = £ Q «n_***. where O = why. >» (*)
a positive integer. (This can be proved by induction as in the
classical case of the theorem, see [13, p.52].)
For two subsets Si, 52 of A we write Si <-> 52 if for every
a G S\, b G 52 we have a <-> b. We also write for an element a G A
and a subset 5 of A, a <-> 5 if a <-> 6, for every 6 G 5.
If 5 <-> 5 we say that the (subset) 5 is commutative. For a
subset 5 of A we set
5' = {x G A : x <-> 5} and call 5' the commutant (or
centralizer) of 5 in A. 5" = (5')' is called the double commutant
of 5.
1.1.8. LEMMA. Let S be a subset of an algebra (or ring) A.
Then
(i) 5' is a subalgebra (subring) of A.
(ii) 5 C 5" and consequently A(S) C 5"; a/so wAen A z's
unital, Ai(5) C 5".
(iii) If S C T C A, wAere T is a sufisei, iAen T" C 5'.
(iv) 5'= 5'".
(v) 5 is commutative iff' S C 5' z/f 5" is commutative.
(vi) // 5 is commutative so are the subalgebras A(S),S".
(vii) // 5 is a maximal commutative subset then S — 5".
PROOF, (i), (ii), (iii) an clear. By (ii) we have 5' C (5')" =
5'". On the other hand, since 5 C 5" (by (ii)), we get; using (iii),
5'" C 5'. Hence 5' = 5'" which is (iv). For (v), we observe that
iff = if and only if.

4
Algebraic Preliminaries
"5 is commutative" iff 5 C 5'. If 5 C 5' then by (iii), (iv),
5" C S' C (S")', so that 5" is commutative. On the other
hand "5" commutative" trivially implies "5 commutative" (since
5 C 5").
For (vi), we note that by (v), 5" is commutative and so also
A(S) C 5" (see (ii)).
Finally, for (vii), we have by (vi), 5 C 5" and 5" is
commutative by (vi). The maximality of 5 implies that 5 = 5".
1.1.9. PROPOSITION. Every commutative subset S of an
algebra A is contained in a maximal' commutative subalgebra
Am{S) °f A such that
S C A{S) C 5" C Am{S).
In particular, each element a of A is contained in a maximal
commutative subalgebra Am(a). If A is unital with unity e then
e G Am(S).
PROOF. By 1.1.8 (vi), A(S),S" are commutative subalge-
bras, and 5 C A(5) C 5". Since the union of any linearly ordered
(with respect to inclusion) family of commutative subalgebras is
a commutative subalgebra we can apply Zorn's lemma to obtain
a maximal commutative subalgebra Am(S) D S".
If A has unity e then Si = 5|J{e} is commutative and hence
Am(5j) D Am{S) 2 S.
By maximality of Am(S) we have Am(Si) — Am(S), e G
Am{Si) = Am(S).
1.1.10. Let A, A" be two algebras (over the same field F).
A mapping <p : A —> A* is called a homomorphism if it is linear,
and multiplicative, i.e. (p(ab) = (p(a)(p(b) for all a,b G A. An
injective or 1 — 1 homomorphism ip is called a monomorphism
and a surjective or onto homomorphism is called an epimorphism.
A homomorphism ip which is both 1 — 1 and onto is called an
' i.e. if Si is a commutative subalgebra of A with Am(S) C S\ then
Si =Am{S).

§ 1. Some Basic Concepts and Results
5
isomorphism (of A onto A*). Sometimes we use the term
"isomorphism into" for a monomorphism and "homomorphism onto"
for an epimorphism.
1.1.11. Given two algebras Ai, A2 (over F) we have the direct
product algebra A = A\ x Ai :
A= {{ai,a2) : ax G Ai,a2 G A2}.
The algebra operations on A are given by:
(01,02) + (61,62) = (01 + 61,02+62)
A(oi,o2) = (Aoi,Ao2)
(01,02)(61,62) = (0161,0262)
where 01, 61 G Ai; 02, 62 G A2; \ <E F.
1.1.12. An algebra A (over F) can be extended canonically
into a unital algebra Ai called the unitization of A. We take for
Ai the cartesian product linear space (over F) given by:
Ai = F x A = {(A, a) : A G F, a G A}.
Write (1,0) = ei,(0,o) = d. Then:
(A, a) = Aei + d, (/i, 6) = /zei + 6.
Define multiplication in Ai by:
(Aei + d)(/iei + 6) = A/zei + A6 + /id + d6.
It is easy to check that under this multiplication Ai is an algebra
and that the map a ^~> a is a monomorphism. We identify d with
a and write.
(A, a) = Aei + o, so that Ai = Fe\ + A. We observe that A
as a linear subspace of Ai has codimt 1. Further it is clear from
the definition of multiplication in Ai that A is both a left as well
as a right ideal of Ai.
If B is a subalgebra of a unital A with its unity e ¢. B then
B\ = Fe + B is unital subalgebra of A called the unitization of
' codim = codimension = d'im(AifA).

6
Algebraic Preliminaries
B in A.
We note that the above construction for A\ can be carried
out even when A has a unity e; of course e\ 7^ e (since e\ ^ A).
Finally, we remark that a ring R has a unitization .¾.
It is given by
i?! = 1 t xR = ~lel + R
with
(mei + a)(nei + 6) = mnei + m& + na + a& (m, n G Z).
1.1.13. Let i? be a ring and ' 0 ' denote the binary operation,
called circle operation, on R given by
aob=a + b + ab (0,6 Gi2).
We denote R with this binary (multiplication) operation by S0 —
S0(R);S0 is a multiplicative system.
1.1.14. LEMMA. Let R be unital, with unity e, and S0 as
defined above. Denote by S = S(R) the underlying multiplicative
semi-group structure of R. Then the map.
t = t(S) :ieSm-eeS0
is an isomorphism of S onto S0 (as multiplicative structures).
Hence S0 is a semi-group with '0 ' as the identity element.
PROOF. Clearly t is bijective. Further
r(a) 0 r(&) = (a — e) 0 (6 — e) = ab — e = r(a&).
1.1.15. COROLLARY. Let R be any ring. Then S0 = S0(R)
is a semi-group with '0' as identity.
PROOF. Consider the unitization i?i of R. Then e\ + R —
{ei + x : x G R}, under multiplication, is a subsemi-group 5*
of 5(i?i). By considering the restriction of the isomorphism (see
1.1.14), t : S(Ri) —> S0(Ri) to 5* we get the desired conclusion
for S0.
1.1.16. If 006 = 0 then a is called a l.q.i.(=left quasi-
t Z = {0,±1,±2,...} (the ring of integers).

§ 1. Some Basic Concepts and Results
7
inverse) of b and b a r.q.i. {=right quasi-inverse) of a; also then
b (respy. a) is said to be l.q. (respy. r.q.) invertible. If a is both
l.q. invertible and r.q. invertible it is called q. invertible or q. regular
(the meanings of l.q. regular and r.q. regular are clear).
1.1.17. LEMMA, (a) If a G R has a l.q.i. a\ and a r.q.i. a'r
then a\ = a'r ~ a' (say) and a' is q.i. of a. Moreover, a <-> a'.
(b) The set Gq of q. invertible elements of R is a group under
the multiplication 'o '; Gq is a subsemi-group of S0.
PROOF, (i) The proof of the first statement is similar to that
of 1.1.4 (a); for the second we note that
a o a = 0 = a o a => aa = a a.
(ii) The proof is similar to that of 1.1.4 (b).
1.1.18. LEMMA. Suppose that a is an invertible
(respy. q. invertible) element of a unital ring (respy. ring) R, x G R
and x <-> a. Then x <-> a-1 (respy. x <-> a'). In particular a-1
(respy. a') G {a}" (the double commutant).
PROOF. za_1 = (a~1a)xa~1 = a~1(ax)a~1 = a~1(xa)a~1 =
a 1x. Similarly, x o a' = a' o x.
1.1.19. COROLLARY. Let Am be a maximal,
commutative subalgebra of an algebra A. If b G Am is invertible (respy.
q. invertible) in A then b~l (respy. b') G Am.
PROOF. By 1.1.18, 6^1 ^ Am and so 5 = AmU{&-1} is
commutative. By 1.1.8 (vi), the subalgebra A(S) is commutative.
Since A(S) 353 Am, by maximality of Am, Am = S = A(S),
so that b~l G 5 C Am. Similarly 6' G Am.
1.1.20. LEMMA. Let R be a unital ring with unity e. Then
an element a of R has a l.q.i. (respy. r.q.i.) b iff e + a has e + 6
as a l.i. (respy. r.i.). In particular a is q. invertible iff e + a is
invertible and then we have (e + a)-1 = e + a', a' being the q.i.
of a. Moreover, t~ : a ^ e + a is an isomorphism of Gq onto
G{.
PROOF. The statements follow from the identity

8
Algebraic Preliminaries
(e + a)(e + 6) = e + (a o 6).
1.1.21. LEMMA. If A\ is the unitization of an algebra (or
ring) A and a G A has a l.q.i. (or a r.q.i.) b\ in A\ then b\ G A.
Hence a is q. invertible in A\ iff its q. invertible in A.
PROOF. Write &i = ae\ + b, where b G A and a G F or 1
according as A is an algebra over F or a ring. Then
ao&i = 0=>a + aei + 6 + aa + a6 = 0=>a = 0=>6i = 6.
1.1.22. Remark. Related to o -operation in R is the
operation denoted by x and given by a x b ~ a + & — ab (a,b G R). This
operation is again associative and has other similar properties of
o. If Sx denotes the multiplicative system in R corresponding
to x , then the map a ^ -a is easily seen to be an isomorphism
of S0 onto Sx . The operations o, x are mutually connected by:
a o b — —(—a x -6); a x b = —( — a o -&).
The operation x has been introduced by Hille (following a
suggestion of Jacobson) and called by him as cross-product. Some
authors like Rickart, Bonsall-Duncan adopt the definition of the
cross-product for the circle operation.
1.1.23. LEMMA. Every np. (=nilpotent) element a of a ring
R is q. invertible with
a' = -a + a2 + • ■ ■ ± ak~l (ak = 0, ak~l ^ 0). (*)
Further, if R is unital with unity e, then e + a is invertible with
(e + a)'1 = a-a2 ^ ±ak~1. (**)
PROOF. It is straightforward to check that a' as defined in
(*) is the q.i. of a, and (e + a)-1 as defined in (**) is the inverse
of e + a.
1.1.24. LEMMA. If <p is a homomorphism of a ring R then
cp preserves o -operation and hence also I. (or r.) q. invertibility.

§ 1. Some Basic Concepts and Results
9
If R is unital with unity e then <p(e) is the unity of <p(R) and
>p preserves I. or r. invertibility.
Proof.
(p(aob) = <p(a + b+ ab) = (p(a) + (p(b) + (p(a)(p(b)
= <p(a)o<p(b) (a-be R).
Further, since (p(0) = 0, a o 6 = 0 => <p(a) o <p(b) ~ 0,
completing the proof of the first statement. The second statement is an
immediate consequence of <p being a homomorphism.
1.1.25. LEMMA. In a ring, if a o b (respy. b o a) = a and a
is l.q. (respy. r.q.) invertible then 6 = 0.
PROOF. Suppose that a o b ~ a. Then
b — 0 o 6 = (a[ o a) o b = a\ o (a o 6) = a\ o a = 0.
Similarly, boa = a=>b = 0.
1.1.26. LEMMA. Let A be an algebra and u G A an idem-
potent. Then:
(i) For any A ^ — 1,A« is q. invertible with (Au)' = — A(l +
A)~1u. In particular u' — —\u. Also, when e exists, e' =
-\e.
(ii) If —u is q. invertible then u = 0. Hence, in a unital A, —e
is not q. invertible.
(iii) If A has unity e then e + u is invertible with (e + u)"1 =
e-\u and (Ae)"1 = A_1e (A ^ 0).
PROOF, (i) Au - A(l + A)_1u - A2(l + A)_1« = 0, whence
(Au)' = -A(l + A)"1u.
(ii) (— u) o u — —u + u — u2 — —u. Hence, by 1.1.25, u = 0.
(iii) (e + u)_1 = e + u' = e - \u. Since (Ae)(A_1e) = e and
Ae^ \-1e,(\e)-1 = A"^.
1.1.27. PROPOSITION, (a) In any ring R we have the
identities:
(i) ba o (— ba — bxa) = —b(ab o x)a

10
Algebraic Preliminaries
(ii) ( — 6a — bxa) o ba = — b(x o ab)a
(iii) — a2 o 6 = a o (—a o 6)
(iv) b o — a2 = (6 o — a) o a
where a,b,x G i?.
(b) /n a unital ring R with unity e we have:
(v) (6za + e)(e — 6a) = 6x(e — a6)a + e — 6a
(vi) (e — ba)(bxa + e) = 6(e — ab)xa + e -- ba
where a,b,x G i?.
Proof, (i)
LHS t = 6a + (—6a — bxa) + 6a(—6a — 6xa)
= — 6(x + a6 + a6x)a = — 6(a6 o x)a = RHS * .
(ii) Similar to (i).
(iii)
RHS = a + (-ao6) + 0(-006)
= a — a + 6 — a6+ a[—a + 6 — a6)
= -a2 + 6-a26=-a2o6= LHS.
(iv) Similar to (iii).
(v) LHS = bx(a — aba) + e — ba = bx[e — a6)a + e — 6a = RHS.
(vi) Similar to (v).
1.1.28. COROLLARY, (i) a6 is l.q. (respy. r.q.) invertible iff
ba is l.q. (respy. r.q.) invertible. In particular, ab is q. invertible
iff ba is q. invertible.
(ii) e — ab is I. (respy. r.) invertible iff e — ba is I. (respy.
r.) invertible. In particular, e — ab is invertible iff e — ba is
invertible.
PROOF, (i) By taking x— (ab)\ (respy. (o6)J.) in (ii) (respy.
(i)) we conclude that "a6 is l.q. invertible (respy. r.q. invertible)"
=>■ "6a is l.q.invertible (respy.r.q.invertible)". This plus
symmetry consideration proves (i).
(ii) Taking x = (e — ab)^ (respy. (e — ab)~l in (v) (respy. (vi))
of 1.1.27, and using symmetry we get the desired conclusions.
t LHS = Left Hand Side; RHS= Right Hand Side.

§ 1. Some Basic Concepts and Results
11
1.1.29. LEMMA. In a unital ring R with unity e, if ab
has a r.i. (respy. I.i.) c then be (respy. ca) is a r.i. (respy. Li.)
of a (respy. b). Similarly, in any ring R, if a o b has a r.q.i.
(respy. l.q.i.) c then boc (respy. c o a) is a r.q.i.(respy.l.q.i.) of
a (respy.) b.
PROOF. Clear.
1.1.30. COROLLARY. The elements ab,ba are invertible iff
a,b are invertible. In particular, if a <-> b and ab invertible then
a,b are invertible.
Similarly, aob, boa are q. invertible iff a, b are q. invertible,
and when a <-> b, aob is q. invertible iff a,b are q. invertible.
PROOF. If a,b an invertible then as already seen, ab is
invertible with 6~1a~1 as its inverse. On the other hand if ab,ba
are invertible than by applying 1.1.29, 1.1.4 (a) we conclude that
a,b are invertible. Hence the statements concerning invertible
elements. The proof of the statements concerning q. invertible
elements is similar.
1.1.31. PROPOSITION. Let R be a ring and x,a<E R. If x',
(a + x' a)\ and (a + ax')'r exist then (x + a)' exists and
(x + a) = (a + x a)i o x — x o (a + ax)r.
PROOF. We have
(a + x'a)\ o x' o (x + a) — (a + x'a)\ o (x' + x + a + x'x + x'a)
= (a + x a)\ o (x o x + a + x'a)
= (a + x'a)'i o (a + x'a) = 0
(since x' o x = 0)
Similarly,
(x + a) o x' o (a + ax')'r = (x + a + x' + xx' + ax') o (a + ax)'r
= (a + ax') o (a + ax)'r = 0.
It follows that x + a is both l.q.invertible and r.q.invertible and
so q. invertible. The required conclusions are now clear.
1.1.32. COROLLARY. With the hypothesis in 1.1.31 we have
(x + a)' - x = (a + x'a)\ + (a + x a)\x' = (a + x a)'r + x'(a + a;'o)^.

12
Algebraic Preliminaries
1.1.33. COROLLARY. If x <-> a and x',(a + ax')' exist, then
(x + a)' exists with
(x + a)' = x o (a + ax')' = (a + ax')' o x .
PROOF. Since x <-> a, by 1.1.18, x' <-> a, so that
(a + x'a)'i = (a + ax')| = (a + ax')'r = (a + x'a)r.
The required result now follows by 1.1.31.
§ 2. Ideals and Radical
1.2.1. Let R be a ring. If I is both a l.(=left) ideal and a
r.(=right) ideal of R it is called a bi-ideal. We will use the term
ideal for denoting any one of these: a 1. ideal, a r. ideal or a bi-
ideal. When R is commutative or anti-commutative, there is, of
course, no distinction between a Lor a r.ideal and the term ideal
has an unambiguous meaning.
If a ring R is an algebra (over a field F) then we have also
the algebra ideals i.e. ring ideals which are also linear subspaces.
While every algebraic ideal is a ring ideal the reverse is not always
true. Thus, if the 1-dim real linear space R is given the trivial
algebra structure (R2 t — {0}) the subset 1 of R is a ring ideal
which is not an algebra ideal. This distinction vanishes for a unital
algebra A since we have the relation Xa ~ (Ae)a (A E F;a E A;
e — unity of A ).
Let R be a ring (or algebra). Recall that an element o^O of
R is called a /.z.d.(=left zero divisor) if there is an element 6^0
with ab — 0. Similarly, a ^ 0 is a r.z.d.(=right zero divisor) if
there is a c^O with ca ~ 0.
Given a subset 5 of R the left annihilator Ai(S) and right
annihilator Ar{S) are defined by:
Ai(S) = {x E R : xa = 0, V a E S}
Ar{S) = {ye R:ay = 0,V ae S}.
' In a ring R, for two subsets Si,£2 we write 5i52 = {xy :i£Si,i/6
S2}; also for a subset S,S2~SS.

§ 2. Ideals and Radical
13
If S = {a} then we write Ai(a), Ar(a) for the corresponding
annihilator ideals. If 5 = R we denote the two annihilators by
Ai and Ar-
A ring R (respy. algebra A) is called a ring (respy. algebra)
with trivial multiplication if R2 (respy. A2) = {0}.
1.2.4. LEMMA, (i) Ai(S) (respy. Ar(S) is a I. (respy. r.) ideal
of R.
(ii) If I is a I. (respy. r.) ideal of R then Ai(I) (respy. Ar(I))
is a bi-ideal.
(iii) Ai,Ar are nilpotent bi-ideals: A2 = A2 = {0}.
PROOF, (i) Clear.
(ii) Let I be a 1. ideal, x E Ai(I), y E A and a El.
Then xy.a = x.ya = 0, xy E Ai(I), proving Ai(I) is a bi-ideal.
Similarly, Ar(I) is a bi-ideal.
(iii) If a,b E A\ then ab E aA = {0}, so that A2 = {0}.
Similarly, A2 = {0}.
1.2.5. PROPOSITION. If a ring R (respy. algebra A) ^ {0}
has no I.-ideals ^ R, {0} (respy.no I.(algebra) ideals ^ A, {0}
then R (respy. A) is' either a division ring (respy. division
algebra) or a ring (respy. algebra) with trivial multiplication and with
underlying additive group of R (respy. linear space of A) of prime
order (respy. of dim l). If R (respy. A) is unital then R
(respy. A) is a division ring (respy. algebra).
PROOF. Assume that R (respy. A) has a l.z.d. a, so that
ab = 0 with a ^ 0,b ^ 0. Then Ix = At(b) ^ {0} so that the
hypothesis implies that 7; = R (respy. A). Thus
Rb={0} (respy. A6={0}).
Since R (respy. A) is a r.ideal its right annihilator 7 is a bi-
ideal with b E I, and since 7 ^ {0} we conclude that I — R
(respy. A). Therefore R2 (respy. A2) = {0}. It follows that the
additive subgroup of R (respy. subspace of A) is cyclic of prime
order (respy. of dim 1).
' The same conclusions hold if a similar condition is put on r. ideals
instead of 1. ideals.

14
Algebraic Preliminaries
Next suppose that R (respy. A) has no l.z.d., so that R* =
R\{0} (respy. A* = A\{0}) ' is closed under multiplication. If
a G R* (respy. a G A*) then Ra (respy. Aa) is a 1.ideal 7^ {0}.
Hence, by our hypothesis,
Ra — R( respy. Aa ~ A) (*)
It follows that there is an element e /0 with ea — a. Then
(e2 — e)a = a - a = 0, so that e2 = e since R (respy. A) has no
l.z.d.. For arbitrary x G R* (respy. A*),
e{ex — x) — ex — ex = 0 and hence ex = x, so that e is
a 1. unity of R* (respy. A*). Again, by (*), there is an element
6 G R* (respy. 6 G A*) such that ba = e, so that a has a 1. inverse
6. It follows that R* (respy. A*) is a group '' and R (respy. A)
is a division ring (respy. algebra). Finally, if R (respy. A) is unital
then clearly R2 (respy. A2) 7^ {0} and R (respy. A) is a division
ring (respy.algebra).
1.2.6. A 1. ideal (respy. r. ideal) I of an algebra (or a ring) A
is called regular or modular if there is an element u in A such
that
xu — x (respy. ux — x) G I for all x G A.
The condition can be expressed briefly by writing A{u — 1)
(respy. (u - 1)A) C I.
The element u is called a relative r. unity (respy. relative
/.unity) for 7; u is not unique (see 1.2.8). If u is both a relative
1. unity and a relative r. unity it is called a relative bi-unity or just
relative unity. We also use sometimes the word relative unity for
a relative r or 1. unity. The precise sense of the usage will be clear
from the context. When A is commutative, relative r. unity and
1. unity concepts coincide and relative unity has an unambiguous
meaning.
1.2.7. Remark. In a unital A every ideal I is regular, with
unity e as a relative unity (since xe — x = ex — x = 0 <E I). On
' If S,T are two sets, we denote the difference set consisting of all
elements in S which are not on T by S\T.
'> A semi-group with a 1. unity in which every element has a 1. inverse is
a group.

§ 2. Ideals and Radical
15
the other hand if A has trivial multiplication then no ideal ^ A
is regular (since xu - x, ux — x = —x for all x.)
In the ring of even integer 21, the (principal) ideal 61 is
regular (with relative unity u = 4) while the ideal 41 is not regular.
1.2.8. PROPOSITION, (a) If u is a relative r. (respy. I.) unity
for a regular I. (respy. r.) ideal I and a G I then u + a is a relative
r. (respy. I.) unity for I.
(b) Let u,v be relative r. (respy. I.) unities for I. Then uv,
and hence un (n = 1,2,---), are relative r. (respy. I.) unities for
I. Further,
x(u — v)(respy. (u — v)x) G I (x G A).
(c) For a regular bi-ideal I, every relative r. or I. unity for I
is a relative (bi-) unity. Moreover, if u,v are relative unities then
u — v G I. Finally, u is a relative unity for I iff u# = u + I is
a unity of the quotient A* = A/1.
PROOF, (a) Let 7 be a 1.ideal. Then
x(u + a) — x = (xu — x)+xa<El + I=I (xG A).
The proof when 7 is a r. ideal is similar.
(b) Assume that 7 is a 1. ideal. Since
xuv — x = xu.v — xu + xu — x<El+I=I
uv is a relative r. unity for 7. Taking v = u and using induction
we get un is a relative r. unit for 7.
Further,
x(u — v) — xu — x — (xv — x) G 7 + 7 = 7.
The corresponding results when 7 is a r. ideal are proved similarly.
(c) Suppose that u (respy. v ) is a relative r. (respy. 1.) unity
for 7. Then
u — v = (vu — v) — (vu — u) G 7 + 7 = 7.
Therefore, by (a), u — v + (u — v) is a relative 1. unity.

16
Algebraic Preliminaries
Similarly, v is a relative r. unity. Finally, observe that for
x G A,
XU — X, UX — X G I iff 2TU^ = I* , U^X^ — I* ,
where ihi* is the canonical quotient homomorphism.
1.2.9. LEMMA. Lei A 6e on algebra (or a ring) and I a
regular I. (respy. r.) ideal with relative unity u. Then:
(i) If u G I then I = A
(ii) If I 7^ A iAen — u is not l.q. (respy. r.q.) invertible
(iii) Anj/ /. (respy. r.) ideal J containing I is regular with u as
a relative unity for J.
PROOF. It is enough to prove the results when 7 is a 1.ideal.
(i) Since u G I, for any x G A, xu G I and hence x = x — xu +
xu G I + 7 = I, whence I = A.
(ii) Suppose that — u is l.q. invertible with a as its l.q.i.. Then
a — u — au — 0, so that u — a — au G I, and so by (i),
I = A - a contradiction.
(iii) Obvious.
1.2.10. LEMMA (Krull). Let I 7^ A 6e a regular ideal. Then
there is a maximal regular ideal M of the same type (I., r. or bi-)
as I with I C M.
PROOF. If u is a relative unity for I, by 1.2.9(1), u ¢. I.
Apply Zorn's lemma, to the poset ' of ideals (all of the same type
as I) yt A, to obtain M (note that if {Ia} is any chain of ideals
in the poset then u ^ {JIa> since u ^ any Ia).
1.2.11. COROLLARY. In a unital A every ideal ^ A is
contained in a maximal ideal.
1.2.12. LEMMA. Let A be unital with unity e. Then:
For definition see 2.1.1 .

§ 2. Ideals and Radical
17
(i) If I 7^ A is an idea/ iAen e ¢ I.
(ii) // I\ (respy. Ir) is a I. (respy. r.) ideal ^ A then I\ (re-
spy. Ir cannot contain any I. (respy. r.) invertible element of
A.
(iii) Aa = A (respy. aA = A) iff a is a I. (respy. r.) invertible
element.
PROOF, (i) Since e is a relative unity for I this follows from
1.2.9 (i).
(ii) If a G I\ is 1. invertible then e = af a G /;, so that 7; = A.
Similarly Ir = A if Ir contains a r. invertible element.
(iii) If Aa = A then there is a 6 G A with ba = e, whence a
is 1. invertible. Conversely, if a is 1. invertible then since a G Aa
we must have Aa — A by (ii). Similarly we can prove a A = A iff
a is r. invertible.
1.2.13. LEMMA. In a unital A an element a is
I. (respy. r.) invertible iff a is not contained in any I. ideal
(respy. r. ideal) ^ A.
PROOF. This is obtained by combining (ii), (iii) of 1.2.12
(noting that Aa (respy. a A) is a 1. (respy. r.) ideal).
1.2.14. COROLLARY. An element a G A is I. (respy. r.)
invertible iff a does not belong to any maximal I. (respy. r.) ideal of
A. In particular, when A is commutative, a is invertible iff it
does not belong to any maximal ideal.
PROOF. This follows from 1.2.13, 1.2.10.
1.2.15. LEMMA, (a) If I is a regular I. (respy. r.) ideal of A
and J a regular bi-ideal of A then If] J is a regular I. (respy. r.)
ideal of A
(b) If I is a bi-ideal of A and J a regular bi-ideal of I then
J is a bi-ideal of A. Further, J is regular if I is regular.
PROOF, (a) Let 7 be a regular 1. ideal with relative r. unity
u and J a regular bi-ideal with relative unity v. Write w —
v + u — vu. Then, for x G A, where
xw — x = xu — x — x(vu — v) G 7 + 7 = 7. (1)

18
Algebraic Preliminaries
Also,
xw — x = xv — x — (xv — x)u G J + Ju = J. (2)
It follows from (1), (2) that If] J is 1. regular with w as as a
relative r. unity.
(b) Let u be a relative unity for J in I. If x G A then
ux G I and hence, if a G J then uxa G J. Also xa G I, so that
u.xa — xa G J. Therefore xa = uxa — (uxa — xa) G J + J = J,
proving J is a 1. ideal. Similarly, J is a r. ideal. Assume now that
I is also regular with relative unity v. Set w = u + v — vu. Then
xw - x = xv - x - (xv — x)u G J (since ct-iGI). Thus J is
regular with relative unity w.
1.2.16. LEMMA. Let <p: A—* A* be an epimorphism Then:
(i) For any ideal I of A, I* = <p(I) is an ideal of A* of the
same type (1., r. or bi.) as I.
(ii) If I is regular with relative unity u then I* is regular with
relative unity u* — <p(u).
PROOF, (i) Clear.
(ii) The regularity of I* follows from the identity (p(x)(p(u) —
<p(x) = (p(xu — x) when I is a 1. ideal and from the analogous
identity when I is a r. ideal.
1.2.17. LEMMA. Let <p : A —> A* be a homomorphism (not
necessarily epimorphism). Then:
(i) For an ideal I* of A*, I = cp~1(I* ) is an ideal of A of the
same type.
(ii) If I* is regular with a relative unity u* belonging to <p(A),
then I is regular with any u G ^?~1(u*) as a relative unity.
(iii) If ip is an epimorphism and M* is a maximal regular ideal
of A* then M = (p~1(M*) is a maximal regular ideal of A.
PROOF, (i), (ii): Clear.
(iii) It is enough to observe that the map
I* h-> <p~\V) (I* an ideal of A*)

§ 2. Ideals and Radical
19
is a bijection between the ideals of A* and the ideals of A
containing ker (p.
1.2.18. Remark. The condition 'V e <p{A)" in (ii) of
1.2.17 cannot be omitted. This is shown by the following
counterexample. A is an algebra (over F) with trivial multiplication
and Ai its unitization and <p the natural injection A —> A\. If
a G A and I = Fa then I is an ideal of both A\ and A. As an
ideal of A1;7 is regular (since Ai is unital) but as an ideal of A
it is not regular (see 1.2.7).
1.2.19. LEMMA. Let A be an algebra (or ring) and a G A.
Write
Il(a) = {xa + x : x G A}, Ir(o) = {aa; + ^ •' x e ^}-
TAen Ii(a) (respy. Ir(a)) is a regular I. (respy. r.) ideal with —a
as a relative r. (respy. I.) unity.
PROOF. It is easily seen that Ii(a) is a 1. ideal and Ir(a) is a
r.ideal. Further, since x( — a) — x — (-x)a — x, the regularity for
h(a) follows. The regularity for /r(a) also follows similarly.
1.2.20. COROLLARY. aGA is l.q.(respy.r.q.) invertible iff
k(a) (respy. Ir(a)) = A.
PROOF. If h(a) — A, then since — a G Ii(a) there is a b G A
with
-a = ba + 6, whence boa — 0, so that a is l.q.invertible.
Conversely, if a has a l.q.i. 6. then we have
-a = ba + be k(a), so that, by 1.2.19,
k (a) = A.
The proof of the assertion for Ir(o) is similar.
1.2.21. LEMMA, (cf. [22, p.173]) (a) a e A is l.q.
(respy. r.q.) invertible iff to each maximal regular I. (respy. r.) ideal Mi
(respy. Mr) there is an element b possibly depending on Mi
(respy. Mr) such that bo a G M; (respy. a o b G Mr)
(b) a G A z's /.g. (respy. r.q.) invertible iff -a is not a relative
r. (respy. I.) unity for any maximal regular I. (respy. r.) ideal of A.
PROOF, (a) If a has a l.q.i. b then b o a = 0 G M;. Next

20
Algebraic Preliminaries
assume that the stated condition is satisfied for all M;. If a is
not l.q. invertible then by 1.2.20, k(a) ^ A. By 1.2.10, there is
an Mi with h(a) CM;. By our assumption we can find a 6 with
boa G Mi. It follows that
-a = b+ba - (boa) G It(a) + M; C Mi + M; = Mj.
By 1.2.19, -a is a relative unity for Ii(a) and so also for M;.
Hence, by 1.2.9(i), Mi — A - which is impossible. So a is
l.q. invertible, as required
The proof of the statement concerning r.q. invertible is similar
(b) The statement to be proved for l.q. invertibility is clearly
equivalent to: a is not l.q. invertible iff -a is a relative r. unity
of some maximal regular left ideal M;.
If —a is not l.q. invertible then by 1.2.20, Ii(a) ^ A and so
h(a) C (some) M;. By 1.2.19, -a is a relative unity for Ii(a)
and so also for M;. Conversely, if —a is a relative r.unity for
some Mi then by 1.2.9(ii), a — -(—a) is not l.q.i.
The proof of the statement for r.q. invertibility is similar.
1.2.22. A 1. or r. ideal I is called q. invertible if every element
of I is q. invertible. Denote the intersection of all maximal regular
1. ideals (respy. the intersection of all maximal regular r. ideals) of
A by \[A (respy. \/A). If there are no maximal regular 1.ideal
(respy. r. ideal) in A we define \J~A~ (\/A) = A.
1.2.23. LEMMA, (a) If every element of a I. (respy. r.) ideal
I is l.q. (respy. r.q.) then I is q. invertible.
(b) The image, under an epimorphism, of a q.i. ideal is a
q.i. ideal (of the same type).
PROOF, (a) It suffices to prove the result when I is a 1, ideal.
For a G I, let a\ be a l.q.i. Since a\oa — 0 we have a\ =
— a — ojo G I. By our hypothesis, a\ has a l.q.i. b : boa\ — 0. By
1.1.17 (a) - applied to a\ - we get a = 6, which implies that a is
q. invertible with a' = a\.
(b) If ip : A —> A* is an epimorphism and I an ideal then by
1.2.16, 7* = (p(I) is an ideal of A* of the same type as I. By
1.1.24, I* is q. invertible.
1.2.24. PROPOSITION, (a) \J~A~ (respy. <fA)isa q. invertible

§2. Ideals and Radical
21
I. (respy. r.) ideal containing every q. invertible I. (respy. r.) ideal of
A. In particular, every element of \j~A or \/A is q. invertible.
(b) \f~A — \J~A — \f~A (say); \J~A is a bi-ideal containing
every q. invertible I. or r. ideal of A.
(c) \f~A contains every nil - in particular nilpotent - I. or
r. ideal of A.
(d) // u^O is an idempotent of A then u ^ v A. In
particular, when A is unital with unity e, e ^ \f~A.
PROOF, (a) Since \J~A = f]Mh if a G \J~A C Mx then aoa =
a + a + a2 G M;, whence by 1.1.21 (a), a is l.q. invertible. So by
1.2.23 (a) \j~A is q. invertible. Let 7; be any q. invertible 1. ideal
of A. If I\ 2 \j~A then there is an M; with /; 2 Mi, so that we
have: (*)M; + 7; = A. If u be the relative r. unity for M; then
by (*) we have
-u=a + b (a£ Mi,b <E Ii).
Since b G I\ its q.i. b' exists and therefore
u=-a-b = -a- (-6'- &'&) = -a + b' + b'(~u-a)
= -a + (b' -b'u) -b'ae Mi,
which is impossible (since M; ^ A). Hence 7; C M;, as required.
The corresponding result for \/A is proved similarly.
(b) For a G \/A and x G A we claim that the principal 1. ideal
{ax}i — Aax + lax is g. invertible. If y is any element of this
ideal then
y = bax + nai = (6a + na)x (b G A, n G Z).
Since a G \/A, j/o = z(&a + na) G \/A and hence j/o is ?•
invertible. By 1.1.28(i), y is g. invertible, so that the ideal {ax}l
is q. invertible. By (a), {ax}i C -^A. In particular ax G \/A,
whence \/A is a bi-ideal. Similarly, \/A is a bi-ideal. Since
\/A (respy. yA) is a q.i.l. (respy. r.) ideal we must have by (a),
</AC J/a (respy. -J/A C \/A). Hence ^A=-^A.
(c) By 1.1.23, every nil ideal is g. invertible and so contained
in ^A (by (b)).

22
Algebraic Preliminaries
(d) If u G \[A then — u G \/A, and — u is q. invertible. But
then by 1.1.26(ii), u = 0 - a contradiction.
1.2.25. The bi-ideal \f~A~ is called the Jacobson radical or
radical of A. Following Neumark [22,p.l73] an element of \f~A is
called an essentially nilpotent element of A. In view of 1.1.24,
an element a is essentially nilpotent iff the principal 1. ideal
{a}i = Aa + la (or equivalently, the principal r. ideal {a}r) is
q. invertible. Further, if A is commutative then every nilpotent
element a is essentially nilpotent (since {a}; = {a}r is a nil
ideal).
1.2.26. LEMMA. If <p : A —> A* is an epimorphism then
<p{\/A) C y/A*.
PROOF. By 1.1.24, £>(\/A) is a q. invertible bi-ideal of A*,
whence by 1.2.24 (b), <p{y/A) C y/A*.
1.2.27. An algebra A in which \f~A~ = A is called a radical
algebra. Note that an algebra is a radical algebra iff every
element is q. invertible. Any algebra Ao, with trivial multiplication
(A0 = {0}) is a radical algebra (since every a G Ao is q. invertible
with a' = —a ).
An algebra A 7^ {0} is called s.s. (=semi-simple) if \f~A~ —
{0}
1.2.28. LEMMA. In a s.s algebra A we have
Ai = Ar = {0}.
Proof. By 1.2.4 (ill), 1.2.24 (c), AuAr c y/A = {0}.
§ 3. Characters and Hypermaximal ideals
1.3.1. Let A be an algebra over a field F. A homomorphism
X of A onto F (as a 1-dim algebra) is called a character of
^)X_1(0) = kerx is called the kernel of x-
We denote by A = A(A) the set of all characters of A; A
may be empty.

§ 3. Characters and Hypermaximal ideals 23
Let X be a linear space over F. A linear map xp : X —> F is
called a linear functional.
1.3.2. LEMMA. Let A, A* be two algebras over the same
field F and <p : A —> A* an epimorphism. If x* e A (A*) then
x = x*o<pe A(A).
PROOF. Clear.
1.3.3. LEMMA. A homomorphism x '• A —> F is a character
iff it is non-trivial (i.e. x^O )•
PROOF. If x i1 0 there is an element ao £ A with x{ao) =
a/0. If /3 G 71 then x(^a_lao) = /?> proving x 's surjective,
whence X is a character. The converse is trivial.
1.3.4. LEMMA. A subspace B of a LS(= linear space) A
is of codim 1 iff it is the kernel xj)~l(0) of a non-trivial linear
functional xj} on A.
PROOF. If \\> ^ 0 there is an a0 <E A with ip(a0) = A^O.
Write A_1ao = u; then if)(u) = 1. For x G A we can write
x = xj}(x)u + y, where y = x — if>(x)u G kerV> = B (say). It
follows that A = Fu + B, so that codim 5=1. Conversely, if
codim 5=1, there is an ao G A\B such that A = Fao + 5. If
x G A and x = Aao + 6, define if>(x) = A. It is clear that t/; is a
linear functional on A with kerV> = B.
1.3.5. LEMMA. A regular I. or r. ideal 7, of codim 1, of an
algebra A is a bi-ideal, which is moreever, maximal as a I. or a
r. ideal, or also, as a bi-ideal.
PROOF. Let 7 be a codim 1 regular 1.ideal with u as a
relative r. unity. Since codim 7 = 1 we have Fu + 7 = A. If
y G A then y = au + b, with a G F, b G 7. Therefore, if a G 7,
ay = a(au + 6) = a(au - a) + aa + ab G 7,
whence 7 is a r. ideal so a bi-ideal. The proof of this when 7
is a r. ideal is similar. The maximality assertions are immediate
consequences of the codim 1 property of 7.
'0' denotes the zero-functional mapping everything to 0.

24
Algebraic Preliminaries
1.3.6. DEFINITION. A codim 1 regular ideal 7 of A is called
a hypermaximal ideal; by 1.3.5 every hypermaximal ideal is a bi-
ideal.
1.3.7. If an algebra A has hypermaximal ideals we denote
their intersection by \J~A. We set \J~A~ = A if A has no
hypermaximal ideal. We call \/A~ the hyper-radical of A, and we say
that A is h.s.s. (=hyper semi-simple) if \/A~ = {0}.
1.3.8. Lemma. >/ac \/a.
PROOF. By 1.3.5 every hypermaximal ideal is maximal
regular 1. ideal. The inclusion relation now follows from 1.2.24 (b) and
the definition of \j~A.
1.3.9. PROPOSITION. The hypermaximal ideals of an algebra
A are precisely the kernels Mx = kerx of characters \ °f A.
Moreover, the correspondence x *~* Mx is a bijection between the
set A of characters and the set DJl of hypermaximal ideals.
PROOF. Since x is a homomorphism, Mx is a bi-ideal, and
since x is surjective there is a u G A with x(u) = 1- Further,
for any x G A, we have (*) x{xu ~~ x) = x{x) ' 1 — x{x) = 0>
whence Mx is regular with u as relative unity. Also, by 1.3.4,
codim Mx = 1, whence Mx is hypermaximal. If M is any
hypermaximal ideal the quotient A* = A/M has dim 1. If u is
relative unity for M, by 1.2.8 (c), u# = u + M is unity of A*.
If x# G A* then x# = axu# where ax G F, since dim A* = 1.
The map x = XM '• x i~^ otx is clearly a character of A with
kerx = M = Mx. Finally, it is clear from the construction of xM
from M that MY = M2 => XMl = Xm2- Hence x ^ Mx is a
bijection.
1.3.10. COROLLARY, (i) The relative unities for M = Mx
are precisely the elements u in A satisfying x{u) = 1- ^n
particular, if A has unity e then x(e) = 1 and further x(a) 7^ 0 for
any invertible a.
(ii) The set of all relative unities (= bi-unities) for Mx is the
unity coset E of A* = A/Mx.
PROOF, (i) It is clear from the relation (*) in the proof of
1.3.9 that any u with x(u) = 1 is a relative unity for M = Mx.

§3. Characters and Hypermaximal ideals 25
Conversely, let v be any relative unity for M. Choose any element
a0 G A\M; then x(ao) ^ 0. Since a0v - a0 G M we get
xMxW =x(a0),x(«) = 1-
If a is any invertible element then
X(a)x(a_1) = X(e) = 1, whence x(a) 7^ 0.
(ii) Since M = Mx is a bi-ideal, by 1.2.8, E = u* is the set
of relative unities for M, u being a relative unity for M. Also, by
1.2.8 (c), E is the unity coset of A#.
1.3.11. LEMMA. Every character \ °f A vanishes on y/A.
PROOF. By 1.3.8, \/~A C yJA~ C kerx, hence the result.
1.3.12. Remark. If A is a radical algebra then A(A) =
0 (if A = A(A) 7^ 0,x e A then \/A C kerx C A, so
that A is not a radical algebra). Again, for the matrix
algebra A = Mn(F)(n > 1), A(A) = 0, since A being simple has no
hypermaximal ideals.
1.3.13. Examples of hypermaximal ideals:
(a) Let G be a finite group of order n and A = F[G] the
group algebra of G over a field F of characteristic 0. Write
a = ^2 9 > where the sum in A is over all elements g of G;
a G A. Clearly, ag = A. for each g G G, whence a2 = na.
If we write u = -a, then u2 = u. Writing v = e — u, it is
easy to see that M — Av is hypermaximal (since it has a
1-dim direct summand Au = Fu).
(b) Let -t1(G) be the group algebra of a locally compact Haus-
dorff group G. Consider the functional
A(/) = f f(t)dt
Jg
where the integral is the (left invariant) Haar integral.
Clearly A is linear. Further
Hf*9) = J f*g{t)dt = J (/ f{s)g{s^1t)ds\ dt

26
Algebraic Preliminaries
f f(s)ds I g(s~lt)dt
J f(s)ds j g{8-H)d{8-H) = A(f)A(g),
where we have used the left invariance of the Haar measure.
Finally, A ^ 0. For, if U is a nucleus t of G with U
compact, and to G G\U, we can, by local compactness of G,
choose a positive continuous / with f(to) = 0, f{i) = 1 on
U. Then
A(/) = [ f{t)dt > f f{t)dt = w(u) > 0,
Jg Ju
where //, denotes the left Haar measure of G.
Thus A is a character and M = ker A is a hypermaximal
ideal (M is called the augmentation ideal).
(c) Let F be a field and Fs denote the algebra, under pointwise
operations, of all F -valued of functions / = /(s) on a set
S. If sq £ S then the map Xs0 '• f ^ /(so) is easily seen to
be a character and Ms = kerx3 = {/ G Fs : /(so) = 0)}
is a hypermaximal ideal.
1.3.14. LEMMA. Let J be a bi-ideal of A. If M is a
hypermaximal ideal of J then M is a bi-ideal of A. If J is regular in
A then M is also regular in A.
PROOF. These follow from 1.2.15 (b) since a hypermaximal
ideal is a regular bi-ideal.
The first assertion can also be proved directly in the following
way.
Let x De the character of J determined by M : M = ker%-
Take u G J with x(u) = 1. If x G A, a G M C J then xa G J
and
X{xa) = x(«)x(za) = x{uxa) = x(«z)x(<i) = 0.
Therefore xa G M, similarly ax G M. Thus M is a bi-ideal
of A.
for definition see 2.1.3.

§ 3. Characters and Hypermaximal ideals 27
1.3.15. PROPOSITION. Let A be an algebra and J a bi-ideal
of A. Then:
(i) Every character x °f J satisfies the condition x{xa) =
x(ax) (x G A, a G J).
(ii) x can be uniquely extended to a character x °f A.
PROOF. If M = kerx, and u an element of J with x{u) ~
1, then M is a hypermaximal ideal of J with u as a relative
unity. Take x G A. Then ux, xu G J and by hypermaximality of
M we have the relations.
ux = m + Qti; xu = m] + /?u (m, mi G M) (l)
Then
uxu = mu + au ; uiw = umi + f3u (2)
since mu,timi G M we obtain from (2)
a = ax{u2) = x(au2) = x{uxu) = x(/?«2) = /?•
So (1) gives
X{ux) = a =: /3 = x{xu).
Now extend x on ^ to X on A by setting
x{x) = x(«z) = X(z«) (z G A).
Clearly x is linear. Further
x(zj/) = X(«zj/)x(u) = x{ux.yu) = x{ux)x{yu)
= x(«z)x(«J/) = x(z)x(j/)-
This proves that x 1S a character of A which is an extension
of x- Further, if x is any character of A extending x then
xx (*) = xx (z)x(u) = xx (a;)x1 («) = xx (*«) = x(z«) = x(x),
which shows that the extension x is unique. Finally,
X{xa) = £(xa) = x(*)x(a) = x(a)x(*) = x(a*) = xM

28
Algebraic Preliminaries
where x G A, a G J.
1.3.16. COROLLARY. Every hypermaximal ideal M of J
satisfies the symmetry condition for x G A, a G J,
xa G M O ax G M.
PROOF. This follows from 1.3.15 (i) since M = ker% for
some character x °f -^-
1.3.17. COROLLARY. Every character x °f an algebra A
can be extended uniquely to a character xi of its unitization A\.
PROOF. If x\ G Ai,x1 = Aei+z (A G F,x G a) then Xi(zi) =
A + x{x) 's easily checked to yield, a character of xi °f A. The
uniqueness of xi follows from 1.3.15 (ii) (taking J = A) or can
also be seen directly using the fact that for any character x' °f
Ai we have x'(ei) = 1.
1.3.18. LEMMA. Let A = Ai © A2 be a direct sum of subal-
gebras Ai,A2. If A = A(A),Ay = A(Aj) (j = 1,2) then A is
iAe disjoint union
A = AJIJA2 0/ subsets A°j such that
A^\Aj = Aj (i.e. if x^ A? then X\Aj G Ay.
PROOF. Let x e A and x(u) = 1 (« G A). Since u = ui + u2
with Uj G Ay, and U1U2 = u2u\ = 0 (since A1A2 = A2A1 =
A\ fl A2 = {0}) we obtain
1 = x(«) = x(«i) + X(«2), x(«i)x(«2) = 0.
From these equations it follows that precisely one of x(ui)>x(u2)
is 1 and the other 0. If x(ui) — l>x(u2) = 0 then for y G
A2, uiy = 0, so that
0 = x(«ij/) = x(«i)x(j/) = x{y)-
This shows that x e A°. Similarly, if x(u2) = 1 then x e A§.
1.3.19. LEMMA. (cf.[24,p.233]). Let A be a unital algebra
and x a linear functional with x(e) = 1.
The following two statements are equivalent:

§3. Characters and Hypermaximal ideals 29
(i) x *s a character.
(ii) M = ker x is closed for squares, i.e. x G M => x2 G M.
PROOF, (i) => (ii), since M is an ideal by 1.3.9.
To prove that (ii) => (i), assume that (ii) holds. Since X is a
linear functional ^ 0, codim M = 1. For x,y G A we have
x = x(z)e + a, j/ = x(j/)e + & (o,6gM).
Computing from these relations the product xy and substituting
it for xy in x{xv) and simplifying we get
x(xv) = x{x)x{y) + x{ai>) (1)
Taking y = x,b = a in (l) and using hypotheses (ii) we obtain
X{x2) = x(z)2 for all x G A (2)
Replacing in (2), x by x + j/ we obtain x((x + J/)2) = (x(x) +
x(j/)) which reduces to
X(xj/ + yx) = 2x(x)x(j/) (3)
It follows that:
x or y G M => xj/ + yx G M (4)
Now we have the identity
(xy- yx)2 + (xy + yx)2 - 2((xy)2+ (yx)2) = 2(xyx.y + y.xyx) (5)
where x, y G A. If j/ G M, then by (4), xy + yx as well as the
end expression in (5) belong to M. Therefore, by applying x to
both sides of (5) and by using linearity of x and relations (2),(4)
we obtain
xy - yx G M (6)
From (4), (6) we conclude that if y G M then xy G M
This means that in (1) we must have x(a^) — 0, so that (1)
reduces to
x{xy) = x{x)x{y)
proving x is a character, as desired.

30
Algebraic Preliminaries
§ 4. Extension of Ideals
1.4.1. Let J be a bi-ideal of an algebra A. For a 1. ideal J;
of J we set
Ji = {ae A: JaC J,}. (*)
Similarly, for a r. ideal Jr of J we set
Jr = {a G A : aJ C Jr}. (**)
1.4.2. PROPOSITION, (i) J; is a I. ideal of A with Jt C J;;Jr
is a r. idea/ o/ A wii/i Jr C Jr.
(ii) //" 7' is a regular I.(respy. r.) ideal of J with relative
r. (respy. I.) unity u, then I' is also regular with u as a
relative r. (respy. I.) unity. Further, if I' ^ J then J <2 /', hence in
particular I' ^ A.
(iii) If I is a I. (respy. r.) ideal of A and I' = J f] I then
I C /'.
(iv) If M' is a maximal regular I. (respy. r.) ideal of J then
M' is a maximal regular I. (respy. r.) ideal of A such that M' —
J f] M'. Also, if M' is a maximal regular bi-ideal of J then M'
is a maximal regular bi-ideal of A.
(v) If M is a maximal regular I. (respy. r.) ideal of A such that
J 2 M then M has a relative r. (respy. I.) unity j belonging to
J; further M' ~ J f]M is a maximal regular I. (respy. r.) ideal of
J with j as a relative r. (respy. I.) unity and M = M'. Finally,
if M is a maximal regular bi-ideal of A with J <£ M then M' =
J f] M is a maximal regular bi-ideal of J.
PROOF. We shall prove only the results for 1.ideals.
(i) Clearly J\ is a linear subspace of A. For a G Ji,x G A we
have J ■ xa = Jx ■ a C J a C Ji:xa G J;, so that J; is a
1. ideal of A.
(ii) If x G A and a G J then ax G J. By regularity for I',ax-
axu G /'. This implies that J(x — xu) C 7'; whence x—xu G
/',/' is regular. Further, by 1.2.8 (b) u2 is a relative r. unity
of /', so that if I' ^ J then u2 ¢. V. It follows that
Ju g I',u ¢. /', so that J % V.

§ 4. Extension of Ideals
31
(iii) Let 7 be 1. regular with relative r. unity u. If x G 7 then
Jx C Ax C 7 and Jx C J (since J is abi-ideal). Therefore
Jx e Jf)I = I',x<E V, so that 7 C 7'.
(iv) Assume that M' is a maximal regular 1. ideal of J with
r. unity u. By (ii), M' is regular with relative r unity u.
If 7 is a 1. ideal of A, I ^ A, with I D M' D M' then 7 is
regular with u as relative r. unity and u ^ I (since I ^ A).
Further, Jf]IDM', and since u$ J(\I, J f] 1^ J. By
maximality of M', Jf]I = M'. Since 7 3 M' D M' we
get Jf]M' — M'. For x G 7, xu = x - (x - xu) G 7
(using regularity of 7); also xu G J (since u G J). Thus
xu G J f] I = -M'- It follows that Jxu C JM' C M', whence
xu G M'. Consequently x = xu-(xu-x) G M'. This proves
7 = M',M' is maximal.
(v) Let M be a maximal regular 1. ideal with relative r. unity u.
Since J g M, by maximality of M we have J + M = A.
Therefore u has a representation (*) u = j -\- m (j G
J,m £ M), so that for any x G A
x - xj' = x — x(u - m) = x - xu + xm G M.
Thus j £ J is a relative r. unity for M. By restricting x to
J we get x - xj <E J f) M = M' showing that M' is regular
(in J) with relative r. unity j. Since u ^ M it follows
from (*) that j ¢ M,j ¢ M',M' ^ J. Consid er now a
1. ideal V of J with V ^ J,f 2 M'. Then ADfD M'.
Also, by (iii), M C M'. By maximality of M we obtain
I' = M' = M. It follows that M' = Jf)M = J f| /' 2 I',
so that 7' = M' and M' is maximal. Finally, by (iv),
M' = J f| M' = J f| M.
1.4.3. PROPOSITION. 7ei J be a bi-ideal of an
algebra A. Then there is a bijection between the maximal regular
I. (respy. r.) ideals M of A with J g M, and the maximal regular
I. (respy. r.) ideals M' of J given by
M' = jf]M,M = M'.

32
Algebraic Preliminaries
In particular, M' is hypermaximal iff M is hypermaximal.
Finally, the map M —> M' = Jf]M is a bisection between the
maximal regular bi-ideals of A and those of J.
PROOF. The first statement follows from 1.4.2 ((iv),(v)). To
prove the particular case statement assume that M is
hypermaximal and x the character determining it. Let \' De the
character of J obtained by restriction of x- Then it is clear that
M' = J H M = ker x' and hence M' is a hypermaximal ideal of
J. Conversely, if M' is hypermaximal ideal of J and \' the
character determining it then x' can be extended, by 1.3.15, uniquely
to a character X = x' of A. If M = ker x then M is
hypermaximal and clearly we have Jf]M = M'. Finally, by 1.4.2 ((iv),
(v)) M' is maximal regular bi-ideal iff M is a maximal regular
bi-ideal.
1.4.4. Corollary. \/j = Jf]y/A.
Proof.
v/j = f]M' = f](jf]M')
= jf](f]M') = jf]VA
(since in the computation of the right hand end term above the
maximal regular 1. ideals of A which contain J can be clearly
dropped).
1.4.5. COROLLARY. If A\ is the unitization of an algebra A
then
\fA = Af]y/Ai.
PROOF. By 1.4.4 (since A is a bi-ideal of Ai).
1.4.6. Let A be an algebra and Ai its unitization. Let
/; (respy. Ir) be a regular 1. (respy. r.) ideal of A with relative
r. (respy. 1.) unity u. By taking Ai for A and J = A in 1.4.1, we
can form 7; (respy. 7r). We also set
h ~ {%i £ Ai : x\u G Ii};Ir — {xi £ ^ '• uxi G 7r}.
1.4.7. PROPOSITION, (i) /; (respy. 7,.) is a I. (respy. r.) ideal
of Ai with
IiQIi^ k- 7r C 7r C 7r. (*)

§4. Extension of Ideals
33
Further, if I\ (respy. 7r) ^ A then Ii (respy. Ir) ^ A\.
(ii) Given a I. (respy. r.) ideal I\ of A\ with I\ <2 A,
the ideal I = A f] I\ is regular and we can choose a relative
r. (respy. I.) unity u for I such that e\ — u G I\
(iii) If I is a regular I. (respy. r.) ideal ^ A with relative
r. (respy. I.) unity u then
I - I + Ai(ei - u) (respy. I + (ex - u)Ai), (**)
I = Af] I. Further, if I\ ^ A is a I. (respy. r.) ideal of A\ with
I = A H h then h C I.
(iv) For a regular bi-ideal I with a relative (bi-) unity u we
have
I = I + Ai(ei - u) = 7 + (ei - u)A = I + F(ei - u),
and I is bi-ideal.
(v) If M is a maximal regular I. (respy. r.) ideal of A then M
is a maximal regular I. (respy. r.) ideal of A\ and M = M, M =
Af]M.
PROOF. As before we will prove all results only for 1.ideals.
(i) It is clear from the definition of 7; that it is a subspace
of A\. Further, if xi G 7; (so that x\u G I\) and j/i =
aei + V (y £ ^), we have
yixiu = axiu + y • x\u G I\ + I\ = 7;.
It follows that y\X\ G 7;, showing that /; is a 1. ideal of A\.
If x\ G I\ and y G A then y\xu G I\. Since yx\ G A and
I\ is regular we get
j/izi = yx]_u ~ (yxxu - yxx) G 7;,
whence Ax\ C 7;, so that x\ G 7;, and /;,C 7;. Also, if
h ^ A then u ^ 7;,ei ^ 7;, and 7; ^ A\.
(ii) Let 7i be a 1. ideal and choose an element.
ai = Aei + a (A G F, A 7^ 0, a G A) in 7X\A.

34
Algebraic Preliminaries
Then u = -X~la G A, and for any x G A we have
xu — x = — A~ xai G A[| 7i = 7,
so that u is a relative r. unity for I\. Also,
e\ — u = A_1ai G 7i.
(iii) If xi = Aei + x G Ai(x G A), then
(xi — xiu)u = X!ti — xiu • u G 7 = 7; (since xiu G A).
It follows that xi — xiu G 7;, so that Ai(ei — u) C 7;. This
inclusion with (first half of) (*) yields: 7; + Ai(e — u) C 7;.
To prove the reverse inclusion relation consider an xi G 7;
(so that xiu G I\). Then
xi = xiu + xi — X!ti G 7; + A]{e\ — u),
completing the proof of the relation (**). Further, it is clear
that 7; C Af]li. On the other hand, if a G A, a G 7; then
au G 7; and so a G 7; (since u is a relative r. unity of 7;).
Thus 7; = Af|7;. Finally, consider the ideal 7^ By (ii) we
can choose for 7 = Af] I\ a relative r. unity u such that
ei - u G 7i. If 3/1 = Ae! + y G 7i then
j/iu = Au + j/u G A.
Also,
J/i« = !/i - J/i(ei - u) G 7i + 7i = 7i.
So j/iu G Af|7i = 7, whence j/i G 7. Thus, 7i C 7, as
required.
(iv) If xi G Ai, xi = Aei + x (x G A) then
xi(e! — u) = A(ei - u) + x - xu G ^(ei — u) + 7
(ei — u)xi = A(ei - u) + x - ux G ^(ei — u) + 7
Hence 7 + Ai(ei - u) = 7 + 71^ - u) = 7 + (ei - u)A\ =
/(say). Clearly 7 is a bi-ideal.

§ 4. Extension of Ideals
35
(v) By (*) we have M C M. Again, by 1.4.2 (iv), M = Af]M.
It now follows, by (iii), that M C M. Combining the two
inclusions we get M ~ M, and M = Af|M = Af| A7-
1.4.8. COROLLARY. TTie correspondence /x h-> / = Af|^i
is infective over each of the following sets of ideals of A\ : the
set of all bi-ideals I\ <2 A; iAe set of maximal I. ideals M\ % A;
the set of maximal r. ideals M\ <2 A. Also, every maximal regular
I. (respy. r.) ideal M of A is of the form M ~ Af] M, where
M = {zi G A : x\u (respy. uxi) G M}.
PROOF. Suppose h,Ji <2 A are bi-ideals of Ai with
AfUl = A(Vi = I (say). If 7 = A, clearly 7X = Jx = Ai
(since codim A = 1). We may therefore assume that I ^ A. By
1.4.7 (ii) we can choose relative r. unities u,v for 7 such that
ei - u G h,ei - v G Ji- By 1.2.8 (c), v - u G 7 C Jx. If
£i = ^ei + J/ (A G 71, j/ G A) is an element of 7i then we can
write:
xi = A(ei - v) + A(v - u) + Au + y, (1)
Au + j/ = xiu - (yu - j/). (2)
It follows from (2) that
Au + y G x\u + 7 C 7i + 7 = 7i( since xiu G 7i)
so that
Au + j/G Af|7! = 7C Jv (3)
Since e! - v,v - u G Ji, it follows from (1),(3) that x\ G J±.
Thus, 7i C J1; and by symmetry considerations we conclude that
7i = Ji.
The injectiveness over the set of maximal
regular 1. (respy. r.) ideals M follows from 1.4.3, 1.4.7(v). The final
assertion concerning the form of M is a consequence of 1.4.7(v)
and 1.4.2 ((iii), (iv)).
1.4.9. COROLLARY, (a) A is a hypermaximal ideal of A\.
(b) The correspondence M\ h-> M = Af]Mi is a bisection
between the hypermaximal ideals M\ ^ A of Ai and the
hypermaximal ideals M of A; moreover M\ — M.

36
Algebraic Preliminaries
(c) ^A = Af]</M-
PROOF, (a) This follows from the construction of A\ (see
1.1.12).
(b) By 1.4.7 (v), M = M, M = Af]M. The bijectivity of
the map M h-> M = Mi is ensured by 1.4.3 .
(c) This follows from (b) and the definitions of \/A, \fA\.
1.4.10. Remark. The character of A\ determined by the hy-
permaximal ideal A is denoted by \ and called the distinguished
or canonical character of A\. It is given by
X0(Ae + x) = A (xeA).
It follows from 1.4.9 and 1.3.17 that there is a bijection \ ~~* Xi
between the sets of characters A and Ai\{xo}, where \i is the
unique extension of \ t° ^1-
1.4.11. PROPOSITION. Let A be a radical algebra and Ai
its unitization. Then:
(i) no ideal I ^ A is regular;
(ii) y/A^A;
(iii) Ai is a local algebra having A as its unique maximal
ideal.
PROOF, (i) If A has a regular 1. (or r.) ideal ^ A then there
is a maximal regular 1. (or r.) ideal M, whence \/A C M ^ A,
contradicting A is a radical algebra.
(i), (iii): If A\ has a maximal 1. (or r.) ideal M\ ^ A then
M = A P| Mi is a regular ideal of A, 7^ A, contradicting (i). On
the other hand, A is a maximal (1. orr. or bi-ideal) of A\. Thus
\f~A\ ~ A and A is the unique maximal ideal of A\.
§ 5. Regular Representation and Primitive Ideal
1.5.1. Let A be an algebra over a field F and X a linear
space over F. A homomorphism <p : A —+ £ (X) (=the algebra of
linear endomorphisms < of X) is called a linear representation
i.e. linear maps of X into itself.

§ 5. Regular Representation and Primitive Ideal 37
or just a representation of A in X. It is well-known that a
representation <p of A gives rise to an A-module structure on
X : a.x = <p(a)x. Conversely, if X is an A-module it yields a
natural representation a h-> /a, where /a is the linear map on X
given by
la : x h-> ax (a G A, x G X)
In particular we have for an algebra A the representation.
£ : a h-> /a(a G A), where /a is now given by
/a : x h-> ax (x G A).
£ is called the (left) regular representation of A
The kernel of the homomorphism C is given by ker L ~ {a G
A : a A = 0} =the left annihilator A\. L is faithful (i.e. is
a monomorphism) iff A\ = {0}. In particular, by 1.2.28, L is
faithful whenever A is s.s..
1.5.2. If /; is a regular 1. ideal of A then the regular
representation C induces a representation C^ in the quotient
space A* = A//;. £# is given by : £# :ay->lf, with
lf(x + Ii) = ax + h(x + A).£# is called the regular representation
of A in A*.
The representation £# is called irreducible if A* is a simple '
(£*—) module. For £# to be irreducible it is a necessary and
sufficient that I be a maximal regular 1. ideal of A.
If Z# is the regular representation of A in A* = A/I, we
write
(/j : A) = ker £# = {a G A : if = 0} = {a G A : aA C /J
Similarly, we write for a regular r. ideal /r,
(/r:A) = {aeA:AaC /r}.
1.5.3. LEMMA. Lei I be a regular I. (or r.) irfea/ o/ A. TAen
T A module E over a ring R is called a simple R -module if RE ^ {0}
and has only E and { 0 } as its submodules.

38
Algebraic Preliminaries
(I : A) is a bi-ideal of A with J ~ (I : A) C 7; also J is the
largest bi-ideal (of A) contained in I. If I is a regular bi-ideal
then (I : A) = 7.
PROOF. That J = (7 : A) is a bi-ideal is clear. Assume now
that 7 is a regular 1. ideal. Then for x G J, xA C 7, so that in
particular xu G 7, where u is a relative right unity for 7.
x = x - xu + i« G 7, J C I.
Further, if J' is a bi-ideal of A with J' C 7 then J'A C
J' C 7, whence J' C (7 : A) = J. The proofs of these statements
when 7 is a r. ideal are similar. Finally, when 7 itself is a bi-ideal
we have clearly (7 : A) = J = I.
1.5.4. Let A be an algebra (or even a ring). An ideal P is
called I. (=left) primitive if there is a maximal regular 1. ideal M;
of A with P = (Mi : A) Similarly, if there is a maximal r. ideal
Mr of A with P = (Mr : A) then P is called r. primitive. By
1.5.3 a Lor r. primitive ideal is a bi-ideal In general a 1. primitive
ideal of a ring need not be r. primitive. ' If an ideal P is both 1.
and r. primitive then it is called bi-primitive. If A is commutative
then all these concepts evidently coincide.
In the sequel we shall use the term primitive for 1. primitive.
We call an algebra (or ring) A primitive if the zero ideal {0}
is primitive; equivalently, if there is a maximal regular 1. ideal M;
with {0} = (Mi : A).
1.5.5. LEMMA. Let A, A* be algebras (or rings) and <p :
A —* A* be an epimorphism. If P is a primitive ideal of A with
ker cp C P, then <p(P) is primitive.
PROOF. If P = (M : A), where M is a maximal regular
1. ideal of A, then <p(M) is a maximal regular 1. ideal of A* and
<p(P) = (<p(M) : A), whence <p(M) is primitive.
1.5.6. COROLLARY. If P is a primitive ideal of A and 9 :
A —+ A/P = A* the canonical homomorphism, then 6(P) = {0*}
t See [4',pp.473- 75

§ 5. Regular Representation and Primitive Ideal 39
is primitive.
1.5.7. PROPOSITION. If A is an algebra (or ring) and A ^
\/A, then
^A = f|(M, : A) = f|(Mr : A)
where Mi (respy. Mr) runs through all maximal regular
I. (respy. r.) ideals of A. In particular A is the intersection of
all (I.) primitive ideals of A as well as the intersection of all
r. primitive ideals.
PROOF. By 1.2.22, 1.2.24 (b) we have y/A C Mu whence by
1.5.3, \J~A C (Mi : A). Also, since M; is regular we have (M; :
A) C Mh Therefore
\/A~ C f](Mi : A) C p| Mi = \[A.
so that \/~A= f](Mi : A), Similarly, \f~A= f](Mr : A).
1.5.8. LEMMA (cf. [27, p.144]). A maximal regular I. (respy.
r.) ring ideal of an algebra A is an algebra ideal. Further, every
primitive ring ideal of A is an algebra ideal. Hence the algebra
radical of A coincides with the ring radical of A.
PROOF. Let M be a maximal regular l.ring ideal of A and
write Mi = {x E A; Ax C M}. Clearly, Mi is a 1. algebra ideal
and M C Mi. Further, if u is a relative r. unity for M then
u ¢ Mi. (since u E M\ => u2 E M => M = A, by 1.2.8 (b), 1.2.9
(i)). By ,maximality of M we conclude that Mi = M, whence
M is an algebra ideal. Similarly, the proof when M is a r. ideal.
Now if P is a primitive ring ideal of A then P = (A : M)
where M is a maximal regular 1.ideal. If a E P,a E F then
aaA — a.aA C a A C M,
so that aa E P, where P is an algebra ideal.
1.5.9. LEMMA. (Jacobson). Every primitive ideal P is
prime. <
' A bi-ideal P in a ring R is called prime or a prime ideal if P ^ R
and for any bi-ideals /, J IJCP=>ICPotJCP.

40
Algebraic Preliminaries
PROOF, (cf. 21, p.54). Suppose that IJ C P = (M : R) and
J 2 P. Then JR <£. M and hence, by maximality of M, JR +
M = R. It follows that
IR C I(JR +M)CIJ+MCP + M = M
so that I C (M : R) = P.
1.5.10. PROPOSITION, (a) Every maximal regular bi-ideal
M is bi-primitive, in particular prime. Furthermore, the quotient
A# = A/M is unital and simple. If A is commutative then A#
is a division algebra.
(b) If A is commutative then the primitive ideals of A are
precisely the maximal regular ideals.
PROOF, (a) Let M; be a maximal regular 1. ideal with M C
Mi. Then
M C (Mi : A) C Mi.
Since (M; : A) is a bi-ideal, by maximality of M, M ~ (Mi :
A), whence M is (1.) primitive. Similarly it is r. primitive.
Further, the bijection, between the bi-ideals of A# and the bi-ideals
of A containing M, shows that A^ is simple. A^ is further
unital since M is regular. Finally, when A is commutative it
follows from 1.2.5 that A^ is a division algebra.
(b) In view of (a) we have only to show that a primitive ideal
I is maximal regular. Since I is primitive there is a maximal
regular ideal M with I = (M : A) But since now M is a bi-
ideal, by 1.5.3, (M : A) = M. Thus, I = M and I is maximal
regular.
1.5.11. The intersection of all maximal regular bi-ideals of an
algebra A is called the strong radical and is denoted by yA; if A
has no maximal regular bi-ideal we define A to be \f~A~ : \f~A~ = A.
1.5.12. PROPOSITION. In any algebra A we have
y/AC^AC VA.
PROOF. The first inclusion relation follows since every
maximal regular bi-ideal is (by 1.5.10 (a)) primitive; the second is a

§5. Regular Representation and Primitive Ideal 41
consequence of every hypermaximal ideal being (by 1.3.5) a
maximal regular bi-ideal.
1.5.13. Let A be an algebra. An A-module X is said to
be cyclic, with generator xq, if xq G X and Axq = X. Also, an
A-module X is said to be irreducible if (i) Ax ^ {0} and (ii) the
only submodules of X are { 0 } and X.
If X is an A-module, for xq £ X we write ker 2¾ = {a G A :
axo = 0}. Similarly, for ao G A we write kerao = {x G x : aox =
0}. Clearly ker 2¾ is a 1.ideal of A and kerao a submodule of
X.
1.5.14. LEMMA. Let X be an A-module. Then:
(i) If X is cyclic with generator xq then ker 2¾ is a regular
I. ideal of A.
(ii) If X is irreducible then it is cyclic with any non-zero element
xq as generator. Further, M = kerxo is a maximal regular
I. ideal of A.
PROOF, (i) Since X = Axq there is an element u G A with
xq = uxq. Then (a - auo)xo = axo ~ «^o = 0, so that a - au G
ker xq whence ker xq is regular with u as a relative r. unity.
(ii) The set X0 = {x G X : Axq = {0}} is clearly a submodule
of X. By irreducibility of X we have Xq = {0} or X. But
condition (i) in the definition of irreducibility rules out Xq = X
and so we must have Xq = {0}. Thus, if xq G X, xq ^ 0 then
Axq ^ {0} and so Axq = X, proving X is cyclic with generator
0¾. Also, by (i), M = kerxo is a regular 1. ideal. It remains to
prove that M is maximal. If L is a 1. ideal of A with L D M,
we can choose an element b G L\M, and then bxo ^ 0. It follows
that Abxo = X, whence there is an ao G A with aobxo = xo.
But then, for any a G A,
a — aaob G kera;o — M C L. (*)
Since b G L and L is a 1. ideal we conclude from (*) that a G L,
which means L = A, proving M is maximal.
1.5.15. If A is an algebra and /; a 1. ideal of A, then the

42
Algebraic Preliminaries
quotient A# = Aj 1\ is canonically a left A-module:
a(x + /;) = ax + I\ (a,x G A).
The corresponding representation £# is given by, C* :a^> if,
where if is the linear transformations on A* such that lf(x +
h) = ax + Ii (cf.1.5.2). C* is faithful (i.e. 1-1) if (h : A) = {0}.
1.5.16. LEMMA. Let I — h be a regular I. ideal of an algebra.
Then:
(i) The quotient module A* = A/1 is cyclic.
(ii) A$ is irreducible iff I is maximal.
PROOF, (i) If u is a relative r.unity for I then x + I =
xu + I = x(u + I), so that A# is cyclic with u + I as generator.
(ii) If 7r : A —> A^ is the canonical module homomorphism
given by x^x + I, then clearly /k~1{Xq), where Xq ranges
through all submodules of X = A*, are precisely the 1. ideals
of A containing I. Hence the irreducibility of X is equivalent to
the maximality of I.
1.5.17. LEMMA. Let X be a cyclic A-module with generator
xq and I = ker xq. Then
$ : a -\-1 i-> axQ
is a module isomorphism of A# = A/I onto X.
PROOF. $ is well-defined since a\ + I = a-i + I => a\XQ =
a2XQ. Since every element of X has the form axQ, $ is surjective.
Finally, $ is a module homomorphism since
*(6(a + I)) = $(6a + 1) = baxQ = 6$(a + I)
1.5.18. Let A be an algebra and X an A-module. Write
V = V{X) = the set of all A-endomorphisms (i.e. endomorphism
T of the additive group of X satisfying Tax = aTx for all a G
A). Clearly V is a unital algebra, with the identity map of X as
the unity element.
1.5.19. LEMMA (Schur). For an irreducible A-module V is
a division algebra.
PROOF. If T e D,T 7^ 0 then TX,kerX are submodules

§ 6. Real and Complex Algebras
43
with TX ^ {0},kerT ^ X. By irreducibility of X, TX =
X,kerT = {0}, so that T is invertible and T~l G P.
1.5.20. Let X be an irreducible A-module and V — D{X).
If x\, • ■ ■, xn G X these vectors are said to be D -independent if
for any T\, ■ ■ ■ ,Tn G D,
TlXl + ■■■ + Tnxn = 0 => 7\ = ■ ■ ■ = Tn = 0.
Since D is a division algebra, X can be regarded as a linear
space over D. Then clearly D -independence is the same as linear
independence in this linear space.
1.5.21. THEOREM (Jacobson's Density Theorem). Given V -
independent vectors x\, ■ ■ ■, xn and arbitrary vectors j/i, ■ ■ ■, yn all
in irreducible module X, there is an a in A with axj = j/y(l ^
j ^ n).
PROOF. See [4, p.123].
§ 6. Real and Complex Algebras
1.6.1. Let K denote either the real field R or the complex
field C. An algebra over K is called real or complex according as
K is R or C. Every complex algebra A becomes a real algebra
A'"*' if we restrict the scalars for A to R. A real algebra A is
said to have a complex structure if A can be made into a complex
algebra A^ such that (aIc1)W = A.
1.6.2. LEMMA. A real algebra A has a complex structure
iff A admits a linear endomorphism J such that: (*) J(xy) =
(Jx)y + x(Jy); J2 — —I, where I is the identity map of A.
PROOF. If A is a complex algebra, J defined by Jx = ix
(i — \J- 1) satisfies the above condition (*). Conversely, if there is
a J satisfying (*), a complex structure can be defined by setting
(a + i/3)x = ax + /3Jx (a,/3 <ER,x e A).
1.6.3. COROLLARY. A real unital algebra with unity e has

44
Algebraic Preliminaries
a complex structure iff there is an element j in the centre of A
such that j = — e.
PROOF. If A is complex we can take j — ie. Conversely, if
an element j satisfying the above condition exists then Jx = jx
will define a linear endomorphism satisfying (*) of 1.6.2.
1.6.4. Let A be a real linear space and A the set of ordered
pairs (x,y) with x, y G A. We denote (x,y) by x + iy, where
i = \/-T. For z — (x, y) = x + iy, we write Rez ~ x, Imz = y
The set A becomes a complex linear space if we define addition
and multiplication in A by:
(xi + ij/i) + (x2 + iy2) = (xi + x2) + i{yi + y2) (1)
(a + t/3) (x + iy) = {ax - /3y) + i(ay + fix) (2)
where Xj, yj (j = 1, 2),x, y G A; a, (3 G R.
We identify ieA with x + iO — (x,0).
In particular 0 = 0 + iO = (0,0).
For z <E A,z = x + iy, we set z = x — iy. Then it is straight
forward to check that the map uj : z ^-^ oj{z) = z (called
conjugation) has the properties:
z\ + z2 — zi + z~2; Xz = Xz; z= z
(zi,z2,z G A, A G C).
If now A is a real algebra then A can be made into an algebra
by defining multiplication in A via the equation
z\z2 = {xxx2 - J/1J/2) + i{xiy2 + x2yi) (3)
fa = Xj + iyi {j= 1,2) e A).
The complex algebra A is called the complexification of the
real algebra A. Also it is easy to check that conjugation preserves
multiplication: Z\Z2 = z\z2. For any subset S C Al we write S
for oj(S). We also write
Re S = { Re z : z G §}, Im S = { Im z : z G 5}.
We call a subset 5 self-conjugate if 5 = 5.
1.6.5. PROPOSITION, (a) A is a subalgebra of (A)M.

§6. Real and Complex Algebras
45
(b) A is commutative iff A is commutative.
(c) A is unital iff A is unital and then both A, A have the
same unity element.
PROOF, (a) Clear.
(b) This follows from (3) of 1.6.4.
(c) If A has a unity e it is easily checked that e is also a unity
of A. On the other hand if e is a unity of A then it is easily seen
that the element e = Re e is a unity of A and consequently also
of A. By uniqueness of the unity, e = e, completing the proof.
1.6.6. LEMMA. Suppose that a,b,c,d G A and
z — (a + ib) o [c + id) in A.
Then
z — [a — ib) o [c — id)
PROOF. Write zx — a + ib,z2 = c + id. Then
z ~ Zi o z2 = {zi + z2 + zxz2) - Zl + Z2 + zxz2
= zxoz2.
1.6.7. LEMMA. If c + id [respy. a + ib) has a + ib [re-
spy, c + id) as l.q.i. respy. r.q.i) then c - id [respy. a — ib) has
a — ib [respy. c-id) as l.q.i. [respy. r.q.i). In particular, a-ib is
q. invertible iff a + ib is q. invertible and then we have [a - ib)' =
c - id if [a + ib)' = c + id.
PROOF. This follows from 1.6.6, by taking z = 0 (noting that
0 = 0)
1.6.8. LEMMA. [a + id)oc = 0 [respy. co[a-\-ib) = 0) => aoc
[respy. c o a) ~ 0, where a,b,c are elements of A.
PROOF. This is obtained by equating real parts on both sides
of the equation (after expanding).
1.6.9. LEMMA. Let A be unital. Then a — ib is invertible in
A iff a + ib is invertible in A, and then we have
[a - ib)"1 = c - id if [a + ib)'1 = c + id.

46
Algebraic Preliminaries
PROOF. If 2:12:2 — e, then 2:12:2 = ^1¾ = e = e. The lemma
now readily follows.
1.6.10. LEMMA. Let A be a real algebra and A its complex-
ification. Then:
(i) An element a G A is q. invertible in A iff it is q. invertible
in A.
(ii) If A is unital, then a G A is invertible in A iff it is
invertible in A.
PROOF, (i) since A C A it is enough to prove that "a is
q. invertible in A" => "a is q. invertible in A". Suppose that
a 0 (c + id) = 0 = (c + id) 0 a.
By 1.6.8, aoc = 0 = coa, ' proving a is q. invertible in A.
(ii) The proof is similar to that of (i)
1.6.11. LEMMA. If I is a I. (respy. r.) ideal of A then uj(I)
is a I. (respy. r.) ideal of A. Further, if I is regular with relative
r. (respy. I.) unity u then uj(I) is regular with r. (respy. I.) unity
uj(u).
PROOF. Easy consequences of the properties of oj.
1.6.12. PROPOSITION. Let A be a real algebra and A its
complexification. Then:
(i) If S is a subspace of A, S = S + iS is a subspace of A.
(ii) If I is a I. (respy. r.) ideal of A, I ~ I + ii is a
I. (respy. r.) ideal of A. Moreover, I is regular whenever I
is regular and every relative r. (respy. I.) unity for I is also a
relative r. (respy. 1.) unity for I. If M is a maximal regular
I. (respy. r.) ideal of A then M = M + iM is a maximal
regular I. (respy. r.) ideal of A. Finally, M is hypermaximal
whenever M is hypermaximal.
' Since c is also a q. inverse of a in A, by uniqueness of q.i., c = c + id,
or d = 0.

§ 6. Real and Complex Algebras
47
(iii) If S is a subspace of A then Re S = Im S ~ S (say) is a
subspace of A such that S C S + iS,Af\S C 5. Further,
S = S + iS iff S is self-conjugate, and then S ~ Af]S.
(iv) If I is a I. (respy. r.) ideal of A, I = Re I is a
I. (respy. r.) ideal of A. Also, if I is regular with
relative r. (respy. I.) unity u then I is regular with relative
r. (respy. I.) unity u = Re u.
(v) Every self-conjugate regular I. (respy. r.) ideal I of A has a
relative r. (respy. I.) unity u belonging to A, which is also a
relative r. (respy. I.) unity for I = Re I = Af] I. Further, if
I ^ A then I ^ A. Finally, the correspondence
I h-> I + il ~ I (say)
is a bijection between the set of regular I. (respy. r.) ideals of
A and the set of self-conjugate regular I. (respy. r.) ideals of
A;I=Af]I.
PROOF, (i) This follows from (1),(2) of 1.6.4.
(ii) The first statement is a consequence of (l), (3) of 1.6.4 . If
I is a regular 1. ideal with a relative r. unity u then
(x + iy)u — (x + iy) = xu — x + i(yu — y) G I + il ~ I,
proving u is also a relative r. unity for I (the proof of the
corresponding result when I is a r. ideal is similar). If M is a regular
1. ideal of A with a relative r. unity u, then as seen above u is
a relative r. unity for M = M + iM. Since u ¢. M,u ¢. M and
so M yt A. By 1.2.10 there is a maximal regular 1. ideal Mi of
A with M C Mi. It follows that M C Re Mi, whence by the
maximality of M we have M = Re Mi. So
Mi C Re Mi + i (Im Mi) C M + iM = M,
whence Mi = M and M is maximal (the proof of this when M
is a r. ideal is similar).
Suppose now that M is hypermaximal. Then M is (maximal)
regular. If z — x + iy G A (x, y G A) and u is a relative unity
for M, we have
x = au + m, y = /3u + mi ,

48
Algebraic Preliminaries
where Q,j3eR and m, mi G M. Therefore
z = (a + i(3)u + (m + j'mi),
so that A = Cu + M, whence M is hypermaximal.
(iii) It follows from (1),(2) of 1.6.4 and the definition of Re 5
that it is a subspace of A. Also, since for z G A, we have Re z —
Im iz, Im z = Re (—iz) it follows that Re 5 = Im S = S
(say). Further, since z ~ Re z — i Im z we have S C 5 + z'5.
Clearly, A (^^ C 5. Finally, if 5 is self-conjugate and z £ 5
then z G 5,
Re z = -(z + z) <E S
so that 5, and hence, i5 C 5, whence 5 = S -\-iS,S = APIS'-
(iv) By (iii), 7 is a subspace of A. Also, it follows from (3) of
1.6.4 that it is a 1. (respy. r.) ideal. Assume now that 7 is a regular
1. ideal with u = u + iv (u,v G A) as a relative r. unity. Then,
for x G A, we have
x(u + iv) — x ~ xu — x + ixv G 7,
so that xu — x £ I = Re 7, proving I is regular with u = Re u
as a relative r. unity. The proof of the corresponding result when
7 is a r. ideal is similar.
(v) Since 7 is self-conjugate, by (iii),
I = I + iI,I= Re 7,7= Ap|7.
By (iv), 7 is regular and by (ii), any relative r.unity for 7 is a
relative r. unity for 7 is a relative r. unity for 7. The bijectiveness
of the map 7 h-> 7 + ii is immediate.
1.6.13. LEMMA. Let I be a regular I. [respy. r.) ideal of A —
A + iA. Then I = Re 7 = A iff I has a relative r. (respy. I.)
unity of the form iv(v G A).
PROOF. It suffices to prove the result when 7 is a 1. ideal.
Assume that u = iv is a relative r. unity for 7. By 1.6.12 (iv),
Re u = 0 is a relative r. unity for 7. Since 0 G 7, 7 = A.
Conversely, assume that 7 = A. Let u = u + ivq be a relative
r. unity for 7. Since uGA=7= Re 7 we must have u + iy0 G I

§6. Real and Complex Algebras
49
for some j/o £ A. Then, for any z = x + iy G A and v = vo — J/o
we have
22v — z — zi(vo — yQ) — z = z(u + ivo) — z — z(u + ij/o) G / + / = /
whence z v is a relative r. unity for /.
1.6.14. Let A be a real algebra and A its complexification.
If / is a functional (= real-valued function) on A, the extension
/ of / given by:
/(* + ») = /(*) + «/(») fc.yeA)
is called the canonical extension of / (to A). It is straightforward
to verify that / is a linear functional (respy. character) of A if /
is a linear functional (respy. character) of A
A character x of A is called real if x ~ x|A is real-valued
(and then X is a character of A). Note that the canonical
extension x of a character x °f A is real.
1.6.15. PROPOSITION, (a) Every real character x of A
satisfies: x(z) = x(z) {z G A)_
(b) A character x °f A is real iff kerx is self-conjugate.
(c) There is a bisection between the set of all characters of A
and the set of all real characters of A, given by \ ^ X> where x
is a character of A and x its canonical extension to A.
Proof, (a)
X{z) = x{x - iy) = X(x) - ix(y)
= (x(x) + *x(y)) = x(z)
(b) If x is real then it follows from (a) that kerx is self-
conjugate. Conversely, if kerx = M is self-conjugate then by
1.6.12 (iii), M = M + iM, where M = Re M. Therefore, by
1.6.12(v), M has a relative unity for M. By 1.3.9, M is hyper-
maximal. It follows that if x G A then
x = (a + i/3)u + mi + irn.2, for some Q,j3sR; mi,rri2 G M.
By taking real parts we get from the above equation
x — au + mi, whence x(z) = a G R,

50
Algebraic Preliminaries
so that x 1S real.
(c) If x is a real character of A and \ = x\A, then x IS
a character of A and further \ 's the canonical extension of \-
The bijectiveness of the map xHX 's now clear.
1.6.16. PROPOSITION. The complexification A of a real s.s.
algebra is s.s. .
PROOF. Let M be any maximal regular 1.ideal. By 1.6.12
(ii), M = M + iM is a maximal regular 1. ideal of A. If z ~
x + ly e ^4 then z G M, whence x,y G M. Since M is an
arbitrary maximal ideal we obtain
x,j/Gp|M = \/A = {0}. Thus, 2 = 0 and V A = {0}.
1.6.17. DEFINITION. An algebra A over K is called formally
real if it satisfies the condition <
x, y G A and x2 + y2 = 0 => x = y — 0. (*)
A formally real algebra A(^ {0}) is necessarily real. For, if A is
complex, x G A and z 7^ 0 then x + («z)2 = 0, so that it is not
formally real.
1.6.18. REMARK. Any real algebra whose elements are real-
valued function on a set 5 is formally real (f(s)2 + g(s)2 = 0 =>
f(s) = g(s) = 0).
1.6.19. PROPOSITION. Let A be a formally real algebra.
Then A as well as its complexification contain no non-zero nilpo-
tent element.
PROOF. It follows from the definition of formal reality that
0 = x2 = x2 + 02 => x ~ 0(x G A). Now suppose that y G A, yn =
Cj,"-1 ^ 0(n > 3). If n = 2k then y2k = {ykf = 0 => yk = 0,
contradicting yn~x 7^ 0. Similarly, if n = 2A: + 1 then since
j/"+1 = 0 we get yk+l = 0, again contradicting yn~l ^ 0. Thus
A has no non-zero nilpotent element.
' This condition which is weaker than the Artin-Schreier condition for
formal reality in the case of fields has been used by Arens and appears to be
due to him.

§6. Real and Complex Algebras
51
Next suppose that z — x + iy (x, y G A) and z2 = 0. Then
we get x2 — y2 = 0, xy + yx = 0 It follows that
x4 = x2.y2 ~ x(xy)y = x(-yx)y = -(xy)2,
(x2)2 + (xy)2 — 0, whence by formal reality of A, x2 ~ 0, y2 =
x2 ~ 0, so that x,y,z = 0. We can show as above that the
condition "z2 = 0 => z = 0" is sufficient to conclude that A
contains no non-zero nilpotent element.
1.6.20. PROPOSITION, (a) The complexification A of a
commutative real division algebra A is a division algebra iff A is
formally real.
(b) Every commutative real division algebra which is not
formally real has complex structure.
PROOF, (a) Suppose that A is formally real. If z G A,z =
x+iy 7^ 0 then by formal reality of A, x2-\-y2 7^ 0 and so invertible
(since A is a division algebra). Since we have
-- 2 2
zz = zz = x + y
by 1.1.30, z is invertible and A is a division algebra.
Conversely, suppose that A is a division algebra and for some
x, y G A we have x2 + y2 = 0. Then, writing z — x + iy we get
zz = x2 + y — 0,
so that 2=0, since A is a division algebra. Thus, x = y = 0
and A is formally real.
(b) Since A is not formally real there exist in it elements
x,y ^ 0 with x2 + y2 = 0. Writing j = yx'1 = x~ly, we get
e + j2 — 0, or j'2 = -e. Therefore, by 1.6.3, A has complex
structure.
1.6.21. LEMMA. If A is a commutative formally real algebra
and M a maximal regular ideal, than the quotient A* = A/M is
formally real.
PROOF. If x*2 + y*2 = 0* then x2 + y2 C M C M, where
/Vf is the complexification of M; M is also maximal regular. Since
(x + iy)(x - zy) = x2 + j/2 G M,

52
Algebraic Preliminaries
M is prime (by 1.5.9, 1.5.10 (a)) and M is self-conjugate we
conclude that x ± iy G M, whence
x,y eM,x# =0#,y* = 0*,
proving A# is formally real.
§ 7. Spectrum and Quasi-spectrum
1.7.1. DEFINITION. Let A be a unital algebra over a field
F. For x G A set
&a{x) = {A G F : x — Ae is not invertible }
where e is the unity of A. The set a(x) = oa{x) of scalars is
called the spectrum of x and its set-complement p(x) = F\a(x)
the resolvent set of x;a(x) or p(x) can be empty.
1.7.2. LEMMA. Let A be a unital algebra. If x G A is
invertible, then either a(x) = <r(x_1) = 0, or a(x) ^ 0 and
o{x-l) = {\~1 : Ag<j(x)}.
PROOF. Since x and (hence) x~ are invertible we have
0 G ^(x),0 G ^(x-1). If A(t^ 0) G ^(x) then there is a y such that
j/(x — Ae) = e = (x — Xe)y.
This can be rewritten as
y{-Xx)(x-1 - A_1e) = e = (x_1 - A"1e)(-Ax)j/
which shows that x_1 — A_1e has a 1. inverse as well as a r. inverse
and hence (by 1.1.4) it is invertible. Thus A-1 G p(x~1). Further,
since (x-1)-1 = x we conclude that
A(^O)G^x) iff A"1 G^(x"1).
The lemma now follows by passing to the set-complements of
^(x)^(x-1) in F.
1.7.3. PROPOSITION. Let A be a unital algebra over F with
e as its unity and x G A be such that <r(x) 7^ 0. Then:

§ 7. Spectrum and Quasi-spectrum
53
(i) o(0) = {0};
00 *(«) = {!};
(iii) 0(//2) =//0(2)(//G F);
(iv) o(//e + 2) =// + 0(2),(//G F);
(v) For any polynomial P over F, we have
a(p(x)) D P(a(x))l (*)
When F is algebraically closed - in particular, when F = C
- there is equality in (*), i.e. we have
o(P(x)) = P(a(x)). (**)
PROOF. Since 0 - Ae = -Ae is invertible for A 7^ 0, and
e - Ae = (1 - A)e is invertible for A / 1 we obtain (i),(ii). For(iii),
the identity //2 - Ae = //(2 - //_1Ae),(// /0,AeF) shows that
A G 0(//2) iff //_12 G 0(2). Also, if // = 0, (iii) follows from (i).
Thus we have (iii). Next, (iv) follows from the identity
(//e + 2) - Ae = 2 - (A - //)e.
It remains to prove(v). If P — ao is a constant polynomial then
(*) holds since both sides reduce to {ao}. Next assume that P
is nonconstant. If A G 0(2) then y = 2 - Ae is not invertible. So,
by 1.2.12 (iii), one at least of Ay, yA 7^ A, whence there is a 1. or
r. ideal J(= Ay or yA) with y G J, J 7^ A. If
P(2) = a„2" + ■ ■ ■ + a0e (otj G F)
zk = xk'1 + Xxk-2 + --- + Xk'1e,
- (2 - Xe)zk = yzk = zky G J, where y = 2 - Ae,
re
P(x) - P(X)e = 2 ak(xk - A*e) e J.
jfc=i
t We write P(tr(z)) for {P(A) : A G a(x)}
and
then
xk - A*e
so that

54
Algebraic Preliminaries
Since J ^ A,P(x) - P(X)e is not invertible and P{\) G a(P(x)),
proving (*).
Assume now that F is algebraically closed and that P —
P(X) is a polynomial of degree n > 0, over F. Then we have the
factorization P(X) - A = a(X - //i) ■ ■ ■ (X - //„), where a,fij G
F and fij are the zeros of the polynomial P — X. It follows that
P(fij) = X (j — 1, ■ ■ ■, n). Further we have the factorization
j/ = P(x) — Xe — a(x - //ie) ■ ■ ■ (i - //„e).
If A G a(P(x)), y is not invertible, whence by virtue of 1.1.30,
some x - \i^e is not invertible. It follows that //¾ G a{x) and
A = P{nk) G P(ff(a;)). Thus, a{P{x)) C P(ff(a;)). Combining
this with (*) we get (**). It remains to deal with the case where
deg P = 0. In this case, if P(X) — aQ (say), (**) holds since as
noted in the proof of the first assertion in (v), each side of (**)
reduces to {c«o}.
1.7.4. LEMMA. Let A be unital, with unity e, and x G A.
Then A(^ 0) G a(x) iff —A_1x is not q. invertible
PROOF. The identity
x- Xe= -A(e - A_1x) (x G A, A G F, X ^ 0)
and 1.1.20 show that A G p(x) iff —X~1x is q. invertible. The
required result now follows (by taking the negation of the statements
on both sides of "iff").
1.7.5. DEFINITION. Let A be an algebra over F, not
necessarily without a unity, and Ai its unitization. For x G A we
write
CT'(z) = cta(z) = oAl{x)
and call o'{x) the quasi-spectrum of x\ its complement
p'(x) = F\a'(x)
is called the quasi-resolvent set of x.
1.7.6. LEMMA. 0 G a'{x) for any x G A, so iAai <r'(a;) is
always nonempty
PROOF. If possible let x have an inverse y + Aei (j/ G A, A G

§ 7. Spectrum and Quasi-spectrum
55
F) in A\. Then e\ ~ x(y + Aei) = xy + \x E A which is
impossible. Hence 0 E aA^x) = a'(x).
1.7.7. Proposition, (i) a'(0) ~ 0; (ii) a'(nx) = na'(x)(n e
F); (iii) // P is a constant-free polynomial over F then
a'(P(x)) D P(a'(x))
with the inclusion relation becoming equality when F is
algebraically closed, so that then
o'(P(x)) = P(o'(x)).
PROOF. These follow from 1.7.3 since for any element a in
A we have c'(a) = &Ai(a)-
1.7.8. LEMMA (Kaplansky). Let A be an algebra x E A.
Then A E a'(x)\{0} iff -X~lx is not q.invertible.
PROOF. Since
x - Aex = -A(ei - X'1x) (A ^ 0)
where e\ is the unity of the unitization A\ of A, we obtain:
A(t^ 0) E aax & x - Aei is not invertible in A\
•O- (ei - A- x) is not invertible in A\
•O- —\~1x is not q.invertible in A\ (by 1.1.20)
•O- — A~ x is not q.invertible in A (by 1.1.21).
1.7.9. COROLLARY. If u j^ 0 is an idempotent then <r'(u) =
{0,1}. Further, if A is unital, with unity e, and u/0,e then
a(u) = {0,1}.
PROOF. The first statement follows from 1.7.8, 1.1.26 (i). For
the second we note that by (**) of 1.7.21, we have
a(u)\J{0} = o'(u)={0,l}.
If 0 ^ <t(u) then u is invertible and since u2 = u we get u = e,
contradicting the assumption on u. Thus 0 E c(«),
*(u) = a(u)\J{0} = a'(u) = {0,l}.

56
Algebraic Preliminaries
1.7.10. LEMMA. If u is a relative unity for a regular I. or
r. ideal I ^ A then 1 G c'(u).
PROOF. By 1.2.9 (ii) -u is not q.invertible, whence by 1.7.8,
1 Gff'(u).
1.7.11. COROLLARY. If x *s an1) character of A and x G A
then x{x) ^ a'{x)-
PROOF. Since always 0 G (r'{x) we may assume that x{x) ~
A^O. Then x(A_1x) = 1, so that by 1.3.10, u = A_1x is a
relative (bi-) unity of Mx, whence by 1.7.10, 1 G o'{u) = a'(X~1x) =
A~V(z), so that Ag<t'(z).
1.7.12. LEMMA. Assume that x G A is q. invertible with
q.i. x'. Then
*V) =(-1^ = ^^)}.
PROOF. First note that 0 G <r'(x) and -j— = 0 G a'(x').
Since x is q. invertible, by 1.7.8, -1^ c(a;). For A/0 we have
the (easily verifiable) identity:
x' o (-A"1!) = (-A"1*) o x' = ii^ar'.
A
It follows, by 1.7.8 and 1.1.30, that -^ G <j'(x) iff A G a'(x),
completing the proof.
1.7.13. DEFINITION. An element x in A is called quasi-
nilpotent or q. nilpotent if <r'(a;) = {0}. The set of q. nilpotent
elements of A is denoted by Aqn;0 G A9" (by 1.7.7 (i)).
1.7.14. Remark. In the algebra Mn(F) of all nxn matrices
over a field F, a q. nilpotent element is nothing but a nilpotent
matrix. This is a consequence of the fact that a matrix is nilpotent
iff all its eiginvalues are 0.
1.7.15. PROPOSITION. Let A be an algebra. Then:
(i) An element x in A is q. nilpotent iff Xx is q. invertible
for every A G F, in particular, a q. nilpotent element is
q. invertible.

§ 7. Spectrum and Quasi-spectrum
57
(ii) A nilpotent or essentially nilpotent element is q. nilpotent.
(iii) \[A C Aqn C tyA.
PROOF, (i) This follows from 1.7.8, 1.7.13.
(ii) If x is nilpotent so is Xx, and hence by 1.1.23, Xx is
q.invertible, whence by (i), x is q.nilpotent. Again, if x is
essentially nilpotent then by definition x G \/A, and hence also
Xx G yf~A, Xx is q.invertible (by 1.2.24 (a)) so that by (i) x is
q. nilpotent.
(iii) The first inclusion follows from (ii). For the second, we
note that if x G Aqn, o'(x) = {0}. If x G A, x(z) G o'(x) (by
1.7.11) so that x(x) — 0; whence x G v^A.
1.7.16. Remark. Every element of a radical algebra A (A =
\/A) is q.nilpotent, so that A = A9". On the other hand, in
the one-dimensional algebra F, 0 is the only q. nilpotent element
(since, by 1.7.20, a'(X) = a(X) |J{0} = {A,0}).
1.7.17. DEFINITION. An algebra A is called quasi-semi-
simple or g.s.s. if 0 is the only q. nilpotent element of A, i.e.
A"n = {0}.
1.7.18. Remark. In view of 1.7.15 (iii), every q.s.s. algebra is
s.s. Note that the matrix algebra Mn(F) is s.s. (actually simple)
but not q.s.s. when n > 2 (since there are non-zero nilpotent
matrices in Mn(F),n > 2).
1.7.19. LEMMA. Let A,B be algebras [over the same field),
<p : A —+ B a homomorphism and x G A. Then we have:
PaW C P'B{<P(*)Y, *b(*>(*)) £ cta(*)- (*)
Pa(x) C />b(^(x)); <tb(^(x)) C <7A(x) (**)
whenever A, B are unital and B has as unity p(e), e being the
unity of A.
PROOF. If A G p'A(x) then -A_1x is q. invertible and so by
1.1.24,
-\~1<p(x) = <p(-\-1x)

58
Algebraic Preliminaries
is q.invertible, so that A G p'B[(p[x)). This proves the first
inclusion relation in (*) and the second one follows by taking set-
complements of both sides in F.
For proving (**), we observe that by using the first relation
in (*) of 1.7.21, the first conclusion in (*) above can be written
as
Pa(x)\{0} C Pb(p(*))\{0}- (*')
If 0 G Pa{x), x is invertible and then <p(x) is also invertible, and
0 G pb(<p(x)). The inclusions in (**) now readily follow from
(*')•
1.7.20. COROLLARY. Let A be a subalgebra of an algebra B
and x G A. Then we have:
p'A(x)tp'B(xy,a'B(x)Ca'A(x). (*)
Pa{x) C pb{x)\ <tb{x) C aA(x) (**)
whenever B is unital and A is a subunital > algebra of B.
PROOF. This follows from 1.7.19 by taking <p to be the
inclusion map A —> B.
1.7.21. PROPOSITION. If A is a unital algebra over F and
x G A, then:
p'(x) = p(x)\{0}; (*)
a'(x) = a(x)\J{0}. (**)
PROOF. Let A\ be the unitization of A and ei,e the unities
of Ai, A respectively. If A G Pay{x) then for some y G A, // G F
we have the equations
(x - Aei)(j/ + //ei) = ei = (y + //ei)(x - Aei). (1)
Multiplying the terms in (1) by e from the left and then by e
from the right we obtain
(x-Xe)(y + fie) = e = (y + fie)(x-Xe). (2)
It follows from (2) that A G Pa{x). Thus,
T i.e. if e is the unity of B then e E A.

§ 7. Spectrum and Quasi-spectrum
59
p'{x) = paAx) ^ Pa[x) = p{x).
On the other hand, if A G Pa{x),\ 7^ 0 then since
x ~ Xe — -A(e - \~lx),
~X~lx is q. invertible in A and hence also in A1; whence A G
^'(x). Further 0 ^ p^i(x)j f°r otherwise either of the equations
(1) with A = 0 shows that e\ G A, which is impossible. This
completes the proof of (*), and (**) follows from (*) by taking
set-complements in F.
1.7.22. PROPOSITION. Let x,y be elements of an algebra A.
Then we have
a'(xy) = a'{yx), (*)
^2/)11(0) = ^)11(°} (**)
whenever A is unital.
PROOF. First assume that A is unital with unity e. If A G
o(xy), A^O then xy - Ae, and hence also
(A-1x)j/-e=-(e-(A-1x)j/)
is not invertible. By 1.1.28 (ii), 1.1.14 (c), -(e - y{X'1x)) =
y\~lx — e is not invertible, so that A G a{yx). Thus
4*J/)U(°K ^)11(0)-
Interchanging x, y we get the reverse inclusion. By combining the
two inclusions we obtain (**). To obtain (*) it is enough to apply
(**) to the unitization A\ of A (remember that for any a G A,
OGff'(i)).
1.7.23. LEMMA. Let A be a unital algebra (over F) with
unity e, and x a linear functional on A such that x(e) = 1.
Then the following conditions are equivalent
(i) Every element x G ker% is not invertible, or equivalently,
x{x) 7^ 0 for every invertible element x.
(ii) x(x) ea(x).

60
Algebraic Preliminaries
PROOF, (i) => (ii). If x{x) = A then x - Ae G kerx 7^ A,
whence it is not invertible, so that A G &(x)-
(ii) => (i). If x G kerx then 0 = x{x) Q a{x)i whence x is not
invertible.
1.7.24. LEMMA. Let A be a unital algebra and x a character
of A. Then x{x) £ °(x) (x£ A).
PROOF. If x(x) — * tnen z - Ae G kerx = M 7^ A and
therefore x ~ Ae is not invertible, and so A G &(x)-
1.7.25. LEMMA. Suppose that A is an algebra over F, a G
A and A G F. If there is a non-zero element b G A such that
ab = A6, then A G ^'(a).
PROOF. In the unitization A\ of A the above condition can
be rewritten as
(a - Aei)6 = 0.
Since 6^0, a - Aei is not invertible and consequently A G a'(a).
1.7.26. PROPOSITION. Let A be an algebra and a G A.
TAen we have:
(i) TVie double commutant Aq = {a}" *s a commutative
subalgebra of A containing a such that
ffA0(a) = <7A(«)- (*)
(ii) H
(a) is a maximal commutative subalgebra of A,
containing a then
°Am(a) = °M- (**)
(iii) If A is unital with unity e then
<TA„(a) = °Am{a) = ffA(a)- (***)
PROOF. First let A have a unity e. By 1.1.8 (vi), A0 is a
commutative subalgebra and clearly e G Ao. If a - Ae has an
inverse 6 in A then by 1.1.18
b <-> {a - Ae}' = {a}', whence 6 G {a}" = Aq.

§ 8. Extended Spectrum and Extended Quasi-spectrum 61
Therefore Pa{o) Q PA0{a), so that by (**) of 1.7.20, PA{a) —
PAn(a)- Also, we have (see 1.1.9) Aq C Am C A. It follows (using
1.7.20) that pa(o-) = PAm{a) — Pa{°)- Therefore, by taking set-
complements in F we obtain (***).
Next let A be non-unital. By applying, (* * *) to the subal-
gebras (Ao)i — Aq + Fe^, (Am)i = Am + Fe\ of the unitization
A\ of A we obtain (*) and (**).
1.7.27. PROPOSITION. Let A be an algebra and B a
q. inverse closed' subalgebra of A.
Then
a'B{x)=aA{x) (xeB).
Similarly, if A is unital and B a inverse closed subunital algebra
of A then
aB(x)=aA{x) (are A).
PROOF. The first assertion follows from 1.7.8, B being
q. inverse closed in A. The second is a consequence of B being
inverse closed in A and the definition of the spectrum.
§ 8. Extended Spectrum and Extended Quasi-spectrum
1.8.1. DEFINITION. Let A be a unital real algebra, A its
complexification and x G A. We write
a(x) = aA{x) ~ o^{x)
and call o[x) the extended spectrum of x. Similarly, writing
ff'(x) = a,^,(x), where (A)i denotes the unitization of A, we
call cr'(x) the extended quasi-spectrum of x; its set-complement
p'(x) ~ K\ct'(x) is called the extended quasi-resolvent set of x.
1.8.2. LEMMA. If A is a real algebra and x G A then
*A(*) = *A(*)riR- (*)
' A subalgebra (respy. subunital algebra) B of A is called q. inverse
(respy. inverse) closed if for any x G B, the q. inverse x' (respy. inverse x~l)
of 2 in A exists => x' (respy. x~l) efl.

62
Algebraic Preliminaries
If A is unital we have also
aA(x)=aAf]R. (**)
In particular, g'a{x) ~ **'a{x)> aA{x) Q &a(x)-
PROOF. We have X(^ 0) e a'A(x) f]R ^ a'A(x) f]R iff -A-1*
is not q.invertible in A iff ~\~~1x is not q.invertible in A iff
A G p'a(x)- This prove (*) (since '0' clearly belongs to both sides
of (*)). The equality (**) follows by applying (*) to A\ and
using 1.7.21.
1.8.3. PROPOSITION. Let A be a real algebra and x G A.
Then g'{x) is a symmetric* subset of C; also when A is unital,
o(x) is symmetric.
PROOF. We have
A(A ^ 0) G g'(x) iff —X~lx is not q.invertible in A
iff (-A"1*) = -Tlx
is not q.invertible in A (see 1.6.7.)
iff AGct'(x),
and 0 = 0, where bar denotes the conjugation in A (see 1.6.4).
Thus g'(x) is symmetric. The symmetry of g(x) is an immediate
consequence of (**) of 1.7.21 and the symmetry of g'(x).
1.8.4. PROPOSITION. Let A be a real algebra, A its com-
plexification, and x G A. Then:
(i) If \ = a+i(3 j^O (a,/3 <ER),-\~1x is q. invertible in A iff
x'\\2?x is q- invertible in A.
(ii) If A is unital with unity e, then x — Ae is invertible in A
iff {x ~ ae)2 + P2e is invertible in A.
PROOF. Write z = — X~1x; then z = — X^1x where bar
denotes the conjugation in A. By 1.6.7, z is q.invertible iff z is
q.invertible. Since z <-> z, by 1.1.30, z o z is q.invertible iff z
(hence z ) is q. invertible. Since
T A subset S of C is said to be symmetric if A G S => A G 5, where A
denotes the complex conjugate of A.

§ 8. Extended Spectrum and Extended Quasi-spectrum 63
x2 - 2ax
Z°~Z= 1112 '
(i) follows. The result (ii) can be proved similarly by using the
identity
(x - Xe)(x - Ae) = (x - ae)2 + /32e
and 1.1.30.
1.8.5. COROLLARY. If A = a + ij3(a,/3 e R), A 7^ 0 iAen
A G cr'(x) iff y = (x2 — 2ax)(a2 + /92)-1 is not q. invertible in A.
Similarly, when A is unital, A G &(x) iff J/ is not q.invertible in
A, and 0 G <7(z) iff x is not invertible in A.
PROOF. The first assertion is an immediate consequence of
1.8.4 (i). Since g'(x) = g(x) U{0} the second assertion follows
from the first together with the observation that if x is not
invertible in A it is also not invertible in A ((xi + iyi)x = e =
x(xi + iyi) => x\x = e = xx\, where x, xi, j/i £ A).
1.8.6. PROPOSITION. (Rickart). Let A be a complex algebra
and AlKJ denote A as a real algebra. Then, for x G A, we have
where bar denotes complex conjugation.
PROOF. We denote the general element of AM by x+jy,
where j2 = -1; n,y G AM = A (as a set). By 1.6.10, if
x G A = AM has a q.i. in AM, it has also a q.i. in A. So,
by 1.7.8, a'A(x) C <7A|K](a;) and hence by 1.8.3 .
To complete the proof it is enogh to show that
v'A[*](z) Ca'A(x)\Ja'A(x).
Suppose that

64
Algebraic Preliminaries
Then fix, fix are q.invertible. Write
(fix)' = y, (fix)' = j/i.
If // = 7 + ^ (7,6 G R), then in AM we have
/*x = 7X + jSx, fix — 7X — j'6x.
It follows from 1.6.8 that we have
(72;)' = (fix)' = y = (/2a;)' = j/i.
Therefore fix ~ y' = y[ = fix, whence 6 = 0, fi = 7. Thus
//x = 7a; G A^l = A, (//x)' = y G A
so that fix is q.invertible in A. It follows that A G o^in),
completing the proof.
1.8.7. DEFINITION. An element x of a real algebra A is
called extended quasi-nilpotent or exi. </. nilpotent if 0^4(2:) = {0},
i.e. if x is q. nilpotent in A.
1.8.8. Remark. By (**) 1.8.2, every ext.q. nilpotent
element is q. nilpotent. On the other hand, in the real algebra
A — Ok', the element i is q. nilpotent but not ext. q. nilpotent.
For, a'A(i) — 0a(O U{0} = {0} since a^(i) = 0 (z — a being
invertible in A for every real a}. But, by 1.8.7,
*a(0 = *c(0U°£(0= {°>±*"}-
1.8.9. LEMMA. Let A be a real algebra. Then every element
of v A is ext. q. nilpotent.
PROOF. Take x G \J~A and let A = a + i/3 ^ 0. Then, y/~A
being a bi-ideal, we have y = (x2 — 2ax)(a2 + /32)~1 G y/A, so that
y is q.invertible. It follows from 1.8.5 that A ¢. &'(%), whence
<r'(x) = {0}, as desired.
1.8.10. DEFINITION. Let A be an algebra over K and x G
A. The spectral radius r(x) is defined by
r(x) — ta(x) — sup{|A| : A G a'(x)}.

§ 8. Extended Spectrum and Extended Quasi-spectrum 65
If A is unital and a{x) ^ 0 then we also have
r{x) = sup{|A| : A G a(x)}.
Similarly, we have the extended spectral radius
r{x) = sup{|A| : A G a'{x)}
for an element x of a real algebra A. When A is unital and
o(x) /i we have
f{x) = sup{|A| : A G ^(z)}-
1.8.11. Remark. Clearly we have the inequalities:
0 < r{x) < oo; r(x) < f(x).
1.8.12. LEMMA. Let A be an algebra over K, and x G A.
Then:
(i) r(0) = 0.
(ii) r(fix) = \n\r(x)((i G K); in particular, r(—x) = r(x).
(iii) r(x") = r(x)n, provided A is a complex algebra.
(iv) 7/u^O is an idempotent of A then r(u) = 1; m particular,
r(e) =1( whenever unity e exists).
(v) r(xj/) = r(j/x) (x,j/GA).
Proof, (i) By 1.7.7 (i), <j'(0) = 0 so that r(0) = 0.
(ii) By 1.7.7 (ii), a'((j.x) = //<t'(x) and hence the result,
(iii) By 1.7.7 (iii), a'(xn) = [a'(x)]n, hence the result,
(iv) By 1.7.9, a'(u) = {0,1}, whence the result.
(v) An immediate consequence of (*) of 1.7.22.
1.8.13. DEFINITION. Let A be a unital complex algebra

66
Algebraic Preliminaries
with unity e. A is called a Liouville algebra' if for every element
x G A, x ¢. Ce, r(x) = oo. If A has no unity then A is called
a Liouville algebra if for every x 7^ 0, r(x) = oo.
1.8.14. EXAMPLE. The algebra A = C[X] of all complex
polynomials is a Liouville algebra. Here if x G A, x ^ Ce then
<t(x) = C, r(x) = oo.
1.8.15. PROPOSITION. Every Liouville algebra A is q.s.s.,
in particular s.s..
PROOF. If A is unital and x (£ Ce then r(x) ~ oo and x is
not q.nilpotent. Also, if x = Ae, a(x) — {A} and r(x) = 0 =>
A = 0. Thus A is q.s.s.. When A is not unital, by definition
of the Liouville algebra, if x ^ 0, r(x) = oo and so x is not
q.nilpotent. Thus A is q.s.s.
1.8.16. PROPOSITION. Let A be a unital algebra over K
such that the spectrum g(x) of every element x of A is
nonempty and bounded. Then the Heisenberg commutator equation
[x, y] = xy — yx = fie (e = unity of A)
has no solution, for any // G K\{0}.
PROOF. Suppose, to the contrary, there are elements x,y G A
with
xy = yx + fie for some //.
It follows that
a{xy)=a{yx + fie)~fi + a(yx). (1)
By 1.7.22, we have
<K*y)U{°} = ^)11(0)- (2)
From (1), (2) we obtain
{fi + a(yx)}\J{0} = a(yx)\J{0}. (3)
T This notion appears to have been first studied by Birtel [51].

§ 9. Strictly Real Algebra
67
If 0 G <r(yx) then (2) becomes
{n + a(yx)}{J{0}=a{yx). (4)
It follows that
// + 0 = // e a(yx).
By repeatedly using (4) we get n\i G a(yx)(n ^ 1). But this
contradicts the boundedness of a(yx) since |n//| = n|//| —+ oo
(as n —> oo). Therefore 0 ^ <r(j/a;). By a similar argument (with
x,y interchanged and // replaced by —//) we also get 0 ^ <t(xj/).
Thus, the equation (2) becomes just
a(xy) ~ a(yx). (2')
So (1) can be rewritten as
a(xy) = (A + a(yx). (1')
Since a(xy) ^ 0 there is a A(^ 0) in a[xy).
Then, by (1'), // + A G a{xy). Once again by (1'),
// + (// + A) = 2// + A G a(xy),
and so on. Thus, n// + A G <t(xj/) for all n > 0. Since
|n// + A| > n|//| — |A| —+ oo, asn-t oo,
once again we are led to contradict the boundedness of a(xy).
Thus the Heisenberg equation has no solutions as asserted.
1.8.17. Remark. The result 1.8.16 is an algebraic
generalization of the well-known fact that the Heisenberg commutation
relation of quantum mechanics: PQ — QP — ^j (h denoting
Planck's constant) has no bounded operator solution.
§ 9. Strictly Real Algebras
1.9.1. DEFINITION. An element x of a real algebra A is
called strictly real if its extended quasi-spectrum is real: a'(x) C R;
by 1.8.2, this condition is equivalent to: cr'(x) = g'{%)-

68
Algebraic Preliminaries
If A is unital the "strictly real" condition can also be stated
as: &{x) C R, or, g(x) = (f(x).
An algebra A is called strictly real if all its elements are strictly
real.
1.9.2. Remark. In any real algebra A the element 0 or
more generally, a nilpotent element x is strictly real (by 1.7.15
(ii), x is q.nilpotent in A, so a'A%) = {0}). Again, if A is
unital with unity e, the elements Ae (A G R) are strictly real
(a'A(Xe) = Xa'A(e) = A).
1.9.3. PROPOSITION, (a) Every real radical algebra A - in
particular real algebra Aq with trivial multiplication - is strictly
real.
(b) Aq is not formally real.
PROOF, (a) Recall that every element a of a radical
algebra is q.invertible (cf. 1.2.27). It follows from 1.8.4 (i), 1.7.8 that
&'a{x) = {*-*}' whence A is strictly real.
(b) This follows from 1.6.19 since every element of Aq is
nilpotent.
1.9.4. Remark. In Aq we have an example of a strictly real
algebra which is not formally real.
1.9.5. LEMMA. The unitization A\ of a real algebra A is
strictly real iff A is strictly real.
PROOF. If A is strictly real, and x\ G A\,
xi = Ae + x (x G A, A G R)
then a(xi) = a(\e + x) = A + a(x) C R (since a(x) C R, A G R),
proving A\ is strictly real.
Conversely, if A\ is strictly real and x G A we have
*'a{x) = a>A(x) = a(A)1(x) = aA1(x) = °aAx) ^ R>
which proves that A is strictly real.
1.9.6. PROPOSITION. A real algebra A is strictly real iff for
any x G A, x2 is q. invertible.
PROOF. Suppose that A is strictly real. By 1.9.5 A\ is

§ 9. Strictly Real Algebra
69
strictly real, whence
It follows that v'(x2) ~ atfa^x2) = atA)lix)2 ^t °' wnence
— 1 ¢- a(x2), so that x2 is q. invertible in A and so also in A
(by 1.6.10 (ii)).
Conversely, suppose that A satisfies that condition for all x in
A. We have to show that cr'(z) C R. If possible let o'(x) contain
a number a + i/3 with /3^0 (a,/?GR). Write
y = (ax2 - (a2 - /?3)x)//?(a2 + /?2); then j/ e A.
By 1.7.7 (iii), a'(y) 3 {a(a + t/3)2 - (a2 - /32){a + i/3)}//3{a2 +
/?2) = i, so that o'{y2) 9 z'2 =-1. It follows that y2 is not
q. invertible in A and so also not q. invertible in A, contradicting
our supposition on A. Thus, g'(x) C R, as desired.
1.9.7. COROLLARY. Let A be unital with unity e. Then A
is strictly real iff for each x G A, e + x2 is invertible. ''
PROOF. This is an immediate consequence of 1.1.20 and 1.9.6 .
1.9.8. COROLLARY. Every epimorphic image of a strictly real
algebra is strictly real.
PROOF. An immediate consequence of 1.1.24 and 1.9.6.
1.9.9. GELFAND'S EXAMPLE OF A NON-STRICTLY REAL
ALGEBRA
The algebra consists of all real-valued functions /= f{t) on
[—1,1] which are holomorphically extendable to the closed unit
disc. It is not strictly real since the function l/(i2 + 1) does not
belong to the algebra, as its analytic extension l/(z2 + l) has
poles at dzi.
t i.e. if A e i?'(z2) then A > 0.
TT The condition "e + z2 is invertible for all x G A " was used by Gelfand
[7 , p.147] to define the notion of strict reality for unital commutative real
Banach algebras; strictly real Banach algebras were called by him just real
Banach algebras.

70
Algebraic Preliminaries
1.9.10. Remark. The algebra in 1.9.9 is however formally
real (see 1.6.18). Thus we have here an example of a formally real
algebra which is not strictly real (cf. 1.9.4).
1.9.11. LEMMA. Let Am be a maximal commutative subal-
gebra of a strictly real algebra. Then Am is strictly real.
PROOF. If x e Am C A. By 1.1.19, (x2)' e Am, whence x2
is q. invertible in Am, Affl is strictly real.
1.9.12. Remark. A real algebra A which has a complex
structure is not strictly real. To see this, we may assume, by
1.9.5, that A is unital. Since
a = e + (ie)2 = 0, (*)
a is not invertible and so by 1.9.7, A is not strictly real. Thus in
particular C as a real algebra is not strictly real. The relation (*)
also shows that the real algebra H of Hamilton quaternions is not
strictly real (note here however that H has no complex structure).
1.9.13. PROPOSITION. Every strictly real q.s.s. algebra A is
formally real.
PROOF. For, suppose that x2 + y2 = 0 (x, y e A).
Then
^) = -^)- (*)
By strict reality, a'~(x2) = o'Ax)2 > 0, a'~{y2) > 0. It follows
from (*) that
a'-(x)2 = (f'Ay)2 = 0, whence
°a(x) = a'*(x) = ®>aA(y) = ^a^V) ~ *-*• ^mce ^ is q-s-s- we
must have x ~ y ~ 0, proving that A is formally real.
1.9.14. PROPOSITION. A formally real division algebra is
strictly real. Conversely, a strictly real commutative division
algebra is formally real.
PROOF. Let A be a formally real division algebra and x e A.
Then by formal reality e-\-x2 — e2-\-x2 ^ 0, whence it is invertible.
So, by 1.9.7, A is strictly real.

§ 9. Strictly Real Algebra
71
Suppose now that A is a strictly real commutative division
algebra and x, y G A, x2 + y2 = 0. If x ^ 0 then we have
e + aTV =e+(x"'1j/)2 = 0,
contradicting strict reality. Therefore x = 0 and similarly j/ = 0.
Thus A is formally real.
1.9.15. PROPOSITION. (Kaplansky [8',p.405]). A primitive
strictly real algebra A is a division algebra.
PROOF. Since A is primitive there is a maximal regular
1. ideal Afj with (Mj : A) = {0} (see 1.5.4). Then X = A/Mt is
a faithful A-module (see 1.5.15) which is further irreducible (by
1.5.16 (ii)). By 1.5.19 we have the division algebra P = D(X).
Denote by dim X the dimension of X as a linear space over V.
If dim X > 1 there are two V -independent vectors x, y G X. By
density theorem (1.5.21) there is an a G A with ax = y, ay = —x.
Then a2x = ay — —x. Since A is strictly real, a2 has a q.i.
b : a2 + b + 6a2 = 0. It follows that
a2x + bx + ba2x = 0, i.e. ~ x + bx — bx — 0,
i.e. x = 0-impossible. So dim X — 1 and X is a division algebra.
Since X is a faithful A-module, A is a division algebra.
1.9.16. PROPOSITION. Let A be a strictly real algebra and
A its complexification. Then:
(i) If I 7^ A is a regular I. (respy. r.) ideal of A then I = Re I
is a regular I. (respy. r.) ideal of A with I ^ A.
(ii) If M is a maximal regular I. (respy. r.) ideal of A and
M = Re M then M is a maximal regular I. (respy. r.) ideal
of A, and M ~ M + iM, M = Af]M; in particular M
is self-conjugate.
(iii) The correspondence M >—> M = M + iM is a bisection
between the set of maximal regular I. (respy. r.) ideals of A
and those of A.
PROOF, (i) In view of 1.6.12 (iv) we have only to show that
I 7^ A. For this, by 1.6.13, it is enough to show that I does

72
Algebraic Preliminaries
not have any relative r. (respy. 1.) unity of the form iv (v G A).
Suppose that I has a relative r. (respy. 1.) unity of the form iv.
Then, by 1.2.8. (6), ~v2 = (iv)2 is a relative r. (respy. 1.) unity for
I. Since I ^ A, by 1.2.9, v2 is not q.invertible in A. On the
other hand, since A is strictly real v2 is q.invertible in A and so
also in A. This contradiction proves that I has no relative unity
of the form iv, and (so) I ^ A.
(ii) By 1.6.12 (iii), M C M+iM. Let u = u+iv (u,v G A) be
a relative r. (respy. 1.) unity for M. By 1.6.12 (iv), u is a relative
r. (respy. 1.) unity for M. Since, by (i), M ^ A, u ¢. M whence
u ¢. M + iM, M + iM ^ A, so that by maximality of M we
have M — M + iM. By 1.6.12 (iii), M is self-conjugate.
(iii) If M is a maximal regular 1. (respy. r.) ideal then by 1.6.12
(ii), M + iM is a maximal regular 1. (respy. r.) ideal of A. Also,
by (ii) above, every maximal regular 1. (respy. r.) ideal of A is of
the form M + iM. The bijection assertion in now clear.
1.9.17. COROLLARY. Every maximal regular (/. orr.) ideal of
A is self-conjugate. Hence is self-conjugate. Also, y/A =
AflV7!.
PROOF. By 1.9.15 (ii), every maximal regular (Lor r.) ideal
of A is of the form M ~ M + iM whence M is self-conjugate.
bince \/A = p|M, Va is self-conjugate. Finally,
y/A = f]M = f](Af]M) = Af)(f)M) = Af)\[A.
1.9.18. PROPOSITION. Let A be a strictly real algebra A its
complexification. Denote by A, A the set of characters of A, A
respectively. Then the map
X !—> X, where x zs the canonical extension of x is a bijection
between A and A.
PROOF. In view of 1.6.14 (c), it is enough to prove that every
character x of A is real. But this readily follows from 1.6.15 (b)
and 1.9.17.

CHAPTER II
TOPOLOGICAL PRELIMINARIES
§ 1. Topological Groups and Linear Spaces
2.1.1. Recall first the notion of a poset. A set A together
with an ordering relation -<, between certain pairs of its elements,
which is reflexive, transitive, and anti-symmetric or proper (a -<
6,6 -< a => a = 6) is called a poset (or partially ordered set). A is
called directed (above) if for any a,/3 G A there is a 7 G A with
a,(3 -< 7.
Let 5 be a set and (xa)(a G A) a net in 5 (i.e. xa G 5 for
each a G A, where A is a directed set) A net (xai), (a' G A')
in 5 is called a subnet' ofa net (xa) in 5 if A' C A (so that
(xai) is a subset of (xa)) and for each a G A there is an a' G i?'
with a -< a'.
Let 5 be a topological space. A net (xa) in 5 is said to
converge to x in 5, in symbols xa —> x, if for any neighbourhood
U of x there is a /? = P(U) in i? such that xa £ U for all a >-
(5 **. It is clear that if a net xa —> x then every subnet xa( —> x.
Further, in a Hausdorff space a convergent net has a unique limit,
i.e., if xa —> x,y then x = y. (If xa —> x and j/ / x we can
choose neighbourhoods J7, V of x, j/ respectively with U f)V = 0.
Since xa —> x there is an ao such that {xa : a > cxq) C J7 and
so disjoint with V, whence xa /> j/ (i.e. xa does not converge
to J/.))
Suppose now a net (xa) in 5 does not converge to x : xa /> x.
Then there is a neighbourhood J7o of x such that for each a there
is an a' with a' > a,xa> <^ [/q- The xa> as a varies clearly form
a subnet (xa<) which is disjoint with Uq. It follows that no subnet
of (xai) converges to x. Thus we have the following useful result:
(*) xa —+ x iff every subnet (xa<) of (xa) has a subnet xan —> x.
T We have adopted a restricted definition for the subnet since that would
suffice for our purposes. For a more general definition see [l6,p.70].
tt i.e. p < a.

74
Topological Preliminaries
2.1.2. Remark. In terms of net convergence the continuity
of a map f : S —+ S1, where S,S' are topological spaces, can be
expressed by the condition: xa —> x in 5 => f(xa) —> f(x) in 5'.
2.1.3. A group G together with a topology on it is called a
topological group or TG if the maps
m
" • it ii\ i—k <rit i" • <r —^ o"
: (x, y) h-> xj/, i* : x-+ x (x, y G G)
are continuous. Any neighbourhood of the identity element e of
G is called a nucleus.
If 5 is a subset of G we write 5_1 = {a-1 : a G 5}. We call
5 symmetric if 5_1 = 5. Also, for two subsets Si,S2 of G we
write
5i52 = {ab : a G 5i, 6 G 52}.
2.1.4. PROPOSITION. Let G be a TG. Then:
(i) The maps
la : x 1—> ax, ra : x 1—> xa, 1* : 1 h x_1
where a,xGG, are homeomorphisms.
(ii) /f 5 is a subset of G and a G G iAen aS = aS, Sa =
5a, 5_1 = (5)~ , where bar denotes closure.
(iii) For anj/ open subset 0 of G and subset S of G, 05
50 are open.
(iv) //" J7 is an open nuc/eust and a G G iAen aU,Ua are
open neighbourhoods of a, ana" evej/ open neighbourhood of a has
these forms.
(v) If U is an open nucleus so is U~l and V = Uf] U~l is
an open symmetric nucleus such that V C U.
(vi) Any open nucleus U contains a symmetric open nucleus
W with W2 C U. Any such W has the property that its closure.
W C U. Hence every TG is regular, ft
PROOF. (i) It follows from the definition of a TG that
la,ra,i are continuous. Since l~l — la-i,lal is continuous and
f i.e. a nucleus which is also an open set.
TT A topological space S is called regular if for every point s in 5 any
neighbourhood of s contains the closure of some neighbourhood of s.

§ 1. Topological Groups and Linear Spaces 75
hence la is a homeomorphism. Similarly ra is a homeomorphism.
Finally, since (i^)2 = l,i&~1 = z*, so that i* is a
homeomorphism.
(ii) This is a consequence of (i).
(iii) Since la is a homeomorphism la(0) = aO is open. Hence
SO = |Ja aO(a G 5) is open. Similarly OS is open.
(iv) These follow from the homeomorphism property of the
maps la,ra.
(v) By homeomorphism property of i#,J/_1 is an open
nucleus. Hence V is an open nucleus which moreover is clearly
symmetric.
(vi) The first assertion follows from the continuity of the
map m^ at (0,0) and (v). To prove the second assertion take
an element b G W, where bar denotes closure. We must have
WPl&W 7^ 0j whence there are elements w,wi G W with w =
bu>i Therefore 6 = ww^1 G WW'1 = W2 C V. This proves that
W C V. It follows that for any element a G G, aW — aW C all,
whence G is regular.
2.1.5. LEMMA. Let G be a TG with identity element e Then:
(i) e = {e} is the intersection of all nuclei of G and e is a
normal subgroup of G.
(ii) For x G G, x = xe.
(iii) G is Ti iff e = e.
PROOF. If J7 is any nucleus, by 2.1.4(vi) there is a nucleus
W with W C U. Thus e C W C {/. Ifag'e there is a symmetric
nucleus V with e (£Va. Then a-1 ¢7, and so by symmetry of
V,a (£V. Therefore e is the intersection of all nuclei. If a,b G e
then a6 G ae = ae = a C e, a^1 G e_1 = e_1 = e. Further, if
x & G then £^1¾ = x_1ea; = e. Thus e is a normal subgroup,
completing the proof of (1).
We have x — xe = xe which is (ii), and (iii) follows from (ii).
2.1.6. LEMMA. A Ti group G is T3 , in particular it is T2
( - Hausdorff).
PROOF. By 2.1.4(vi), G is regular TY and so T3 (by
definition of T3). Suppose that a,6 G G and a^b. By 7\-property
there is a neighbourhood V of a with b ^V. By regularity choose

76
Topological Preliminaries
a neighbourhood Va of a with Va C V. Set Vb = G\Va. Then V4
is a neighbourhood of 6. Clearly, Vaf\Vf, = 9, whence G is Ti.
2.1.7. THEOREM. (Birkhoff-Kakutani). Every first countable
TG G admits a one-sided invariant [for multiplication) semi-
metric d which induces the topology of G. Further, the semi-
metric d is a metric iff G is Hausdorff.
PROOF. See [21, pp.34-36].
2.1.8. PROPOSITION. (Zelazko). Let G be a group endowed
with a complete metric topology such that la,ra are continuous for
each a G G. Then the map x \—> x~l is continuous.
PROOF. First we show that if a sequence yn —> e (in G) then
j/"1 —> e. For establishing this it is enough to show that (j/„) has
a subsequence (yk = ynk) such that yk —> e (see 2.1.1). Define
inductively the subsequence (y^) of (j/„) as follows.
Set 2/T = j/i- Suppose that yi,---y~k has been chosen such
that
d{Pr,Pr+i) < ^TT,of(pry;1,pr+iy71) < ^y (1)
for r = 1,2, ••-,£- 1, s = 1,2,••-,&, where pr ~ yT • • • j/7
(product). Then we can choose j/jt+i so 'na' (1) *s satisfied for
r = 1, • • •, k, s = 1, • • •, k + 1. This choice is possible by taking
njfe+i sufficiently large, since there are only a finite number of
inequalities to be satisfied, yn —> e, and multiplication on the left
or right is continuous. From(l), by using the triangle inequality
for d we obtain
. 1/,11 \ 2 1
dfrri^.Pr+iyr1) < ^ (2)
It follows by the completeness of d that
pk -* p (say), (3)
PkVs1 ~* 9s (say).
(4)

§ 1. Topological Groups and Linear Spaces 77
Then
q„ = lim pky~l = (lim pk)y~l = py'1. (5)
k—»oo K—»oo
Further, by using (2),(3),(5), we get
rf(p,Pt)<2-*, 4¾.¾¾1^-^ (6)
It follows that
d(p,qs) < d(p,psyj1) + d(qs,psy;1) = d(p,ps-i) + d(qs,psyj1)
1 1 1
< ^—r + ^7 <
2*-i 2s 2s-2'
whence qs —> p. Therefore, by (5),
Next, if xn —> xo in G then x„Xq —> e, so that xox"1 =
(x„Xq x)_1 —> e, whence x"1 —> Xq . Thus the map x
continuous, proving the proposition.
x 1 is
2.1.9. The letter K will denote as before C or R. A linear
space or LS X over K is called a topological linear space or TLS
if X is equipped with a topology such that:
(Tl) The map (x, y) i—> x + y of I X X -> I is continuous.
(T2) The map (A,x) h-» Ax of K X X —> X is continuous.
By taking A = — 1 in (T2) we obtain
(T3) The map x i—> — x is continuous.
In terms of net convergence the continuity conditions (Tl)-
(T3) can also be put in the forms:
(Tl') If xa —> x, yp —> y in X then xa + j/^ —> x + j/ in X.
(T2') If Aa —> A in K, and xp —> x in X then A^x^ —> Ax
(in X)
(T3') If xa —> x in X then — xa —> —x.
In a TLS a neighbourhood of 0 is called a nucleus.
2.1.10. LEMMA. For each A G K, the map x i—> Ax (x G X)
is continuous, and, for each A 7^ 0 ii is a homeomorphism.
PROOF. The first half of the statement follows from (T2). For

78
Topological Preliminaries
the second half we observe that if m> denote the map x i—> Ax,
then for A 7^ 0, m^~ = m^-i and so m^ is also continuous.
2.1.11. Remark. While every group G is a TG under either
the discrete ' or the indiscrete > topology, a LS X (over K) is
a TLS under the indiscrete topology but not under the discrete
topology (if x 7^ 0 in X then -i/0 and so -x -/* 0 = 0 • x,
though £ -> 0).
2.1.12. THEOREM (Tychnoff). Every n-dimensional Haus-
dorff TLS is linearly homeomorphic to Kn.
PROOF. See [14, p. 13].
2.1.13. LEMMA. The underlying additive group of a TLS
X is a TG. Further, 0 = {0} is a closed subspace of X ■ X is
Hausdorff iff {0} is closed.
PROOF. The first statement follows from (Ti), (T3) of 2.1.9.
For the second statement we note that, by 2.1.5(1), 0 is a closed
subgroup. That it is a subspace follows from the continuity of
scalar multiplication (A0 C A0 = 0). Finally, the last statement
follows from 2.1.5(iii), 2.1.6.
2.1.14. LEMMA. Every TLS X is path connected and hence
connected.
PROOF. For x,y £ X, {(1 - A)x + Xy : 0 < A < 1} is a path
joining x and y.
2.1.15. A subset 5 of LS X is called symmetric if x G
5 => —x G S (S = ~S). It is called balanced if x G 5, A G
K, |A| < 1 => Ax G 5. Note that a balanced set is symmetric and
that if 5 is a balanced set and |A| < |//| then A5 tt c //5 (Ax =
(j.(X(j.~~1)x, |A//_1| < 1). A subset S is called absorbing if to each
x G X there is a real number e = ex > 0 such that Ax G 5 for
all A with 0 < |A| < e. Trivially the set X is absorbing; on the
' The discrete topology on a set S is that in which every subset is open
and the indiscrete topology that in which empty set and S are the only open
subsets.
'' If 5 is a subset of a LS X and A £ K then XS = {A2 : 2 £ S}.

§ 1. Topological Groups and Linear Spaces 79
other hand {0} can never be absorbing (unlesss X = {0}). Also,
if S is absorbing and a/0 then aS is absorbing ( Xx G aS iff
Xa'1xe S).
2.1.16. LEMMA. Let X be a TLS. Then:
(i) The closure S of a balanced set S in X is balanced.
(ii) Every nucleus U is absorbing.
(iii) Every nucleus U contains a balanced open nucleus as well
as a balanced closed nucleus.
PROOF, (i) This follows from the inclusion XS C XS.
(ii) Since the map (X,x) >—> Xx is continuous at (0,x), there
is an e > 0 and a neighbourhood x + V of x [V a nucleus;
cf.2.1.4(v)) such that for |A| < e and y G x-\-V we have Xy G U.
In particular, Xx £ U for |A| < e whence U is absorbing.
(iii) From the continuity of the map (A, x) t—* Xx at (0,0) we
obtain an e > 0 and an open nucleus V such that if |A| < e and
x G V then Ax G U. Write A£ = {A G K : |A| < e} and W = A£V.
Since W = LM^> when A ranges in Ae\{0},W is open (using
2.1.10 each XV is open). Further, if A G A£, \fi\ < 1 and x <E V
then |//A| = |//| |A| < e. //A G A£ and //Ax G W, proving W is
balanced. Also it is clear that 0 G W C J/. Finally, by 2.1.4 (vi)
there is a nucleus Wi with W\ C J/. By what has been just proved
there is a balanced nucleus Wi C Wj. Then W^2 Q Wi ^ C^ and
W^2 is balanced (by (i)).
2.1.17. PROPOSITION. Every TLS X has a basis U of nuclei
with the following properties:
(i) Each U G U is balanced and absorbing.
(ii) If U1,U2ell, there is a U3 G U with U3 C Ux f] U2-
(iii) If U ell there is a V G U with V + V CU.
(iv) If U ell and X G K iAere is a V G i/ tw'iA AV C U.
Conversely, given a nonempty family U of subsets of a LS X
such that U satisfies (i)-(iii), it determines a unique topology in
X making it a TLS having U as a basis of nuclei.
PROOF. See [28, p.96].
2.1.18. Let X be a TLS and Xq a subspace of X. Then the
quotient LS X* = X/Xq = {x + X0; x G X} carries a natural

80
Topological Preliminaries
topology, viz., the quotient topology: a subset S& C X& is open
iff 7r_1(5#) is open in X, where n is the canonical homomor-
phism x i—> x# = x + Xo.
2.1.19. LEMMA. The quotient space X# is a TLS which is
Hausdorff iff Xq is closed in X. The canonical map ■k is open
and continuous.
PROOF. By 2.1.17 we can choose a base U of nuclei of X.
Write U* = { all subsets U* of X* such that jr-1^*) G U}.
Then it is easy to see that 11% has the properties (i)-(iv) of 2.1.17,
whence by this proposition X^ is a TLS. Further, by 2.1.13, X&
is Hausdorff iff 0* = Xo is closed. By definition of the quotient
topology, 7r is continuous. It is also open as can be easily seen by
using 2.1.4 (iii).
2.1.20. A subset 5 of a TLS X is called topologically
bounded or t. bounded or just bounded if for every nucleus U there
is a A = X(U) ^ 0 in K such that 5 C XU.
2.1.21. LEMMA, (i) Every subset of a bounded set is bounded.
(ii) The closure S of a bounded set S is bounded.
(iii) If S is a bounded set then to each nucleus U there is a
positive real number e = c(U) such that S C //[/ for every // G K
with |//1 ^ e. In particular, there is a positive integer n such that
S C nU.
PROOF, (i) Clear (if S C XU,S' C S then 5' C XU).
(ii) Given a nucleus U there is a nucleus V with V C U (see
2.1.4(vi)). Since 5 is bounded there is a // with 5 C fiV. It
follows that 5 C fiV = fj.V C //[/, whence 5 is bounded.
(iii) Choose a balanced nucleus V C U. By boundedness of 5,
there is a A^O with 5 C XV, so that if x G 5, x = Xv (v G V).
Set 6= |A|. If |//| > |A| then |//_1A| < l,//_1Av G V since V is
balanced), so that
x = Xv = // • //- Xv G //V C //[/.
Also, 5 C nU for any integer n > e.
2.1.22. THEOREM (Mazur-Kolmogorov). yi su6sei S of a
TLS X is bounded iff for each sequence (xn) in S and sequence

§ 1. Topological Groups and Linear Spaces 81
(A„) in K with Xn —> 0, we have Xnxn —+ 0.
PROOF. Suppose that 5 is bounded. By 2.1.21 (iii) there is
an e > 0 with 5 C (j,U for |//| > e. Write r/ = e_1 and a = //-1.
Then aS C J7 for |a| < r?. Since there is an integer N such that
|A„| < r/ for n^ N, it follows that Xnxn G A„5 C J7 for n > TV,
whence Xnxn —> 0. Conversely, if 5 is not bounded there is a
nucleus U such that 5 ^ AC/ for any A ^ 0 in K. Therefore, in
particular, 5 $Z nU{n = 1,2,---). It follows that we can choose
xn in 5 such that ^ ¢ £/". Then clearly ^/»0 (though ^ -^ 0)
violating the stated condition, which completes the proof.
2.1.23. COROLLARY. For S to be bounded it is sufficient
that the following condition is satisfied:
For any xn G 5, > 0.
n
2.1.24. DEFINITION. A TLS is said to be locally bounded if
it has a bounded nucleus.
Note that if U is a bounded nucleus so is every nucleus V C U
(see 2.1.21(0).
2.1.25. PROPOSITION. Let X be a locally bounded TLS and
U a bounded nucleus of X. If //„ G K\{0} and //„ —> 0 then
{finU} is a basis of nuclei. Thus, X is first countable and hence
semi-metrizable. In particular, every locally bounded Hausdorff
TLS is metrizable.
PROOF. Suppose that V is any nucleus of X. Since U is
bounded, there is, by 2.1.21 (iii), an e > 0 such that U C yV for
all // £ K with \n\ > e. Since fin —> 0 we can choose a sufficiently
large n such that \nn\~l > e. The U C n~lV or finU C V. The
statements regarding semi-metrizability and metrizability follow
by applying 2.1.7 to the underlying additive TG of X.
2.1.26. LEMMA. Let X, X* be TLS's and T : X -> X* a
linear transformation. If T is continuous at some point xq then
it is continuous everywhere.
Proof. Assume that T is continuous at xo, and that xa —> x.
Then xa ~ x + x0 —> x0, so that Txa - Tx + Tx0 —> Tx0, whence

82
Topological Preliminaries
J. JC Q r _t JO.
2.1.27. LEMMA. If T : X —> X* is a continuous (or
more generally, sequentially continuous) linear transformation and
S(C X) a bounded set so is T(S).
Proof. If xn G 5 then ^- —> 0, so that
1^ = T (^] -> T(0) = 0,
n \ n J
proving (by 2.1.23) that T(S) is bounded.
2.1.28. Let X, X* TLS's and T : X -* X* a linear
transformation. T is said to be t. bounded or (sometimes) just bounded
if it carries bounded sets to bounded sets. By 2.1.27, a continuous
linear transformation is t. bounded.
2.1.29. PROPOSITION. Let X be a first countable (or equiv-
alently semi-metrizable) TLS and X* any TLS. Then a linear
transformation T : X —> X* is t. bounded iff it is continuous.
PROOF. In view of remark in 2.1.28 we have to prove only
the "if" part. Let T be bounded. If T is not continuous it is
not continuous at 0 (by 2.1.26) so that there is a nucleus U* such
that T~ (U*) is not a nucleus. Since X is first countable it has
a countable decreasing sequence of open nuclei as basis: U^ I>
U2 2 ■■■■ By 2.1.10,¾1 is an open nucleus. Since T'^U*) is
not a nucleus we have: ^ <2 T~1(J7'") (n = 1,2,---). This means
that there are elements xn e Un such that ^- ¢. T~l(U*) (n =
1,2,---). Since {Un} is a decreasing basis and xn G Un, xn —> 0
and so xn is bounded (see 2.3.7. (a)). Since T is bounded (Txn)
is bounded. This means by 2.1.21, that T (^-) = ^ -> 0, so
that ^En. e u* for n > N. But then ^ G T~l(U*) (n > N),
contradicting the choice of xn. Hence the proposition.
2.1.30. LEMMA. Let X be a TLS and f a linear functional'
on X. Then f is continuous iff ker / is closed.
PROOF. It suffices to prove the "if" part. We may assume
i.e. a function with values in K.

§ 1. Topological Groups and Linear Spaces 83
that / 7^ 0, ker / is closed. Then, for any e > 0, the translate
Xq = {x G X : f(x) = e} of ker / is closed, so that U = X\Xq
is an open nucleus. By 2.1.16 (iii), there is a balanced nucleus
V C U. We claim that for any x G V, \f{x)\ < e. If not there
is a x0 G V with 6 = \f(x0)\ > £■ Write y0 = (e/S)x0. Then
|/(j/o)| = e, so that /(j/o) = f0 where 5 £ K, |0| = 1. Hence
f(9yQ) = e, so that z0 = Oyo G Xo. On the other hand, since
xq G V and V is balanced,
Ve
0
since j < 1. Therefore 2r0 G Xofl^, contradicting Xofl^ = 0-
Thus, \f(x)\ <e(x<EV) proving / is continuous.
2.1.31. LEMMA. Let X be a TLS with {Ua} as a basis of
nuclei and f a continuous functional on X with /(0) = 0. Then,
given C > 0, there is a Uq = Uaa such that \f(x)\ < C for all
x G Uq.
PROOF. Set G = {x e X : |/(x)| < C}. Since /(0) = 0,
0 G G and so by continuity of /,G is an open nucleus. {Ua}
being a basis we must have G 2 some Uao which implies the
lemma.
oo
2.1.32. Let X be a TLS. A series \]xn in X is said to
n=l
converge to x if the sequence (s„) of partial sums sn — 11 + --- +
xn converges to x in X : sn —+ x. More generally, a generalized
series J2aeA xa, where A is any indexing set, is said to converge
to x if the net sp, where franges through all finite subsets of A
and sp = 2^, xa, converges to x; x is called the generalized sum
aeF
of the xa 's.
00
2.1.33. LEMMA. If N xn converges in X then xn —> 0.
n=l
PROOF. Given a nucleus U, choose a symmetric nucleus W
00
with W + W C U. Suppose that ^xn = x. Then there is an N0
n=l

84
Topological Preliminaries
N
such that ^2 xn ~ x G W for N > N0. It follows that
n=l
XN + 1 = f XZ ^rz — ^ ) — f 5T ^rz — ^ j
G W + W CU for N > N0.
It follows that xn —> 0.
§ 2. Topological Algebras
2.2.1. DEFINITION. An algebra A over K is called a
topological algebra or a TA if it is equipped with a topology such that:
(TA1) The map (x, y) >—> x + y of A X A —> is continuous.
(TA2) The map (A, x) i—> Ax of K X A —> A is continuous.
(TA3) The map (x, j/) i—> xy of A X A —> A. is continuous.
In view of (TA1), (TA2) every TA is a TLS. Also, the condition
(TA3) can be expressed in terms of net convergence as:
xa -+ x, yp -+ y => Xaj/0 -+ xy.
2.2.2. DEFINITION. An algebra A is called a weaA; topological
algebra or a WTA if the underlying linear space of A is a TLS
and further we have:
(TA3') The maps la : x >—> ax, ra : x i—> xa (x, a G A) are
continuous for all a.
Clearly (TA3) .-> (TA3'), so that a TA is a WTA.
Using 2.1.26, it is easy to see that in terms of net convergence
the condition (TA 3') can be expressed as
(TA3") If xa —> 0 then axa —> 0, xaa —> 0 for each a G A.
2.2.3. LEMMA. For a WTA A to be a TA it is necessary and
sufficient that
(*) the map (x,y) >—> xj/ is continuous at (0,0).
PROOF. We have only to prove the sufficiency of the condition
(*). For this it is enough to show that in an A satisfying (*),
(TA3) holds.

§ 2. Topological Algebras
85
Assume now that (*) is satisfied, and in A xa —+ xq, yp —+
j/o- By (TA3") we have
zo(j//3 - J/o) -> 0; (1)
(xa - x0)y0 -+ 0. (2)
Again, by (*) we have
{xa - xo)(y/3 - y0)-* 0. (3)
From (1),(2),(3) we obtain (by adding the terms)
zaJ//3 - zoj/o -+ 0, or ,xayp -+ x0y0.
Thus,(TA3) holds and A is TA.
2.2.4. Examples, (i) Every algebra over K is a TA under
the indiscrete topology.
(ii) The algebra A = K5 of all K - valued functions
(algebra operations being point (or coordinate)-wise is a TA under
pointwise net convergence : i.e., if fa,f G A then fa —> / if
fa{s) -~> f{s)^s £ S; this topology is called weak topology or
topology of simple convergence. A is a commutative TA.
(iii) Let f) be a Hilbert space and B = B(.f)) the algebra of
all bounded l.o. t 's on fi. B is an algebra which not commutative
if dim :f) > 1. Under the weak or strong operator topology B is a
WTA but not a TA (since multiplication is not jointly continuous;
see [22, p.448]). The weak or strong operator topology is defined
vie net convergence as follows. Ta —> T in the weak operator
topology if (Tax,y) —+ (Tx,y) for every x, y G Sj, where {■)
denotes the inner product of ft. Similarly, Ta —> T in the strong
operator topology if \\Tax~Tx\\ —> 0 for every x G Sj, where ||-||
is the norm induced by the inner product: ||x|| = (1,1)2.
In the norm topology, B is a TA; Tn —> T in the norm
topology if \\Tn - T\\ -♦ 0 (for definition of \\T\\(T G B), see
3.5.1).
t
l.o.=linear operator.

86
Topological Preliminaries
2.2.5. LEMMA. A WTA or a TA is a TG under addition.
Under multiplication a TA A is a TSG. *
PROOF. The first statement follows from 2.1.12; the second
is a consequence of condition (TA3) of 2.2.1.
2.2.6. LEMMA. In a unital TA A, the groups Gq,Gf are
semi-topological'' groups and the map
t~1 : a E Gq i—> e + a E G,
is a t. isomorphism. *
PROOF. By (TA1), (TA3), Gq and G, are semi-topological
groups. Also, by 1.1.20, r_1 is an isomorphism. That r_1 is
topological follows from the fact that translation is a homeomor-
phism (see 2.1.4(i)).
2.2.7. LEMMA. In a TA (or even WTA) A,0 = {0} is a
closed bi-ideal of A.
PROOF. By 2.1.13, 0 is a closed subspace of A. Further,
from the continuity of I. or r. multiplication in A we obtain
xO C ^0 = 0, Oj/ C 0
which show that 0 is a bi-ideal, completing the proof.
2.2.8. LEMMA (a) Any subalgebra of a TA (respy. WTA) is a
TA (respy. WTA).
(b) Any direct product or direct sum of TA 's (respy. WTA 's)
is a TA (respy. WTA).
PROOF, (a) Clear.
' TSG= topological semi-group, i.e., a semi-group with a topology under
which multiplication is continuous.
>> A semi-topological group is a group with a topology under which
multiplication continuous (inversion may not be continuous).
T t. isomorphism = topological isomorphism, i.e. an (algebraic)
isomorphism which is also a homeomorphism.

§ 2. Topological Algebras
87
(b) Since the operations of the direct product are coordinate-
wise and a direct sum is a subalgebra of the direct product, the
statements follow.
2.2.9. PROPOSITION. The unitization Ai of a TA
(respy. WTA) A is a TA (respy. WTA) under the product topology
of A\ — K x A. Further, A is a closed bi-ideal of A\ and A\ is
Hausdorff whenever A is Hausdorff.
PROOF. Clearly A\ is a TLS in either case. Suppose now
that A is a TA and that in A\ we have
zia = Aaei + xa -+ xi - Aei + x
3/1/3 = M/3ei + J//3 -* J/i = Mei + y
where x, y G A and \a,(Ap, A,// G K. Then
Xa -+ A,///3 -+ n,xa -+^,2//3-+ y.
It follows that
ZlaJ/l/3 = Aa///3ei +Xay/3 +^¾ + xayp
-+ A//ei + Aj/ + //x +xj/= xij/i,
showing A is a TA.
Next suppose that A is a WTA, and yia = nae\ + ya —+
Mei + V = !/i- Then /^(j/ia) = A//aei + Aj/a + [Aax + xj/a -+
^i(Mei+J/) = ^i(j/l)- This shows that /II is continuous. Similarly
rXl is continuous. Hence A\ is a WTA.
We have already noted that A is a bi-ideal of A\ (see 1.1.12).
That it is closed in A\ is clear from the definition of the topology
of A\. Finally, if A is Hausdorff, {0} is closed in A and so also
in Ai (since A is closed in A\) and consequently A\ is also
Hausdorff (by 2.1.5,2.1.6).
2.2.10. PROPOSITION. The complexification A of a real
TA (respy. WTA) A is a TA (respy. WTA) under the topology of
Ax A.
PROOF. The proof is on the same lines as the proof of the
first part of 2.2.9.
2.2.11. PROPOSITION. Let A be a TA and I a bi-ideal of

88
Topological Preliminaries
A. Then the quotient algebra A# = A/I is a TA relative to the
quotient topology. Moreover, A^ is Hausdorff iff I is closed.
PROOF. In view of 2.1.19, it is enough to prove that
multiplication in A* is continuous. Let U# be an open neighbourhood
of x^yft = [xy)&. Then U = 7r_1(J7^), where n is the
canonical map A —> A*, is an open neighbourhood of xy. Choose open
neighbourhoods V,W, of x, y respy. such that VW C U. Then,
since n is open (by virtue of 2.1.19), V* = k(V),W# = n(W)
are open neighbourhoods of x#,y# respy. with V#W# C {/#,
proving the continuity of multiplication in A*.
2.2.12. COROLLARY. Let A be a TA. Then A/0 is a
Hausdorff TA.
PROOF. This is an immediate consequence of 2.2.7 and 2.2.11.
2.2.13. THEOREM. Every WTA has a basis U of nuclei
satisfying (i)-(iv) of 2.1.17 and
(v) Given U G U and a G A there are V, W G U such that
aV CU, WaC U.
Conversely, if U is a nonempty family of subsets of an algebra
A satisfying (i)-(iii) (o/2.1.17.), and (v) [above) then U is a basis
of nuclei of a unique topology on A making it a WTA.
PROOF. By 2.1.17 we can choose a basis U of nuclei satisfying
(i)-(iv). Then U also satisfies (v) since la,ra are continuous for
each a G A.
For the converse we observe that by 2.1.17, A is a TLS. To
prove that A is a WTA it remains to show that each la and each
ra are continuous. By 2.1.26 it is enough to prove that la,ra are
continuous at 0. But this is precisely what condition (v) expresses.
2.2.14. THEOREM. Every TA A has a basis U of nuclei
satisfying (i)-(iv) of 2.1.17 (v) o/2.2.13, and also
(vi) Given a U G U there is a V G U such that V2 C U.
Conversely, every algebra A together with a family U of
subsets satisfying (i)-(iii) of 2.1.17, (v) of 2.2.13, and (vi) above is a
TA under a unique topology having U as a basis of nuclei.
PROOF. If A is a TA then it is a WTA and so by 2.2.13, has

§ 2. Topological Algebras
89
a basis U of nuclei satisfying (i)-(v). This also satisfies (vi) since,
multiplication in A is continuous.
The converse statement follows from the converse part of
2.2.13, and 2.2.3.
2.2.15. Remark. In terms of convergence of nets the
conditions (v),(vi) above take the forms:
If xa —> 0 then axa —> 0, xaa —> 0; (v )
If xa —> 0, yp —> 0 then zaj//3 —> 0. (vi')
2.2.16. LEMMA. Let A be a TA. We have:
(i) If in A,xn —> 0 and (j/„) is bounded then xnyn —> 0,
j/„z„ -+ 0.
(ii) If S,T C A are bounded subsets so is ST.
PROOF, (i) Given a nucleus J7 choose a nucleus V such that
V2 C U. Since (j/„) is bounded then is a scalar A ^ 0 such that
{Vn} ^ AV. Since xn —> 0 we can find TV such that i„ e A_1V
for n ^ N. Therefore
xnyn e A" V • XV = V2 C C/ for n > TV,
whence xnyn —> 0.
(ii) Any sequence in ST is of the form (xnyn) where
(xn),(yn) are sequences in S,T respy.. Since {xn} C 5 is
bounded, ^- -> 0 (by 2.1.22). Since {yn} C T is bounded,
by result (i), =^ -> 0, whence by 2.1.23, ST is bounded.
2.2.17. LEMMA. In a Hausdorff, TA (respy.unital TA) A, if
oo oo
the series ^(-l)"x" (respy. ^(e - x)n) converges then x is
ri=l n=0
g invertible (respy. invertible) and its q inverse x' (respy. inverse
x_1) is given by
oo oo
*'= 2(-1)^(^ = ;>>-*)")•
n= 1 n=0

90
Topological Preliminaries
PROOF. Write y = J2(~l)nxn. Then x o y = x +
n=l
oo
^(-l)"(x" + x"+1) = lim (-1)^^+1 =0 (by 2.1.33).
ra=l
Similarly, j/ o x = 0. Therefore x' = y, proving the assertion
concerning q invertibility. Again, if we set e — x = z, by 2.1.33,
zn —> 0. Therefore we have
oo
(e — z) V] z" = lim(e — zn+ ) = e, whence
oo oo
(e - ^)-1 = £ *», x-1 = (e - z)"1 = J> - *)",
completing the proof.
2.2.18. In a TA A we denote by Ac = AC(A) the set of
all continuous characters; A = A(A) will denote the set of all
characters. Of course Ac (or even A) can be empty.
We set
■^4_{ {nx_1(0):xeAc} if Ac 7^0
V 1 A if Ac = 0.
Since C\/A~ is the intersection of all closed hypermaximal ideals
(of A) it is closed and we have clearly the inclusion relation
y/ActyAC VA (*)
2.2.19. DEFINITION. Following Michael, [20, p.48] we call a
TA A functionally continuous if every character of A is
continuous, i.e. if Ac = A.
In a functionally continuous algebra we have
Va = cVa. (**)

§ 3. Completions of Topological Linear spaces 91
§ 3. Completions of Topological Linear spaces and
Topological Algebras
2.3.1. DEFINITION. Let X be a TLS. Two nets (xa),(yp)
in X are said to be equivalent, in symbols, (xa) ~ (yp) if they
satisfy the condition:
(*) Given a nucleus U of X there are indices ao,/?o such
that xa — yp G U for a > cxo,/3 > /¾.
A net (xa) is called a Cauchy net or a C-net if we have (xa) ~
(xa), i.e. if for each nucleus J7 there is an ot@ such that xa xp G
U for a,f3 > cto. A TLS X is called complete if every C-net in
X converges to some element in it.
2.3.2. LEMMA. The notion of equivalence of nets in X has
the properties:
(i) If (xa) ~ (yp) then (yp) ~ (xa).
(ii) 7/ (xa) ~ (yp),(yp) ~ (^) iAen (xa) ~ (2^).
(iii) 7/ (xa) ~ (j/^) and (xa/) z's a subnet of (xa) then (xai) ~
(iv) 7ei (xa) 6e a net and (x) a principal net. Then (xa) ~ (x)
(v) 7/ (¾) ~ (yp) and A e K then (Xxa) ~ (Ajz/j), m particular
{-xa) ~ (-J//3).
(vi) 7/ (ia) ~ (j/^), (2^) ~ (us) iAen (ia - z*,) ~ (j/^ - us).
PROOF, (i) Given a nucleus U select a symmetric nucleus
V C U. Since (xa) ~ (yp) there are 0:0,/¾ such that xa - yp G
V(a >- a0,/? >- /¾) whence
yp - xa = -(xa - yp) £ -V =V C U (a > a0,/3 > /30).
(ii) Given a nucleus U select a nucleus V with V +V C U,
and choose ao,/?o,/?i,7i such that
z<* - ff/s e^(a >- a0,/3 > /30),yp - z^ eV(/3> £1,7 > 71).

92
Topological Preliminaries
Then, for a > c«o,7 > 71, we have
xa - zn — xa - up + up - z~, e V + V C [/( where /?' >- /?0, A)-
(iii) Given U, choose ao,/?o such that xa — yp G U[a >
oto,j3 > /3o). Since (xai) is a subnet there is an a'Q > olq. Then
xa' - J//3 £ ^(a' > ®'o,P > Po), whence (xa<) ~ (2//3).
(iv) Clear.
(v) Given U, by 2.1.17(iv), there is a V with AV C U. Choose
oto,Po such that xa — yp £ V(a > ao,P > Po). Then
Axa - Xyp = A(xa - 2//3) e XV CU.
(vi) Select a symmetric V with V + V C (7. By hypothesis
there are indices ao,/?o>7o,£o such that
*a ~ J//3 e^(a > a0,p > Po),z1 - us 6^(7 > 70,6 >- £0)-
Then (xa - 2^) - (yp - us) = xa - yp - fa - us) <EV +V C U
(a > a0,p > /?o,7 > 10,6 > S0).
2.3.3. COROLLARY, (i) If (xa) is a C-net in X and (xa>)
a subnet then it is a C-net such that (xai) ~ (xa).
(ii) If (xa) is a C-net in X so is (Xxa); in particular (— xa)
is a C-net.
(iii) If (xa), (yp) are C-nets so is (xa — yp).
PROOF, (i) From (xa) ~ (xa), by 2.3.2 (iii), (xai) ~ (xa)
and also by 2.8.2.((i), (iii)) we get (xai) ~ (xai).
(ii) Follows from (v) of 2.3.2.
(iii) Follows from (vi) of 2.3.2.
2.3.4. LEMMA. Let X,X* be TLS's and T : X -> X* a
continuous linear transformation. Let (xa),(yp) be nets in X
with (xa) ~ (2//3). Then (Txa) ~ (Txp). In particular, (Txa) is
a C-net whenever (xa) is a C-net.
PROOF. Given a nucleus U* of X*, there is, by continuity of
T a nucleus U of X with T(U) C U*. Since (xa) ~ (yp) there

§ 3. Completions of Topological Linear spaces 93
are indices ao,/?o such that xa - yp G U(a > oto,/3 > /3o). Then
Txa - Typ = T(xa - yp) G T(U) C U*,
proving (Txa) ~ (Typ).
2.3.5. DEFINITION. A net (xa) inaTLS X is called bounded
if it is bounded as a set (see 2.1.20). A net (xa) in X is called
essentially bounded if given a nucleus U of X there is an index
ao = oto(U) and a scalar A = X(U) ^ 0) such that
{xa : a > ao} C AC/.
Trivially, a bounded net is essentially bounded.
2.3.6. PROPOSITION. Every C-net - in particular a
convergent net - (xa) is essentially bounded.
PROOF. Given a nucleus U choose a balanced nucleus V
such that V + V C U. Since (xa) is a C-net there is an ao such
that xa — xat) G V for a > ao, so that
za e xau +V (a > a0)
Since V is absorbing (by 2.1.16(ii)) and balanced (by choice) we
can choose A > 1 such that xao G XV. Then
proving that (xa) is essentially bounded.
2.3.7. PROPOSITION. (a) Every essentially bounded
sequence - in particular, a C-sequence or a convergent sequence - is
bounded.
(b) Not every convergent net is bounded.
PROOF, (a) Given a nucleus U choose a balanced nucleus V
with V C U. Since (xn) is essentially bounded there is an integer
N > 1 and a scalar Ai such that {xn : n > N} C XiV. Using
the absorption and balanced properties of V we can choose a A2
with
Xje\2V (j = i,...,N).

94 Topological Preliminaries
Set A = max (|Ai|, |A2|). Then
K) aye xu,
proving the boundedness of the sequence (xn).
(b) It is enough to construct a convergent net in R which is
not bounded. Write
A = {ai,012,----,01,/32,---}
and define a partial ordering in A by
/3i -< /¾ • • •! 0im < /3n(m = 1, 2, • • •; n = m, m + 1, • • ■).
Then A is a directed set (as can be easily verified). Define a
net z-y (7 G A) in R by setting
1
xa„ = n,xpn = -.
n
Then x1 —> 0, but (x7) is unbounded in R (since -xan = 1 /» 0).
2.3.8. PROPOSITION. Let A be a TA. Then we have:
(i) Let (xa),(yp), (27),(uj) 6e essentially bounded nets in A
such that
(xa)~ (^),(»/s)~(u«).
TAen (xayp) ~ (^tij).
(ii) // (0:^),(2/^) are C-nets then so is [xayp).
(iii) If xa —+ 0 and (j/^) is essentially bounded then xayp —> 0,
J//3Xa -+ 0.
PROOF, (i) Given a nucleus J/(of A) we can find, using
2.1.17(iii), 2.2.14(vi), a nucleus V with
V2+V2CU. (1)

§ 3. Completions of Topological Linear spaces 95
By essential boundedness of the nets we have scalars, A, // 7^ 0
such that
{y/3 : /3 > /?i} C XV, {z1:1> 71} Q I&. (2)
Since (xa) ~ (2^), (j/^) ~ (us) we have ia-27e JV(a > ao,7 >-
72),2//3 ~ us E jV(/3 > /32,S > S2) Choose /¾ >- £1,/¾ and
7o > 71,72- Then
£c*J//3 - ^u« = (¾ - ^-7)2//3 + 2-7(1//3 - us)
G |v -AV + /iV --V =V2+V2 C C/
A //
(a > aQ,p> /?0,7 >- 7o,# >- M
proving (2^2//3) ~ (z-,us).
(ii) Since by hypothesis (xa) ~ (xa), (j/^) ~ (1//3) and by 2.3.6,
C-nets are essentially bounded we conclude using(i) {xayp) ~
(za3//j) i.e. (xayp) is a C-net.
(iii) Given a nucleus U, choose a nucleus V such that V2 C U.
Since (j/^) is essentially bounded there is a A^O and a /¾ with
{j//3 : /3 > /¾} C AV. Since xa —> 0 we have xa G A-1^ for
a >- ao- It follows that
xay/3eV2 CU (a >a0,/3> j30).
Hence xayp —> 0. Similarly, j//3Xa —> 0.
2.3.9. Let X be a TLS. The relation ~ between nets in X is
not only symmetric and transitive but also reflexive when confined
to C-nets. So ~ is an equivalence relation in the usual sense on
the class of all C-nets (xa) in X. Denote by X the set of the
resulting equivalence classes [(ia)]. To each subset 5 of X we
associate a subset S of X given by
5 = { [(ia)] G X : for some representative (xa) of the class
we have i„eS for all a} .
If (x) is a principal net the corresponding class (denoted by)
[x] = [(x)] is called a principal class.
2.3.10. LEMMA. X is a LS and the map j : in [x] is

96
Topological Preliminaries
linear. It is infective iff X is Hausdorff.
PROOF. If x = [(xa)],y = [{yp)} G X,X G K we define linear
operations in X by:
x + y = [{xa)] + [(yp)] = [(xa + y0)]
Xx = X[(xa)} = [(Xxa)}.
These operations are well-defined in view of 2.3.2. The
linearity of j is an immediate consequence of the definition of linear
operations:
[x + y} = [x} + [y], [Xx] = X[x\.
Finally, if X is Hausdorff and x, y G X, x ^ y there is a nucleus
U such that x — y (£ U, so that (x) -/- (y), [x] ^ [y\. On the other
hand, if X is not Hausdorff it is not 7\ (see 2.1.6). Therefore
there is an x ^0 with x G 0. \i U is any nucleus then by 2.1.5(i),
x G 0 C U, whence (x) ~ (0), [x] = [0], so that j is not injective.
2.3.11. LEMMA, (i) If Si C S2(C X) then Si C S2.
(n) AS = AS.
(iii) If x G S then [x] G S.
(iv) If [x] G S iAen x £ S, where bar denotes closure in X.
(v) 7/ (xa) is a C-net such that xa G S /or a >- ai iAen
[(¾)] e 5
PROOF, (i) - (iii): Clear.
(iv) If [x] G S there is a net (xa) ~ (x) with i„eS (for all
a). By 2.3.2 (iv), xa —> x, whence x £ S.
(v) Since (xa : a >- ai) is a subnet of (xa) it is, by 2.3.2(iv),
equivalent to (xa). Hence the result.
2.3.12. LEMMA. If U is a nucleus of X then U is an
absorbing subset of X. If U is balanced so is U.
PROOF. Choose a balanced nucleus V with V +V C U.
Suppose that [{3//5}] G X. Then yp - ypi G V((3,/3' > /30). Since
V, as a nucleus, is absorbing, ypn G XV for some A > 1. It follows

§ 3. Completions of Topological Linear spaces 97
that for (3 > /¾
yp = (2//3 - »/9o) + »/Jo ey + AycAy + AV=A(^ + y)c AC/.
Hence, by 2.3.11 (v), [{j//?}] £ AC/, proving U is absorbing. Now
assume that U is balanced. If [(xa)] G £7 with xa G C/, then since
C is balanced, Axa G C(|A| < 1), so that A[(xa)] = [(Axa)] G U,
proving U is balanced.
2.3.13. THEOREM. X can be made into a TLS such that:
(i) j : x i—> [i] is continuous; if X is Hausdorff j is a home-
omorphism.
(ii) j(X) is dense in X.
(iii) X is complete.
(iv) X is Hausdorff.
(v) X has the following universal mapping property: given
any complete Hausdorff TLS X* and continuous linear map <p :
X —> X*, there is a unique continuous linear map (p : X —> X*
sucA iAai <p — (p o j.
PROOF. Choose a basis £/ of nuclei of X satisfying (i)-(iv)
of 2.1.17. By 2.3.12, the family U = {U : U G 11} of subsets
of X satisfies (i) of 2.1.17. Further, by monotonicity of the (set)
map 5^5 (see 2.3.11(1)) it satisfies (ii) of 2.1.17. Finally,
given U G U we can choose a V G U with V +V C [/. Select
[(la)], [(j//?)] e ^- We may assume that (xa),(yp) C V. Then
[(**) + (J//?)] = [fca + 2//3)] £ ^
since xa + yp G C/. Thus C/
satisfies (iii) of 2.1.17. We can now apply the converse part of
2.1.17 to conclude that X is a TLS with U as a basis of nuclei.
It remains to prove the statements (i)-(v) above. By definition
of U we have j(U) C U, whence j is continuous. Let now X
be Hausdorff. Then, by 2.3.10, j is 1-1. Let V be any nucleus
of X and choose a nucleus W of X with W C V. If [1] G W
then x G PF C V. It follows that j~l{W) C V, whence j'-1 is
continuous. This completes the proof of (i). For proof of (ii) it is
enough to observe that j(xa) = [xa] —> [(ia)].
To prove (iii), Let af7 = [(arL^x)] be a C-net in X. Therefore,

98
Topological Preliminaries
given a nucleus U oi X, for each 7 there is a 7^ such that
£7 - £y G U for 7,7' > iu, i.e., we have 2^(-,) ~~ xa(y) e ^
for all 0:(7),a(7') with 7,7' >- 7^. This mean that the net (with
index 7 ) of principal classes
([«W)~^(^) = «(7j)-
By considering (a^rn) as a ne^ indexed by pairs {"1,U) with the
ordering
(1,U)^(1',U')i{1^1',U'CU
it is clear that it converges to the element [(x7,^,)] of X. So
af7 —> [(zL^i)], completing the proof of (iii).
For proving (iv) take x = [(xa)] 7^ [0]. Then xa /» 0. So we
may assume that there is a nucleus U of X with the
representative (xa) of x disjoint with it. Choose a symmetric nucleus W
with W + W CU. We claim that W $ [(xa)}. For, if [(xa)} G W
then there is a C-net (j/^) G W with (xa) ~ (2//3)- It follows that
there are xai,yai with za' — j/^i G W. Then
xai = xai - ypi + ypi <E W + W C U
contradicting U f)(xa) = 0- Thus, W is a nucleus of X not
containing x, whence X is 7\ and so Hausdorff.
Finally, for proving (v), take an element [{za}] G X. By 2.3.4,
(<p(xa)) is a C-net in X* and so converges uniquely to x* G X*.
Set
£([{*<*}]) =z*-
This is well-defined, since if (xa) ~ (j/^) then (^(xa)) ~
(^(j//?)), so that ^(3//5) -> x*. Also,
V?(a;) = <£([x]) =(Soj'(i).
Finally, if t/> is a continuous linear map with V ° J — 'P ° J then
clearly ip = 92 on the dense subspace j'(X) and hence everywhere.

§ 3. Completions of Topological Linear spaces 99
This completes the proof.
2.3.14. THEOREM. If A is a TA then its completion A (as
a TLS) is a complete TA.
PROOF. By 2.3.13, A is a complete TLS. So we have only to
show that A is a TA. If x = [(za)],y = [(j//s)] are two elements
of A then, by 2.3.8(ii), (xayp) is a C-net and so determines an
element [(x^j/^)] of A. We define multiplication in A by setting
xy = [(xayp)] (that this product is well-defined is assured by
2.3.8). It is easy to see that under this multiplication A is an
algebra.
To prove that A is a TA, consider a nucleus U and an element
a G A. Since la is continuous there is a nucleus V with aV C U.
Suppose now that x1 = [(arL^)] —+ 0 in A. Then there is a 70
such that for 7 > 7o,^L \ £ V if a{l) > some 0^(7). It follows
that
ax11 , G aV C U
a (-7) -
whence
ax1 —>0. (1)
Similarly, using continuity of ra we get
£70 -► 0 (2)
The conclusions (1),(2) imply that A is a WTA. Further, since A
is a TA, given a nucleus U, there is a nucleus V such that
V2 C U. (3)
Assume now that
^ = [(^w)]-o.w = [(»S(o)]-"°-
Then we have for some 70, S0, ^l^,yS^s) e V2 C U,
whence x^yg —+ 0, completing the proof that A is a TA.

CHAPTER III
SOME TYPES OF TOPOLOGICAL
ALGEBRAS
§ 1. Quarter-norms
3.1.1. DEFINITION. A real-valued function p = p{x) on an
additive abelian group X is called a subadditive or sad. functional
if it satisfies:
(Ql) p(0)=0;
(Q2) p(-x) = p(x)(x e X);
(Q3) p{x + y) < p{x) + p{y) (x, y e X).
If p is a sad. functional then we have also
(Q4) p(x) > 0 (xeX)
(0 = p(0) = p(x - x) < p(x) + p(-x) = p(x) + p(z) = 2p(x)).
If p satisfies
(Q5) p(x) = 0 => x = 0
then p is called faithful.
3.1.2. LEMMA. Let X be an additive abelian group and p a
sad. functional on X. Then:
(i) p(nx) < \n\p(x) for any integer n.
(ii) \p(x)-p(y)\ ^p(x-y).
(iii) If p(a) = 0 then p(x + a) = p(x) for all i6l
(iv) dp(x,j/) = p{x — y) is a semi-metric < on X which is
invariant for translations:
dp(x + a,y + a) = dp(x,y) (x, y, a G X).
Also, dp is a metric iff p is faithful.
' d is called a semi-metric or pseudo-metric if it satisfies all metric
axioms except: d(x,y) = 0 => x = y.

§ 1. Quarter-norms
101
If d is any semi-metric on X invariant for translations then
p(x) = d(x,0) is a sad. functional.
PROOF, (i) If n is positive we have p(nx) < p{{n- l)x + x) <
p((n — l)x) + p(x) from which the inequality follows for positive
n by induction. If n is negative then p(nx) = p(— n. — x) <
— np(—x) = — np(x) = |n|p(x), completing the proof for all n.
(ii) p(x) =p{x- y + y) < p(x - j/) + p(j/). Interchanging x, y
we get p(y) < p(j/ - z) + p(z) = p(x - y) + p(z)- The required
inequality now follows.
(iii) p(x + a) < p(z) -j- p(a) = p(z) + 0 = p(x). Also p(z) =
p(x + a ~ a) < p(x + a) + p( — a) = p(x + a). Hence the required
equality.
(iv) By (Q4), dp(x,y) > 0 and by (Ql), dp(x,x) = 0. The
symmetry and triangle inequality properties for dp follow from
(Q2), (Q3). Further, we have clearly dp(x + a, y + a) = dp{x,y).
Finally, if p is faithful then dp(x, y) = p(x — y) = 0 => x = y, so
that p is a metric. Conversely, if dp is a metric then p is faithful
(p(x) = dp(x,0) = 0 => x = 0).
Suppose now d is an invariant semi-metric and p(x) = d(x,0).
Then p(0) = rf(0,0) =0,
p( — x) = a!(—z,0) = d(x — x,x) = d(0,x) = d(x,0) = p(x).
Finally,
p(x+j/) = d(x+y,0)=d(x,-y)^d(x,0) + d(0-y)
< p(x)+p(-j/) = p(x)+p(j/).
3.1.3. LEMMA. Lei X 6e an additive abelian TG and p a
sad. functional on X. If p is continuous at 0 then it is uniformly
continuous on X.
PROOF. Let {Ua} be a basis of nuclei of X. Then, for x G X,
{x + Ua} is a basis of neighbourhoods of x. Suppose now that p
is continuous at 0. Then, given e > 0 there is a nucleus Ua such
that p(a) < e when a G Ua. Therefore, for any x <E X,
x + a G x + Ua, \p(x + a) - p(x)\ < p(a) < e
which show that p is uniformly continuous on X.
3.1.4. The topology induced on X by the semi-metric dp is

102
Some Types of Topological Algebras
called the p-topology and X with the p-topology is denoted by
(X,p). A neighbourhood basis for a point xo in the p-topology
is given by the family of sets {M(xo,e) : e > 0}, where
M(x0, e) = {x G X : p{x - x0) < e}.
Since
M(xo,ei) C M(x0,€2) for ei < e2
it follows that the subfamily {M(xo,~) : n = 1,2,...} is also a
neighbourhood basis at xq, whence the p-topology is first
countable. It is clear that a sequence xn —> x in the p-topology iff
p(xn — x) —> 0. We call the p-topology complete if it is complete
with respect to dp. As already seen in 3.1.2 (iv), dp is a metric
if p is faithful.
Two sad. functionals pi,p2 on X are said to be equivalent, in
symbols, p\ ~ p2 if both of them induce the same topology on X.
3.1.5. LEMMA. Let p be a sad. functional on an additive
abelian group X. Then:
(i) X is a TG under the p -topology;
(ii) p is a continuous function on (X,p);
(iii) kerp ={i6l: p[x) = 0} is a closed subgroup and kerp =
0
(iv) If pi ~ p2 then kerpi = kerp2. In particular pi is faithful
iff Vi *s faithful.
PROOF, (i) It follows from (Q3) that the map (x, y) h-> x + y
is a continuous and from (Q2) the map x i—> —x is continuous. So
X is a TG.
(ii) The continuity of p follows from 3.1.2 (ii).
(iii) If x, y G kerp then p(x — y) < p(x) + p(j/) = 0 + 0 = 0,
whence x — y G kerp. So kerp is a subgroup. Further, it is
closed since kerp = p_1(0) and p is continuous. Since 0 G kerp
and kerp is closed, 0 C kerp. On the other hand, if p{x) = 0,
p(0 - x) = p(-x) = p(z) = 0, so that 0,0, > x, x G 0. Thus,
0 = kerp.

§ 1. Quarter-norms
103
(iv) By (iii), kerpi = 0 = kerp2-
3.1.6. The set 6 of all sad. functionals on X can be partially
ordered by: p < q if p(x) < q(x) for all x e X (p, q G S).
3.1.7. LEMMA, (a) If p is a sad. functional and t > 0 then
tp is a sad. functional.
(b) If pj(j = 1,---,71) are sad. Junctionals on X then so are
p = pH \-pn, q = Pi V--- Vp„ t (?(x) = maxpj(x))
PROOF, (a) Clear.
(b) That p is sad. is clear. To prove that q is sad. it is clearly
enough to show that it satisfies (Q3). Now, by definition of q, we
have for given x, y G X, an index j with q(x + y) — pj(x + y).
Then
Pj{* + V) < Pj(z) + Pi(») < ?(*) + ?(»)»
so that ?(x + j/) < g(x) + ?(j/).
3.1.8. DEFINITION. Let X be a LS and p a sad.functional
on the underlying additive group of X. Then p is called a quarter-
norm if it satisfies.
(Q6) If (xn) is a sequence in X, x G X, p(xn — x) —> 0 and
A„ —> A in K, then p(A„x„ — Ax) —> 0 (i.e. if in the p-topology
x„ —> x and A„ —+ A in K then Xnxn —> Ax).
The condition (Q6) can be split up into three subconditions:
(Q6a) If p(xn) -♦ 0, An -♦ 0 then p(A„x„) -♦ 0.
(Q6b) If p(xn -x) -+0 then p(Ax„ - Ax) -+ 0, A G K.
(Q6c) If A„ -+ A then p(A„x - Ax) -^ 0, x G X.
That (Q6) implies each of these subconditions is clear.
Conversely, it follows from the identity
A„x„ - Ax = (An - A)(x„ - x) + Ax„ - Ax + A„x - Ax (*)
that the subconditions can be replaced by (Q6).
3.1.9. Remark. If p is a quarter-norm on X, define p* on
X by: p*(x) = p(x) or 1 according a p(x) < 1 or p(x) > 1.
Then it is easy to see that p* is also a quarter-norm; p* is called
a reduced quarter-norm, the reduced form of p. It is clear that
' q is the lattice sum (see 8.6.7) with respect to the partial order defined
in 3.1.6.

104
Some Types of Topological Algebras
p* ~ p and that p* is bounded (as a functional): p*(x) < 1 for
all x in X, or briefly p* < 1.
3.1.10. LEMMA. Let p be a quarter-norm on X. Then p** =
p/l+p is a quarter-norm which is also bounded: p** < 1. Further,
p ~ p.
PROOF. For proving p** is a quarter-norm only the subad-
ditivity has to be checked (since the rest of the properties are
clear). If p[x + y) = 0 then p**(x + y) = 0 and then the sub-
additivity holds trivially. Next assume that p[x + y) > 0. Then
0 < p(x + y) < p(z) + p(j/), whence
p(x + y) ' p(x)+p(y)'
so that
p**(x + y) = 1 < l = p(*) + p(y)
1 i + PlAjJ 1 + fi^m i + pW + pW
The boundedness of p** is evident from its definition. Also, the
definition of p** shows that p**{xn — x) —> 0 iff p(x„ — x) —> 0,
whence p** ~ p, completing the proof.
3.1.11. PROPOSITION. lei X be a LS over K. yi
sad. functional p on X is a quarter-norm iff X is a TLS
under the p -topology.
PROOF. This follows from 3.1.5 (i) and the observation that
the condition (Q6) is equivalent to the continuity of the map
(A,x) i—> Ax in the p-topology.
3.1.12. COROLLARY. A TLS X is quarter-normed iff it is
first countable.
PROOF. Since the p-topology is first countable (see 3.1.4) we
have only to prove the "if" part. If X is first countable then by
2.1.7 it has an invariant (with respect to addition) semi-metric d.

§ 1. Quarter-norms
105
By 3.1.2, p(x) = d(x,0) is a sad. functional which is, by 3.1.11, a
quarter-norm, so that X is quarter-normed.
3.1.13. Remark. It can be shown easily that if X = (X,p) is
a quarter-normed LS then it is complete as a TLS iff it is complete
with respect to p (i.e. with respect to the semi-metric dp). Such
a quarter-norm p is called a complete quarter-norm.
3.1.14. Let X be a LS over K and p a faithful quarter-
norm. Then X with the p-topology is called a pre- (F) space.
The quarter-norm in this case is called a (F) norm and will be
usually denoted in the sequel by | • |. If the (F) norm | - | is
complete then X is called a (F) space. Note that a (F) space
is a complete metric linear space.
3.1.15. THEOREM (Open mapping theorem). If T is a
continuous linear transformation of a (F) space X onto a (F) space
X* then T is open [i.e. T maps every open set of X on an open
set of X*).
PROOF. See [22, p.48].
3.1.16. THEOREM (Closed graph theorem). Let X,X* be
(F) spaces. If T : X —> X* is a linear map which is closed then
it is continuous.
(A map T : X —+ X* is called closed if xn —> x in X and
Txn -tj in X* => y = Tx.)
PROOF. See [22, p.50].
3.1.17. Let P be a family of quarter-norms p on a LS X.
Then the P -topology on X is that topology in which a net xa —>
x iff p(xa — i) —> 0 for each p G P. It is easy to see that X is a
TLS under the P -topology. If the family P consists of a single p
then the P -topology reduces to the p-topology defined in 3.1.4.
A neighbourhood basis for a point xo in X, in the P -
topology, is given by the family of all sets of the form
M{xo : pi,- ■ ■ ,Pn,e) = {x G X : pj(x - x,) < e,j = 1, ■ ■ •, n}
where p1} ■ ■ ■ ,pn G P and e > 0. We write X = (X,P) to denote
X with the P -topology.

106
Some Types of Topological Algebras
Two families ?\,?i of quarter-norms on X are said to be
equivalent, in symbols, P\ ~ ?2 if the /i-topology and the P2 -
topology are the same: (X, Pi) = (X, P2) (cf. equivalence of 2
sad.'s in 3.1.4).
3.1.18. LEMMA. If pj(j = 1,---,71) are quarter-norms on
X so are
P = PlH \-pn, ? = PlV---Vp„
Further, p ~ {pi, • ■ •, pn} ~ ?.
PROOF. By 3.1.7 (b), p and g are sad.functionals. The
verification that they are also quarter-norms is straightforward.
Further, since
p(xa - x) -+ 0 <=>• pj(xa - x)-+0 (j =1,---,n)
•<=>■ g(xa — i) —> 0
the equivalence statement follows.
3.1.19. LEMMA, (a) If p,p* are quarter-norms on X such
that there are constants Ci,C2 > 0 satisfying
Cip < p* < C2p (*)
iAen p ~ p*.
(b) 7/ p is a quarter-norm and t > 0 then tp is a quarter-
norm with tp ~ p.
PROOF, (a) This is an immediate consequence of the
inequalities
Cip{xa - x) < p*(xa - x) < C^Zq, - z).
(b) That tp is a quarter-norm is clear. The equivalence
assertion follows from (a) (taking p* = tp, C\ = C2 = t).
3.1.20. If P is any family of quarter-norms (or more generally
sad. functionals) on X we write
ker P = nkerp (p e P).
A net (xa) is called a C-net with respect to a quarter-norm p if
given e > 0 there is an c*o = c*o(e) such that p(xa — xp) < e for

§ 1. Quarter-norms
107
all a, (3 >- ao. It is called a C-net with respect to a family P of
quarter-norms if it is a C-net with respect to each p in P. X is
called P -complete if every C-net with respect to P converges in
X.
3.1.21. LEMMA. Let X = (X,P) be a TLS, where P is a
family of quarter-norms on X. Then:
(i) B€p = {x G X : p(x) < e} is a symmetric nucleus of X,
for each p G P.
(ii) 0= {0} = ker P, where bar denotes closure in X.
(iii) X is Hausdorff iff ker P = {0}.
(iv) X is a complete TLS iff it is P -complete.
(v) If f i-s a continuous functional on X with /(0) = 0 then
ker P C ker f.
PROOF, (i) By definition of P -topology, Bep is a nucleus of
X; further it is symmetric since p( — x) = p(x)(x G X).
(ii) Since p(0) = 0, for any p, 0 G ker P. Now ker P is closed
by virtue of 3.1.5 (iii). So, 0 C ker P. If x G ker P, p(0 - x) =
p(—x) = p{x) = 0 (Vp), so that 0,0,--- —> x, whence x G 0.
Therefore 0 = ker P.
(iii) This follows from (ii) and 2.1.13.
(iv) This is clear from the definitions of the C-net and P -
topology.
(v) If x G ker P = 0, we have 0,0,--- —> x. By continuity
of /, the sequence /(0),/(0),--- -> f(x). Since /(0) = 0, we
get 0,0,--- —> /(z). By uniqueness of limit property in K, we
conclude that f(x) = 0, completing the proof.
3.1.22. PROPOSITION. Let X = (X,p) be a quarter-normed
LS and Xq a subspace of X. Write X& = X/Xq and define p^
on X# by
p*(x + X0) = inf{p(z + a) : a <E X0}.
Then:
(i) The canonical map n : x <—* x + Xq = x^ = 7r(x) satisfies
p#(nx) < p(z).

108
Some Types of Topological Algebras
(ii) The functional p# is a quarter-norm.
(iii) The topology of the quotient space X^ is induced by p*.
(iv) p# is faithful iff Xq is closed in X.
(v) If p is complete so is p*.
PROOF, (i) Evident,
(ii) We have
p*(x + y + X0) < p(x + y + a + b) (a,beX0)
< p(x + a) + p(y + 6), so that
p*(x + y + X0) < p#(x + X0)+p#(j/ + X0).
Since 0 £ X0 and p(0) = 0, we get p#(0#) = 0.
Again p(x) = p(-z) => p(z*) = p(-x#), where a;* = x + Xq.
Thus p# is a sad. functional. Suppose now that x% —> x^, A„ —>
A. Since
p#(x# _ x#) = p#((Xn - x)*) -* 0,
using the definition of p#, we can first fix xn G jr-1^*), a; G
7r_1(x*) and then choose suitably an G Xo such that
p(x„ + a„-x) <p*(x* -x*)+- (n = 1,2,---).
71
So p(x„ + a„ — x) —> 0. Since p is a quarter-norm we obtain
p(A„x„ + Xnan - Ax) -+ 0,
whence by using the inequality in (i) we get p#(A„x# —Ax*) —> 0,
which shows p* is a quarter-norm.
(iii) Consider the ball B* = B*(x*,e) = {y* G A# :
p (j/ —x*) < e}. We shall show that B* is open in the quotient
topology. Now
n-1{B*) = 7r-1(B*(x*,e))= (J B(x+a,e)
aex0
and hence open in A. Since, by 2.1.18, it is an open map, B# =
n(n~1(B^)) is open in the quotient topology. It follows that every

§ 1. Quarter-norms
109
open set of (A*,p*) is open in the quotient topology. Conversely,
let G# be an open set in the quotient topology and x# G G#.
By continuity of n, 7r_1(G*) is open in A and so we have: (*)
7r_1( — G*) 3 B(x,e) for any x G 7r-1(x*) and some e = ex > 0.
If j/# G B#(x#) then p*((j/ - a;)*) < e, whence p(j/ -x+ a) < e
for some a <E X0. Then j/+ a G B(x,e) C 7r_1(G#) (using, (*)),
and j/# = 7r(j/+a) G G*, whence G# is open in the p# -topology.
(iv) Now p^(x + Xo) = 0 <=>■ x G Xo where bar denotes closure
in the p-topology. It follows that p^ is faithful iff x G Xq => x G
Xo, i.e. iff X0 = Xq.
(v) Suppose that p is complete and (x„) is a C -sequence of
X#. We can find a subsequence (a;*) with
P [xnk ~ Xnk+l) < ^fc-
Choose inductively x^- <E X with 7t(xjj.) = x# and p(xjt,Xjt+i) <
p-. Then (xjt) is a C -sequence of X and so x^ —> x (say) in X
(since p is complete). By continuity of 7r,
and since (a;*) is a subsequence of the C -sequence (a;*) we have
also x£ —> a;*, proving p# is complete.
3.1.23. DEFINITION. Let X be a TLS of the form X =
oo
(X,P), where P is a family of quarter-norms. A series V] x„ in
ri=l
oo
X is said to absolutely converge if Y_, pC^m) converges for each
ri=l
p<E P.
3.1.24. LEMMA. Let X = (X, P) be sequentially complete
( in particular, complete). Then every absolutely convergent series
/ ^ xn converges in X.
n=l
PROOF. This follows from the inequality.
p(x„+i + ■ ■ ■ + xn+k) < p(x„+1) + ■ ■ ■ + p(x„+jfe)(p G P).

110
Some Types of Topological Algebras
3.1.25 LEMMA. Let T : X —> Y be a continuous
transformation [not necessarily linear) with TO = 0, where X = [X, P),
Y = (y,Q) o.re TLS's and P,Q families of quarter-norms. Then
for x G ker P, qp{Tx) = 0 for all qp e Q.
PROOF. By continuity of T,
Tx,Tx, > T0 = 0.
So qp(Tx) —> ?/3(0) = 0, whence qp(Tx) = 0.
§2. p-Seminorms
3.2.1. DEFINITION. Let X be a LS over K and p a real
number such that 0 < p < 1. A real-valued function p = p[x) on
X is called a p- < seminorm if it satisfies:
(Q3) p(x + y) < p(x) + p(y) for all x, y G X;
(Q7) p(Xx) = |A|^p(a:) for all x<E X,\e K.
If in the above, ^=1 then p is called a seminorm. (Q7)
is called p-modulus homogeneity [modulus homogeneity if p = 1)
condition and ^> is called the homogeneity index of p.
A p -seminorm (respy. seminorm) p is called a p -
norm(respy. norm) if p is faithful (i.e. satisfies (Q5) of 3.1.1).
3.2.2. Remark. The homogenity index of a p 7^ 0 ft is
uniquely determined; for, if p is both ^-seminormed and p'-
seminormed then for an xq with p(xq) ^ 0 we have p(2xo) =
2pp(xo) = 2P p(xo), whence p = p'.
3.2.3 LEMMA. If Pj(j = 1,2,---,71) are p-seminorms on X
then so are
P-Pi-\ \~Pn, ? = PiV---Vp„.
[q defined as in 3.1.7).
PROOF. Straightforward and so omitted.
' In literature the letter p has been used in the place of our p. We have
adopted p to avoid conflict with p used by us with a different connotation.
'' i.e. p is not identically 0.

§ 2. p -Seminorms
111
3.2.4. PROPOSITION. Every p-seminorm p, on X is a
quarter-norm. In particular it satisfies (Ql), (Q2), (Q4) of 3.1.1
and also (i), (ii) of 3.1.2.
PROOF. By taking in (Q7), A = 0,-1 successively we get
(Q1),(Q2), so that p is a sad. fnctional To show that it is a quarter-
norm we have to prove (Q6). Using the identity (*) of 3.1.8, (Q3)
and (Q7) we get
p(Xnxn - Xx) < |A„ - X\pp(xn -x) + \X\pp(xn -x) + \Xn - X\pp(x),
from which (Q6) readily follows.
3.2.5. LEMMA. Let X be a TLS, p a p-seminorm on X
and F = {iGl: p[x) < 1}. Then p is continuous iff 0 G V°(=
int V)
PROOF. If p is continuous then since p(0) = 0 there is
an open nucleus U such that if x G U then p[x) < 1. This
implies that 0 G U C V°. Conversely, if 0 G V° then, using
i _ i
2.1.10, we get eeV° is an open nucleus. If x G V°, j/ = ex then
p(j/) = ep(x) < e. Hence p is continuous at 0 and so everywhere
(by 3.1.3).
3.2.6. If p is a ^-seminorm on a LS X then the resulting
TLS, X = (X,p) (cf. 3.1.4) is called a p -seminormed LS. If p is
a p-norm then X is called a p -normed LS. We generally denote
a p-norm by the symbol || ■ || instead of p. Further, if p = || ■ ||
is complete then X is called a p -Banach space (a Banach space
if p = 1). A ^-seminorm p and a ^'-seminorm p' are said to be
equivalent, p ~ p1, if they are equivalent as sad functionals (i.e.
if they induce the same topology).
3.2.7. LEMMA. Let X = (X,p) be a p-seminormed LS.
Then we have:
(i) If Br = {x G X : p(x) < r},Br = {x G X : p(x) < r}
iAen Br is the (topological) closure of Br in X, where r G
R, r > 0. Further more, Br (respy. Br) is a balanced open
[respy. closed) nucleus of X.
(ii) Let (r„) be a sequence of positive numbers with rn —> 0. If

112
Some Types of Topological Algebras
Vn — Brn,Un = BTn then {Vn},{Un} are bases of nuclei.
PROOF, (i) Trivially, Br C Br. By continuity of p (see
3.1.5(iii)) Br is closed. For x G Br, set xn = (1 — -^i)'PX- Since
pK) < (1 - —rr)r < r (« = 1,2,---)
re + l
z„ G Br and clearly i„ —> i. It follows that Br is the closure of
Br
(ii) If re is sufficiently large then we have
Vn,UnCB£ = M(0;p,e)
Hence the result.
3.2.8. LEMMA. A p -seminormed LS X — (X,p) is locally
connected.
PROOF. Since {Br : r > 0} is a basis of nuclei, it is enough
i
to show that (each) Br is connected. But Br = f B\ is home-
omorphic to B\ (see 2.1.10), so that it suffices to prove that B\
is connected. Now B\ is path-connected (and hence connected),
since if x G B\ then p(x) < 1 and {tx : 0 < t < 1} is a path in
B\ joining 0 and x.
3.2.9. LEMMA. If p is a p -seminorm (0 < p < 1) on a LS
I
X (over K) and 0 < p' < p. Then q = p f is a p' -seminorm with
q ~ p. In particular, if p is a seminorm, pp is a p -seminorm
with pp ~ p.
PROOF. Write ^ = t, so that 0 < t < 1, and q = p*. If
£, y G X then
?(z + y) = P(z + y)'<(p(a0 + p(y))'<p(a0' + p(y)'
(since 0 < i < 1)
< ?(z) + ?(j/)-
Also, if A G K,
?(Ax) = p(\Xy = |A|"*p(a:)* = |A|"'?(x).

§2. p -Seminorms
113
Finally, it is clear that q{xn-x) = p{xn-x)t -> 0 iff p(xn-x) —> 0,
proving q ~ p.
3.2.10. PROPOSITION. Let (X,p),(X*,p*) berespy.a p-
seminormed LS and a p' -seminormed LS. Let T : X —+ X* be a
linear transformation. Then T is continuous iff there is a constant
C > 0 such that
p*(Tx)^Cp{xy'/>> {xeX). (*)
PROOF. Suppose (*) holds. Then
p*{Txa - Tx) = p*{T(xa - x)) < Cp(xa - xy'l»,
whence T is continuous. Conversely, suppose that T is
continuous. By continuity of T at 0, given e > 0, we can find 8 > 0
such that
p"(Tx) < e whenever p[x) < 6. (**)
For any x with p(x) ^ 0, write y = 8llpx/p(x)«. Then p(j/) = 6,
so that by (**) we have p*(Ty) ^ e, which reduces to
p*{Tx) < Cp(xY'!'', with C = 6/«"'/p. (*)
Suppose next p(x) = 0, then for any integer n, p[nx) = 0, so
that by (**),
p*[nTx) < e, i.e. np p*{Tx) < e.
Since n is arbitrary we conclude that p*(Tx) = 0, so that the
inequality (*) holds in this case as well. The proof is complete.
3.2.11. COROLLARY. Let p,p* be respectively a p-seminorm
and a p" -seminorm on a LS X. Then
(i) p* is continuous in the p -topology iff there is a constant
C > 0 such that
p*
p* < Cp f . (*)
(ii) p* ~ p iff there are constants C,C* > 0 such that
p* < Cp~p~, p < C*p*pw. (**)

114
Some Types of Topological Algebras
(iii) Suppose that p* = p. Then p* ~ p iff there are constants
Ci,C2 > 0 such that
Cip < p* < C2p. (* * *)
(iv) If p is a p -seminorm on X and C > 0, then so is p* = Cp\
further, p* ~ p.
PROOF, (i) This follows from 3.2.10 by taking X* = X and
T = lx (the identity map)
(ii) This follows from (i) (by considering also the continuity
condition for p in the p* -topology).
(iii) This follows from (**) (taking C\ = 1/C*,C2 = C).
(iv) Clear.
3.2.12. DEFINITION. Let p be a ^-seminorm on a LS X.
A subset 5 of X is said to be bounded with respect to p or
p-bounded if there is a constant M > 0 such that
p[x) < M for all x G 5, i.e. p|5 is bounded.
If p = || ■ || is a ^-norm then instead of p-bounded we also
say norm bounded or n.bounded.
3.2.13. PROPOSITION. Let X = (X,p) be a p-seminormed
LS. A subset S of X is bounded [i.e. t. bounded) iff it is p-
bounded.
PROOF. Assume that 5 is bounded. Since B = {x G X :
p(x) < 1} is a nucleus of X, there is a Ao 7^ 0 such that 5 C XqB.
If x G 5, x = Aoa (a G B) then
p(x) = |Ao|pp(a) < \XQ\P so that p|5 is bounded .
Conversely, assume that p(x) < M for all x G 5. Let V be any
nucleus. Then we can choose e > 0 such that
B€ = {x : p(x) <e}CV.
Write T) = e/M. If x G 5 then
1 e
p(rj f x) = »7p(a;) < ~r7^ = ei

§ 2. p -Seminorms
115
so that rjpS C Be, or S C r) p B€ C XV, where A = r) p. This
proves 5 is bounded.
3.2.14. COROLLARY. Every p-seminormed LS X is locally
bounded.
PROOF. B\ is a bounded nucleus for X.
oo
3.2.15. COROLLARY. If ^2,xn is a convergent series in
X = (X,p), then the sets {xi,X2, • • •} and {si,S2,- • •} where
n
sn = 2_^xii are P -bounded.
3 = 1
PROOF. Since the sequences (s„) of partial sums sn =
n
yj Xj converges, by 2.3.7(a), (s„) is bounded and so, by 3.2.13
i=i
{si)s2)'-'} *s p-bounded. Suppose that p(sn) ^ M for all n.
Then
P{xn) = P{sn - sn-i) < p(sn) + p(s„_i) < 2M,
whence {xi,X2,- • •} is bounded.
3.2.16. THEOREM (Banach-Steinhauss). Let X be a p -
Banach space and {Ya : a G A} be a family of p -normed linear
spaces and Ta : X —> Ya a family of continuous linear
transformations. If for each x G X there is a Cx > 0 with ||Taa;|| < Cx
for all a then there is a C > 0 such that: \\Ta\\ t < C for all a.
PROOF. Tt For each positive integer n, write Xn = {x G X :
\\Ta(x)\\ < n for all a}. The continuity of the Ta and the fact
that || ■ || is a /)-norm (see 3.5.7) imply that Xn is closed in X.
Furthermore
oo
\j Xn = X{ since x G Xn if n > Cx).
n=l
' \\Ta\\ is the same as the bound \Ta\ of Ta defined in 3.5.1.
TT Follows closely the exposition of the proof of the theorem for Banach
spaces given by Simmons [26, p.239]

116
Some Types of Topological Algebras
Since X is a complete metric space, by Baire's category theorem
there is an Xno ^ 0. It follows that Xno contain a ball So =
-Bo = {z E X : \\x - io|| < r0}
Therefore ||Ta(x)|| < no for all x E So, and all a or briefly
\\Ta{S0)\\ < n0 for all a. (*)
Clearly S = j is the closed unit ball.
r,
o
If x E S, x — 1— (y E SQ), then
ro
||Ta(*)|| = WT°(y) ~ T°(*°)W s ^+^ = ^ = C( say )
(using (*) and noting xo E So). By 3.5.2, ||Ta|| < C, completing
the proof.
§ 3. Quarter-normed Algebras; (F) Algebras
3.3.1. DEFINITION. Let A be an algebra (over K) and p
a quarter-norm on A as a LS. Then (A,p) is called a quarter-
normed algebra if p satisfies the condition
(QA) If p(xn - x), p(yn - y) -+ 0 then.
p{xnVn " xv) ~> 0 (here (xn), (yn) are sequences of elements
and x, y elements, of A).
It is clear that (QA) is precisely the condition needed to make
A a TA under the p -topology. More generally, if P is any family
of quarter-norms on A with each p E P satisfying (QA) then
A is a TA under the P topology; A with the topology will be
referred to as a P -algebra.
3.3.2. PROPOSITION (Arens). Let A be a WTA whose
topology is induced by a complete quarter-norm p. Then A is
a [complete) quarter-normed algebra.
PROOF. It is clearly sufficient to prove that A is a TA (under
the p-topology). Set
Un = {x E A : p(x) < -} (n =1,2,---)-
n

§3. Quarter-norm Algebra; (F) Algebras 117
Then Un is closed (since p is continuous) and symmetric (since
p( — x) = p(x)). Thus, by virture of 3.1.21 (i), {Un} is a basis of
closed symmetric nuclei. We shall now show that for any closed
symmetric nucleus U and element x G A there is an integer no
such that
xUno CU + U.
Write
An = {x G A : xUn C 17}.
Then A„ is closed. For, if xn G A„ and x„ —> x in A then
for any y £ Un, xny —> xj/ (since ry is continuous), so that
xy £ U (U being closed), whence x G A„ and An is closed.
Since lx : j/ i—> xy (x G A) is continuous at y = 0 there is
an n such that x?7„ C U, so that x G A„. This means that
oo
M A„ = A. Since A is complete semi-metric it is Baire (i.e. of
ri=l
the second category). ' Consequently there is an no such that
Ano 2 B = {x G X : p(x - x0) < r0}
for some xo G X and ro > 0. If x G A and p(x) < ro then
j/ = x + x0GBC A„0;
in particular x0 = 0 + x0 G A„0. It follows that
xUno = j/C/„n - x0Uno CU+U.
Given any nucleus V, we can find, using 2.1.14 (vi), 2.1.16 (iii), a
closed balanced (hence symmetric) nucleus U with U + U C V.
Therefore
stf„o CU+UCV.
This means that if
p(x) <r0,p(y) < — (i.e. j/G C/„0)
no
t See [16, pp.200-1]

118
Some Types of Topological Algebras
then xy 6 xUno C V, proving that the map (x,y) i—> xy is
continuous at (0,0), whence by 2.2.3, A is a TA.
3.3.3. DEFINITION. Let 5 be a (multiplicative) semi-group
and p a non-negative real-valued function on S. The function p
is called submultiplicative or sm. if
p(x, y) < p{x)p(y) for all x,y £ S.
The function p is called almost submultiplicative or a.sm. if there
is a constant C > 0 such that
p(z, j/) < Cp(x)p(y) (x, yeS).
The smallest constant C satisfying the above inequality is denoted
by \p\, and we have:
p(xy) < \p\p(x)p(y).
It is clear that p is sm. iff \p\ < 1. Also, for any a.sm. p, if
p*(x) — \p\p[x) then clearly p* is sm.
3.3.4. LEMMA. (Zelazko). Let p on S be sm. Then, for any
x G 5 either p(xn) > 1 for all n > 1, or p(z") —> 0.
PROOF. Suppose that p(xno) < 1 for some integer no > 0.
Then
p(xkn") ^ p(xnn)k-+0 ask-+oo.
Write M = max{p(x") : 1 < n < no}. Given e > 0, choose TVo
such that for k ^ N0, p(xkno) < jg. Write N = N0n0. Then for
n > TV, n = gno + r, with q ^ N and r < n0. Hence
p(x")<p(x^')p(x')<^.M = e,
and p(xn) -+ 0.
3.3.5. DEFINITION. Let p be a non-negative real-valued
function on the semi-group 5. Set
vp{x) = limsupp(x")" ; fp(x) = supp(x")" .
n—»oo n

§3. Quarter-norm Algebra; (F) Algebras 119
Then clearly
0 < vp(x) < i/p(x) < oo.
Moreover, it is evident that i'p(x) < oo iff Vp(x) < oo.
3.3.6. LEMMA. (Gelfandt). For an a. sm. p we have
vv(x) = lim p(xn)" < oo (x G 5).
n—»oo
If p is sm. then we have also up{x) = inf„p(x")" .
PROOF. Assume first that p is sm. and set
c = inf p(xn)" (^ p(x) < oo).
n
Then, given e > 0, there is an integer re > 0 such that
p(xk)$ < c+e. (1)
For any integer n > 0, we write
n = q(n)k+r(n) (2)
where q(n),r(n) are non-negative integers with 0 < r(n) < re.
Then
I = iM+ r(n)
re n nre
so that
q(n) 1
— > — as n —> oo (3)
nre
(since r(n)/nre < -). Using (2) and sm. property of p we get
p[xn)n = (p(x^W).xrW))^ < p(xk)qJ^p(x)rJ^. (4)
Using (1), we have, for sufficiently large n,
c < p(x")" < p^i + e < c+2e (using (1)). (5)
t
He obtained the result for a sm. norm.

120
Some Types of Topological Algebras
Therefore lim p(xn)" = c.
n—»00 v
Now suppose that p is a.sm. and write p* = \p\p. Then p*
is sm. and so vv*(x) = lim p*(xn)" exists. But
y v ' n—»00 v
lim p*(a;")« = lim |p|«p(a;")« = lim p(xn)n
n—»00 ra—»00 ra—»00
since |p|« —> 1 and so up(x) = lim p(x")" < 00.
ra—»00
3.3.7. PROPOSITION. Let p t 6e an a.sm. function on S.
Then v(x) = 1^(2;) (x G 5) /las iAe following properties:
(i) 0 < ^(z) < |p|p(x); in particular p[x) = 0 => v(x) = 0.
(ii) (a) v(xy) = v(yx) (b) 1/(2^) = 1/(2:)4.
(iii) If x <-> y then v(xy) < v(x)v{y).
(iv) //" u is an idempotent with p(u) > 0 then v{u) = 1.
(v) If S has unity*' e and p 7^ 0 then v{e) = 1, and for any
invertible x G 5, 1/(2) > 0.
(vi) If p is sm. then:
0 < u{x) < p(x), and v(x) = p(z) iff p(x2) = p(z)2 i/f p(z") =
p(z)" (/or integers n > 1).
PROOF, (i) We have p(z") < |p|"_1p(a;)", whence
v(x) = lim p(x")" < lim |p|1_«p(a;) = Iplpfx).
ra—»00 ra—»00 « \ / v
(ii) (a) By observing that
(xy)n = xiyx^-^in > 2)
we obtain
P((xy)n) < |p|2p(^M(^)"_1)p(j/) (1)
T As always we assume p^O.
TT i.e. 5 is a monoid.

§3. Quarter-norm Algebra; (F) Algebras 121
so that
p(Mn)" < \p\"P{x)^\p{yx)n-1]«p(y)«. (2)
If p(x) or p(y) = 0 then it follows from (1) that
p{{xy)n) =0,v{xy) =0,v{yx)=0
(interchanging x, y)) so that in this case we have v(xy) = v(yx).
Next consider the case p(x),p(y) > 0. By allowing n —+ oo in
(2) we obtain v{xy) < v{yx), so that by symmetry consideration
we conclude that v(xy) = v(yx).
(b) v(xk) = \imnp(xkn)^ = \imn(p(xkn)-t)k = i/(a:)*.
(iii) Since x <-> j/, (zj/)" = x"j/". Therefore
i/(xj/) = limp(x"j/")" < lim|p|"p(x")p(j/")"
< v(x)v(y).
(iv) i^(u) = lim„p(u")« = lim„p(u)" = 1 (since p(u) > 0).
(v) Since p ^ 0 there is an xq in 5 with p{xq) > 0. Then 0 <
p(xo) = p(exo) ^ \p\p(e)p(xo), whence p(e) > 0. Taking u = e in
(iv) we get i^(e) = 1. Again since 1 = v{e) < v{x)v(x~l),v{x) >
0.
(vi) The inequality herein follows from (1), since p being sm.,
\p\ < 1. We now assume that p(x2) — p(x)2. By iteration we get
p(x2k)=p(xf {k =1,2,-).
For arbitrary integer n > 1, choose integer k such that n <
2k and write 2k = n + m(m > 0). Then
p(*)>(*r" = p(*)2' = p(x2k) < p(*>(*m)
< p(x>(xr
whence p(z)" < p(z"), so that p(x)n = p(xn). Hence it follows
from its definition that v{x) = p(x). Conversely, if v(x) = p{x)
then
p(x)n = V(xr = V(xn)^P(xn),

122
Some Types of Topological Algebras
so that p(xn) = p(x)n. This completes the proof of the
proposition.
3.3.8. If A is an algebra, by considering the underlying
multiplicative semi-group of A we can speak of an a.sm. or a
sm. quarter-norm or p-seminormon A.
3.3.9. LEMMA. Let p be an a.sm. quarter-norm on an algebra
A. Then (A, P) is a quarter-normed algebra. More generally if
P is a family of a.sm. quarter-norms on A then (A, P) is a
P -algebra.
PROOF. Using the identity
XnVn - xy = (xn - x)(yn - y) + (xn - x)y + x(yn - y) (*)
and a.sm. property of p we get
p(xnyn-xy) < \p\{p(xn-x)p(yn-y)+p(xn-x)p(y)+p(x)(p(yn)-p(y))}. (**)
It follows from (**) that p satisfies (QA) of 3.3.1, whence (A,p)
is a TA. Similarly, since each p € P satisfies (**), (A, P) is a
TA.
3.3.10. LEMMA. Let A be an algebra. Then we have:
(i) If p is an a.sm. quarter-norm (respy. p-seminorm) then
tp{t > 0) is also an a.sm. quarter-norm [respy. p -
seminorm) with t\tp\ = \p\, tp ~ p.
(ii) If p is a sm. quarter-norm (respj/. p -seminorm) and t ) 1
then tp is also a sm. quarter-norm [respy. p -seminorm).
(iii) If p is a.sm. then p* = \p\p is sm. and p* ~ p.
PROOF, (i) Clearly p' = tp is a quarter-norm (respy. p-
seminorm). Also,
p'(xy) - tp(xy) < t\p\p(x)p(y) < t~1\p\p'(x)p'(y).
So we have \p'\ < £-1|p|- Similarly, by considering p — t~lp' we
get \p\ < t\p'\- Combining the two inequalities we obtain \p\ =
t\tp\. Also, by 3.1.19 (b), tp ~ p.
(ii) Since 1)1 we have t2 }> t so that tp is sm.

§ 3. Quarter-norm Algebra; (F) Algebras 123
(iii) By (i) p* is a quarter-norm (respy. ^-seminorm). Further,
p"[xy) = \p\p[xy) < \p\2p(x)p(y) =p*(x)p*(y).
Finally, by 3.1.19 (b), p" ~ p.
3.3.11. LEMMA. If pj(j = 1,---,71) are a.sm. quarter-
norms [respy. p -seminorms) then q = p\ V • • • V pn is also an
a.sm. quarter-norm [respy. p-seminorm) with \q\ <max|pj|.
3
PROOF. By 3.1.18 (respy.3.2.3) q is a quarter-norm
(respy. ^-seminorm). Further, if C = max|pj| then
PjM < \Pj\Pjix)Pj{y) < Cp,ix)pj(y) [j = 1, ■ • ■, n)
whence q[xy) < Cq[x)q[y), so that \q\ ^ C
3.3.12. PROPOSITION. (Zelazko). Let A-[A,p) be a unital
quarter-normed algebra with p a.sm. Then A is locally bounded.
PROOF. Write U = Bi = {x e A : p[x) < 1}.
If xneU, A„ e K with A„ —> 0, then
p(Kxn) = Pn[Ke-xn) < \p\p[Xne)p[xn) < \p\p[Xne) -+ 0.
Hence, by 2.1.22, U is bounded and so A is locally bounded.
3.3.13. DEFINITION. A quarter-normed algebra A - [A,p)
is called a pre- [F) algebra if the underlying LS of A is a pre- [F)
space (i.e. if p is faithful). A pre- [F) algebra is called a [F)
algebra if p is complete.
If A is a pre- [F) or [F) algebra its quarter-norm p will be
usually denoted by | ■ | and called a [F) norm (cf. remark in
3.1.14). If the [F) norm is sm. we call A a sm. pre- [F) algebra
or sm. [F) algebra as the case may be.
3.3.14. Examples of [F) algebras
(i) Let K°° denote the algebra of all infinite sequences x = [an)
of elements an from K, the operations being coordinate-
wise, i.e. if x = (a„), y = (/?„) and A e K then
X+y = (an + Pn), xy= (<*nPn), ^X = (\an).

124
Some Types of Topological Algebras
El |ara|"
2" 1 , i ii '
where \a\ denotes the norm or absolute value of a in K(= R
or C). It can be shown that (K°°, | • |oo) is a commutative
unital (F) algebra over K.
(ii) Denote by H°° the real algebra of all sequences x = (qn) of
elements from H (with coordinate-wise operations), where
H stands for the algebra of Hamilton quaternions; H = R +
Ri + Rj + Rk, t2,j2,k2 = -1, ij = k = -ji, jk = -kj,
ki = —ik.
Define \x\qq as in example (i) with qn replacing an, and
\qn\ = (al + a\ + a\ + a§)5 if qn = a0 + axi + a2j + a3k.
Then (H°°, \ • |oo) is a real unital (F) algebra which is not
commutative.
(iii) Denote by £ the algebra (under pointwise operations)
of all entire functions / = f(z) of a complex variable
z; £ is a unital complex algebra. The following metric
| \g introduced by V.G.Iyer is an (F) metric on £ : if
OO
/ = f(z) = Yl01"2"' 1-^1° = suP{lao|,|Q!„|«,n ^ 1} (see
[6', Theorem 1, Remark 2]). It is known that the
convergence under this metric is the same as uniform
convergence over compacta in C (ibid, Theorem 3). The algebra
£ = (£ ,\ • \g) is a unital (F) algebra. £ is not locally
bounded (ibid, Theorem 2) and hence, by 3.3.2, the (F)
norm \ • \g is not a.sm.. Finally, £ is separable since the
set of polynomials with complex t rational t coefficients is
dense in £.
(iv) The entire functions form an algebra also with respect to
pointwise linear operations and Hadamard multiplication: if
OO OO
/ = 2_^ otnzn, g = 2. Pn2" the Hadamard product / x g
is defined by
i.e. having rational real and imaginary parts.

§ 3. Quarter-norm Algebra; (F) Algebras 125
/ x 9 = ]L anPnZn-
(Note that pointwise multiplication of / and g corresponds
to Cauchy multiplication of the associated power series.)
We denote the above algebra by f (x) . f (x) has no unity
element (since 1 + z + z2 H is not entire). Also, we have
clearly
I/ x </|g < \f\c\g\c-
So £(x) = (£(x\\ • \g) is a sm. (F) algebra which again is
not locally bounded (note that both £ ,£^x' have the same
topology),
(v) Let M = M([0,1]) be the algebra of (equivalence
classes) of almost everywhere defined Lebesgue
measurable complex functions on [0,1]. For / G M, if we set
fi \f(t)\
1/1 = / ,., ,, dt then it is an (F) norm and M is an
'" A) 1 + 1/(01 l
(F) algebra. The norm or metric convergence in M is the
same as convergence in measure (see [28, pp. 116-17]).
3.3.15. LEMMA. The unitization A\ of a quarter-normed
algebra (A,p) is canonically a quarter-normed algebra (Ai,pi)
with
Pi(Aei + x) = |A| + p(x) (x e A, A G K),so that pi(ei) = 1. (*)
Further, A\ is a pre- (F) algebra or (F) algebra according as A
is a pre- (F) algebra or (F) algebra.
PROOF. We already know that A\ is a TA under the product
topology of K x A (see 2.2.9). It remains to see that pi (defined
as in (*)) induces the topology of A\. But this readily follows
since we have pi(Ae! + a;„ - (Aei + z)) = |A„ - A| +p(xn - x). If p
is faithful, and pi(Aei + x) = 0 then |A| = 0, p(x) = 0, so that
A = 0, x = 0, whence pi is faithful. Again when p is complete,
it is easy to see, using (*), that p\ is complete.
3.3.16. Remark. p\ may fail to be sm. when p is sm.
A counter example is provided by the unitization (£^)1 of the

126
Some Types of Topological Algebras
algebra £^x' of 3.3.14 (iv). If pi = | ■ |i then pi cannot be sm.,
for in that case by 3.3.12 (£^x')i is locally bounded whence also
£(*> which is impossible (see 3.3.14 (iv)). So | ■ | is not sm. (nor
even a.sm.).
3.3.17. PROPOSITION. The metric completion A of a sm.
pre-(F) algebra A = (A,p) is canonically a sm. (F) algebra
A = (A,P).
PROOF. If x,y G A and x — limz„, y = limj/„ (xn,yn£A)
we define
x + y = lim(z„ + yn), xy~limxnyn, Xx — \imXxn.
The above limits exist since the sequences defining them are C-
sequences, as can be verified by using the sa. and sm. properties
of p as well as the property (Q6b) of p (given in 3.1.8). We
extend p to p on A by defining p{x) = limp(x„) where xn G A,
xn —> x; the limit exists since p(xn) is a C-sequence (as can be
seen using 3.1.2 (ii)). It is straightforward to verify that p is a
(F) norm. Further, p is sm. since
p(xy) = lim p{xnyn) < lim p(xn)p(yn) < p(x)p(y).
n—»oo n—»oo
This complete the proof.
3.3.18. LEMMA. (A, | ■ |) be a sm. pre-(F) algebra Then:
(i) An x G A with \x\ < 1 is q. invertible iff the series
oo
/^(-l)nxn converges and then the q. inverse x' is given
n=l
by x> = j2(-irx». h
n=l
In particular, if A is complete [i.e. A is an (F) algebra)
then every x with \x\ < 1 is q. invertible with its q. inverse
given by (*).
(ii) Let A be unital with unity e. Then an element x G A with
oo
|e — x\ < 1 is invertible iff the series 2_](e — x)n converges,
n=0

§ 3. Quarter-norm Algebra; (F) Algebras 127
and then the inverse x 1 is given by
oo
x~x = 52(e-x)n. (**)
In particular, if A is an {F) algebra every x with \e—x\ < 1
is invertible with x~l given by (**).
PROOF, (i) In view of 2.2.17 it is enough to prove the "only
if" part, i.e. if x' exists then the series ^2{—l)nxn converges to
x'. Since \x\ < 1, by 3.3.4, |x"| -> 0, so that xn -> 0 in A.
Therefore
N N
x°^2(-l)nxn = (~i)NxN, whence, ^{-l)nxn = x'o{-l)NxN
n~1 n~l
oo
Making N —> oo we get ^(-l)"x" = x' o 0 = x'.
n=l
Now assume that |-j is complete. Since \x\ < 1 the numerical
oo oo
series N |x|" converges, whence the series Y_]( —l)"x" converges
n=1 ra=1
absolutely and consequently, by 3.1.24, it converges and so has x'
for its limit.
(ii) The proofs are similar to that of (i).
3.3.19. Proposition. Let A = (A, | ■ |) be a sm. (F)
algebra-in particular a p -Banach algebra-and v — v\.\ . If v(x) <
1 {in particular if \x\ < 1) then x is q. invertible with
oo
^ = 23(-1)^, w
where the series converges absolutely. Moreover, if x is
q. invertible and \y — x\ < (1 + |a;'|)_1 then y is q. invertible.
Hence, every q. invertible x has an open {ball) neighbourhood
B{x, (1 + |x|)_1) consisting of q. invertible elements.
PROOF. Since i/{x) < 1 we can choose rj such that v{x) <
r) < 1. Then \xn\n < r) < 1 for n > N. This implies that
oo oo
^3 |x"| < ^3 ?y" < oo (since r] < 1).

128
Some Types of Topological Algebras
oo
If follows that the series V,( —l)"x" converges absolutely and
n=l
oo
hence also in A (by 3.1.24). By 3.3.18 (i), x'= ]T(-l)"x" is
n=l
q. inverse of x.
Assume that x G A is q. invertible and y G A with \y — x\ <
(1 + |x'|)_1. Using x' o x = 0 we have
x' o y = x' + y + x'y = (—x - x'x) + y + x'j/ = j/ — x + x'(j/ — x).
Therefore
|z' ° y\ < \y — x\ + I^'IIj/ — *| = (i + I^'DIj/ — x\ < i.
Similarly, |j/ox'| < 1. It follows that x'oy, yox' are q.invertible,
whence by 1.1.30 that y is q. invertible, as required.
3.3.20. COROLLARY. Let A=(A, | ■ |) be a unital sm. (F)
algebra with unity e. If v{x) < 1 (in particular \x\ < 1) then:
(i) e ± x are invertible with
oo
oo
(e-x)^1 = ^x" (**)
the series in (*),(**) converging absolutely.
(ii) If v(e — y) < 1 (in particular \e — y\ < 1) then y is invertible
with oo
j/_1 = J2(e~ y)n- (***)
PROOF. The representation (*) follows from (*) of 3.3.19 by
using the fact that e + x is invertible iff x is q. invertible. The
representation (**) is, of course, got from (*) by replacing x by
—x. Finally the representation (***) is got from (**) by putting
x = e — y.

§ 3. Quarter-norm Algebra; (F) Algebras 129
3.3.21. COROLLARY. If x e A is invertible and
\y — x\ < jz^1^1, then y is invertible with
oo
y'1 = E[*~1(* -vW*'1-
Hence, every invertible x has an open neighbourhood B(x, la;-1!-1)
comprising invertible elements.
PROOF. Since
\e — x~ y\ = \x~ (x — y)\ < \x~ \ \x — y\ < 1
x~1y is invertible (by 3.3.20 (ii)), and we have
oo oo
y~lx = (x-'y)-1 = ^(e - x~lyY = ^[x^x - y)]n.
n=0 n=0
Hence
oo
y-1 = ^.1^-yW*"1-
n=0
3.3.22. PROPOSITION. Let p be an a.sm. quarter-norm on
an algebra A and x, y q. invertible elements of A with q. inverses
x', y' respy. Then:
p{y'-x')[l-{l + \p\p{x'))\p\p{y-x)] < {l + \p\p{x'))2p(y-x). (*)
p{y' - x')[l - (1 +p{x'))p{y - x)} < (1 +p{x'))2p{y - x) if \p\ < 1.
PROOF, t Write y = x + a, y' = x' + b. Then 0 = y o
y' = (x + a) o [x' + 6) which, on using x o x' = 0, reduces to
a + b + xb + ax' + ab = 0.
Therefore
ioi = i- a- ax' — ab.
' The principle underlying the proof is essentially due to Arens who
obtained (**) of 3.3.23 for normed algebras.

130
Some Types of Topological Algebras
By pre-multiplying (with respect to o operation) by x' we
get
b = x' o (x — a — ax' — ab) = —a — ax' — ab — x'a — x'ax — x ab,
using d'oi = 0. It follows that
p(b) < p(a) + \p\p(a)p(x') + \p\p{a)p(b) + \p\p(x')p(a)
+ \p\2\p{x')2p{a) + \p\2p{x')p{a)p{b),
so that
p(6)[l - \p\p{a) - \p\2p{x')p{a)} ^ p{a) + 2\p\p{a)p{x')
+ \p\2p{x')2p{a)
= (l + \p\p(x'))2p(a)
which is same as (*). When |p| < 1 it is clear that the inequality
remains unchanged if we drop the terms \p\, \p\2, whence we obtain
(**).
3.3.23. PROPOSITION. If A— (A,p), where p is an a.sm.
quater-norm, is unital and x,y G A are invertible then we have:
p(y-1 - x-1)^ - \p\2p(y - x)p(x-1)] < \p\2p{x-1)2p(y - x) (*)
p(y~1 - x1)^ - p(y - x)p(x~1)) < p(x~1)2p(y-x) if \p\ < 1. (**)
PROOF. The proof of (*) is similar to that of 3.3.22 and
even simpler. Writing a = y — x,b — y~1 — x~1 we get e =
yy~1 = (x + a)(x~1 + 6) which reduces to xb = —ax~l — ab, i.e.
6 = — x~1ax~1 — x~1ab. Applying p to both sides, using a.sm.
property of p, and rearranging terms we get (*). Once again
(**) follows from (*) by here omitting the term |p|2 in (*).
3.3.24. COROLLARY (Bonsall-Duncan). For a sm. p, if
p(y - x) < |p(x-1)-1 then
p(y~1 - x-1) < 2^-1)2^ - x).
Proof. We have 1 — p(j/ — x)p(x~1) > 1 — | = |, whence the
required inequality follows from (**) of 3.3.23.

§4. p-Seminormed Algebras; p-Banach Algebras 131
§4. p -Seminormed Algebras; p-Banach Algebras
3.4.1. LEMMA. If p is an a.sm. p -seminorm on an algebra
A and q any p -seminorm on A with q ~ p. Then q is also
a.sm. .
PROOF. By virtue of 3.2.11 (iii), there are constants Ci,C2 >
0 with Cip ^ q ^ C2p.
Therefore
q(xy) < C2p{xy) < C2\p\p{x)p(y) < C2\p\C^2q(x)q(y),
whence q is a.sm. .
3.4.2. LEMMA. Let p be a p -seminorm on an algebra A
such that the map
m# : (x,y) i—> xy is continuous at (0,0) in the p-topology,
Then:
(i) ker p is a closed bi-ideal of A;
(ii) p is a.sm. .
PROOF, (i) By 2.1.13, 3.1.21 (ii), kerp is a closed subspace of
A. It remains to show that it is a bi-ideal. Since m^ is continuous
at (0,0), given e > 0, there is a 8 > 0 such that
P(*)»P(y) < S=>p(xy) < e. (1)
For a given y G A we can choose n such that
p(f) = Tp'W < * (2)
n n?
(the choice being possible since p > 0).
If p(x) = 0 then
p[nx) = npp(x) = 0. (3)
From (1), (2), (3) we obtain
p(xy) = p(nx- -) ^ e.

132
Some Types of Topological Algebras
Since e is arbitrary, p(xy) = 0; similarly p(yx) = 0. This
completes the proof.
(ii) Suppose that x, y G A, p(x),p(y) ^ 0. Writing
- \ - - -
xi ~ 8p/p(x)p, yi = 8p/p{y)p
we find that p{x\) = p(j/i) = 8, whence by (1), p(xy) < e which
reduces to
p(xy) < Cp(x)p(y) (4)
where C = e/82. If p{x) or p(y) = 0, then by (i), p(xy) = 0,
so that (4) holds trivially. Thus (4) holds for all x, y and p is
a.sm. .
3.4.3. COROLLARY. Let p be a p-seminorm on an algebra
A. Then A is a TA under the p -topology iff p is a.sm. .
PROOF. The "only if part clearly follows from 3.4.2. The
"if part is a consequence of 3.3.9.
3.4.4. A TA A whose topology is induced by a ^-seminorm p
is called a p -seminormed algebra (semi-normed algebra if p = 1)
and we write A = (A,p). A complete ^-seminormed algebra is
called a p -semi-Banach algebra (semi-Banach algebra if p = 1).
In view of 3.4.3 the p in (A, p) must be necessarily a.sm. . Such
a conclusion is not possible if p were only a quarter-norm (cf.
3.3.16).
Since, by 3.3.10, p* = \p\p is a sm. ^-seminorm with p* ~ p,
we can assume when necessary, without any loss of generality, that
the ^-seminorm p of (A,p) is sm. .
The meaning of a ^-normed or a ^-Banach algebra is clear.
Of course, when p = l, a ^-normed (respy. ^-Banach) algebra is
just a normed (respy. Banach) algebra.
Note that a ^-normed algebra is a pre- (F) algebra and a
^-Banach algebra an (F) algebra.
3.4.5. LEMMA. Let (A,p) be a p-seminormed algebra. Then
v = up has the property
v{Xx) = \X\"v(x).
PROOF. v(Xx) = limnp(\nxn)" = lim„ \X\fip(xn)n =

§ 4. p -Seminormed Algebras; p -Banach Algebras 133
\X\"v(x).
3.4.6. Examples of ^-seminormed algebras and p-
Banach algebras.
(i) Let 5 be any set. The set Ks of all K-valued functions /
on 5, under pointwise operations, is a unital commutative
algebra over K. Fix a point so in 5 and define po(f) =
|/(so)|. Then po is a sm.semi-norm on Ks and (K",po)
is a sm. semi-Banach algebra. For 0 < p ^ 1, if we set,
Pq(J) = \f(so)\p then (K",po) is a ^-semi-Banach algebra.
(ii) K" is a Banach algebra under the norm
\\x\\ = |Ax [ H |A„|, where x = (Ai, ■ ■ ■, A„) G K".
(iii) The set H , where H is the 4-dim real algebra of Hamilton
quaternions, is under pointwise-operations an algebra over
R which is not commutative. Defining, as above, po(f) =
|/(so)|,where |-| is now the quaternion norm (see 3.314(ii))
we obtain a real ^-semi-Banach algebra (Hs,Pq) which is
not commutative.
(iv) If B = B{S, K) denotes the set of all bounded K -valued
functions on 5 then B is a Banach algebra over K under
the sup (=supremum) norm: ||/||oo = supseS |/(0)|(/ G B).
It is commutative. Also (B, \\ ■ ||go) is a p-Banach algebra
(the p-norm property of || • ||^ follows from 3.2.9).
Similarly, we have the real ^-Banach algebra B(S,H) of all
bounded H-valued functions with /) -norm || • H^ : ||/||£o =
sups \f(s)\p, where |-| denotes the absolute value in H (see
3.3.14(ii)) This algebra is not commutative.
oo
(v) All sequence (x„) of elements in K such that y^|x„|p (0 <
n=l
p < 1) form under coordinate-wise operations a
commutative ^-Banach algebra lp = lp(K) under the p-norm
\\x\\p = T,n\xn\p, which is sm. (\xy\p = £„ WnVnY =
Hn l^filll/raK ^ IklUlj/llp)- More generally, if 5 is any set,
denote by lp(S) the set of all K-valued functions / on 5

134
Some Types of Topological Algebras
such that the generalized sum Yls \f{s)\P < °°- Then lp(S)
is a ^-Banach algebra under the norm \\f\\p = Yla \f{s)\p>
which again is sm.: ||/ff||p ^ ||/IWIff||*>-
(vi) Let Ll ~ 2/1[0,l] denote the space of (equivalent classes)
of absolutely Lebesgue integrable functions on [0,1]. L1
is a Banach space under the norm ||/|| = /0 \f\(t)dt =
/o 1/(01^- ^1 *s cl°sed for a multiplication '*' called
convolution defined as follows:
f*g(s)= [" f(s-t)g{t)dt
Jo
( f * g exists for almost all s by Fubini's theorem and so
f*geL1).
It is straightforward to check that the operation * is
associative and commutative. Thus L1 is a commutative Banach
algebra; we have ||/*ff|| < ||/||||ff||- (See [10,p 17]).
(vii) Let 5 be a compact Hausdorff space C(S) = C(S, K) the
algebra of K-valued continuous functions on 5. Then C(S)
is a Banach algebra under the 'sup' norm. If 5 is a locally
compact Hausdorff space, denote by Co(S) the algebra of
K-valued continuous functions / which vanish at oo (i.e.
/ has the property that for any e > 0 there is a compact
set K = Ke such that / = 0 on S\K). Cq(S) is also
a Banach algebra under the sup norm; of course, if 5 is
compact, Cq(S) = C(S).
If 5 is locally compact Hausdorff and 5oo is its one-point
compactification then it is easy to see that C(S^) is the
unitization of Co(S).
3.4.7. PROPOSITION (Zelazko). Let G be a discrete TG.
Then the set L<> = LP(G) = L<>(G, K), 0 < p < 1, of all K -valued
functions x = x[s)[s G G) such that Yis \X{S)\P < °°> 2S an ahe-
bra [over K) under pointwise linear operations and convolution as
multiplication: if x,y G Lp then its convolution product is defined
by x * y(s) = Yltx{t)y{t~ls)- ^or s G G, set xs(t) = 1 or 0
according as s = t or s ^ t Then xs G Lp, and the map sm,
is 1-1 and xs*xt = xst (so that G is multiplicatively embedded in

§4. p -Seminormed Algebras; p -Banach Algebras 135
LP(G)). Moreover, Lp is a p-Banach algebra under the p-norm
\\x\\p = 52\x(s)\p-
a
PROOF. That Lp is a LS and \\-\\p is a p-norm are clear (the
subadditivity of || ■ \\p depends on the inequality (s-\-t)p < sp + tp,
if s, t > 0, 0 < p < 1). If x, y G Lp then
II* * y\\P = EI E^M'-1*)!' < E^(*)riy(*_1*)lp
at a,t
t a t
^¾ ll^llpll y\\p-
It follows that Lp is closed for convolution and that || ■ \\p is sm.
The proof of completeness of Lp is straightforward (and will be
omitted). Thus Lp is a p-Banach algebra. Further it is clear that
xs G Lp and sm, is 1 — 1. Finally, by a simple computation,
%a * %t — xat •
3.4.8. COROLLARY. Lp(G) is commutative iff G is
commutative.
PROOF. If G is commutative then
**j/(s) = Ex(i)j/(rls) = E yC^'MO
t t
Y^y{u)x(su 1) = Y^,y{u)x{u 1s) = y*x(s)
u
On the other hand, if Lp is commutative we have xst ~ xs * xt =
xt*xs = Xts, st = ts (since sm, is 1-1) and G is commutative.
3.4.9. Remark. Zelazko [31, p. 12, Theorem 29] has shown
that if G is a locally compact group then for 0 < p < 1,2/ =
LP(G) the space of (left) Haar measurable functions x(s) on G
such that f \x[s)\pds < oo is closed for convolution multiplication
iff G is discrete. Thus, under convolution, Lp is a ^-Banach
algebra only when G is discrete. L1, on the other hand, is always
a Banach algebra (under convolution).
3.4.10. PROPOSITION. Let Wp denote the set of all
continuous complex-valued 2n -periodic functions f = f[t) on U having

136
Some Types of Topological Algebras
oo oo
a Fourier expansion f = Y^/(n)emi with \2\f(n)\P < °°- ^en
—oo —oo
W is under pointwise linear and multiplication operations a
commutative p -Banach algebra [the Wiener-Zelazko algebra) with p -
oo
norm \\x\\p = ^2\f(n)W>.
— oo
PROOF. Clearly W can be identified with the algebra
LP(1,C) = Lp(l) = l"(l). By 3.4.7, Lp(l) is ^-Banach which
moreover, by 3.4.8 is commutative (since T is commutative).
3.4.11. LEMMA. Let A be an algebra [over K) and A\
its unitization. Then a p -seminorm p on A has a canonical
extension p\ on A\ with pi(ei) = 1. If p is a.sm. so is pi
and |pi| = max{l,|p|}. In particular, p\ is sm. when p is sm.
Further, p\ is faithful or complete according as p is faithful or
complete.
PROOF. Define pi by setting for x\ E A\,x\ = Aei +
x (x E A) Pi(zi) = |A|P + p(x). Clearly pi(ei) = 1 and pi is
a ^-seminorm on A\. If p is a.sm. and j/i = \ie\ + y E A\ then
Pi(ziJ/i) = \Xn\p + p(Xy + nx + xy)
< |AnM|" + |A|>p(y) + \fi\"p(x) + \p\p(x)p(y). (*)
If C = max(l,|p|) then (*) implies
Pifciji) <C(\\\' + p(x))(\ii\''+p(y)) = Cp1(x1)p1(y1) (**)
whence |pi| < C. On the other hand, \p\ < |pi| and 1 = pi(ei) =
Pi(ei) ^ |pi|p(ei)2 — |pi|> so tnat |pi| ^ ^- Hence |pi| — C —
max(l, |p|)). If p is sm. then \p\ < 1, so that |jpx| = max(l, |p|) =
1, whence pi is sm.. Finally, that pi is faithful (respy.complete)
when p is faithful (respy. complete) follows as in the case of (F)
norms (see proof of 3.3.15).
3.4.12. Remark. Note that the canonical extension of a p-
seminorm p differs from the canonical extension of p as a quarter-
norm (except when ^=1).
3.4.13. DEFINITION. A ^-seminorm p on an algebra A is

§ 4. p -Seminormed Algebras; p -Banach Algebras 137
called normalized if p is sm., and p(e) = 1 whenever A has a
unity e.
3.4.14. LEMMA. Let (Aj,pj)(j = 1,---,71) be p-
seminormed algebras. Then the product TA A = A\ x ■ • • x An
[under coordinate-wise operations) is a p -seminormed algebra
A = (A, q) where q is given by:
for x = (x\, ■ ■ ■, xn) G A, q(x) = maxpj(xj).
3
Moreover, \q\ ^ max|py|. In particular q is sm. if each pj
is sm. If the Aj are unital and pj normalized then A is unital
with q normalized.
PROOF. It is clear from the definition of q that a sequence in
A converges with respect to q iff their associated coordinate
sequences converge in the respective factor spaces. But this precisely
means that the q -topology is the same as the product topology.
Further, if x = (xi, ■ ■ ■, xn), y = (j/i, ■ ■ ■, yn) are in A then
q(xy) = maxpj(xjyj) < max |p-|p-(a:-)pj{yj) < Cq(x)q(y)
where C = max|pj|, so that \q\ < C, as required. If pj are sm.
then \pj\ < 1, whence \q\ < C = max|pj| ^ 1, so q is sm.
Finally, if ey is the unity of Aj (j = 1,---, n) then e = (ei, • • •, en)
is unity of A and q(e) = maxpj(ej) = 1 (since each Pj(ej) = l).
3.4.15. PROPOSITION. Let p be a quarter-norm on an
algebra A and I be a bi-ideal of A. Write A$ = A/I. Define p^
on A^ by
p*(x + I) = inf{p(z + a) : a e I}.
Then:
(i) p# is a quarter-norm on A*.
(ii) If p is a p -seminorm so is p*.
(iii) If p is a.sm. then so is p* with \p*\ ^ |p|; hence p^ is
sm. whenever p is sm. .

138
Some Types of Topological Algebras
(iv) If A is unital so is A*; if I 7^ A [bar denoting closure)
and p is normalized then so is p*.
PROOF, (i) This follows from 3.1.22.
(ii) It is enough to check modulus homogenity condition for
p*. Now
p*{Xx + 1) = inf{p(Ax + a) : a G 1} < |A|'p#(* + I), (l)
If A 7^ 0 then
p*(x + 1) = p#(\-1\x + I) < \X-1\pp*(Xx + I). (2)
From (1),(2) we get the homogenity condition (when A^ 0).
When A = 0 the homogenity condition holds trivially since both
sides are 0.
(iii) This follows from the inequality:
p*((x + I)(y + I)) < p((x + a)(y + b)) ^ \p\p(x + a)p(y + b)(a,b G I).
(iv) Let A have unity e. Then e# = e + I is the unity of
A*. Assume now that p is normalized, so that p(e) = 1 and p
is sm. (consequently also p^ is sm.) Since / 7^ A, e ¢ / and so
p#(e#) 7^ 0. Since p#(e#) = p#(e# ■ e#) ^ p#(e#)2 we conclude
that p#(e#) ^ 1. On the other hand, p#(e#) ^ p(e) = 1. Thus,
p#(e#) = 1, as required.
3.4.16. COROLLARY. If (A,P) is a p -seminormed algebra
then the quotient A& is p -normed iff I is closed. In particular,
A/kerp is a p-normed algebra with
p*(x*) = p{x) (x G A, x* = x + ker p).
PROOF. The first statement follows from 3.1.22(iv). For the
second it suffices to note that if a G kerp then by 3.1.2(iii), p(x +
a) = p(x).
3.4.17. PROPOSITION. The completion of a p-normed
algebra (A,p) is a p-Banach algebra (A,p).
PROOF. We may assume that p is sm. The construction of
A and p is exactly as in 3.3.17. That p is now a ^-seminorm is

§ 4. p -Seminormed Algebras; p -Banach Algebras 139
an easy consequence of the definition of p : if xn G A, x G A and
xn —> £ then p(Ax) = lim p(Ax„) = lim lAKpfz^) = lAKpfx).
n—»00 n—»00
3.4.18. PROPOSITION. lei A = (A,p) 6e a real p-normed
algebra which has a complex structure and let the resulting complex
algebra A^c' be denoted by A. Set for x G A(= A),
p{x) = sup p{el6x).
O<0<2?r
Then (A,p) is a complex p-seminormed algebra with p ^
Pi \P\ ^ \p\- In particular, if p is sm. so is p.
PROOF. First note that p{x) < p(x), since e,ex = x when
9 — 0. Further we have
p(x + y) = supp(e'"'(a: + j/)) < sup(p(e'"'a:) + p(e'"'y))
^ supp(e'"z) + supp(e'"j/) = p(x) +p(y).
e e
Also, for a. G R,
p(e'ax) = supp^^x) = p(x).
e
For A G C we write A = |A|etQr and we get
p{\X\eiax) = supp(|A|e,V';r) = |A|',supp(e,Va;r)
e e
= \X\pp(eiax) = |A|^(x).
Finally,
p(xy) = supp(etexy) < sup |p|p(e'"x)p(j/)
0 0
< \p\P(x)p(y) ** \p\P(x)p(y),
whence \p\ < |p|.

140
Some Types of Topological Algebras
§ 5. Bounded Linear Transformations on
p-seminormed LS's
3.5.1. DEFINITION. Let X = (X,p), X* = {X*,p*) be
respy. a ^-seminormed and a p* -seminormed LS's (with p ^ 0)
and T : X —> X* a linear transformation. T is said to be n.
bounded (= norm bounded) or sometimes just bounded if there is
a constant C > 0 such that
p*(Tx) <Cp(x)V for allxeX (*)
The smallest C satisfying (*) is denoted by \T\ and is called the
bound of T; we have
p*{Tx) < |T|p(x)T (x G X). (**)
We use ||T|| for |T| when the LS's X,X* are ^-normed and
p* -normed.
3.5.2. PROPOSITION. For a linear transformation T, write
p*
C\ = sup{p*(Tx)/p(x) f :p{x) > 0},
C2 = snp{p*(Tx) :p(x) ^ 1},
C3 = sup{p*(Tx):p{x) = l}.
If T is bounded, then \T\ = C\ = C2 = C3.
PROOF. We see from (**) of 3.5.1 that p(x) = 0 => p*(Tx) =
0. It follows that p*{Tx) < Cip{x)~e~ for all x, whence \T\ < Ci.
On the other hand, (**) shows that if p(x) > 0, p*(Tx)/p(x) e <
\T\, so that Ci ^ \T\. Thus |T| = Cx.
It is clear from the definitions that C3 < C\. To prove the
reverse inequality assume that p[x) > 0, y = x/p{x) e. Then
P{y) = 1, P*{Tx)/p(x)£p~ = p*{Ty), proving d < C3, so that
Ci = C3.
Finally, assume that 0 < p[x) ^ 1. Then
p*{Tx) <c p*{Tx)/p{x)^ = p*(Ty) $ C3,

§5. Bounded Linear Transformations on p -seminormed LS's 141
where y = x/p{x)~e, p(y) = 1. The inequality p*(Tx) < C$ is
also satisfied when p(x) = 0 (since then p*(Tx) = 0), whence
C*2 < C3. Since trivially C3 < C2 we conclude that C2 = C3,
completing the proof of the proposition.
3.5.3. COROLLARY, yl linear transformation T is bounded
iff C3 < 00.
PROOF. If T is bounded then by 3.5.2, C3 = \T\ < 00.
Conversely, assume now that C3 < 00. We shall show that this
implies C2 < 00.
Suppose to the contrary C2 = 00. Then there is a sequence
(xn) G X with p(xn) < 1, p*(Txn) —> 00. If there are an infinity
of these xn, say xni (n' = 1,2,---) with 0 < p(xni) ^. 1, then
setting yni = xn,/p{xn,yp we obtain p(yn>) = 1 and
£__
P*{Tyn,) = p*(Txn,)/p(xn>) " > p*{Txn>) -* 00,
which contradicts that C3 < 00. On the other hand,if there are
only a finite number of xn with 0 < p(xn) ^ 1 then we may
assume after omitting these that p(xn) = 0 for all n, p*[Txn) —>
00. Since p 7^ 0 there is a y with p(y) — 1. Set zn = xn — y;
then by 3.1.2 (iii), p(zn) = 1. Since
\p*(Txn) - p*{Ty)\ < p*(Txn - Ty) = p*(T2r„)
and p"(Txn) —> 00, we get p*(Tz„) —> 00, again contradicting
that C3 < 00. Therefore C2 < 00. It follows from this that for
all x with 0 < p[x) we have
p*(Tx) < C2p{x) f (***).
1
If p(x) = 0 then p(n ^4 x) = 0, and since C2 < 00 we get
np*(Tx)^C2 (n =1,2,---).
This clearly implies that p*(Tx) = 0, so that (* * *) holds for all
x and T is bounded.
3.5.4. COROLLARY. For T to be bounded it is sufficient that
there exist constants C, D > 0 such that
p(x)^C=>p*(Tx)^D. (*)

142
Some Types of Topological Algebras
PROOF. Suppose that (*) holds. If p(x) = 1, then p{C"x) =
C, so that by (*), we have
p*(TCLpx) < D, whence p*(Tx) < C~^D.
Hence C3 (in the notation of 3.5.2) < 00, whence by 3.5.3, T is
bounded.
3.5.5. PROPOSITION. Let T : X -> X* be a linear
transformation, where X — {X,p) is a p-seminormed LS and
X* = [X*,p") a p* -seminormed LS. Then the following
statements are equivalent.
(i) T is n. bounded.
(ii) T is continuous.
(iii) T is t. bounded.
PROOF. If T is (n.) bounded then we have
p*{Txa - Tx) = p*{T(xa - x)) < \T\p^(xa - x)
whence T is continuous. Conversely, if T is continuous; given
e > 0 there is a 8 > 0 such that p*(Tx) < e whenever p(x) ^ 6.
Hence, by 3.5.4, T is (n.) bounded. Thus (i) <-> (ii). Since
(X,p) is first countable the equivalence of (ii) and (iii) follows
from 2.1.29. (We can also easily deduce directly the equivalence
of (i) and (iii), using 3.2.13).
3.5.6. COROLLARY. Let p,p* be respectively a p-seminorm
and a p" -seminorm on the same LS X. Then p ~ p* iff there
are contants C,C* > 0 such that
p* < Cp p , p ^ C*p* .

§5. Bounded Linear Transformations on p -seminormed LS's 143
PROOF. The above conditions clearly express the bounded-
ness of theidentitly maps: (X,p) —+ {X,p*) and (X,p*) —> (X,p),
whence by 3.5.5 they are continuous and so p ~ p*.
3.5.7. PROPOSITION. Let X= (X,p) t be a p-seminormed
LS. Then the bounded linear operators T on X form a unital p -
seminormed algebra B = B(X) with the bound \T\ of T as the
p -seminorm. Moreover, \-\ is sm. and satisfies \I\ = 1 (7 being
the identity operator). Finally, if p is a p-norm or a complete
p -norm then so is | ■ |.
PROOF. That |-| is a p -seminorm follows from its definition
and the ^-seminorm properties of p. To prove that | • | is sm.
We observe that
p(TiT2x) < \Ti\p(T2x) < |Ti||T2|p(x),
so that \TiT2 < |ri||r2|. Trivially, |7| = 1. Further, if p is a
p-norm then \T\ = 0 => p(Tx) = 0 => Tx = 0 (for all x G X) =>
T = 0, so that | • | is also a ^-norm.
Assume now that p is also complete. If (T„) in B is a C-
sequence then we have
\\Tn - Tm\\ < e (n,m^ n). (1)
This clearly implies
\\Tnx - Tmz|| < e\\x\\ (n,m^N). (2)
If follows from (2) that (Tnx) is a C -sequence in A and let us
write Tx = \imTnx (the limit existing since p is complete). Then
T is linear. For, if Ty = UmTny then Tx+Ty = \\m(Tnx+Tny) =
\iu\Tn[x+y) = T[x+y). By uniqueness of limit property [X being
Hausdorff) we get T(x + y) = Tx + Ty. Similarly, TXx = XTx.
Finally, by allowing, m —> oo in (l) we get ||T„-T|| ^ e (n ^ N),
so that limT„ = T, and B is complete.
3.5.8. PROPOSTION. Let A= (A,p) be a p -seminormed
We always assume p^O.

144
Some Types of Topological Algebras
algebra and B = B(A) (see 3.5.7) the p -seminormed algebra of
all bounded linear operators on the underlying seminormed LS of
A. Then the left regular representation I : x —> lx is a continuous
homomorphism of A into B. Also, \lx\ < |p|p(x); in particular,
if p is sm., then \lx\ ^ p(z).
The map I is injective iff the annihilator ideal Ai = {0}. In
the unital case we have p(x) < Kz|p(e)> whence I is a t.
isomorphism. Besides, if p(e) = 1 then p[x) = \lx\.
PROOF. That / is a homomorphism is easily verifiable and in
fact a standard result in algebra. Since
p(izy) = p(xy) $ |p|p(*)p(y)
we obtain
\lx\ < sup{p(zj/) : p(y) < 1} ^ |p|p(z), i-e.
1^1 ^ \p\p(x) (1)
Hence lx G B and I is bounded and so continuous (by 3.5.5). If
p is sm. then \p\ < 1, so that (l) gives
|/1|<p(x). (2)
The injectiveness (or faithfulness) conlusions are clear. Further,
in the unital case we have
p(x)=p(lxe)^\lx\p(e), (3)
so that l"1 is bounded and so continuous. Thus in this case I is
a t. isomorphism. Finally, if p(e) = 1, (3) gives p(x) ^ \lx\ which
together with (2) yields \lx\ = p(x).
3.5.9. THEOREM (Bonsall-Duncan t). Let A= (A,p) be a
p -seminormed algebra and S a bounded {multiplicative) subsemi-
group of A. Then we can find a sm. p -seminorm q such that
(i) q ~ p,
They confine themselves [4, p.18] to the case where p is a sm.norm.

§ 5. Bounded Linear Transformations on p -seminormed LS's 145
(ii) q(s) ^ 1 for all s in S. Further, if p is a p -norm so
is q and if A has unity e then q(e) = 1 [so that q is
normalized).
PROOF. By replacing A by its unitization A\ (if
necessary) we may assume that A has a unity e. Write 51 = S\J{e}.
Clearly, 51 is a subsemigroup of A which is also bounded, say,
p(s) ^ M for all s G 51. Define
pi(x) = sup{p(sx) : s G 51}.
Since
p(sx) ^ \p\p{s)p(x) < \p\Mp(x),
it follows that pi(x) < oo and further
Pl^Cp (C=\p\M). (1)
It is easy to see that pi is a ^-seminorm. Since e G S we obtain:
p(x)=p(ex)^pi(x). (2)
From (1), (2) we obtain:
Pi ~ P- (3)
Further
pi(xy) = sup{p(sxy) : s G 51} < \p\ sup{p(sx) : s G S }p(y)
< \p\pi(x)p(y) < \p\pi{x)pi{y) (using (2)).
Setting
q{x) = \lx\l = sup{pi(xy) : p1(y) < 1}
we find that
q(x) < \pi\pi{x). (4)
Since
q{xy) = \izy\i = \lxly\i < Kz|iK«|i = q{x)q{y),
q is sm. If we write a = e/p1'p{e), then pi(a) = 1 and g(x) ^
pi(xa) = pi{x)/pi{e), so that
pl(x)^Pl(e)q(x). (5)

146
Some Types of Topological Algebras
From (3), (4), (5) we conclude that q ~ pi ~ p, which proves (i).
To prove (ii), we observe that if s G 51,
pi(sx) = sup{p(isx) : t G 51} < pi (a;) (6)
(since ts G 51). It follows that
q(s) = sup{pi(sj/) : pi(j/) < 1) ^ sup{pi(j/) : pi(j/) < 1}
(using (6))
< 1. (7)
In particular, q{e) ^ 1. But q{e) = q(e2) < ?(e)2, whence q{e) >
1. Therefore q[e) = 1, as desired.
Finally, by (3), pi is faithful when p is faithful and then, by
virtue of 3.5.7, q = \lx\i is a ^-norm.
3.5.10. Corollary
(i) v(x) ~ vp(x) = inf{g(x) : q is a sm. p -seminorm with q ~
(ii) If A has unity e then v{x) = m.i{q{x) : q is a sm. p -
seminorm with q ~ p, q[e) = 1}.
PROOF, (i) Since v(x) < q(x) for all q we get
v{x) ^ inf q(x) = t (say).
Suppose that v{x) < t, so that v(x)/t < 1. Since ^j —+ 1 (as
moo we can choose a sufficiently large n that we have
^M<-?_<i. (i)
i n+1 K '
Set j/ = ((n + l)/ni)?x.
Then q(y) = ((n + l)/ni)g(x), so that we have
mfg(y)=^mfg(«)=^±!t=^±!. (2)
9 ni 9 ni n
Now, by 3.4.5,
n +1,. n + 1 i/(z) n + 1 n +1 .
i/(y = -—v{x) = —— .-^ < —— • 1 = —— = mf q{y)
nt n t n n i
(3)

§5. Bounded Linear Transformations on p -seminormed LS's 147
where in the last step we have used (2).
Again,
n + 1 n + 1 v{x) n + 1 n .
^y) = ——u(x) = — L^< . =1 (using (1)).
nt n t n n + 1
Since v(y) = limp(j/")" < 1 we get p(yn) < 1 for all n ^ No,
n
whence it folows that the semigroup 5 = {yn : n = 1,2,---}
is p-bounded. By 3.5.9, there is a sm. ^-seminorm qo ~ p and
?o(j/) < 1 < ^-^) contradicting (2). Therefore we must have
i^(x) = i, proving (i).
(ii) This can be proved in exactly the same way as (i).
3.5.11. PROPOSITION, (a) Let X be a p-seminormed LS
and X* a p* -Banach space. Let Xq be a dense subspace and
T : Xo —> X" a continuous linear tranformation. Then T can be
uniquely extended to a continuous linear transformation
f : X -> X* with \f\ = \T\.
(b) Suppose that A is a p -seminormed algebra, Aq a dense
subalgebra of A and A* a p -Banach algebra. Then every
continuous homomorphism ip : Aq —+ B can be uniquely extended to
a continuous homomorphism <p : A —> B with \<p\ = \<p\.
PROOF, (a) Let X = (X,p), X* = (X*,p*). Suppose that
x E X, xn —+ x (xn E X0). Then
£__
P*(Txn - Txm) ^ \T\p(xn - xm) p -+ 0,
so by completeness of Y, Txn —> x*. Define Tx = x*. By using
the uniqueness of limit property of convergent sequences in X*
(which is Hausdorff) it is easy to verify that T is well-defined (i.e.
independent of the particular choice of the sequence xn —> x) and
linear. Further we have
p*(fx) = limp*{Txn) < |T|p(x„)V < |T|p(x)V,
so that \T\ < \T\. On the other hand, T being an extension of T,
\T\ < \T\. Thus \T\ = \T\. The continuity if T is an immediate

148
Some Types of Topological Algebras
consequence of its boundedness. Finally, the uniqueness of T is
also clear.
(b) Extend <p (as a linear transformation) to p on A as in
(a); then <p is continuous linear. We have to show that <p is a
homomorphism. If x,y G A, (z„), (j/„) G A0, xn —> x, yn —> y
then £>(xj/) = lim<p(xnyn) = lim^(x„)^(j/„) = ^(x)^(j/),
proving <p is a homomorphism and completing the proof.
§ 6. Topological Algebras with Inverses
3.6.1. DEFINITION. Let A be an algebra (over K ) and a G
A. Denote by 1° the map x>—>aox [x G A). Similarly r° is the
map x i—> x o a.
3.6.2. LEMMA. lei A be a WTA. Then /°,r° are
continuous. If a is q. invertible then l°a,T^ are homeomorphisms of A.
PROOF. Since /a(z) = a + x + lax(la : x i—> ax) the continuity
of /° follows from the continuity of addition and the continuity of
Ia- The continuity of r° follows similarly.
If a is q. invertible with a' as q.i, so that a'oa = o = aoa',
then
Therefore, /° is invertible with its inverse (/^)-1 = /°/ continuous.
It follows that /° is a homeomorphism. Similarly, since (r°)~ =
r°(, r° is a homeomorphism.
3.6.3. LEMMA. If A is a unital WTA and a G A is invertible
then la,ra are linear homeomorphisms.
PROOF. Here we have l~l = la-i,r~l = ra-\ and hence the
result.
3.6.4. DEFINITION. A TA A is called a C algebra or a
continuous algebra if the map x t—> x' of Gq —> Gq is continuous,
where G9 denotes the group of q. invertible elements of A.
3.6.5. PROPOSITION, (a) A TA A is a C algebra iff Gq is
a TG.

§6. Topological Algebras with Inverses 149
(b) A unital TA A is a C algebra iff its group G,- of invertible
elements is a TG.
(c) In a unital C algebra the groups Gq,G{ are TG's and the
map
t"1 : x eGq -> e + z G G;
is a t. isomorphism.
PROOF, (a) Since A is a TA the map (x,y) h-> xy is always
continuous. It follows that Gq is a TG iff the map x <—* x' is
continuous, i.e. iff A is a C algebra.
(b) By 1.1.20, y = e+x e G,- iff x e Gq and j/_1 = (e+x)-1 =
e + x'. It follows that G,- is a TG iff y i—> j/_1 is continuous iff
x i—> x' is continuous iff G9 is a TG.
(c) The first half of the statement follows from (a),(b). Also,
r_1 : x i—> e + x is an isomorphism of Gq onto G,-. Again, J", J"-1,
being translations in A, are continuous. Hence the second half of
the statement.
3.6.6. LEMMA. Every quarter-normed algebra (A,p) with an
a.sm. p is a C algebra. In particular, any p -seminormed algebra
is a C algebra.
PROOF. The first statement is an immediate consequence of
the inequality (*) of 3.3.22. The second statement folllows from
the first since, by 3.4.3, the ^-seminorm p of a ^-seminormed
algebra is always a.sm..
3.6.7. DEFINITION. A TA A is called a Q algebra or
algebra with q .inverses if there is an open neighbourhood U(0) of 0
consisting of q. invertible elements. A unital TA A is called an I
algebra or algebra with inverses if there is an open neighbourhood
U[e) of e consisting of invertible elements.
3.6.8. LEMMA (Michael). Let A be a TA. Then:
(i) If S is a balanced subset of A comprising q. invertible
elements then r(s) < 1 for every s in S, where r{s) denotes
the spectral radius of s.
(ii) Write 6 = {x e A : r(x) ^ 1}. Then A is a Q algebra iff
int e = 6° ^ 0.

150
Some Types of Topological Algebras
PROOF, (i) Suppose that r(s) > 1 for some s G S. Then
there is a A G a(s) with |A| > 1. By 1.7.8, s0 = -\~1s is not
q. invertible. But since |A-1| < l,s G 5 and 5 is balanced we
must have so G S, so that so is q. invertible. This contradiction
proves (i).
(ii) First assume that A is a Q algebra with [/(0) a nucleus
comprising q. invertible elements. Choose a balanced open
nucleus V C [/(0). By(i), V C 6 so that 6° ^ 0. Conversely,
assume that in A we have 6° ^ 0. Then U = \&° ^ 0 and
open. If x G U then 2x G 6, so that by 1.8.11(ii), r(x) < | < 1.
It follows that -1 ^ ^'(x) whence, by 1.7.8, x is g. invertible.
Thus every x <E U is g. invertible and A is a Q algebra.
3.6.9. LEMMA. If A is a Q (respy. I) algebra then each q.
invertible (respy. invertible) element x has an open neighbourhood.
U°(x) = /£(17(0)) ( respy. U(x) = lx(U(e))
comprising q. invertible (respy. invertible) elements. Hence the set Gq
(respy. G{) of q. invertible (respy. invertible) elements of a Q
(respy. I) algebra A is open.
PROOF. Since Gq (respy. G,-) is a group we have U°(x) =
x o [/(0) C G,- (respy. U(x) = xU(e) C G,-) Further, U°(x)
(respy. U(x)) is open since /° (respy. lx ) is, by 3.6.2 (respy.3.6.3) a
homeomorphism.
3.6.10. PROPOSITION. For a TA (respy.unital TA) A to
be a Q (respy. I) algebra it is necessary and sufficient that Gq
(respy. G,) is open. In particular, a TA which is a division algebra
is an I algebra iff it is Hausdorff.
PROOF. The "necessity" follows from 3.6.9 and the
"sufficiency" is immediate from the definition of a Q (respy. I) algebra
(since 0 G Gq,e G G, ). Since in a division algebra G,- = A\{0}
the assertion concerning TA's which are division algebras is clear.
3.6.11. COROLLARY. Let A be a TA under two topologies
T\,T<i- If r2 is finer than t\ and (A, ti) is a Q algebra (respy. I

§6. Topological Algebras with Inverses 151
algebra) then (A,T2) is also a Q algebra [respy. I algebra)
Proof. Clear ( Gq t\ -open => Gq r2 -open).
3.6.12. LEMMA. Let A be a unital TA with unity e. If there
is an open neighbourhood U(0) of 0 consisting of q. invertible
elements then U(e) > = U(0) + e is an open neighbourhood of e
consisting of invertible elements. Conversely, if V{e) is an open
neighbourhood of e consisting of invertible elements then V(0) =
V(e) — e is an open neighbourhood of 0 consisting of q. invertible
elements.
PROOF. By 1.1.20, x G A is q. invertible iff x + e is
invertible. Moreover, the translations n-n + e,jnj-e of A are
homeomorphisms. The statements of the lemma are now clear.
3.6.13. COROLLARY. A unital algebra A is an I algebra iff
it is a Q algebra.
3.6.14. LEMMA. A Q algebra A is commutative iff Gq is
commutative. Also, an I algebra A is commutative iff Gi is
commutative.
PROOF. Suffices to prove the "if parts. Suppose that G
is commutative and x, y G A. Since -,^ —> 0 and Gq is open
i
n' n
there are integers ni,n2 such that —, — G G„. Then — <-> -^-,
whence x <-> y, A is commutative. Again, if G,- is commutative
so is Gq since Gq isomorphic to G,- (by 1.1.20). This completes
the proof.
3.6.15. Remark. The above lemma can fail if the
algebra is not a Q algebra. For example, consider the algebra
P* = P*(X,Y) of all polynomials P = P{X,Y) over K in two
non-commuting variables X, Y. This algebra can be normed by:
||P|| = the maximum of the absolute values of the coefficients of
the monomials of P. It is easy to check this gives a norm that
is sm.: ||PQ|| ^ ||P||||Q||. It is clear that the group of invertible
elements G,- = K\{0} which is commutative, but P* is not
commutative since XY ^ YX. Note here that G,- is not open in P*
t i.e., U(e) = {x + e :x eU(0)}.

152
Some Types of Topological Algebras
so that P* is not a Q algebra.
3.6.16. THEOREM (Arens* -Banach). A (F) algebra A =
(A, | • |) is a C algebra iff its group Gq is a Gg > set of A.
PROOF (cf.[31,p.4]). First assume that A is a C algebra, so
that the map x i—> x' is continuous on Gq. (*)
For n = 1, 2, • • • define
Wn = {x G Gq : 3 an r) = r)(x,n) > 0 such that if y,z €
Gq, \y — x\ < rj,\z — x\ < r) then \y' — z'\ < -}, where the bar in
Gq denotes closure. It is easy to see, using continuity of the map
(*), that Wn is open in Gq. Also it is clear that Wn D Gq, so
that W = f]nWn D Gq. Thus, W is a Gs in G and G^ as a
close subset of the metric space A is a Gs of A. It follows that
W is a Gs of A. We shall complete the proof of the "only if"
part by showing that Gq = W.
Suppose that x G W C Gq,xn G Gq and xn —> a;. Then
I x'n — x'm I "^ *-* as n >m "~* °° > so ^ha^ x'n "^ J/ e ^ (since A is
complete). It follows that x o j/ = (lim x„) o (lim x^) = lim(x„ o
x'n) = 0. Similarly yox = 0. So y = x',x £ Gq, proves W = Gq.
It remains to prove the 'if part and assume therefore that Gq
is a Gs of A. Since A is a complete metric space, Gq is
topological^ complete and consequently there is in Gq an equivalent
metric under which it is complete. By applying proposition 2.1.8
to Gq (with this complete metric) we conclude that inversion is
continuous in Gq, completing the proof.
3.6.17. COROLLARY. A unital (F) algebra A is a C algebra
iff the group G,- of its invertible elements is a Gs of A.
PROOF. It suffices to observe that the map x >—> e + x is a
homeomorphism of A which carries Gq onto G,-.
3.6.18. COROLLARY. An (F) algebra which is a Q algebra
is also a C algebra.
' Arens proved (essentially) only the "if part of the theorem and that
under the additional assumption that Gq is seperable.
TT i.e. Gq is the intersection of a countable family of open sets of A.

§6. Topological Algebras with Inverses 153
PROOF. Since an open set is a Gg this follows from 3.6.16.
3.6.19. Remark. There exist Q algebras which are not C
algebras. For instance, the algebra 9ft of germs of meromorphic
functions of one complex variable in the neighbourhoods of 0 in
C is a commutative division algebra. This can be topologized so
as to make it into a complete Hausdorff locally convex TA. It is
known that 9ft is not a C algebra (see [29, p.136]). On the other
hand since it is a Hausdorff division algebra, by 3.6.10 it is a Q
algebra.
There are also C algebras which are not Q algebra. For
example, if Bi is the closed unit ball in K denote by P the
algebra of the restriction of polynomials in one variable to Bi
with coefficients from K. P is a normed algebra under the sup
norm. By 3.6.6, P is a C algebra. On the other hand since
the invertible elements in P are just the non-zero constants, P is
clearly not an I (and hence also not a Q) algebra.
3.6.20. DEFINITION. A Q (respy. I) algebra is called a
CQ (respy. CI) algebra or algebra with continuous q. inverses (re-
spy, continuous inverses) if the map
im' (respy. ih> x~1) from U(0) (respy. U(0)) into A is
continuous, where U(0) (respy. U(e)) as in 3.6.7.
3.6.21. LEMMA. An algebra A is CQ (respy. CI) iff it is
a C algebra and a Q (respy. I) algebra.
PROOF. The "if part follows from 3.6.5((a),(b)) For the "only
if part, in view of 3.6.13, it is enough to prove that a CQ algebra
is a C algebra. Let A be a CQ algebra and U(0) as in 3.6.7. If
y G U(0) then xoy G U°(x) (as defined in 3.6.9) and (xoy)' = y'o
x'. The map xoy \—> (xoy)' is continuous since it is the composite
of the continuous maps ioj/h y = /°/(x o j/), y i—> y\ y' \—> y' o x'.
This proves that A is a C algebra.
3.6.22. LEMMA. Every CI algebra is a CQ algebra.
Conversely, every unital CQ algebra is a CI algebra.
PROOF. In view of 3.6.21, 3.6.13 it is enough to prove that
the map x i—> x' (x G U(0)) is continuous iff the map (e + x) i—>

154
Some Types of Topological Algebras
(e + x) 1 (x G U(0)) is continuous. But this is immediate since
(e + z)_1 = e + x'.
3.6.23. PROPOSITION, (a) A sm.(F) algebra is a CQ
algebra.
(b) Every p -Banach algebra is a CQ algebra and a unital
p -Banach algebra a CI algebra.
PROOF, (a) By virtue of 3.3.18, a sm. (F) algebra is a Q
algebra and hence by 3.6.18, 3.6.21 it is a CQ algebra.
(b) By the remark in 3.4.4 we may assume that the norm of the
^-Banach is sm. and the first half of the required result follows
from (a). The second half follows from the first in view of 3.6.13,
3.6.21.
3.6.24. LEMMA. Let B be a q. inverse-closed' subalgebra
(respy. inverse-closed subunital'' algebra) A. If A is a Q (re-
spy. I) algebra then so is B; if A is CQ (respy. CI) then so is
B.
PROOF. If Gq(B),Gq(A) denote the groups of q. invertible
elements of B, A respy. then we have clearly the relation Gq(B) =
Bf)Gq(A) Hence the statements regarding Q and CQ properties
of B. The I and CI properties of B follow similarly from the
corresponding relation G{(B) = Bf]G{(A).
3.6.25. PROPOSITION. Let A=(A,p) beasm.(F) algebra
which is a Q algebra. If B is a closed subalgebra of A then B
is also a Q algebra.
PROOF. Since A is a Q algebra we can choose an open
neighbourhood U(0) of 0 in A such that x G U(0) => x is q.
invertible and p(x) < 1. Suppose that x G Uo = Bf]U(0). By
oo
3.3.18, x' exists and is given by x' = Y^( —l)"x". Since xn G B
n=l
and B is a closed subspace, x' G B, so that x' G Uo- Thus, Uo
is an open
' i.e. if x G B has q .i. x' (respy. inverse x l) in A then x'
(respy. x~l ) e B.
'' For definition see 1.1.6.

§ 6. Topological Algebras with Inverses 155
neighbourhood of 0 in B, comprising q. invertible elements and
B is a Q algebra.
3.6.26. LEMMA. Let A, A* be TA's and <p : A —> A* a
continuous open epimorphism. Then A* is a Q [respy. CQ) algebra
provided A is a Q (respy. CQ) algebra.
PROOF. If A is a Q algebra then Gq is open in A and
consequently G* = <p(Gq) is an open neighbourhood of 0 in A*.
By 1.1.24, every element y G G* is q. invertible with y' G G*.
Hence A* is a Q algebra.
Next suppose that A is a CQ algebra. Then as above
G* = <p(Gq) is an neighbourhood of 0 in A* . For j/o £ G*,
choose an open neighbourhood V of y'0 with V' C G*. Set
U' = (p~1(V1)f]Gq. Then U' is an open set, so that U = {x G
A : x' G U'} is an open neighbourhood of x0, where (p(x0) — j/o-
It follows that V = <p(U) is an open neighbourhood of y such
that q*(V) C V', where q* is the map y h-> j/' (j/ e Gq(A*)).
This proves the continuity of q# at j/q an arbitary point of G*,
whence A* is a CQ algebra.
3.6.27. COROLLARY. In a Q (respy. CQ) algebra A, if I
is a bi-ideal the quotient algebra A& = A/I is a Q [respy. CQ)
algebra.
PROOF. This follows from 3.6.26 since the canonical map
7r : A —> A^ is an open continuous epimorphism.
3.6.28. PROPOSITION. Let A be a TA and Ai its
unitization. Then A\ is a Q [respy. CQ) algebra iff A is a Q
(respy. CQ) algebra.
PROOF. First assume that A is a Q algebra and U an open
neighbourhood of 0 comprising q. invertible elements. Choose a
balanced open neighbourhood V of 0 with V +V C U. Then, for
0 ^ |//| < 1, we have
(1 + n)V C V + nV C V + V C U.
Choose fj, such that 0 < |//| < \. Then |1 +//| > 1 - |//| > |.
Writing A = 1(1+//)-1 we get |A| < 1, so that for x G V,Xx G V.

156
Some Types of Topological Algebras
Since
y = x(l + //)-1 = 2Xx e 2XV C 17
it follows that y is g. invertible. Therefore, e\ + j/ is invertible
and hence also
(l + //)(ei + j/) = (l+//)ei+x.
is invertible. It follows that
Vi = {(l+/i)ci+a::0< |//| < 2'iey}
is an open neighbourhood of e\ comprising invertible elements,
whence A\ is an I algebra and so also a Q algebra.
Conversely, assume that A\ is an I algebra; then it is a Q
algebra and so, by 3.6.10, the group G* of q. invertible elements
of A\ is open. If Gq denotes the group of q. invertible elements
it is clear (see 1.1.21) that Gq = Af]Gq, so that Gq is open in
A. By 3.6.10, A is a Q algebra.
It remains to prove that A\ is a CQ algebra iff A is a CQ
algebra. But for this, by 3.6.21 and the result already proved it
is enough to show that A\ is a C algebra iff A is a C algebra.
Now by virtue of 1.1.20, j/i = Ae + x (A e K,x G A) in Ai is
invertible (i.e. j/i £ G\) iff A ^ 0 and A_1a; £ Gq, and then we
have the relation
j/f = A~~1(ei + (A_1x)'), where dash denotes q.i.. Again, if
x G Gq then e\ + x G Gj and (ei + x)"1 = e± + x'. It follows
from these relations that the map j/i i—> j/J" in Gj is continuous,
which completes the proof of the proposition.
3.6.29. PROPOSITION. The complexification A of a
commutative Q algebra A is a Q algebra.
PROOF. Let U be a nucleus (= a neighbourhood of 0)
comprising q. invertible elements. Then by 2.1.17(iii) (applied twice)
and 2.2.14(vi) we can choose nuclei V, W such that
V +V +V +V CU, W2 CV, W CV.
If x, y in W then we have
u = 2x + x2 +y2 eW +W +W2 +W2 CV +V +V +V CU

§6. Topological Algebras with Inverses 157
so that u is q. invertible in A and hence also in A. But then
since
(x + iy) o (x — iy) = u, x + iy is also q. invertible in A.
It follows that W = W + iW is a nucleus of A, comprising q.
invertible elements, so that A is a Q algebra.
3.6.30. COROLLARY. The complexification A of a
commutative real I algebra A is an I algebra.
PROOF. Since, by 3.6.13, a unital algebra is an I algebra iff
it is a Q algebra, the corollary follows from the proposition.
3.6.31. PROPOSITION. The complexification A of a
commutative real CI algebra A is a CI algebra.
PROOF. In view of 3.6.21, 3.6.30 it is enough to prove that the
map (*) z i—> z"1 of G,- onto itself is continuous where G,- is the
group of invertible elements of A. Suppose that za,z G G,,za —>
z,za - xa +iya,z = x + iy, where xa,ya,x,y G A. By 1.6.7,
z~^ = xa — iya and z = x — iy lie in G,- and hence the products
a„ = zaz~^ — x\ + y2,a = zz = x2 + y2 also lie in G,-. But
aa,a G A, whence by 1.6.10, aa,a G G,-. It follows that
*al = za(x2a + yl)"1 = zaa~l -> za~l = z(x2 + y2)"1 = z~l,
proving the continuity of the map (*).
3.6.32. COROLLARY. The complexification A of a real
commutative CQ algebra A is a CQ algebra.
PROOF. By 3.6.28, the unitization A1 of A is a CQ, and
hence by 3.6.22, a CI algebra. By 3.6.31, A"i = (A)i is a CI
algebra and hence CQ. So, by 3.6.28, A is CQ.
3.6.33. WILLIAMSON'S ALGEBRA. The field C(X) of
rational functions over C can be embedded in the algebra M of
example (V) of 3.3.14, by identifying a rational function P(X)/Q(X)
( P, Q polynomials over C ) with the almost everywhere defined
function P{t)/Q{t) in M. The topology thereby inherited from
M makes C[X) into a Hausdorff topological field (more precisely,
a commuttive Hausdorff division algebra in which inversion is
continuous). In particular, C(X) is a CI algebra as also a pre- (F)

158
Some Types of Topological Algebras
algebra. This method of topologizing C(X) is due to Williamson
[ 14' ,p.73l]; we call C(X) with this topology as Williamson's
algebra and denote it by W.
§ 7. Topological Zero Divisors
3.7.1. DEFINITION. Let A be an algebra. Write Glq (re-
spy. Gq) = the set of l.q. invertible (respy. r.q. invertible elements)
of A. Then Gq = Glqf]Grq. Write
Slq = A\Glq, Srq = A\Gq, Sq = Slq\JSrq.
The elements of Slq, Sq, Sq are called respectively l.q. singular,
r.q.singular, and q. singular. When A is unital, we have also G\
(respy. G[) = the set of 1. invertible (respy. r. invertible) elements
of A. Then G, = G\f\Gr.
Now write
Sl = A\G\, Sr = A\Gr{,Sbi = Slf]Sr, S = Sl\JSr.
The elements of 5', 5r 5*', 5 are called respectively I. singular,
r. singular, bi-singular, and singular.
3.7.2. LEMMA. Let A be any algebra. Then \J~A C Gq. If
A is unital then \J~A C 5*'.
PROOF. The first inclusion follows from 1.2.24 ((a), (b)).
Let now A be unital. Then by 1.2.2.4, \f~A is a bi-ideal with
e ¢. \/A. So, no element of \/A is 1. or r. invertible, whence
y/A C Sbi.
3.7.3. DEFINITION. (Arens). Let A be a TA. An element
x in A is called a left (respy. right) topological zero divisor < , or
a l.t.z.d. (respy. r.t.z.d.) if there is a closed set F such that
' The notion t.z.d. under the name generalized zero divisor was first
introduced by Shilov for commutative Banach algebras [ 7' , p.208]; his definition
is essentially based on the condition (**) of 3.7.5 .

§ 7. Topological Zero Divisors
159
0 ¢. F and 0 £ xF respy. Fx
where bar denotes closure.
An element which is both a l.t.z.d. and a r.t.z.d. is called
a bi-topological zero divisor or a bi-t.z.d.. Obviously, when A is
commutative every l.t.z.d. or r.t.z.d. is a bi-t.z.d. The same is
true (as can be easily seen) when A is anticommutative.
We denote by 3", 3ri, 3*'' respectively the sets of l.t.z.d's,
r.t.z.d.'s and bi-t.z.d.'s of A.
3.7.4. Remark. In any TA whose topology is not indiscrete,
0 is a bi-t.z.d. For, if x G A\0, t then 0 ¢ x = F (say). But
0GOF = F0 = 0.
3.7.5. LEMMA. Let A be a TA. For an element x G A to be
a l.t.z.d. (respy. r.t.z.d.) it is necessary and sufficient that
3 a net (xa) in A with xa /» 0 but (*)
xxa (respy. xax) —> 0.
If A is first countable then (*) can be replaced by:
3 a sequence (xn) in A with xn -f+ 0, xxn (respy. xnx) —> 0.
(**)
PROOF. Suppose there is a closed set F such that 0 ¢.
F and 0 G xF (respy. Fx). Choose xa G F such that xxa
(respy. xax) —> 0, then the net (xa) clearly satisfies the condition
(*). Conversely, suppose (*) holds. S ince xa -f+ 0 there is a
subnet (xai) such that none of its subnets converge to 0. This
means that if F = {xai} then F is a closed such that 0 ¢. F,
0 G xF (respy. 0 G Fx), whence x is a l.t.z.d. (respy.r.t.z.d.).
This completes the proof of the first statement in the lemma. The
second statement is clear.
3.7.6. PROPOSITION. (a) If x e A is a l.t.z.d.
(respy. r.t.z.d.) and y G A then yx (respy. xy) is a l.t.z.d.
(respy. r.t.z.d.). In particular, if x is a bi-t.z.d. and x <-> y then xy
is a bi-t.z.d.
For an element a £ A we write a for {a}.

160
Some Types of Topological Algebras
(b) If xy is a l.t.z.d. [respy. r.t.z.d.) then x or y is a l.t.z.d.
[respy. r.t.z.d.). In particular, x2 is a l.t.z.d. [respy. r.t.z.d.) => x
is a l.t.z.d. [respy. r.t.z.d.).
PROOF, (a) It suffices to observe that xxa —> 0 (respy. xax —>
0) implies that yxxa —> 0 (respy. xaxy —> 0).
(b) Suppose that xyya —> 0, ya -/* 0. If yya —> 0 then y is
a l.t.z.d. On the other hand, if yya -/* 0, writing xa = yya we
get xa /» 0, xxa —> 0, whence x is a l.t.z.. Similarly the proof
when xy is a r.t.z.d. .
3.7.7. In aunital algebra A a formal sum X+x [X G K,x G A)
can be identified with the element Xe + x. But in the case of an
algebra without unity X + x can be treated only as a formal sum.
The notion of t.z.d. can be extended to formal sums. Thus we
call X + x a l.t.z.d. (respy. r.t.z.d.) if there is a net xa-f+x with
Xxa + xxa = [X + x)xa —> 0 (respy. xa[X + x) —> 0).
3.7.8. LEMMA. Let A be a TA and x G A. If 1 + x is a
l.t.z.d. [respy. r.t.z.d.) then x is I. [respy. r.) q. singular. If A is
unital and x is a l.t.z.d. [respy. r.t.z.d.) then x is I. [respy. r.)
singular.
PROOF. Suppose that xa -f+ 0 and xa + xxa —> 0. Then
x o xa —+ x. If x[ exists then by premultiplying by x\ on both
sides we get xa —> 0 - a contradiction. So x is l.q. singular.
Similarly, when 1 + x is r.t.z.d., x is r.q. singular.
Next let A be unital with unity e. Then if x in A is a l.t.zd.,
writing x = e + y we see that \ + y = e + y = x is a l.t.z.d. So
by above y is l.q. singular, whence x = e + y is 1. singular.
3.7.9. LEMMA. Let A be a Hausdorff TA. Then:
(i) Every l.z.d. [respy. r.z.d ' ) is a l.t.z.d. [respy. r.t.z.d.).
(ii) When A is finite-dimensional, a l.t.z.d. [respy. r.t.z.d.) is
nothing but a l.z.d. [respy. r.z.d.).
(iii) An element x in A is a l.t.z.d. [respy. r.l.z.d.) iff la
[respy. rx) is not a topological monomorphism.
' l.z.d. (respy. r.z.d) = left (respy. right) zero divisor.

§ 7. Topological Zero Divisors
161
PROOF, (i) If xy = 0 and x, y ^ 0 then F = {y} is closed
(since A is Hausdorff), 0 ¢. F and 0 e {0} = 0 = xy = xF, so
that x is a l.t.z.d. . Similarly, y is a r.t.z.d. .
(ii) Since A is finite-dimensional we may assume that it is a
normed algebra (see 3.4.6. (ii), 2.1.12). If x is a l.t.z.d., then by
3.7.13 there is a sequence (xn) with xxn —> 0, 11x„11 = 1. Since
closed unit ball of A is compact (A being homeomorphic to K"),
we can choose a subsequence (j/„) of (x„) with yn —> y. Then
xyn —> 0, xyn —> xy. So xy — 0 and a; is a l.z.d. Similarly, if x
is r.t.z.d. it is r.z.d. .
(iii) If lx is not a monomorphism then x is a l.z.d. and so by
(i) a l.t.z.d. Next suppose that lx is a monomorphism which is
not topological. Then l~l is not continuous (since lx is always
continuous) so that there is a net ya —> 0, but xa = lxlya -/* 0.
Therefore xxa = lxxa = ya —> 0. By 3.7.5, x is a l.t.z.d. This
proves the "if" part. For the "only if" part assume that x is a
l.t.z.d., so that there is a net (xa) such that
xa -f+ 0, xxa —> 0. (*)
If lx is not 1 — 1 it is not a monomorphism, and if it is 1—1, the
condition (*) implies that l~l is not continuous. Thus, lx is not
a topological monomorphism in either case, completing the proof.
(The proof of the corresponding statement concerning r.t.z.d.
is similar.)
3.7.10. Examples of t.z. divisiors
(i) In the Banach algebra C = C ([0,1] K) the function f0(t) =
t is not a zero-divisor. But it is a t.z.d. . To see this, define
/„ G C by: fn(t) = 0 if i > i and = 1 - nt if t < i.
Since 0 < /„ ^ 1 and /„(0) = 1 we have ||/„|| = 1. Also
fofn —+ 0 since ||/o/ra|| < ^- Thus /o is a t.z.d. .
(ii) Write /i = /i(K), the space of all infinite sequences x = (x„)
oo
(xn e K) with ]T| xn\ < oo; l\ is a Banach space under the
n=l
oo
norm ||i|| = ^|x„|. Denote by B = B(/x) the algebra of a
n-l
all bounded linear operators on /^ B is a Banach algebra
(under operator norm). Then l.o's S,T,U defined by
Sx= (x2,x3,---),Tx= (p,Xl,x2,---),Ux= (xlt 0,0,---)

162
Some Types of Topological Algebras
belong to B : ||5|| < 1, ||T|| = 1, \\U\\ < 1. Further ST = I
(7 = identity operator), UT = 0. T as a r.z.d. is a r.t.z.d. .
On the other hand T being l.invertible is not a l.t.z.d. .
3.7.11. DEFINITION. An element x in A is called a
symmetric topological zero divisor or s.t.z.d. if there is a net (xa) in
A with xa —> 0, xxa and xax —> 0. Evidently a s.t.z.d. is a bi-
t.z.d. P.G. Dixon has given an example of an element in a Banach
algebra which is a bi-t.z.d. (actually abi.z.d.) but not s.t.z.d. (see
[4, p. 13, Example 13]).
3.7.12. Remark. It is clear from the definitions that a l.t.z.d.
or a r.t.z.d. i of A continues to be so with respect to any TA B
which is an extension ' of A. If both A, B are unital then by
3.7.8, x is singular in A as well as B. Following Lorch we call
a singularity which continues to be a singularly in any unital
extension a permanent singularity. Thus any t.z.d. is a permanent
singularity. F. Quigley has shown that every unital Banach
algebra A has a unital Banach algebra extension B such that every
singular element of A is a zero-divisor of B (see [23, pp.25-27]).
On the other hand Arens has proved that in commutative unital
Banach Jtlgebra the permanently singular elements are precisely
the t.z.d.'s. For a proof of this theorem (extended to ^-Banach
algebra) and other interesting connection between permanently
singular elements and t.z.d's in TA's see [31, p.32 and pp.112-123].
3.7.13. LEMMA. In a p-normed algebra A = (A, || • ||) an
element x is a l.t.z.d. [respy. r.t.z.d.) iff there is sequence (xn)
in A with \\xn\\ = 1, xxn [respy. xnx) —> 0. In particular, x is
a s.t.z.d. if there is a sequence (xn) with \\xn\\ = 1 and xxn —> 0,
xnx -+ 0.
PROOF. The "if" part of the first statement follows form 3.7.5
(since ||a;„|| = 1 => xn /> 0). For the, "only if" part assume that
x is a l.t.z.d. Since A is first countable, by (**) of 3.7.5 there is
a sequence (xn) with xn /> 0, xxn —> 0. We may assume (after
passing to a subsequence if necessary) that \\xn\\ > r) > 0 for all
' i.e. A is a subalgebra of B and the topology of A is the relative
topology from B.

§ 7. Topological Zero Divisors
163
i
n and some r) > 0. Set j/„ = a;„/||a;n||p - Then \\yn\\ = ||1||, and
xyn -+ 0 since ||xj/„|| = ||xx„||/||a;„|| < ?7_1||a;a;„||, \\xxn\\ -+ 0.
The proof of the "only if" part when x is a r.t.z.d. is similar. The
final assertion of the lemma is clear.
3.7.14. LEMMA. Let A be a p-normed algebra and A its
completion. If x G A is a l.t.z.d., r.t.z.d. or s.t.z.d. of A then it
is accordingly the same of A.
PROOF. We shall prove the result only when x is a l.t.z.d.
(the proofs for the other two being similar). By 3.7.13 there is a
sequence (xn) in A with ||£„|| = 1, xxn —> 0. Choose xn G A
such that \\xn — xn\\ < ^- so that xn — in —> 0 (as n —+ oo).
Then
„ .. ... .. 1 1
II " II -^ I I "II II * * * * I I "^ C% C\ '
In I
so that xn /+ 0. Also, xxn = x(xn — xn) + xxn —> 0. Therefore x
is a l.t.z.d. of A.
3.7.15. PROPOSITION (Rickart). Let A be a p-Banach
algebra and a G A. Then we have:
(i) a is a l.t.z.d. (respy. r.t.z.d.) iff
0Ji(a) = 0 (respy. ur(a) = 0), where
ui(a) = inf" ||ax||/||x||,u;r(a) = inf ||xa||/||x||.
(ii) Suppose that la (respy. ra) is 1—1. Then a is not a l.t.z.d.
(respy. r.t.z.d) iff aA (respy. Aa) is closed.
(iii) If a is not a l.t.z.d. (respy. r.t.z.d.) then aA (respy. Aa) is
closed.
PROOF. As usual we shall prove only all results pertaining to
l.t.z.d. .
(i) If 0Ji(a) = 0, there is a sequence (an) in A which
llaan||/||an|| -* 0. Writing xn - ||a„||~?a„, we get ||z„|| =
1, ||aa;„|| —> 0, whence by 3.7.13, a is a l.t.z.d. Conversely,
if \\xn\\ = 1, 11oa;,,11 —> 0 then clearly W\(a) = 0.

164
Some Types of Topological Algebras
(ii) Since la is 1 — 1, l~l exists and l~1{aA) = A. If aA is
closed then a A, as a closed subspace, is a ^-Banach space.
By the open mapping theorem, l~l is bounded. But this
means that sup ||i||/||oa;|| < oo, so that 0Ji(a) ^ 0, and so
X
by (i), a is not a l.t.z.d. . Conversely, suppose that a is not
a l.t.z.d. and axn —> 6. Then w;(a) > 0, whence \\l^l\\ =
l/oji(a) < oo. Since
H^ri •'-m.H ^¾ ||'a II ll^^n "■''mil * *-*,
as n,m-+ oo, by completeness of A, xn —> x (say). Then
axn —> ax. By Hausdorff property of A, ax = b and aA is
closed.
(iii) This follow from (ii) since if a is not a l.t.z.d. it is also not
a l.z.d. so that la is 1 — 1.
3.7.16. LEMMA. Let A be a p -Banach algebra containing
an element a satisfying the conditions:
(i) There is a sequence (Xn) in K with Xn —+ oo such that
each Xna is r. (respy. I.) q. invertible.
(ii) The element a is not a l.t.z.d. [respy. not a r.t.z.d.).
Then A has a I. [respy. r.) unity u.
PROOF. Write bn = (Xna)'r. Then Xna + bn + Xnabn = 0, so
that
abn = -a - -^- (1)
and hence also
K a bn
in- - -
\bn\\" \\bn\\" K\\bn\\p
(2)
If ||6„|| —+ oo then in (2) the R.H.S. —> 0 and a would be a l.t.z.d.,
contradicting the hypothesis (ii). So we must have ||6„|| < C for
all n, whence it follows from (1) that abn —> —a, -a G aA = aA
(by (ii) above and 3.7.15 (iii)). It follows that a = au for some
u G A. Now for any x G A we have a[ux — x) = 0. Since a is

§ 7. Topological Zero Divisors
165
not a l.z.d., we conclude that ux — x, proving u is a 1. unity of
A.
3.7.17. PROPOSITION. Every element of a radical p -Banach
algebra A is a bi-t.z.d. .
PROOF. Suppose that A contains an element a which is not
a bi-t.z.d. . Since every element of \/A = A is q. invertible, we
can apply 3.7.16 to conclude that A contains a 1. or a r. unity u,
and hence a non-zero idempotent u. But this contradicts 1.2.24.
Hence the proposition.
3.7.18. COROLLARY. If A is a p-Banach algebra then every
element of \/A is a bi-t.z.d. .
PROOF. If a e \/A then by 3.7.17, a is a bi-t.z.d. of \/~A
and so also of A.
3.7.19. Let A be a real TA and A its complexification.
Then the property of an element x in A being a l.t.z.d., r.t.z.d.,
or s.t.z.d. clearly carries over when x is regarded as an element
of A. Further we have
3.7.20. LEMMA, (i) x + ix is a l.t.z.d., r.t.z.d., or s.t.z.d. of
A according as x is a l.t.z.d., r.t.z.d., or s.t.z.d. of A.
(ii) If x G A is a l.t.z.d. (respy. r.t.z.d.) of A then it is
already a l.t.z.d. (respy. r.t.z.d.) of A.
PROOF, (i) Clear.
(ii) Assume that x(xa + iya) —+ 0, xa + iya /» 0(xa, j/„ £ A).
Then either xa -/* 0 or ya -/* 0, while xxa —> 0, yya —> 0.
Therefore x is a l.t.z.d. of A if it is of A. Similarly x is a
r.t.z.d. of A if it is of A.
3.7.21. LEMMA. Let A = (A,\\ ■ ||) be a p-normed
algebra (respy. a unital p -normed algebra) and (xn) a sequence of
q. invertible (respy. invertible) elements of A such that (i) xn —> x
(ii) sup \\x'n\\ (respy. sup \\x~ || = oo, where x'n (respy. a;"1) is
n n
q. inverse (respy. inverse) of xn. Then 1 + x (respy. x) is a
s.t.z.d. .

166
Some Types of Topological Algebras
PROOF. First consider the case where xn are q.invertible.
We assume (as we may) that || • || is sm. and by passing to a
subsequence (if necessary) we can also assume that ||x'J| —+ oo.
i
Define yn = x'^/Ux'JI *■, so that \\yn\\ = 1. Then, by using the
relation xn + x'n + xnx'n = 0, we have
x
(l + x)yn = yn + xyn = yn + {x-xn)yn + xnyn= '1-T + {x-xn)yn,
\\x'n\\P
so that I |T II
ii 11 i \ ii^ i iii ii
||(1 + X)yn\\ < r—r + \\X - Xn\\.
\\Xn\\
Since xn —> x, \\x'n\\ —> oo we get (1 + x)yn —> 0. Similarly,
j/„(l + x) —+ oo. So 1 + x is a s.t.z.d. .
Next, consider the case where xn are invertible. Writing zn =
xn — e, z = x — e we see that zn are q. invertible, zn —> z,
sup \\z'n\\ = sup \\x~ — e\\ = oo.
By the first part (proved above) we have xyn = (e + z)yn =
(1 + z)yn —> 0, and similarly ynx —> 0. Also \\yn\\ = 1. So, by
3.7.13, x is a s.t.z.d. .
3.7.22. LEMMA. Let A be a unital TA, G, its group of
invertible elements and xa, x G G,, xa —> x. Then x"1 —+ x
iff the net (a;"1) is essentially bounded.
PROOF. If x"1 —> x'1 then (x"1) being a C-net is
essentially bounded (by 2.3.6). Conversely, if (a;"1) is essentially
bounded then, by 2.3.8 (iii),
xal{xa - x) —> 0, i.e. xalx —> e.
It follows that
x~ = (x~ x)x~ —> ex"1 = x"1.
3.7.23. PROPOSITION. Let A be a unital Hausdorff complete
TA which is a C algebra. Suppose that xa G G,- and xa —+ x in
A. Then x G G,- i/f (x"1) is a C -nei.
PROOF. If x G G,- then, A being a C-algebra, x^1 —> x"1,
so that (x~l) is a C-net. On the other hand, if we suppose that

§ 7. Topological Zero Divisors
167
(x^1) is a C-net then by completeness of A, xa1 —> y (say)
G A. Since (xa) is convergent it is essentially bounded, and so by
2.3.8 (iii),
M^a1 -!/)-* 0, or, xay -+ e.
Also, xay —> xj/, whence by uniqueness of limits (A being Haus-
dorff) we get xj/ = e. Similarly we can show that yx = e. Thus
x is invertible, x G G,, completing the proof.
3.7.24. PROPOSITION. Let A be a CI (respy. CQ)
algebra, G{ (respy. Gq) its group of invertible (respy. q. invertible)
elements. If
(xn) G G, (respy. Gg) and xn —> x G A
iAen x G G, (respy. Gq) iff x"1 (respy. (x'n)) is bounded.
PROOF. First suppose that xn G G,-, x„ —> x, x G G,-. Then
x^1 —> x"1 (since A is a C algebra) and hence (x~*) is bounded.
Next suppose that xn G G,, x„ —> x G A and (x"1) is bounded.
Then by 2.3.8 (iii),
(x„ - x)xnl -+ 0, xnl(xn - x) -+ 0 i.e. xx"1 -+ e, xnlx -+ e.
Therefore, since A is a / algebra, xx"1 and x~lx are invertible
for sufficiently large n, whence by 1.1.30, x is invertible, x G G,-.
This completes the proof of the part relating to inverses.
To prove the part relating to q. inverses, consider the
unitization A\ of A and observe that
(ei + x,,)"1 - ei + x'„.
It follows that (x'„) is bounded iff ((ei + x^)"1) is bounded.
Therefore, by the part proved, e\ + x G G,- iff ((ei + x^)"1) =
(e! + x'„) is bounded iff (x'n) is bounded, completing the proof.
3.7.25. COROLLARY (Gelfand). Let A be a unital pt.
Banach algebra, G, its group of invertible elements, xn G G,-
and xn —+ x G A. Then x G G,- i/f there is a constant C > 0
T Gelfand considered only the case /) = 1.

168
Some Types of Topological Algebras
such that \\xn || < C for all n, where \\ • \\ denoting the p-norm
of A.
PROOF. Since a unital ^-Banach algebra is a CI algebra (by
3.6.23(b)) and the boundedness in A is the same as the bound-
edness with respect to || • || (by 3.2.13), the corollary follows from
3.7.24.
3.7.26. COROLLARY. In a unital p-Banach algebra A the
limit of a convergent sequence of invertible elements is either an
invertible element or is a s.t.z.d. . In particular, every element of
dGi < is a s.t.z.d. .
PROOF. The first assertion follows by combining 3.7.25 and
3.7.21. The second follows from the first since G,- being open 3G,-
is disjoint with G,-.
3.7.27. COROLLARY. Every element of the radical \J~A~ of a
unital p Banach algebra is a s.t.z.d. (c/. 3.7.17).
PROOF. If a E \J~A~ then since na E \f~A~, na is q.invertible
and consequently (e + na)"1 exists. It follows that (- + a)-1 =
n(e + na)"1 exists, whence by 3.7.26, a = lim„(- + a) is either
invertible or a s.t.z.d. . But a cannot be invertible since a E \M,
so a is a s.t.z.d. as required.
3.7.28. PROPOSITION. Let A be a unital p-Banach algebra,
x E A and o{x) the spectrum of x. If o{x) ^ 0 tt and X E do(x)
then x — Xe is a s.t.z.d. .
PROOF. Since A E da(x) and (f(x) is closed, there is a
sequence A„ E p(x) = K\<r(x) such that A„ —> A. Then x —
Xne E G,, x — Xe ^ Gj. Since x — Xne —> x — Xe it follows that
x — Xe E dGi and consequently, by 3.7.26, x — Xe is a s.t.z.d. .
3.7.29. PROPOSITION (Rickart). Let A=(A, || • ||) be a
' For a subset 5 of a topological space X we denote by dS the frontier
of S, i.e., 35(= dS') — Sf^S , where S' — X\S and bar denotes closure.
>> This condition is satisfied for every element 2 if A is complex or
strictly real. Moreover, whenever 17(2) /(} we have also dcr[x) ^ 0, since by
6.1.2, 17(2) 7^ K, is closed but not open (by connectedness of K).

§ 7. Topological Zero Divisors
169
p -Banach' algebra with ||-|| sm. and xn —+ x in A with xn <-> x
(for all n). Let v — v\y\\. Then:
(i) If A is unital, xn invertible and v(x~l) is bounded then x
is invertible.
(ii) If xn are q. invertible and v(x'n) bounded then x is
q. invertible.
PROOF, (i) Suppose that v(x~l) ^ C. Then we have
v{e-xnlx) = v(x~1xn-x~1x) ^u(x~l)v(xn- x)
(using 3.3.7. (iii))
<; C\\xn-x\\-+0.
It follows that for sufficiently large n, v{e — x~^x) < 1, whence
by 3.3.20 (ii), x~lx is invertible, so that x is also invertible.
(ii) This can be deduced from (i) by passing to the unitization
A1 of A.
3.7.30. PROPOSITION. Let A = (A,\\ • ||) be a unital p-
normed algebra with G, its group of invertible elements. If xn G
Gi,xn —> x in A and x is a t.z.d. then supHx"1]! = oo and x
is a s.t.z.d. .
PROOF. Let A be the completion of A. Then x is a t.z.d. of
A and so not invertible in A, whence by 3.7.26, sup H^1!! = oo
and so by 3.7.21, x is a s.t.z.d. .
3.7.31. COROLLARY. An element x G dGi\Gi is a s.t.z.d.
iff sup \\xn l\\ = oo, where xn G G,- and xn —> x.
PROOF. This follows from 3.7.21, 3.7.30.
3.7.32. PROPOSITION. In a strictly, real unital p-Banach
algebra A every singular element is a bi-t.z.d. and the square of
a singular element a s.t.z.d. .
PROOF. Since x is singular and A is strictly real we have
Rickart considered only Banach algebras (i.e. p = 1 case).

170
Some Types of Topological Algebras
0 G &{x) — &{x) ?= ^> an(l consequently we have 0 G ^(z)2 =
a(x2) -= a(x2). Since a{x) C R, <j(a;2) = a(x)2 > 0. It follows
that 0 G 3<t(x2), so that by 3.7.28, x2 is a s.t.z.d.. Hence, by
3.7.6 (b), x is a bi-t.z.d.
3.7.33. COROLLARY. In (a strictly real) A we have
S(= 5' |JSr) = Sbi = Sl = Sr = 3" = 3ri = ?>bit.
PROOF. It suffices to observe that a bi-t.z.d. is, by 3.7.8, both
1. singular and r. singular.
3.7.34. Remarks. In the Banach algebra of complex valued
continuous functions on the unit interval, with sup norm, every
singular element is a t.z.d. . On the other hand, in the Banach
algebra A of all complex-valued continuous functions / = f(z)
on the closed unit |z| < 1 which are holomorphic on \z\ < 1, the
function /0(2) ~ z (\z\ ^ 1) is singular but not a t.z.d. (see
[10, p.70] or [7's p.29]). Further the real algebra i?W gives an
example of a real Banach algebra containing a singular element
(/0) which is not a t.z.d.; note that AR is not strictly real (since
it has complex structure), so that this example does not contradict
3.7.32.
3.7.35. PROPOSITION. Let B be a complex or strictly real
unital p -Banach algebra and A a closed subunital algebra of B.
Then, for every x G A,
aB{x) C aA(x), (*)
daA(x) C daB(x). (**)
Further, if either (aA{x))° = 0 or Pb{x) is connected we have
aA(x) = aB(x). (* * *)
PROOF. The relation (*) has already been obtained (see
1.7.20). For proving (**) we note that if A G oA(x) then by
3.7.28, x-Xe isas.t.z.d. of A and hence also of B. In particular
x—Xe is singular in B and so A G aB[x). Thus daA(x) CaB(x).
Again, by (**) of 1.7.20, pA(x) C pB{x). Therefore

§ 7. Topological Zero Divisors
171
daA(x) = daA(x)f]aB(x) = (pA(x)f]aA(x))f]aB(x)
= pA(x)f]aB(x) CpB(x)f]aB(x) = daB(x)
which is (**).
Assume now that {aA{x))° = 0. Then
aA(x) = daA(x) C doB(x) C <rB(x)
(since uB(x) is closed). Combining this inclusion with the
inclusion (*) we get (***). Next let pB(x) be connected. Since
Pa{%) ^ Pb{%) we can write
Pb(x) = pA{x)\J(pB(x)\pA(x)) = pA(x)\J(aA(x)\aB(x)) (1)
where |J denotes disjoint union. Since daA(x) C dB(x) every
point of aA[x)\aB{x) is an interior point, i.e. oA{x)\aB(x) is
open. The connectedness of pB(x) and (l) imply that aA{x) =
aB(x), proving (* * *) in this case as well.
3.7.36. COROLLARY. Let A be a closed subunital algebra of
a unital complex p -Banach algebra B and x G A. Then
aA(x) C R iff aB(x) C R
and when these inclusions hold we have actually aA(x) = aB(x).
PROOF. Since always aB(x) C aA(x) it is enough to prove the
"if" part of the "iff" assertion. Assume therefore that <rB(x) C R.
By (**) of 3.7.35, daA{x) C daB(x) C R, and by 6.1.2, aA(x) is
compact. It follows that (aA(x))° ~ 0, whence by 3.7.35, aA(x) =
aB(x), completing the proof.
3.7.37. PROPOSITION. A closed subalgebra A of a strictly
real p -Banach algebra B is strictly real.
PROOF. First assume that B is unital with unity e and set
A\ = A + Re (Ai = A if e G A). Consider the complexification B
of B and write A\ ~ A\ + iA\, A\ is a closed subunital algebra

172
Some Types of Topological Algebras
of B. Now 0^(2) = a^ (x),(t~b(x) = &B{x)- Since B is strictly
°b(x) l= R- By 3-7-36, ^(z) = °aSx) = ^5(1) - R- Tt follows
that Ai, and hence A (by 1.9.5) is strictly real.
If B has no unity, consider its unitization Bi which again is
strictly real. Also A is closed in B\ (since B is closed B\).
Applying the result just obtained above to the unital B\ we conclude
that A is strictly real.
3.7.38. DEFINITION. (Zelazko). Let A be a TA. A pair
(5,T) of subsets of A is called a generalized topological divisor
of zero [31,p.73] or g.t.z.d. if 0 ¢ S, T but 0 e ST, where bar
denotes the closure.
3.7.39. LEMMA. A g.t.z.d. (S,T) with S - {x} (respy. T -
{y}) is a l.t.z.d. x [respy. r.t.z.d.y).
PROOF. Write f = F. Then we have 0 ^ F while 0 e ST =
xT C xT = xF, proving x is a l.t.z.d. . Similarly, y is a r.t.z.d. .
3.7.40. LEMMA. A pair of nets (xa),(yp) such that xa -f+
0, yp -f* 0 and xayp —> 0 determine a g.t.z.d.
PROOF. Since xa /> 0 there exists a nucleus U and a subnet
(xai) of (xa) which lies outside U. It follows that if S = {xai}
then 0 ^ S. Similarly there is a subnet (yp>) of (j/^) such that
0 ¢: T, where T = {j//3'}- Since (xaiypi) is a subnet of (xayp)
and xayp —> 0 we get 0 G ST, proving (S,T) is a g.t.z.d.
3.7.4.1. PROPOSITION (Zelazko). A commutative unital
Hausdorff TA which admits no g.t.z.d. must be a C algebra.
PROOF. Suppose that A is not a C algebra. Then we can
find a net (xa) in G, with
xa -* x0 G d (l);^1 -/* Xq1. (2)
By 3.7.22, (x^1) is not essentially bounded and so by 2.3.6, it is
not a C-net. It follows that there is a nucleus U and subnets
(xp),(xpi) of the net (xa) such that a;7, — x~„ ¢- U, so that
X of Xq -f+ U.
(3)

§ 7. Topological Zero Divisors
173
As subnets of (xa) it follows from (l) that
Xp -+ Xq, Xpl -+ Xq
whence
xpixp —> Xq 7^ 0 (since x0 G G,).
Since A is Hausdorff we have
xpxp /* °- (4)
On the other hand, by using the commutativity of A we get
xpixpfep,1 - x^1) = xp - xpi -+ 0. (5)
From (4), (3), (5) we conclude, by 3.7.40, that A has a g.t.z.d.,
contradicting the hypothesis. Hence A is a C algebra.
3.7.42. Remark. For more information on g.t.z.d. see [31,
pp.73-77].
3.7.43. Remark. For the class of topological algebras called
locally sm. convex (defined in 4.4.11) there is a weaker definition
of t.z.d. due to Michael. For results concerning such t.z.d.'s see
[20, pp.43-48].

CHAPTER IV
LOCALLY PSEUDO-CONVEX SPACES
AND ALGEBRAS
§ 1. p -convexity
4.1.1. DEFINITION. Following Landsberg [9', p.104] a subset
5 of a LS X (over K) is called p-convex, where 0 < p ^ 1, if
x,y,e S; a,/3 <ER,a,/3 > 0 and ap +/3P = 1 (*)
=> ax + /3y G 5.
It is called absolutely p-convex if
x,yeS;a,peK and |q|" + |/3|" s£ 1 (**)
=> ax + /9j/ G 5.
Evidently when p = 1, ^-convex (respy. absolutely ^-convex)
is just convex (respy. absolutely convex) in the usual sense.
Let S be a subset of X. An element of the form V] ajxj =
i
n
2_^otjXj, with xj G 5, a.j > 0 and 22aj ~ ^> *s caUed a ^>~
convex linear combination of elements of 5. The meaning of an
absolutely p- convex linear combination of elements of 5 is clear. For
any subset 5 of X we denote by Cp[S) (respy. |Cp|(S)) the set
of all ^-convex (respy. absolutely ^-convex) linear combination of
elements of 5; Cp(S) (respy. |Cp|(S)) is called the p-convex
[respy. absolutely p-convex) hull of S. Obviously, Cp[S) C |Cp(5)|.
4.1.2. LEMMA, (a) If S is p-convex [respy. absolutely p-
convex) then S = Cp[S) [respy. \Cp\[S)).
(b) For any subset S of X, Cp[S) [respy. \Cp\[S)) is the
smallest p -convex [respy. absolutely p -convex) subset of X
containing S.

§ 1. p -convexity
175
(c) If S is p -convex (respy. absolutely p-convex) so is XS
(AeK).
n
PROOF, (a) Let 5 be ^-convex and x = ^2aixi w^ a/ ^
/= i
0, xj G 5 and (*) E"=i <*/ = 1- We shall show that x G S
by induction on n the length of the combination). If n = 2,
x G 5 by the definition of ^-convexity. Assume now that x G 5
whenever the length is n - 1. For the of- occuring in (*) write
ra— 1 ra— 1
^2 aj = Pp (say)> so that we ^ "a"1/ e ^- Therefore, since
/=i /=i "
ri /n-1 a. \
/3P + a£ = 1, it follows that ]T ayx/ = /? ( £ ~z/ + <*„£„ G 5.
/=i 1/=1 " y
Thus, Cp[S) C 5 whence Cp(S) = S (since always 5 C Cp[S)).
The assertion concerning |Cp|(S) is proved similarly.
(b) It follows from (a) that if Si is a p -convex set with Si ^ S
then Si = Cp(Si) D Cp(S). We will now show that Cp(S) is p-
convex. Suppose that
m ra
/=i *;=i
with0<ay, ft; ^|Qy|"=l, ]T |/?*|'= 1.
/=1 jfc=l
If 0< A, n;\t> + iit> = 1 then
m n
Xx + fiy = J2 XotjXj + J2 vPkVk-
/=i )fc=i
Since
/=1 k=\
we conclude that \x + fiy £ Cp(S) which proves that Cp(S) is p-
convex. The proof of the assertion concerning |Cp|(S) is similar.

176 Locally Pseudo-convex Spaces and Algebras
(c) Clear.
ra n
4.1.3. Remark. If y^ajXj, V]/^/ are absolutely P-
3=1 3=1
convex linear combinations of elements and lA^ + \fi\p < 1, then
n
V^Ac^- + (j,/3j)xj is also an absolutely ^-convex linear combina-
3 = 1
tion of elements.
In fact we have
3 3
< Ed^W + NiA-i")
3
(since 0 < p ^ 1)
On the other hand, the analogous assertion for ^-convex linear
combination can fail if p < 1. For example, take n = 2; ai =
OL2=\\ P\ = \, /?2 = |; A = /i=i; p = i.
Then
/i o c
7i = (Aqi + nPi)? = -^-, 72 = (Aa2 + //^2)2 = —
so that 71 + 72 = ^13-1-5 < 1, showing that 7J1 + 7f £2 is not a
^-convex linear combination.
4.1.4 LEMMA, yl subset S [of a LS X) is absolutely p-
convex iff it is p -convex and balanced. Also, if S is any
balanced subset then CP(S) is absolutely p -convex, so that \Cp\(S) =
CP(S).
PROOF. If 5 is absolutely ^-convex it is trivially ^-convex.
Moreover, it is also balanced since ^1^1=^-^^^1. Conversely,
suppose that 5 is ^-convex and balanced. If Xj G 5, )> 1^/^ ^ 1
3

§ 1. p -convexity
111
then ^\-f\p =h where /3 > 0, /3P = ]T \a3\p ^ 1. We assume
that all \a3\ > 0 and set /33 = oij/\oij\. Then /33x3 E S (since 5
is balanced) and so by ^-convexity
IZ -^-^¾ G S' le- -g IZ °W G 5-
By balanced property of 5 we get \]a3x3 E S, completing the
j
proof of the first assertion. For the second we assume that 5 is
balanced. If a3 > 0, Y^ aj = 1> \M ^ 1 and xj ^= S then
3
A N otjX3=S, ot3(Xx3) E Cp(S) (since Xx3 E S, S being balanced).
j i
Hence CP(S) is balanced and CP(S) = \C\P(S).
4.1.5. LEMMA. If S is p-convex and a,/3 > 0 then
a$S + p$S C (a + /3)h.
If S is convex (i.e. p = 1) then aS + /3S = (a + /3)S.
PROOF. For x,y E S
i i
ai'x + /3py = (a+ /3)"
i i
ae Be
rx^ rV
{a + 3)p (a + /3)
G (a +/3)?S (by ^-convexity).
If ^) = 1 the above gives aS + /3S C (a + /3)S. On the other hand,
since (a + /?)a; = ax + /3y the reverse inclusion relation also holds.
4.1.6. LEMMA. If U is an open subset of a TLS X then
\Cp\(U) is open.
n
PROOF. Consider an element x = y~]a3a3 (a3 E U),
3 = 1

178 Locally Pseudo-convex Spaces and Algebras
otj 7^ 0)/^ \aj\P ^ 1- Since U is open we can choose a balanced
3
open nucleus V such that ay + V C Z7 (j = 1,-- • ,n). Write
5^|ay| := a(> 0). For a G V we have
x + aa = N ayoy + )> | ex j-1 a = )> aj{aj + a7 la/la)-
Since V is balanced, a-" |ay|o G V', ay + ctJ1|ctJ-|a G (7, so that
x + aa G |Cp|((7),a; + aV C |Cp|((7). Since x is an arbitrary
element of |Cp|((7) we conclude that |Cp|((7) is open.
4.1.7. LEMMA. If S is absolutely p -convex (0 < p < 1) and
0 < p' < p then it is also absolutely p' -convex. In particular, if
S is absolutely convex it is absolutely p -convex for every p with
0 < p ^ 1.
PROOF. Suppose that £ la/K' ^ 1. Then
5>yl'= £>/-* < (El^lpy (since ^1)
^ l£ = 1.
Therefore, every absolutely p' -convex linear combination is an
absolutely ^-convex linear combination, whence the lemma.
4.1.8. DEFINITION. Let 5 be an absorbing subset of a LS
X. For x G X, 0 < p ^ 1, set
p(x) = pp(x) = pPts{x) = ps{x)
1
= inf{a > 0 : x G a ? S]
= mi{/3" :/3>0,xej3S}.
The non-negative realvalued function p is called the p-gauge
[gauge if p = 1). The gauge of 5 is also known as the Minkowski
functional of 5.
It is clear from the definition of the p-gauge that we have:
Ps2 <PSl if Si C S2, (*)

§ 1. p -convexity
179
px=0, (**)
Pas = ~Ps (<* e K\{°})- (* * *)
a
4.1.9 Remark. If 0 < p' ^ p < 1 then pp< = p >> , where
4.1.10. PROPOSITION. Let S be an absorbing subset of X.
Then the p -gauge p = ps has the following properties:
(i) p(0) = 0.
(ii) p(Xx) = X"p[x) if A Z 0.
(iii) p(Ax) = |A|''p(a;) for all A G K, provided S is balanced.
(iv) p[x + J/) ^ ^(z) + p(j/) for all x,y £ X provided S is
p -convex.
In particular, p is a p -seminorm if S is absolutely p-convex
[and also absorbing).
i
PROOF, (i) Since 5 is absorbing, OsS and hence OGa'S
for all a > 0, whence p(0) = 0
(ii) x G a^S iff Ax e (\pa)pS(\ > 0). Hence p(Xx) =
X"p(x)
(iii) Assume first that |A| = 1; then A-15 = 5 since 5 is
balanced. It follows that Ax G aeS iff x G a'pX~1S = a^S.
Therefore p(Xx) = p(x). Next for arbitrary A G K, A^O, write
A = \X\fj. where \fi\ = 1. Then, by (ii)
p(Xx) = \X\pp(fj,x) = \X\pp(x) (since |//| = 1).
Finally, the relation clearly holds if A = 0.
(iv) Given e > 0, we can choose a,/3 > 0 such that x G a^S,
i
y G fie S ; a < p(x) + e, (j < p(y) + e. Since, by 4.1.5,
x + y ea^S + /3pS C (a + /?)pS
it follows that
p(x + y) < a + /3 < p(x) + p(y) + 2e.

180 Locally Pseudo-convex Spaces and Algebras
From the arbitrariness of e we conclude that
p(x + y) < p(x)+p(y)
proving the first result of (iv). For the second result we remark
that it follows from this and (iii), since an absolutely ^-convex
set is both ^-convex and balanced.
4.1.11. LEMMA. Let Sj(j = 1,---,71) be absorbing balanced
sets and S = Si |"| • • • |"| S„; then S is absorbing and balanced. If
Pj,p are p -guages of Sj, S respectively then p = pi V • • • Vp„. >
PROOF. That S is absorbing and balanced are easy
consequences of the definitions (see 2.1.15). Since S C Sj we have
p < p. Write </ = pi V • • • V pn. Then q(x) = maxpy(x) ^ p(x).
1 i
Again we have, for some en —> 0,
i. i
x G (pj(x) + en) o Sj C (q(x) + en) > S}-
(since Sy is balanced)
= XSj (say).
It follows that X~1x G S, so that
xe AS = (q(x) + en)pS.
This implies that p(x) ^ q(x) + en, whence making en —> 0 we
get p(x) -< q(x) ^ p(x). Thus q = p, completing the proof.
4.1.12. PROPOSITION. Let S be a subset of a LS X (over
K), which is absolutely p-convex (0 < p ^ 1) and absorbing.
Then:
(i) The p -gauage p = ps is a p -seminorm such that if
Bi = {x<E A: p(x) < 1}, Si = {x e A : p(x) ^ 1}
then Bi C S C Bi; in particular, kerp C S.
(ii) S — Bi or Bi according as S is open or closed in the
p -topology.
' See 3.1.7 (b) for the definition.

§ 1. p -convexity
181
(iii) If A is an algebra and S a [multiplicative) subsemigroup of
A then p is sm..
PROOF, (i) By 4.1.10, p is a ^-seminorm. If x E B\ then
i -I
for some e > 0, p(x) < e < 1, x E f'S . Then e x E S, and
11 I, _ i
since |f| = ^^ < 1 and 5 is balanced, x = f(e ex) E S, so
i
that Bi C 5. Again, if x E S then i£l'S, so that p(z) ^ 1,
whence SCBj.
(ii) Assume first that 5 is open and x E S. Since (1 —)fx —>
x and 5 is open, (1 - -)ex E S for some n > 2, so that p(x) ^
1-- < 1, whence B\ = 5. Next assume that 5 is closed. By
3.2.7, Bi is the closure of B\. Therefore, since by (i), B\ C 5 C
B\ and 5 is closed, it follows that 5 = B\.
(iii) By definition of p, since 5 is balanced, we have for any
e > 0 with 0 < e < 1,
xE(p(x) + e)$S, yE(p(y) + e)?S.
Therefore
iff G [(p(x) + e)(p(ff) + 6)]'S2 C [p{x)p{y) + Ce]'S,
where C is a constant independent e. By making e —> 0 we
conclude that p(xy) ^ p(x)p(y), i.e., p is sm..
4.1.13. PROPOSITION. Let p be a p-seminorm on an
algebra A. Then B\,B\ [as defined in 4.1.12) are absolutely p-
convex and absorbing. Further, Bi is open, B\ is closed in the
p -topology and we have
Pb^Pb^P- (*)
If p is sm. then B\,B\ are subsemigroups.
PROOF. Using the identity p(Xx) = \\\pp{x) it is
straightforward to check that Bi,Bi are absolutely ^-convex. Next, for any
x E A, set Xx = 1 if p(x) = 0 and Xx = (2p(z))~p if p(x) ^ 0.
Then p(\xx) = 0 or ^, so that Xax E B\ C _B1( proving that
Bi, B\ are absorbing. It is further clear from the definition of the

182 Locally Pseudo-convex Spaces and Algebras
p-topology that B\ is open, and B\ is closed, being the closure
of Bi (by 3.2.7.). Again it is immediate from the sm.property
that B\. B\ are subsemigroups whenever p is sm.. It remains to
prove the equalities (*).
Since B\ C B\ we have
PbSx)^PBi(x)- (1)
i
Again, if A > p(x) then x G A^Bi, so that PBj(z) < A, whence
PBi(z) ^P{x). (2)
l
Finally, p(x) < A iff igA'Bi. Hence
pB-i(x) = M{\:\^p(x)}^p(x). (3)
From (1), (2), (3) we obtain (*).
4.1.14. Remark. If p be as in 4.1.12, and r > 0, then we
can show as above (for the r = 1 case) that
Br = {x G A : p(x) < r}, Br = {x G A : p(x) ^ r}
are respy. open absolutely ^-convex and closed absolutely p-
convex. Also, when p is sm., Br, Br are subsemigroups for
0 < r < 1.
4.1.15. PROPOSITION. Let p be a p-seminorm on X, 0 <
p ^ 1, 0 < p' < 1. Then p' = p <> is a p' -seminorm iff the unit
ball Bi — {x : p[x) < 1} is p' -convex. In particular, p' is a
p' -seminorm for all p' with 0 < p' < p.
PROOF. Note first that we have also Bi = {x : p'(x) <
1}. If p' is a ^'-seminorm then, by 4.1.12, B\ is ^'-convex.
Conversely, let B\ be p' convex. Denote by p'B the p' -gauge
_ i
determined by B\. Then p'B (x) < A ■<=>• A ^ x <E Bi <=>•
_ i
p'(A ^x) < 1 iff p'(x) < A. Therefore p' = p'B and consequently

§ 1. p -convexity
183
p' is a p' -seminorm. Then second assertion in the proposition is
an immediate consequence of the first and 4.1.7.
4.1.16. THEOREM (Rickartt). Let A= (A,p) be a real p-
seminormed algebra, 0 < p ^ 1, and A the complexification of
the algebra A. Then A is a p -seminormed algebra. Further, A
is unital, p -normed or p -Banach according as A has the
corresponding property. Moreover, we can choose the p -seminorm p
of A in such a way that the map x \—> x + i.O of A into A is an
isometry [i.e. p[x) = p(x)), p(e) = 1 whenever A has a unity
e, with p[e) — 1, and finally p is sm. if p is sm. .
PROOF (following the method of Bonsall-Duncan [4, p.68]).
Write U = {x G A : p(x) < 1}, U — the absolute ^-convex hull
of U in A. Then U is absorbing in A. To see this, suppose that
z — x + iy 7^ 0 (x, y G A) and
fj," >max{p(x),p(y)}. (1)
Then x/fj.,y/fj, G U, so that
proving U is absorbing.
We next prove that U has the property:
z = x + iy G U => x,y G U. (3)
Assume that z has the representation
jfc=i
Writing
fe=i k
lk = ak + i(3k (at,ft£R)
we get
z = x + iy, where x = ^otkxk, y = Y^PkVk-
' Rickart [23, p.8] obtained the results when A is a real normed algebra.

184 Locally Pseudo-convex Spaces and Algebras
Since |afc|,|/?fc| ^ |7jt| it follows, by the absolute ^-convexity
of U in A, that x,y G U, whence (3) holds.
Denote by p the gauge of U (in A); then p is a ^-seminorm
on A. Writing Bi = {z G A : p(z) < l}, by 4.1.12(i), we get
Bi C {/. (4)
On the other hand, if
n
z= ^XkxkeU (xkeU,Xkec),^\Xk\<> <: l,
k=l k
we can choose e > 0 such that p(xk) < e < 1 for all k . Then
_ i _ i_
e pxk G U, so that e i>z £ U, whence p(z) ^ e < 1. It follows
therefore that
U = BX (5)
and so in particular U is p-bounded. From (1),(2) it follows that
p(z) ^ 2max{p(x),p(y)}. (6)
Again, if A > 0,Az G U then by (3), Ax, Aj/ G J7, so that
p(Xx),p(Xy) < 1, whence
A'max(p(x),p(j,))<l. (7)
For a given e > 0, we can choose the A > 0 such that
3^<P(*) + e- (8)
We conclude from (7),(8), since e is arbitrary, that
max{p(x),p(y)} ^p(z). (9)
The inequalities (6),(9) imply that p induces the topology of A,
and also that p is faithful (respy. complete) when p is faithful
(respy. complete).
If x G A and A G R then it is clear from (3) that
Ax + i0eU iff Ax G U,
whence p(x) = p(x). When A is unital we can, using 3.5.9,
assume that p(e) = 1, so that then we get p(e) = p(e) = 1.

§ 2. Locally Bounded Algebras
185
Finally, let p be sm.; then U is a subsemigroup of A. We
shall show that U is a subsemigroup of A. Suppose now that
z,z' G U, then they have representations
z = Y^AfcZjt, z' = Y^AJa;;, where Xj^xi G J7 and
A: I
EW-EWl"^1- Now
22'= ^AjfeAjxjta;;, and
k,l
DiA*Ajr=(x;iA*r) few
~ 'KH= 1.
Since U is a semigroup, x^xi G U and consequently the above
relations show that zz' G U, so that J7 is a subsemigroup of A,
as desired, completing the proof of the theorem.
§ 2. Locally Bounded Algebras
4.2.1. LEMMA. Let X be a locally bounded TLS and U a
bounded balanced nucleus of X. Then we can find a p,0 < p < 1
such that
U+ U C2~?U. (*)
PROOF. Since U is bounded, by 2.1.25, {^U : n = 1,2,---}
is a basis of nuclei, so that there is a N > 2 such that
U U
Write ^> = log 2/log ./V. Then N = 2~? and (*')=>(*).
4.2.2. LEMMA (Kothe). Let S be a balanced subset of a LS
X such that
S + 5C2pS (1)
where 0 < p < 1. Then
2~i"S + --- + 2~^S CS (2)

186 Locally Pseudo-convex Spaces and Algebras
for all integers kj > 1 such that \] 2 J< 1, and further
/= i
|C,|(S) C 2^5. (3)
PROOF. ([18,p.l65]). To prove (2) it is clearly sufficient to
prove the special case (2') of (2) where we have
E2-*> = i. (*)
/=i
We call /c = max kj, the order of the resolution (*). We prove
/
(2') by induction on k. If k = 1 the relation (2) reduces just
to the hypothesis (1) and so (2) holds for this case. Next assume
that (2') holds for order k > 1. To prove that it holds for order
k + 1, observe that each decomposition (*) of order k + 1 is
obtained from a decomposition of order k, by replacing one (or
more ) of the summands 2~k by 2_(*+1) + 2_(*+1), since it follows
from Yl^~kj — 1 that the summands 2~'*+1) occurs an even
number of times. Thus, to obtain (2') for order k + 1 starting
from (2') for order k we have to replace the summand 2 ^5 by
A:+l k+1 _ __1 __1
2 p S + 2 p S. But this is permissible since 2 *> S + 2 p S C S
(by (1))- This completes the proof of (2') (by induction).
n
It remains to prove (3). Suppose that /^1¾^ < 1 and Xj G
/=i
5. Determine integers kj such that 2~ki ^ \oij\p < 2"ki+1 =
2-2~*>. Then
J2\ai\" < 2^2_A:j ^2-J2\ai\P ^2-1 = 2
/ /
since 5 is balanced,
OijXj G ctjS C \atj\S C 2 p S,
so that
5>ya:ye E2"^5 = 2' |E2"^5 ^2'5 (using(2))>

§ 2. Locally Bounded Algebras
187
proving (3).
4.2.3. PROPOSITION. A TA A (or more generally a TLS) is
p -seminormed (0 < p < 1) iff it has a bounded nucleus U such
that
U + U C 2~?U.
Moreover, when A is semi-normed we can even choose the U so
that it is a subsemigroup of A.
PROOF. If A is a ^-seminormed algebra (A,p) we may
assume that p is sm. (see 3.4.4). By 4.1.13, U = Bi is an
open absolutely ^-convex nucleus which is a subsemigroup. U
is, moreover, bounded (by 3.2.13). Finally, if x, y e U then by
p -convexity,
-^- + -^- e U, so that U + U C ihjj.
2» 2»
Conversely, suppose that U is an open nucleus having the stated
properties. Then, by 4.2.2,
V = \Cp\{U)C2^U (*)
and by 4.1.6, V is an open nucleus. The boundedness of U
together with the inclusion (*) implies that V is bounded. Since
V is absolutely p -convex the gauge p = pv is a p -seminorm (by
4.1.10). By 4.1.12(H), V = B1 = {x e A : p(x) < 1} and hence
(*) gives:
\v C U C V = Bx. (**)
2~p
Since U, V are bounded nuclei it follows from (**) that the
topology of A coincides with the p -topology, whence A is p-
seminormed.
4.2.4. COROLLARY. Every locally bounded * TA is a
p-seminormed algebra for some p,0 < p ^ 1.
' A TA is said to be locally bounded if it is locally bounded as a TLS
(i.e. it has a bounded nucleus).

188 Locally Pseudo-convex Spaces and Algebras
PROOF. This is an immediate consequence of 4.2.1, 4.2.3 .
4.2.5. Remark. Rolewicz [12'] has shown that to each
locally bounded TLS X there is a largest number ^o,0 < po ^ 1,
such that if 0 < p < po then the topology of X can be induced
by a p -seminorm.
4.2.6. COROLLARY. A TA A is a semi-normed algebra iff it
has a bounded convex nucleus U.
PROOF. If A = (A,P) is semi-normed we can take U =
Bi = {x e A : p(x) < 1}. On the other hand, if A has a bounded
convex nucleus U then we have, by 4.1.5,
17 + 17=1-17+1-17= (1+1)17 = 2lU.
By 4.2.3, A is semi-normed.
4.2.7.. COROLLARY (Kolmogorov t). A Hausdorff TA is a
normed algebra iff it has a bounded convex nucleus.
PROOF. This follows from 4.2.6 since the topology induced
by a semi-norm p is Hausdorff iff p is a norm.
4.2.8. PROPOSITION. An infinite direct product of locally
bounded Hausdorff TLS's - in particular such a product of TA's -
cannot be locally bounded.
PROOF. Suppose X = HXa, where {Xa} is an infinite
family of locally bounded Hausdorff TLS's. Denote by ira the factor
projection: X —> Xa. Then sets of the form
where Ua . run through an open basis of Xa., form a basis of
nuclei for X. Fixing a U, let /? be an index with /? ^ ct\,• • • ,an
and x 7^ 0 an element of the space Xp. Then the sequence
[nx){n — 1,2,- • •) of elements clearly belong to U (after suitable
' Actually Kolmogorov proved the following result: A Hausdorff TLS is
normable iff it has a bounded convex nucleus.

§ 3. Locally Pseudo-convex Spaces
189
identification t ). Since - • nx = x ^ 0 this sequence does not
converge to 0 and consequently U is not bounded. Since U is an
arbitrary basis nucleus we conclude that X is not locally bounded.
4.2.9. COROLLARY. A direct product of p -seminormed
algebras is p -seminormed iff it is a 'finite direct product'.
PROOF. If A — Ai x ••• x An, when Aj = (A3-,pj) are p-
seminormed algebras then A is ^-seminormed under q : q(x) =
maXj Pj(x), where x G A, x — (xi, • • • ,xn), xj G Aj (cf. 3.3.11).
Conversely, if a direct product A of p -seminormed algebras
is ^-seminormed then A is locally bounded (by 3.2.14) and hence
by 4.2.8, A is a finite direct product.
§ 3. Locally Pseudo-convex Spaces
4.3.1. DEFINITION. Following Waelbroeck [29,p.4] we call a
subset 5 of a LS X (over K ) pseudo-convex if it is ^-convex
for some p (0 < p ^ 1). A TLS X is called locally pseudo-convex
if it has a basis {Ua} of pseudo-convex nuclei Ua; if Ua is pa -
convex we also say that X is locally {pa} -convex. If all the pa =
(some) p then X is called locally p-convex, and if p = 1 then
X is called locally convex.
A ^-seminorm p is also called a pseudo-seminorm and p is
called the homogenity index of the pseudo-seminorm p (cf. 3.2.1).
4.3.2. HP is a family of pseudo-seminorms on a LS X, the
P -topology (see 3.1.7) makes X a locally pseudo-convex space
(X,P); if P = {pa} and B? = {x G X : pa(x) < r} (r G
R,r > 0) then the family of all finite intersections of the B®
(a,r varying) give a basis of pseudo-convex nuclei for (X, P).
Conversely, if X is any locally pseudo-convex space with {Ua}
as a basis of pseudo-convex nuclei them the gauges pa associated
with the Ua determine a family P ~ {pa} of pseudo-seminorms
Pa-
' . The element xp G Xp is identified with the element x = (xa) in X
such that xa = 0 if a ^ /?, xa — xp if a = f).

190 Locally Pseudo-convex Spaces and Algebras
4.3.3. Let 6* denote the set of all pseudo-seminorms on a
LS X. We introduce an order -< in 6* by writing
I _L I -i.
p -< p if pe < pe' (i.e. p(x)e ^ p''(x) <•' Vz G X) and p ^ p,
where p.p' G 6* are respectively a ^-seminorm and a p' -
seminorm. Clearly, the relation -< is reflexive and transitive. It is
i , i
also anti-symmetric, for, if p -< p', p' -< p then pe = p?, p = p',
so that p = p'. Thus -< is a partial order. We also note that if
p < p' then for 0 < e ^ 1, B? C Bf (since if p'(x) < e then
p(x) ^ p'(x)7 < e).
4.3.4,. LEMMA. The poset G* = (6*,^) is closed for all
finite lattice sums.
PROOF. If pj G 6* and pj is a ^-seminorm (j = 1,---,71)
_£_
set p ~ mm{pj},p'- = ppi ;q{x) = maxp'(i). By 3.2.9, p'- is a
j i
^-seminorm and so by 3.2.3, q is a ^-seminorm, so that q G 6*.
It easily follows from the definition of q that q > pj, and that if
p' G &*,p" > Pj for all j then p' > q. So q is indeed the lattice
sum pi V • • • Vp„.
4.3.5. Lemma. If p3- e e*(y = 1,--- ,n),q = px v ••• vp„,
and 0 < e ^ 1 iAen
Bf C Bf1 f| • • • f) Bf".
PROOF. Since pj ^{we have B« C B?0 (see 4.3.3). The
required result is now clear.
4.3.6. Lemma. If pj,p3 e e*(y = l,••-,«) anrf py ~ pV
iAen
pi V---Vp„~pi V---Vp^.
PROOF. By (**) of 3.2.11 we have
p4 p'j
Pj^C'jp'-3 ,p'j^CjpJ>.

§3. Locally Pseudo-convex Spaces
191
Set C — max C'-, p = min p}-. Then
Pi V ■ • • V p„
Similarly, p'x V • • • V p'n ^ C(pi V • • ■ Vpn), where C ~ max C,-.
Hence the lemma.
4.3.7. DEFINITION. A family P of pseudo-seminorms is
called saturated if it satisfies the condition
Pi,---,pneP =>pi v---vp„e/>. (*)
For any subset P of 6* we set
?={?£6* :? = piV---Vpn.p,- eP}.
The set P clearly satisfies the condition (*) and is called the
saturated closure of P.
4.3.8. LEMMA, (a) P,P induces on X the same topology.
(b) If Pq is a cofinal' subset of P [with respect to the order
in 4.3.3) then Pq induces on X the same topology as P.
(c) If P is a saturated family then the family {Bf : p E P ,e >
0} is a basis of nuclei.
PROOF, (a) It is sufficient to observe that p(xa — x) —> 0 for
each p E P iff q(xa - x) —> 0 for each q E P.
(b) It is enough to note that if p -< po (p E P ,po E Pq) and
B^{xEX : p{x) <r},B0 = {xEX : p0(x) < r'}
En.
where r' = r e , po (respy. p) being the homogenity index of po
(respy p.), then Bo ^ B.
(c) This follows from 4.3.5 and the simple fact that B^ C B*2
if 0 < ri < r2.
max p -J < C' max py
C'(p'iV---Vp').
i.e. given any p £ P there is a p0 G /¾ with p -< p0-

192 Locally Pseudo-convex Spaces and Algebras
4.3.9. Let P be a family of pseudo-seminorms on a LS X. A
subset S of X is said to be P -bounded if the numerical functions
p(x)(p G P) are bounded on S (i.e. p\S is bounded for each p G
P ). The meaning of p -bounded where p is a pseudo-seminorm
is clear.
4.3.10. LEMMA. Let X = {X,P) be a locally pseudo-convex
space. A subset S of X is (t.) bounded iff it is P -bounded.
PROOF. Let S be bounded. If possible let pa\S be not
bounded for some pa in P. Then we can find a sequence (xn) in
i_
S with pa{xn) ^ «• Writing yn = n p<* xn we have: pa(yn) ^ 1-
It follows that yn -f+ 0 in X, contradicting the boundedness of
5.
Conversely, suppose now 5 is such that pa\S is bounded
for all pa. Then, for (xn) in 5,A„ —> 0 we have pa(\nxn) =
\^n\PaPa{xn) ~~> 0, since the sequence is bounded and pa > 0.
Therefore 5 is bounded.
4.3.11. Proposition. Let X = (X,P),Y = (Y,Q) be
locally pseudo-convex spaces, with P saturated. Then, a linear
transformation T : X —> Y is continuous iff for each qp G Q
there is a pa G P and a constant C = Cap > 0 such that
q/3(Tx)^Cpa(x)% (xeX) (*)
where pa,Pp ire respectively the homogenity indices of pa,P/3-
PROOF. The condition (*) clearly implies continuity of T at
0 and and hence also everywhere. Conversely, if T is continuous
then (by continuity at 0), given qp and e > 0 there is a pa and
a 8 > 0 such that
pa{x) ^ S => qp{Tx) ^ e. (**)
_i —
For an x with pa{x) 7^ 0, write xi — [8pa{x)~ )c« x. Then
Pa{x{) — 8, so that by (**) we get qp{Tx\) ^ e which reduces
to (*) above will C = e8 P™ . If pa(x) — 0, we can argue exactly

§ 3. Locally Pseudo-convex Spaces
193
as in the proof of 3.2.10 and show that qp(Tx) = 0, so that (*)
holds trivially in this case.
4.3.12. LEMMA. Let X be a TLS, Y = {Y,Q) a locally
pseudo-convex space, T : X —> Y a linear transformation. For
each qp E Q we set p*p(x) = qp(Tx). Then p„ is a pseudo-
seminorm on X with the same homogenity index as qp. The
transformation T is continuous iff all p*„ are continuous.
PROOF. We have
p*p{x + y) = qp(Tx + Ty) ^ p}{x) + pp{y). Further, if p is the
homogenity index of qp then:
p*p(Xx) = qp(TXx) = qp(XTx) = \X\"p*p(x).
To prove the criterion for continuity of T it is enough, in view of
2.1.26, 3.1.3, to consider the continuity of both at 0. Now T is
continuous at 0 iff: xa —> 0 in X => p*g(xa) = qp(Txa) —> 0 (for
all qp ), iff all pp are continuous at 0. This proves the lemma.
4.3.13. LEMMA. Let X — (X,P) be a locally pseudo-convex
space, where P is saturated. Then:
(i) For any continuous linear functional f on X there is a
pa E P such that f is pa -continuous, i.e. f is a
continuous functional of the pseudo-seminormed linear space
(X,pa).
(ii) If p is any continuous pseudo-seminorm on X then there
is a pa such that
_£_
p < Cpaa , where C is some constant > 0 and p,pa the
homogenity indices of p,pa-
PROOF, (i) By 2.1.31, there is a pa and an e > 0 such that
Pa(x) < €=> \f(x)\ < 1.
It follows from 3.5.4 that
/ : (X,pa) —+ K is n. bounded, whence by 3.5.5, it is
continuous.

194 Locally Pseudo-convex Spaces and Algebras
(ii) The identity map : (X, P) —> (X,p) is continuous by
hypothesis on p. So, by 4.3.11, there is a C > 0 and a pa with
p
p(x) < Cpa(x)"c
as desired.
4.3.14. DEFINITION. Let X = (X,P) be a locally pseudo-
convex TLS - or more generally a TLS with its topology induced
by a family P of quarter-norms. A map
T : PT(C X)-+X
where Dt is the domain of T, is called a contraction if we have
for each pa G P a constant ea with 0 < ea < 1 such that for all
x,y <E Dt we have
pa(Tx - Ty) < eapa(x - j/).
Note that a contraction map is automatically continuous.
4.3.15. THEOREM (Contraction mapping principle). Let
X — (X,P) be a complete [or even sequentially complete) Haus-
dorff TLS, with P a family of p-seminorms (or more generally
quarter-norms). Let T : S —+ S be a contraction, where S is a
closed subset of X. Then T has a unique fixed point x* G S with
x* =limT"xo, where xq is an arbitrary point of S.
PROOF. Define xi = Tx0,xn = Txn-X = Tnx0. If m < n
then
pa(xm - xn) = pa(Tmx0 - Tnx0) ^ (ea)mpa(x0 - xn-m)
^ {ea)m[Pa{xO ~Xi)-\ h Pa(z„-(m-l) ~ Im-l)]
< {ea)mPa{x0-x1)[l + ea +---6^}
< {£a)mPa{xo ~ Xi)-— .
1 €a
It follows that pa(xm - xn) —> 0 as m,n —> oo (for each
a). By completeness of X the sequence xn —> some x* G X;
since xn G S and 5 is closed, x* G S. By continuity of T

§4. Locally Pseudo-convex Algebras
195
we have: Tx* = lim Txn. By the construction of T we have
n~»00
also: lim Txn — lim xn+i = x*. Therefore, since X is Haus-
n—»00 n—»00
dorff, Tx' — x*. Thus x* is a fixed point of T. If Tj/ = y, we
have
pa(z* - y) = pa{Tx* - Ty) ^ eapa(x* - y).
Since ea < 1 we must pa(z* - J/) = 0 for all pa, whence x* = y
(X being Hausdorff).
4.3.16. LEMMA. A locally pseudo-convex space X = {X,P)
is locally connected.
PROOF. Let P be the saturated closure of P, so that we have
also X = (X,J). Since the open balls B" = {x e X : pa(x) <
r}(pa £ P) form a base of nuclei the lemma follows since each B"
is connected (see proof of 3.2.8).
§ 4. Locally Pseudo-convex Algebras
4.4.1. DEFINITION. A TA A is called locally pseudo-convex
if it is locally pseudo-convex as a TLS. This means that a locally
pseudo-convex algebra A is of the form A = (A,P), where P
is a family of pseudo-seminorms. If each pa G P is sm. then A
is called locally sm. pseudo-convex algebra. Again, if each pa is a
semi-norm, A is called a locally* sm. convex algebra.
4.4.2. LEMMA. If A is locally sm. pseudo-convex then it has
a basis consisting of pseudo-convex subsemigroup nuclei.
Conversely, if A is a TA which admits such a basis then it is locally
sm. pseudo-convex.
PROOF. If A = (A, P) with each pa E P sm. then, by
4.1.13, B" = {x G A : pa(x) < r},0 < r < 1, are pseudo-convex
subsemigroup nuclei. On the other hand if A has such a bais
then the associate gauges pa are sm. pseudo-convex (by 4.1.12
((i), (iii)))-
This is the same as locally m. convex algebra in the terminology of
Michael [20).

196 Locally Pseudo-convex Spaces and Algebras
4.4.3. LEMMA. If pi,-"Pn are sm- Pj -seminorms (j =
1,--- , n) then q = pi V ■ ■ ■ V pn is a sm. p -seminorm with p =
min{^y}, where pj is the homogenity index of pj.
_£_
PROOF. By 4.3.4, q(x) = max pj (x)p J , and q is a p-
i
seminorm. Further, we have
_e_ _£__£_
q(xy) = maxpj(xy)''j < max pdx)"'Pj(y)fi = q{x)q{y),
i i
whence the lemma.
4.4.4. PROPOSITION. Let A be an algebra (over K) P a
family of pseudo-seminorms on A, and P the saturated closure
of P. Then A is a TA under the P -topology iff for each, p' G P
there is a po G P and a C > 0 such that
I I
p'(xy) ^ Cp0(x) ? p0(y) r (x,y<EA) (*)
where p',po are respectively the homogenity indices of p',po-
PROOF. The proof is somewhat similar to that of 3.4.3.
First suppose that (*) holds for each p' G P. Assume that in
A,xa -+0:,2//3-+ y. Then from the identity
XaVp - xy = (xa - x)(yp - y) + x(yp - y) + (xa - x)y
we obtain, using (*) and the subadditivity of p', the inequality:
p p
Pc{xa ~ x)»op0(yp - y)po
i i i i \
£- £_ P P I
+Po(x)p«p0(yp - y)po +p0(xa - x)pop0(y)po \ .
It follows that p'(xayp — xy) —> 0, for every p', whence
xay/3 —> xy, and (A, P) is a TA.
Next suppose that (A, P) is a TA. Since the map (x,y) h-> xy
is continuous, given p' G P and e > 0 then is a po £ P and a
8 > 0 such that
Po{x),po{y) ^8=>p'(xy) <c. (1)

§4. Locally Pseudo-convex Algebras
197
1 -L
For x,y G A with po(*),Po(j/) 7^ °, set *! = 6',z/Po(z)''0,
J/i = #^11 y/po{y)p° where po is the homogenity index of po- Then
Po(xi) = 6 = po(j/i), whence by (1), p'(zij/i) < e which reduces
to
p'(zj/) ^CpoOcWotj/)'0 (2)
where C = c/«2"'/p.
It remains to consider the case where po(x) or po(j/) = 0.
Suppose that po(x) = 0. Then po(n2x) = 0 for any integer n < 1.
For arbitrary j/ G A we can choose n (large enough) that
,,y, p'{y) , r
P -)= —7^ O-
Then, by (1), p'(nxj/) = p'(n2x-j//n) < e which gives np p'(xy) ^
e. Since n is arbitralily large we get p'(xy) = 0. Similarly
p'(zj/) = 0 if x is arbitrary and po(y) = 0. Thus (2) holds for all
x, y as required.
4.4.5. COROLLARY. Let A = (A, P) be a locally pseudo-
convex algebra. Then for each p' G P we can find a pseudo-
seminorm p* on A such that
p p
p'(xy) ^p*(x)e*p*(y)p* (x,y<EA), (**)
p" ~ pi V ■ ■ ■ V pn for some pj G P.
4.4.6. COROLLARY. If P is any family of sm.pseudo-
seminorms on an algebra A then under the P -topology A is a
locally (sm.) pseudo-convex algebra.
PROOF. By sm. property of pa G P we have pa(xy) <
Pa{x)pa{y) and the condition (*) of 4.4.4 evidently holds.
4.4.7. LEMMA. Let A = (A, P) be a locally pseudo-convex
algebra. If P = {pa} and po = inf^>a > 0, when pa is the
homogenity index of pa, then A is a locally p$ -pseudoconvex
algebra.
PROOF. By 3.2.9, qa ~ ppa^a is a ^0-seminorm such that

198 Locally Pseudo-convex Spaces and Algebras
oa ~ Pa- Thus A = (A, Q) where Q = {qa} is a family of po-
seminorms, whence A is a locally po -pseudoconvex algebra.
4.4.8. PROPOSITION. The locally {pa} -convex algebras A
are precisely the TA 's of the form A = (A, P), where P is a family
{Pa} of pa-seminorms on A satisfying (*) of 4.4.1. Moreover,
A is Hausdorff iff ker P = {0}.
PROOF. The first statement is essentially a restatement of
4.4.4. The second statement follows from 3.1.21 (iii).
4.4.9. DEFINITION. A family P of pseudo-seminorms on an
algebra A is called well-behaved if P is saturated and further for
each p' G P there is a p G P such that
p p
p'(xy) <p(x) "p(y) " (**)
where p' ,p are respectively a ^'-seminorm and a ^-seminorm.
4.4.10. PROPOSITION. The locally pseudo-convex algebras
are precisely the TA', A of the from A = (A, P), where P is a
well-behaved family of pseudo-seminorms.
PROOF. If P is well-behaved then by (**) of 4.4.5 and (*) of
4.4A, A is a TA (under the p-topology) which by 4.3.2, is locally
pseudo-convex. Next assume that A is a locally pseudo-convex
algebra. Then, by 4.4.4, A = (A, P) with P satisfying (*) of
4.4.4. Let P denote the saturated closure of P and write
P = {p G e* : p ~ q G J}
where 6* denotes the set of all pseudo-seminorms on A. Then,
by 4.3.6, P is saturated. Further, by 4.4.5, P is well-behaved.
4.4.11. LEMMA. Let A= (A,P) be a locally p-convex
algebra such that:
(i) P = {pa} is a well-behaved family;
(ii) suppa(x) < oo for each x G A.
Pa

§ 4. Locally Pseudo-convex Algebras
199
Then p(x) = suppa(x) is a sm. p -seminorm and (A,p) is a p-
Pa
seminormed algebra.
PROOF. Since P is well-behaved, for each pa G P there is a
Pp G P such that
Pa{xy) < pp{x)pp(y) < p(a;)p(y).
Taking the supremum of the first term over all a we get
p(xy) < p(x)p(j/).
4.4.12. PROPOSITION. Let A = (A,P) be a locally pseudo-
convex (respy. locally sm. pseudo-convex) algebra. Then:
(i) Any subalgebra Aq of A under the relative topology is locally
pseudo-convex [respy. locally sm. pseudo-convex).
(ii) If I is a bi-ideal of A and A* = A/1, then A#, under
the quotient topology, is locally pseudo-convex [respy. locally
sm. pseudo-convex).
PROOF, (i) We may assume that P is well-behaved. For
pa G P write pan = pa\Ao (restriction to Ao) and Pq = {pao}-
Then it is clear that (Aq,Po) has the required properties.
(ii) Let pj be defined in terms of pa as in 3.4.15 and write
Ptt = {Pa}. Then (A*,.P*) is the quotient algebra and this has
the required properties (the proof that P^ induces the quotient
topology is similar to that of 3.1.22 (iii))-
4.4.13. PROPOSITION. The unitization Ai or the complexi-
fication A of a locally pseudo-convex or locally sm. pseudo-convex
algebra A is again an algebra of the same type.
PROOF. If A = (A,P) then for each p G P denote by pi
the canonical extension of p to Ai as defined in 3.4.11 and by p
the extension of p to A as defined in the proof of 4.1.16. Write
Pi = {pi-.peP}, P = {p:PeP}.
Then A\ = [A\,Pi), A = (A,P) have the stated properties.
4.4.14. LEMMA. Let A=[A,P) be sequentially complete-in
particular complete-Hausdorff locally sm. p-convex algebra. If for

200 Locally Pseudo-convex Spaces and Algebras
an x G A we have
va{x) < 1 for all va
oo
then x is q.invertible, )> (— l)nxn converges in A and the
n=l
q. inverse
oo
^ = 23(-1)^-
PROOF. As in the proof of 3.3.19, we obtain for each pa,
oo
Y^Pa{xn) < OO.
n=l
So
(N+r \ N+r
J2 (~l)nxn < J2 P°(xn) -^ 0, as 7V,r -^ oo.
n=N J N
oo
This means that ^2(-l)nxn converges absolutely and so also
in A. Exactly as in the proof of 3.3.19 we can show that
oo
£(-1)^ = ^.
ri = l
4.4.15. PROPOSITION. A locally sm. pseudo-convex algebra
[or more generally a TA whose topology is induced by a family of
a. sm. quarter-norms) is a C algebra.
PROOF. Let A = (A,P), where P = {pa} and each pa is
a. sm.. Then, by 3.3.22, each pa satisfies the inequality (*) stated
therein. It follows that if (each) pa{^^ — i) —> 0 (xa,x G Gq)
then pa{x1^ — x') —+ 0, which proves that the map x \—> x' of Gq
is continuous, so that A is a C algebra.
4.4.16. LEMMA. In a locally sm. pseudo-convex algebra
A = (A,P), where P is saturated, for any continuous pseudo-

§ 5. Projective Limit Decompositions
201
seminorm p on A there is a pa €E P such that v£ < v£™ , where
p,pa are homogenity induces of p,pa respectively.
_£_
PROOF. By 4.3.13 (ii) there is a pa with p ^ Cpa" ■
Therefore
vJx) ~ lim (p(xn))n < lim C« lim pa(xn)n
y n—»oo v n—»oo Lra—»00
^ 1/0(1)"«,
i _J_
so that i/pp < Vaa , as required.
§ 5. Projective Limit Decompositions
4.5.1. Let {Aa : a e A} be a family of TA's with the indexing
set A, a poset directed above by -< . Suppose that for each pair
(a,/3) of elements of A, with a -< /3, there is given a continuous
homomorphism ipap : Ap —> Aa, such that
(i) Pan = 1a„
(ii) ^a/3 o cpp1 for a < fi < ^.
Write
A = the Cartesian product TT Aa.
aCA
Under coordinate-wise operations A is an algebra. Set
Aq = {x = (xa) G A : ^0/3(3:/3) = %a whenever a -< /3}.
Aq is called the projective limit of the Aa and we write
Ao=limAa, x — Yunxa (xeAo).
The projective limit Aq has the following universal mapping
property: for any TA B and continuous homomorphisms oja :
S —+ Aa such that ipap 0 ojp = oja (whenever a -< 0), there
is a unique continuous homomorphism 8 : B —> A0 such that
7ra o 5 = aja (a £ A), where na denotes the natural projection
A —+ Aa. (It is easy to see that 6 is given by 6(b) = (oja(6)), for
6e s.)
4.5.2. LEMMA. Aq is a closed subalgebra of A.
PROOF. By 2.2.8 (b), A is a TA, and since <pap are homo-

202 Locally Pseudo-convex Spaces and Algebras
morphisms, Aq is a subalgebra of A. It follows from the
continuity of <pap that Aq is closed in A.
4.5.3. THEOREM (Michael t). Every complete Hausdorff
locally sm. pseudo-convex algebra A is (i. isomorphic to) a projective
limit, limAa of pseudo-Banach algebras'' Aa. In particular,
if A is locally p-convex, then A = limAa, with all Aa as p-
Banach algebras.
Conversely, a projective limit B = \imBa of pseudo-Banach
algebras Ba, is a complete Hausdorff locally sm. pseudo-convex
algebra.
PROOF. Let A = (A,P), where we may assume that P =
{pa} is saturated. Set Aa = A/Na, where Na = kerpa. Then
Aa = (Aa,p#) is a ^a-normed algebra (pa being the homogenity
index of px) whose completion Aa is, by 3.4.17, a pa -Banach
algebra. Make the indexing set A = {&} a directed set (directed
above) by defining a -< /3 if pa < pp, where the latter -< is
the order introduced in 4.3.3 (the directedness of the order is a
consequence of the family P being saturated).
If a -< /?, it is clear that Np C Na, where Np = kerp„,
Na = kerpa. Consequently, the map
ipap : xp = x + Np i—> xa = x + Na (x G A)
is well-defined and is a continuous homomorphism of Ap onto Aa,
which can be extended, by 3.5.11 (b), to a homomorphism <pap
of Ap into Aa. It follows that
{Aa : <pap (a,/3 G a,) a -< ^}
is a projective system of pseudo-Banach algebras which defines the
projective limit
' Michael [20, p.l7.j established the theorem for the particular case of
the locally sm. convex algebras but his method of proof extends to our general
case.
'' A pseudo-Banach algebra is just a p -Banach algebra for some p,
0 < p ^ 1. This algebra is not to be confused with the pseudo-Banach algebra
introduced by Allen-Dales-McClure who use the term for a certain inductive
limit of Banach algebras.

§ 5. Projective Limit Decompositions 203
A0 = limAa C Y\ Aa.
Similarly, {Aa : ipaf}} gives rise to the projective limit
Aq = limAa,
which is identifiable as a subalgebra of Aq. The map ip : A —+ Aq
given by ((p(x))a ~ x + Na = xa is a homomorphism, since each
of the maps ipa : x i—> xa(a E A) is a homomorphism. It is 1-1
since xa = 0 for all a => x E f] Na = {0} (since A is Hausdorff).
Further, ip is continuous since each <pa is continuous.
We now show that ip is "onto". Take any element y = lim ya
in Aq. For any finite subset 5 of A, choose a 3 E A such
that 3 >- a for all a G 5. Take an element xs E A such that
{<p{xs))p = yp\ then (<p(as))a = Va for all x E S. Also, given
finite subsets Si, 52 with Si f] S2 7^ 0, we have
{<p{xSl))a = ya = {<p{xs2))a for all a G 5i P| S2.
This relation implies that
xSl ~ xs2 E Na C J7Q|£ = {x G A : pa(a) < e} (e > 0).
It follows that (xs) - S varying among all the finite subsets of
A - is a C-net. Since A is complete
xs ~> (some) x E A.
Since ip is continuous we obtain
ya = (<p(xs))a {S varying) -+ (<p{x))a = xa,
i.e. ya = xa, so that y = (p(a), proving that <p is "onto".
We next show that (p~l is continuous. Consider a net (x^')
in A such that
<p(xW) -> <p(x) (xEA).
Then by 3.4.16 we have
pa{xW -x)= p*(xW - Xa) -* 0 for all a.

204 Locally Pseudo-convex Spaces and Algebras
This means that x^ —> x in A, proving the continuity of ip~l.
To complete the proof of the first statement of the theorem it
remains to show that Aq = Aq. Now any non-empty open set W
of Ao contains an open set V of the form
V = *J(Gai)()--()^(6^)()^,
where Ga are open sets in Aa . and 7T"1 the natural projection
■kAa —> Aa. (j = 1,---,71) choose an a„+1 G i? with an+i >
otj(j — 1, ■ ■ ■, n). Then we have the homomorphisms
^a,-,an+i : ^n+i ""* Any (j = 1>" " " >raJ-
Set
Then
V?ay,an + l(Gan + 1) ^ Gay (j = 1, ■■-,«)•
Set
u = v()kUg^+1)^v.
Since Attn+1 is dense in Aan+1 we can find x G A so that zan+1 G
Ga„+1. It follows that
<p(x) G £7, so that U 0 A0 7^ 0.
Therefore Ao is dense in Ao- But Ao is complete, being
t. isomorphic to the complete space A and consequently closed.
Thus, Ao = Ao.
The second (or converse) statement follows since every pseudo-
Banach algebra is a complete Hausdorff locally sm. pseudo-convex
algebra and these properties carry over to direct products and to
closed subalgebras.
4.5.4. For descriptive convenience we call a complete
Hausdorff locally sm. pseudo-convex algebra A as a pseudo-Michael
algebra. If it is a locally sm. ^-convex algebra then it will be
referred to as a ^-Michael algebra. Finally, if p = 1, A is called

§ 5. Projective Limit Decompositions
205
a Michael t algebra.
If completeness hypothesis is dropped in any of the above
definitions then the resulting notions will be indicated by employing
pre-Michael instead of Michael in their names
4.5.5. COROLLARY. Every pseudo-pre-Michael algebra is a
dense subalgebra of a projective limit \imBa of pseudo-Banach
algebras Ba.
PROOF. It suffices to apply 4.5.3 to the completion B = A
of A.
4.5.6. PROPOSITION (Michael). Let A=(A,P) be a pseudo-
Michael algebra with projective limit decomposition
A = limAa (Aa a pseudo-Banach algebra). For x G A,
x = X\mxa [xa G Aa) we write x — (xa). Then:
(i) x = (xa) is q. invertible in A iff xa G Aa is q. invertible
(for each a).
(ii) A is unital iff all Aa are unital, and then the unity e =
(ea),ea being unity of Aa.
(iii) u = (ua) is idempotent in A iff for each a, ua is idem-
potent in Aa.
(iv) If A is unital, then an element x — (xa) is invertible iff for
each a, xa is invertible in Aa.
PROOF. (i) If x is q.invertible then, by 1.1.24, xa is
q. invertible in Aa and so also in Aa 2 Aa. Conversely, if for each
a,xa , has q.i. x'a, then (x'a) G A. To see this, note that if /? > a,
(PaP '■ Ap > Aa the element <pap(x'p) is q.i. of ipap{xp) ~ xa, so
that (pap{x'p) = x'a. It follows that (x'a) G limAa = A, whence
there is an element x' G A with (x')a = x'a. Since multiplication
and addition are component-wise it is clear that x' is q.i. of x.
(ii), (iii): These follow at once since multiplication in A is
component-wise.
Such algebras have also been considered by Arens.

206 Locally Pseudo-convex Spaces and Algebras
(iv) The proof is similar to that of (i).
4.5.7. COROLLARY. For x = (xa) e A we have:
(1) *a(*) = Oa«>");
a
(2) rA(x) = suprA (xa);
a '*
(3) 0^(^) = U^a (xa) whenever A is unital.
PROOF. By 1.7.8 A(^ 0) e a'A(x) iff -A_1x is not
q.invertible, i.e. by 4.5.6 (i), iff for some a, -\~1xa is not
q. invertible in Aa iff A G a'~ (xa). This proves (1), and (2) read-
Aa
ily follows from (1).
For (3), assume now that A is unital; then each Aa is unital.
From (1) and 1.7.21 we obtain
M*)U{°> = IK(*«)U{°>-
a
If 0 ^ 0a{x) then x is invertible and consequently each xa is
invertible, so that 0 ¢. (Jo^ {xa). On the other hand, if 0 G
a
(ta{x) then x is not invertible and consequently, some xa is not
invertible so that Oe^ (za). Thus, the equality in (3) holds in
both cases.
4.5.8. COROLLARY. [\<p~l{\JAa) <Z y/A = [^<p~l(\/Aa).
PROOF. By 1.2.17 (i), each <p~l(\JAa) or <p-l(y/7^) is a
bi-ideal. So
are bi-ideals of A. By 1.2.26, <p(\/A) C y/Aa. So \/~A C I. On the
other hand, if x G I then xa e \[Aa and so xa is q. invertible.
Consequently, by 4.5.6 (i), x is q.invertible so that I is q.i.bi-
deal, whence I C \/A. Thus \/A = I. Again, if x e J then

§ 6. Metrizable Locally Pseudo-convex Algebras 207
xa £ v/ Aa, xa is q. invertible and consequently x is q.invertible.
Thus, J is a q.invertible ideal whence J C \f~A~.
§ 6. Metrizable Locally Pseudo-convex Algebras
4.6.1. PROPOSITION. Every first countable locally pseudo-
convex algebra A is semi-metrizable and has the form A = (A, P),
where P = {pi,P2,"--} is a countable well-behaved family of
pseudo-seminorms such that
pl<p2<...\ (*)
fj pj
pAxv) < Pi+i{xVi+lPi+i{.y)'i+l (**)
[j = 1,2,---), where pj, Pj+i are respectively the homogenity
indices of pj,pj+\.
PROOF. That first countability implies semi-metrizability
follows from 2.1.7. Since A is first countable its topology is
induced by a countable family Q ~ {qn} of pseudo-seminorms. Set
p'n = ?i V ■■■ V qn and /" = {p'n}. Then p\ < p'2 < ■ ■ ■ ■ Set
pi = p[. By 4.4.4, there an integer i(l) such that
Z pL
pAxv) =Mxv) < cip'i(i)(x)np'i(i){y)n
where pi,p' are homogenity indices of pi,p^,^ and C\ a constant
_£_
which we may clearly suppose > 1. Set pi = Ci2p'pJ-m- Then
P2 ~ p'iiu and satisfies
Pi{*y) < P2(x)"P2{y)
For p2 we can similarly choose pj-(2)> "with z(2) > 2(1),2, such
that
fa fa
P2(a:») ^C2P' (^)^p' (j/)^2
T Recall that py -< py+i means: p.' ^ pPj,+1 and Py+i ^ py.

208 Locally Pseudo-convex Spaces and Algebras
where we can assume that C<i > C\. Set p3 = C2 2Ptf21 anc^ we
Thus proceeding we get P = {pi,P2>'"} with the desired
properties. It is clear from the way we constructed that P is
equivalent to a cofinal subset of P', whence P ~ P' ~ Q. So
A = (A,P), completing the proof.
4.6.2. DEFINITION. A first countable Hausdorff locally
pseudo-convex algebra is called a pseudo-pre-Frechet algebra or a
pseudo-pre- 3 algebra. If A is also complete it is called a pseudo-
Frechet or a pseudo- 3 algebra.
4.6.3. DEFINITION. A pseudo-pre-3 algebra A is called a
locally sm. pseudo-pre- 3 algebra if its topology can be induced
by a countable family Q ~ {qn} of sm. pn -seminorms for some
pn with 0 < pn < 1, n = 1,2,... .
4.6.4. LEMMA. Every pseudo-pre- 3 algebra A is of the form
A = (A, Q) where Q = {?«} JS o. countable family of pseudo-
seminorms with ker Q = {0}. If A is locally sm. pre-pseudo-%
algebra then we can choose Q such that each q G Q is sm..
PROOF. This follows from 4.6.1, 3.1.21 (iii).
4.6.5. DEFINITION. If in a pseudo-pre- 3 algebra A =
(A, Q), all q £ Q are ^-seminorms for some fixed p (0 < p < 1)
then A is called a p -pre- 3 algebra. If in addition each q is sm. ,
then A is called a locally sm. p -pre- 3 algebra. The meaning of a
p- 3 algebra or a locally sm. p-$ algebra is clear. Finally, when
p = 1, we say simply 3 algebra or locally sm. 3 algebra as the
case may be.
4.6.6. PROPOSITION. The locally sm.pseudo-$ {re-
spy, locally sm. p - 3) algebras are precisely projective limits of
sequences of pseudo-Banach (respy. p -Banach) algebras.
PROOF. This follows from 4.6.4, 4.5.3.
4.6.7. PROPOSITION. Every pseudo-pre- 3 (respy. pseudo- 3)

§6. Metrizable Locally Pseudo-convex Algebras 209
algebra A is a pre- (F) (respy. (F) algebra.
PROOF. Let A = (A, Q), Q = {qn : n = 1,2,---} and
ker Q = {0}. Then
- J_/ qn(x) \
1 ' ^2" Vl+ ?„(*)/
is a (F) norm and A=(A,|-|). Further, |-| is complete whenever
(A, Q) is complete.
4.6.8. Examples of metrizable locally pseudo-convex
algebras
(i) Let (pn) be a sequence of real numbers such that 0 < pn <
1. The set C of all K-valued continuous functions x = x(t)
on R is an algebra (over K) under pointwise operations.
Set
||x||„=sup|x(0|""; then || ■ ||„
is a sm. ^„-seminorm. It can be seen that C = (C,{\\ •
\\n : n = 1,2,---}) is a locally sm. pseudo-3 algebra. If all
pn = p then C is a locally sm. p - 3 algebra. Finally, when
p = 1, we get a locally sm. 3 algebra which we denote by
C(R).
(ii) Let Ii>-IP{K) denote the algebra of all sequences (xW)^=1
of elements of K such that p{x) = V] |a;(,l)|'' < oo, under
n
coordinate-wise operations. Consider the Cartesian power
A = (/')■" = V x l" x ■ ■ ■.
If V = (l/i," ■>!/*,-■■) £ A, Vn £ lp then set pn(y) =
oo
P(Vn) = Yl ^^^1 where Vn = (^)^=1- Then [t is easy
m=l
to check that (A, P) is a ^>-3 algebra, where P = {p„}.
Further, it is locally sm. p- 3 algebra since each pm is sm..

210 Locally Pseudo-convex Spaces and Algebras
(iii) We have already noted in 3.3.14 (example (iii)) that the
algebra £ of all entire functions (under pointwise operations)
is a unital (F) algebra. Now set for / G £,
11/11,,= sup |/(*)|.
|z|<n
Then || ■ ||„ 's are clearly sm. seminorms ' and the family
{|| ■ || : n = 1,2,---} induces the same topology as the (F)
metric || ■ ||g since convergence with respect to either is
uniform convergence over compacta. Thus, (£,{|| ■ \\n '■ n =
1,2,---}) is a locally sm. 3-algebra (over C).
(iv) For 0 < p ^ 1, let U° denote all K-valued measurable
functions x = x(t) (or rather equivalence classes of such
functions) on the unit interval [0, l] such that
j,
11^11^=(/^1^)1^^1 <oo.
Now
xy\\ip)]n = fl\x{t)\n"\y{t)\n"dt
Jo
ljx(t)\^"dty [J* \y(t)\*n>dt
whence we conclude that
/II„m(p)ii..i, .
l2ri '
\*v\M < \\4£\\v\lM
Again, if 0 < r < s, then by applying Holder's inequality
for xpr and 1, with p = -,-K = 1 — - we obtain
p
7
f \x(t)\"r ■ ldt< (f \x(t)\^Uty x (f lp'dt\
which gives
V \x(t)\prdtY < (f \x{t)\psdt
Actually they are norms.

§6. Metrizable Locally Pseudo-convex Algebras 211
It follows that we have
||z||i < ||z||2 < ■■■ (i.e. || a: |f x -< ||z||2 -< ' ■ 0
so that P ~ {|| ■ ||„ : n — 1,2,- ■ ■} is a well-behaved
(countable) family of p -norms and U°p is a p-^ algebra. It is not
a C algebra. To see this, consider elements of the algebra
given by:
gm(t)=[ ^(l + ^-l)") forO<t^^
I 1 for 1/W < t < 1.
Then
9m [)~\ 1 (1/m1/' <t^l).
Since
I
m(l+(m- l)?)>v
as m —+ oo, for each n, we get <7m —> 1. On the other hand
/0
(ff-1 - l)""rfi = 1(m - l)""1 -> oo (if n > 1)
o m
which implies that g^1 /» 1. Thus Lw is not a C algebra.
Therefore, by 4.4.15, it is not locally sm. (p- 3) algebra.
The algebra Lu ~ Lf is a Frechet algebra which may be
called Aren's algebra since this was introduced and studied
first by Arens [l'] who also showed [2'] that it is not a C
algebra.
(v) Let T be any locally compact, a -compact, Hausdorff space.
Then the algebra C = C(T,K), of all the K-valued
continuous functions under the topology of uniform convergence
over compacta, is a locally sm. 5 algebra. (If T = \J Kn,
Kn compact, / e C, \\f\\n = sup{|/(i)| : t e Kn} then
C ~ (C, {|| ■ ||„}). If T is compact C is a Banach algebra.

Locally Pseudo-convex Spaces and Algebras
When T is not compact C may fail to be an I algebra.
For example, take T — R, Kn ~ [-n,n\. Then || ■ ||i <
II " lb ^ """ • Clearly, if
Un<e = {f e C : \\f\\n < e}
then {Un>e : n > l,e > 0} is a basis of nuclei, so that
{C/„,e(l)} = {1 + C/„,J
is a basis of neighbourhoods of 1. Since the function
€X
— eUn>e (xeR),
In
€X
/„,*(*) = 1+- eC/„,e(l)
In
and fne is not invertible since it vanishes at x ~ . It
e
follows that C(R) is not an I algebra.
Let C°° = C°°[0,1] be the algebra of all K -valued infinitely
differentiable functions on the closed interval [0,1], with
pointwise operations. For / G C°° write
H/lloo = SUp |/(t)|,
and set
ii/iu = Eii/W||oo,
jfc=0
where /(*) denotes the A;th derivative of /(/(°) = /). It
follows from the definitions that we have
||/IU<||/|U+l (n = 0,1,2,---).
It can be shown by induction, using Leibnitz's theorem, that
II /11 .- o"+l II /11 II -II
If we set ll/H; = 2"+1||/||„ then || ■ ||; is sm. and || ■ ||; ~
||-||„. It is easy to see that (C°°,{|| ■ ||; : n = 0,1,2,-■ ■}) is

§ 7. Ample Algebras
213
a locally sm. Frechet algebra. Further it is an I algebra. To
see this consider its group G,- of invertible elements; G,- =
{/ G C°° : f 7^ 0 on [0,1]}. We claim that G,- is open.
For, given / G G,- we have by compactness of [0, l], mo =
info<t<i \f(t)\ > 0 choose an e with 0 < e < \m,Q. If g G
C°°, ||ff - /||o < e then
«7(01 > 1/(01 - 1/(0 - 2(01 >m0-f>y>0,
so that g G G,-. Thus G; is open and C°° is an I algebra.
Finally, by 4.4.15, it is a C algebra. Thus C°° is a locally
sm. 5 algebra which is a CI algebra (cf. Example V).
(vii) In 3.6.33 we have seen that the field C(X) of rational
functions carries a topology (Williamson topology) under which
it is a CI (division) algebra W. There is another topology on
C(X) making it a pre- 3 algebra which is also an I algebra.
This topology as well is due to Williamson (for details, see
[14', pp.731-32] or [31, p.83]).
§ 7. Ample Algebras
4.7.1. DEFINITION. A TLS X is called ample if its
(continuous) dual X* separates points of X : given elements x\ ^ X2 in
X there is an / G X* with /(zi) 7^ /(2:2)- The separation
condition is clearly equivalent to the simpler condition: given i/O
in X there is an / G X* with f(x) 7^ 0 (it suffices to observe
that /(h) 7^ f{x2) iff f(Xl - x2) + 0).
A TA A is called ample if it is ample as a TLS.
4.7.2. Remark. Clearly an ample algebra (or TLS) must
necessarily be Hausdorff.
4.7.3. PROPOSITION, (a) Every subalgebra A0 of an ample
algebra A is ample.
(b) A direct product or direct sum of ample algebras in ample.

214 Locally Pseudo-convex Spaces and Algebras
PROOF, (a) If y G Ao, y 7^ 0 then by ampleness of A there
is an / G A* with f(y) ^ 0. Write /0 = /|A0. Then /0 G A*0
and /o(j/) = /(j/) 7^ 0, whence Ao is ample.
(b) Let A = Y\ Aa be a direct product of ample algebras Aa.
If a = (aa) ^0 in A then aao ^ 0 for some ao. Since Aao is
ample we can choose fao G A*a with /ao(aao) 7^ 0. Define / on
A by f[x) = fai^a^), where x = (xa) and 7ra is the projection
x 1—> xa. Clearly / G A* and /(a) = f(aao) 7^ 0, so that A is
ample. Since the direct sum ^Aa is a subalgebra of A, by (a)
it is also ample, completing the proof.
4.7.4. PROPOSITION, (a) The unitization Ai of an ample
algebra A is ample.
(b) The complexification A of an ample real algebra A is
ample.
(c) If an ample real algebra A has complex structure then the
resulting complexal gebra A is ample.
PROOF, (a) Let a\ = ae\ + a [a G A) be a non-zero element
of A\. We have to show that there is a /1 G A\ with f\{a\) 7^ 0.
For the construction of f\ we have to consider two cases.
Case 1: a = 0,ai ~ a. Since A is ample there is an f G A*
with f(a) 7^ 0. Extend / to /1 on Ai by setting for xi — Aei + z
(x G A, A G K) /!(xi) = A+ f(x). Then /x G A\ and /i(ai) =
/(a) ^0.
Case 2: a^0. Set /i(Aei + x) - X. Then /1 G AJ and
/i(a!) = Q^0.
(b) Suppose that z0 G A, z0 = x0 + zj/0 7^ 0 (x0,y0 G A).
Since A is ample, and 2¾ or j/0 ^ 0, there is an /0 G A* with
/o(z0) or /o(j/o) 7^ °- Setting
/0(2) = /o(z) + ifo{y) (z = x + 1 j/)
and f(z) — fo(z) —ifo (iz), it is easy to check that / is C-linear
and / G (A)*. Since f(z0) = 2(f0(x0) + if0(y0)) ^0, A is ample.
(c) For / G A* set /(x) = /(x) - if(ix). Then

§ 7. Ample Algebras
215
f((a + ifi)x) = /((a + i0)x) - if{i{a + i/3)x)
~ af(x) + Pf{ix) - i. - /3f(x) - iaf(ix)
= (a + i/3)[f(x) - if(ix)} = (a + i/3)f(x)
so that / G (A)*. If x0 G A = A and 2¾ / 0 then by ampleness
of A there is an / G A* with /(z0) / °- But then / G (A)* and
/(xo) = /(zo) - iffao) / 0, proving A is ample.
4.7.5. PROPOSITION. For a TA which is a division algebra
to be ample it is sufficient that A* / {0}.
PROOF. Suppose that A is a division TA with A* / {0}.
If / G A*, //0 there is an ao G A with /(ao) / 0. For any
a in A with a / 0 define fa(x) = / (xa_1ao). Since the maps
x 1—> xa_1ao and j/ 1—> /(j/) are continuous, fa is continuous,
fa<EA*. Since /a(a) = /(ao) / 0, A is ample as required.
4.7.6. PROPOSITION. Every locally convex algebra - in
particular a Banach algebra - is ample.
PROOF. This is an immediate consequence of the Hahn-
Banach theorem (see [24, p.59]).
4.7.7. Remark. For 0 < p < 1; there are ^-Banach algebras
which are ample as well as those which are not ample.
For instance the algebra lp = lp(K) (example (v) of 3.4.6) is
an ample ^-Banach algebra. To see this define, for each n > 1,
fn(x) - xn, where x = (xi,x2,- ■ ■ ,xn,- ■ ■) = (xn) G lp. Then it
is easy to see that /„ e (lp)*. If a = (a„) G lp, a / 0 then for
some n, an / 0. Then /„(a) = an / 0, proving lp is ample.
Consider next the space Lp = 2/[0,l] consisting of
(equivalence classes) of K-valued Lebesgue measurable functions / such
that
f \f{t)\pdt < 00.
With
11/11 = /011/(01'*,
Lp is a ^-Banach space. It can be made into a ^-Banach algebra
by introducing in it trivial multiplication: for x,y G Lp, xy ~ 0.

216 Locally Pseudo-convex Spaces and Algebras
Lp is not ample since it is known that '0' is the only continuous
linear functional on Lp (see [24, p.36]).
§ 8. Topological Spectral Radius
4.8.1. DEFINITION. Let A = (A,p) be a ^-seminormed
algebra. Then, by 3.4.3, p is a. sm. and so the limit
vv(x) = lim p(xn)" < oo
v n—»oo v
exists, by 3.3.6. The non-negative real number vp{x) is called the
topological spectral radius of A, the adjective "topological" being
justified by:
4.8.2. LEMMA. If q is a p-seminorm with q ~ p then
PROOF. By 3.4.1, q is also a.sm.. Further, by 3.2.11 (ii) we
have constants Cq,Cp such that
P < Cql, 1 < CvP-
Therefore,
1 i- 1
vp(x) = lim p(xn)n ^ Km Cqn q(xn)n < uAx),
n—»oo ra—»oo *
since lim C„" = 1. Similarly, va(x) < ^(z)- Then vv(x) =
i^p(j/), as desired.
4.8.3. PROPOSITION. TAe topological spectral radius u(x) =
vp(x) possesses, besides the properties (i) - (v) of 3.3.7, the
following:
(vi) u(\x) = \\\pv{x);
(vii) v{x + j/) < ^(z) + u{y) provided x <-> j/.
PROOF. The property (vi) has already been established in
3.4.5 . So we have only to prove (vii). For this it is convenient to
follow the method of Bonsall-Duncan [4, p.71].
We may assume that p is sm.. Choose e > 0 and set
u = (v(x) + e)~~?x, v = (v(y) + e)~~ey.

§ 8. Topological Spectral Radius
217
Since x <-> y we have also u <-> v. Also, by construction
i/(«),i/(v) < 1, so that by 3.3.6, p(un),p(vn) < 1 for all
sufficiently large n. If follows that the semi-groups {un : n =
1,2,---},{v" : n = 1,2,---} are p-bounded. Since u <-> v the
semi-group 5 generated by u,v is
{«*,«', uV :k,j= 1,2,---}.
Since p is sm. we have
p(ukvj) < p(«*)p(vJ').
It follows from the above that 5 is p-bounded. By 3.5.10 we can
find a sm. ^-seminorm q such that q ~ p, ?(«),?(v) < 1. By
4.8.2 we have i/p = i/9 = v (say). Then
q(x) = q((v(x) + e)eu) = (v(x) + e)q(u) < ^(z) + e (since q(u) ^ 1).
Similarly, q(y) < i^(j/) + e. It follows that
f(x + y) < ?(x + !/) < ?(*) + ?(y) < K*) + v{y) + 2e-
From the arbitrariness of e we conclude that v(x + y) ^ ^(z) +
Hv)-
4.8.4 COROLLARY. If A is commutative then v is a
continuous sm. p -seminorm on A= (A,p). Further, vv — v.
PROOF. That v is a sm. ^-seminorm follows from 3.3.7 (iii),
4.8.3 ((vi), (vii)). The continuity of i/ is a consequence of the
inequality in 3.3.7 (i). Finally, vv — v since v[xn) = v{x)n (by
3.3.7 (ii b)).
(2n\ *"
4.8.5. LEMMA, lim = 2.
n->oo \n I
PROOF. Write an = nn{n})~x. Then a„/a„_i = (1- £)/(1-
\Y. Using the well-known result lim (1+ -)" = ez t (z e C) we
get
T We denote the classical exponential function by ez, so that in partic-
ii. ! 1
ular we have e = H 1 1----
1! 2!

218 Locally Pseudo-convex Spaces and Algebras
,. an 1 1
lim = =— = —r = e.
«->°° a„_i lim (1 - -)n e 1
It follows that
Therefore x
- a
lim a,? = lim —— = e. (1)
n->oo n->oo a„_i
,. ,2n\ 2" ,. ( 2n\ \ &
lim ( I = lim , ,
ra-»oo \ n I ra-»oo \(n!)
2n! „,„ / n'
„ i
lim { -^- ■ 22n . —
n->oo ^ (2n)2" \ra!
,. / 2n! \£ /b"
= hm . .„ ■ 2 ■ lim —
n->oo \[2n)in / n->oo \ nl
= e~ ■ 2 ■ e, where we have used (1).
= 2.
4.8.6. THEOREM (Zelazko). Let A be a commutative algebra
(over K, p(^ 0) a sm. p -seminorm on A, v — vp. Then:
(i) The unit ball B\ = {x G A : i>(a;) < 1} is convex.
(ii) ||i|| = v(x)^ is the gauge of B\ and it is a sm. seminorm.
PROOF, (i) By 4.8.4, v is a p-seminorm and consequently,
by 3.2.7 (i), B\ is closed in the v -topology. To show that B\ is
convex it is enough to show that it is midpoint convex. '
Assume first that v(x),v(y) < 1. Then
H 2 (x + v)) = y?v{x + y) = 27^^ +y^"' (1)
n
Since x <-> y, (x + y)n) = )> (£)z yn~k, so that by subadditivity
of p, we have
p((* + yB)<£^Vp(*V-*). (2)
Choose e > 0 such that 1/(1), i/(j/) < e < 1.
t i.e. x,y€~B~i => ^- €~B~i (see [15, p.17]).

§8. Topological Spectral Radius
219
Since v{x),v(y) < 1 we can find Ni such that
P(xn)«,p(yn)" < e> or, p(xn),p(yn) <en<l
for n > Ni. Therefore, by 3.3.4, p(xn),p(yn) —> 0 and
consequently we can find C > 0 such that p(xn),p(yn) < C, for all
n.
Since en —> 0 we can choose N2 such that e"C < 1 for all
n ^ N2. It follows that for n^ N — max(iVi, N2) and arbitrary
p{xnym) < p(x>(j/m) < e"C < 1. (3)
Similarly for arbitrary n and m ^ N,
p(xnym) < 1. (4)
From (2), (3), (4) we obtain for n 3¾ N,
p«*+^ke (T) '*(* v-*) * e (2;) % (2»+1) (2;)'
(5)
since ( A") < ( ") for all k. From the definition of v and (5) we
get
< — Um(2n + l)£(lim |2b) ")>
< — ( lim J V (since lim (2n + 1)^" = 1)
< ^ ■ 2s (using 4.8.5)
< 1,
which proves \{x + y) e B\. Next, if v{x), v{y) < 1, we can
choose a sequence A„ with 0 < A„ < 1 and A„ —> 1. Then
v(\nx), v(\ny) < 1, so that, by the result above, zn — | (A„z +
\ny) e B\. It follows that \{x + j/) = lim2r„ e £1. Thus Bi is
midpoint convex, completing the proof of (i).

220 Locally Pseudo-convex Spaces and Algebras
(ii) Since clearly, B\ = {x G A : \\x\\ < 1}, it follows from
4.1.13 that || ■ || is the gauge of B\. Finally, since p is sm., B\
is a subsemigroup and consequently the gauge || ■ || is sm..
4.8.7. COROLLARY. If the algebra A is a division algebra
then || ■ || is a norm.
PROOF. If x G A, i^O then x is invertible, and so by 3.3.7
(v), ||z|| > 0.
4.8.8. DEFINITION. An element x of ^-seminormed algebra
A = (A,p) is called topologically nilpotent or t. nilpotent if v{x) ~
vp(x) = 0. If A — (A, {px}) is a locally pseudo-convex algebra
then an element x G A is called t.nilpotent if each ua{x) —
Clearly every nilpotent element is t.nilpotent (xn — 0 =>
pa(xn)«=0).
4.8.9. LEMMA. If x is t. nilpotent then
(i) xn -+ 0,
(ii) Xx is t. nilpotent for every A G K.
PROOF, (i) Take any e with 0 < e < 1. Since va(x) —
lim pa(xn)" = 0, we have pa(xn) < en < e, for n > TV. It
ra—»oo v ' \ /
follows that pa(xn) -+0, xn -+ 0.
(ii) By 4.8.3 (vii), i>a(\x) = \X\',ua(x) = 0.
4.8.10. PROPOSITION. In a commutative p -seminormed
algebra A = (A,p) the set I of t.nilpotent elements is a closed
ideal of A.
PROOF. By 4.8.4, 3.3.7 (iii) we have:
v(x+ y) < v(x) +v(y),v(\x) = \X\pu(x),u(xy) < L>(x)i>(y).
Therefore J is an ideal. Moreover, J is closed, as follows from
the continuity of v (see 4.8.4).
4.8.11. PROPOSITION. Let A be a p-Banach algebra and
x G A. Then:
i
r(x) < v(x)e . (*)

§ 8. Topological Spectral Radius
221
If A is real we have also
f(x) ^ v{x)~e. (**)
PROOF. We may assume the norm || ■ || to be sm. and write
v = v.... If |A| > i/(x)p then
v(-\-lx) = |A|-"i/(i) < 1,
so that by 3.3.19, —A_1x is q. invertible. It follows by 1.7.8 that
a'(x) C {A G K : |A| < u(x)^} so that (*) holds.
If A is real we apply (*) to A and deduce (**).
4.8.12. COROLLARY. In any p-Banach algebra a t. nilpotent
element is q. nilpotent.
i
PROOF. Since r(x) < v{x)i>, v(x) = 0 => r(x) = 0 and
hence the result.

CHAPTER V
SOME ANALYSIS
§ 1. Vector-valued Differentiability and Analyticity
5.1.1. DEFINITION. Let X be a Hausdorff TLS and X*
its (continuous) dual. Let G C K be an open set. A function
/ : G —> X is called weakly differentiable if there is a function
g : G -+ X such that for each x* <E X*, the scalar-valued function
x*f(X) = x*(f(X)) has the scalar-valued function x*g(X) as its
derivative in the usual sense; we write g = f'(w> and call f'(w>
the weak derivative of /.
The function / is called strongly differentiable if for each A G
■in, !M ' ((A)
^->A // — A
exists in the topology of X. We denote this limit by /'(A) and
call /' the strong derivative of /.
5.1.2. LEMMA. A strongly differentiable function f is weakly
differentiable and continuous, and moreover, the strong derivative
/' coincides with the weak derivative f'(w>.
PROOF. It follows from the definition of the strong derivative
that
lim (/(//) - /(A)) = lim (// - A)/'(A) = 0,
/J—»A /J—»A
proving / is continuous. Again,
lim ,-/(,)-»•/(*) = lim x, ,fM-tw\ = l7,(A)i
^->a // - A /j->> V // - A /
whence / is weakly differentiable with f'(w> = /'.
5.1.3. DEFINITION. A function / : G —> X is called weafc/j/
analytic if x*f[X) — x*(f(X)) is an analytic function of A on G in
the usual sense, for each x* G X. We call / strongly analytic on G

§ 1. Vector-valued Differentiability and Analyticity 223
if around each point Ao G G C K, there is a neighbourhood {A G
K : | A — Aq I < ro} in which / has a power series representation >
oo
/(A) = X>»(A - Ao)",
n=0
where x„ G X and the series converges in the topology of X.
If X is a complex TLS we will also use the expression weakly
holomorphic (respy. strongly holomorphic) for weak analyticity
(respy. strong analyticity).
5.1.4. LEMMA. A strongly analytic function f is weakly
analytic.
PROOF. Suppose that
oo
/(A)=5>„(A-A0r.
n=0
Then, for each x* G X*, we have
oo
z7(A) = 5>*Or„)(A-A0)n,
n=0
so that / is weakly analytic.
5.1.5. Remark. If X is a complex, Frechet space (i.e. a
complete metrizable locally convex space) then for a function / :
G —> X, weak holomorphy of / => strong holomorphy of / (see
[24, p.79]).
5.1.6. Let X — (X, || ■ ||) be a ^-Banach space and
oo
/(A)=5>„An (x„GX,AGK) (*)
a series in X. Put
' In the representation we find it convenient to write the scalars on the
right side (instead of the customary left side)

224
Some Analysis
ro = lim ||x„||" , i?o = r0 ,R = RQ=r0p (lim = lim sup).
R is called the radius of convergence and {A G K : |A| = R} the
circle ' of convergence of the series (*).
Next let X = [X, {pa}) be a complete Hausdorff locally
pseudo-convex space. Set
i_
Pa
ra = lim pa(xn) n , Ra = raPa ,
ra—»00
i_
R ~ inf Ra ~ inf ra "a .
a a
R again is called the radius of convergence of the series (*) and
{A G K : |A| = R} its circle of convergence.
5.1.7. PROPOSITION, (a) The series (*) converges absolutely
[and hence also in X) for all A with |A| < R, and converges
uniformly for |A| < r, for any r with 0 < r < R.
(b) If \X\ > R the terms of the series are unbounded and
consequently the series diverges [i.e. does not converge).
(c) The term-wise differentiated series got from (*)
00
E n**^-1 (**)
has the same radius of convergence R (as (*)).
PROOF. We will first treat the case where X is a ^-Banach
space [X, || ■ ||). The proofs are all modelled after those of the
corresponding classical results.
00
(a) The absolute-valued series )> ||£ri||(|A|'')" - which is a
numerical series - converges, by the classical Abel's lemma, for
(Al'' < Rq, or equivalently, for |A| < R. Further, this series
T In the real case the circle of convergence is to be interpreted as the
end-points of an interval.

§ 1. Vector-valued Differentiability and Analyticity 225
converges uniformly for |A|P < rp, whence (*) converges
uniformly for |A| < r.
(b) If |A| > R, choose r so that |A| > r > R. Then
I l l -p—II ,|±
rP RP R0 n-oo" ""
and consequently there are arbitrarily large n with
II 111 1 MM1
Fn r > -7» or W > -^,-
It follows that
||x„A"|| = |A""|||x„|| > f ^J -.oo.
Thus the series (*) is not bounded (for |A| > R) and so not
convergent (by 3.2.15).
(c) If we put r'— lim ||(n + l)x„+i|| " , then
r' = lim [{(n + l)^}^^^!^] V = lim ||x„+1||mT = r,
where we have used the well-known result : n« —> 1 as n—>oo.
To extend the above proofs to the case where X is locally
pseudo-convex, we note that if |A| < Ra (for each a,) so that
00
the proof above for (a) shows that V]pa(^n)(|^|''a)" < °o, for
each pa, where X = (X, {pa}). Hence (a) follows for the
locally pseudo-convex case. Further, if |A| > R then there is an
a such that |A| > Ra, whence it can be shown, as above, that
pa(xnXn) —+ 00, so that the series is not pa -bounded. Therefore
the series is once again divergent, proving (b) for the general case.
Finally, if we set
r'a= limpa((n+l)x„+i)«+i
ra—»00
then as in the proof above for ^-Banach spaces we obtain ra = r'a,
whence R' — R, where R' is the radius of convergence of the

226
Some Analysis
series (**).
5.1.8. Proposition. The function
oo
/(a)=y: *»a» (*)
is strongly differentiable inside its circle of convergence and we
have
oo
f'(X)=^xnnXn-1. (**)
n=l
In particularly, /(A) is a continuous function of A.
PROOF. The proof we give is again modelled after the proof
in the classical case (see [25, p.200]). We take xn G X = (X, {pa})
and assume that the radius of convergence of the series (*), R > 0
(if R = 0 there is nothing to prove). Choose a A £ K with
|A| < R. Take a real number r with |A| < r < R. Writing
oo
g(X)^YxnnXn-1,
we have, for |//| < r,
v ~~ A „=i
where /3n = (//" - A")/(// - A) - nXn~K
Clearly /?i = 0, and for n > 2, we get
pn = {n-xy£kxk-lnn-k-1 (2)
jfc=l
(as can be easily checked).
Since |A|, |//| < r, and ^£=1 A; < n2, we obtain from (2)
1/3,,1 < |m - A| ( £ * ) r"~2 < I/* ~ A|n2r"-2 (n £ 2). (3)

§ 2. Exponential and Logarithmic Functions 227
Therefore
oo oo oo
n=l ri=l n=2
(since /?i = 0)
oo
^ \X~n\pa^2Pa(xn)n2^An-2^
i=2
(using (3))
i.e.
Vn=l / n=0
oo
For the series ^ x„+2(ra + 2)2A", we have
n=0
2£a
r^ = limpa(x„+2)"(n +2) - = ]impa(xn)r> = ra.
n—»oo v ra—»00
It follows that R' ~ R. Since r < /? = i?', the series on the RHS
of (4) converges, whence by (1), (4), as // —> A we obtain
/'(A) = *(A)
proving (**). The continuity, of / follows from 5.1.2.
§ 2. Exponential and Logarithmic Functions
5.2.1. DEFINITION. Let A be a unital Hausdorff TA with
unity e. The exponential function in A is defined by
oo
^t(x) = Exp(x)=E^, (x° = e, 0!=1) (*)
n=0
whenever the series on the right converges. We denote by Pe the
domain of E and by Ze its range. Since 0 e Pe, e G Ze both
The Hausdorff assumption ensures that E is well-defined.

228
Some Analysis
De,R-e are non-empty. The inverse function Log is defined on
Re by setting Log y = x if y = E(x). The function Log is in
general many-valued.
We denote the classical exponential function by
E(A)=eA = l + A + ^- + --- (AeK).
In particular, we write
e=l + l + i + -
5.2.2. PROPOSITION. De = A if A is a unital pseudo-
Michael algebra - inparticular a unital p -Banach-algebra;
moreover the series converges absolutely for all x G A.
PROOF. Let A - (A, P),P = {pa}. Since
p° UJ * hF = Un (say)
and "^+1 = (n?i)la —> 0, the series for E(x) converges absolutely
and so also in A for all x G A, by 3.1.24.
5.2.3. PROPOSITION. Let A be a pseudo-Michael algeba - in
particular a p -Banach algebra - with unity e. Then:
(i) E(0) = e; E(e) = ee.
(ii) E{x + y) = E{x)E{y) provided x <-> y.
(iii) E(-x) = E(x)~l.
(iv) If A is complex and u an idempotent then E(2niu) = 0
(•■ = >/=!).
PROOF, (i) Clear.
oo
(ii) E(x + y) = yj \x+v> _ Expanding the summands on
the RHS, using the binomial theorem, and regrouping the terms
(which is permissible because of absolute convergence) we find that
the cofficient of ^-r is

§2. Exponential and Logarithmic Functions 229
(»
X2
+ 2)(n
= e + x
+ 1)
X2
+
+
(n+l)x (n + 2)(n + l)
e+ n + 1 + 2!
= E(x).
Thus,
£(x + j/) = £(x)(e + y+»- + ■■■) = E{x)E{y).
(iii) This follows from (ii) by taking y = -x and using (i).
(iv) E(2niu) = e + {2ni+^~^ + ---}u. The expression in the
bracket above = e2m — 1 = 1 - 1 = 0. So we get E(2niu) = e.
5.2.4. COROLLARY. Let T be a bounded l.o. on a p-Banach
space X. Then
E(T) = I+T+ — + ---
is an invertible bounded l.o. on X.
PROOF. Consider the ^-Banach algebra B{X) and use 5.2.3
(iii). Note that E(T)~l = E(-T).
5.2.5. LEMMA. If A is a Banach algebra with unity e and
normalized norm || ■ ||, then
\\E{x)\\ <E(||*||).
PROOF. Clear.
5.2.6. If A is a TA without unity we can introduce the quasi-
exponential function Eq by
x2
Eq{x)=x+ — + ■■■
which makes sense (whenever there is convergence).
If A\ is the unitization of A and E the exponential function
in A\ then we have the obvious relation
E(x) = ei + Eq(x) (zGA),

230
Some Analysis
where e\ is the unity of A\. Using this relation and 5.2.3 (ii) we
can easily deduce the identity
Eq{x) o Eq(y) = Eq(x + y) whenever x <-> y.
Since Eq(0) = 0 we get Eq(-x) = {Eq(x))'.
5.2.7. DEFINITION. Let A be a TA and Gq its group of
q. invertible elements. The connected component of Gq containing
0 is called the principal component of Gq and we denote it by Gqo.
Similarly, if A has unity e, the connected component of the group
G,- (of invertible elements) containing e is denoted by G,e; G,e
is called the principal component of G,-.
5.2.8. PROPOSITION. In a C algebra A, Gqo is a closed
normal subgroup of Gq. If A has a unity e, G,e is a closed
normal subgroup of G,-.
PROOF. Since A is a C algebra, by 3.6.5 ((a), (b)) Gq and
G,- are TG's. The stated results now follow by applying to these
groups a standard result [21, p.39] in the theory of topological
groups regarding the identity component.
5.2.9. COROLLARY. If A is a p-Banach algebra then Gqo is
a clopen > normal subgroup; if A is, besides unital then G,e is a
clopen normal subgroup of G,\ These results hold, more generally,
when A is a pseudo-Michael Q algebra.
PROOF. In view of 5.2.8 we have only to prove Gq0, G,c are
clopen. By 3.6.23 (b), 3.6.21, 3.6.10 we conclude that Gq,G{ are
open when A is a ^-Banach algebra. Also they are open in the
pseudo-Michael algebra case by virtue of the Q algebra hypothesis
we have in this case. Finally, in all the cases A is locally connected
(see 3.2.8, 4.3.16). Hence the clopen properties of Gq0,G{e.
5.2.10. PROPOSITION. Let A - (A, || ■ ||) be a unital p-
Banach algebra. Then:
oo
(i) The series Logx = — )> -(e — x)n converges absolutely for
all x G A with v[e — x) < I, where v = i/iui.
T clopen=closed and open

§2. Exponential and Logarithmic Functions 231
(ii) E(Logx) = x.
(iii) For each x G A, the function Ex{\) = E(Xx) is analytic
for all A G K. In particular, Ex is entire when K = C.
PROOF. Suppose that i/(e-x)<e<l, then ||(e- x)n\\ < en
for n > (some) N. It follows that
(e-z)" j_ — <e".
11 n " n'"*' ; " n"
Since V, en converges the series for Logx converges absolutely
n=N
for v[e — x) < 1.
The identity (ii) can be checked, as in the classical case, by
substituting the series for Log x in each summand of the series
*(Log*)=E^
ra=0
and simplifying (the needed steps for the simplication are justified
because of absolute convergence).
Finally, we have
n=0 U-
Thus radius of convergence for the above power series in A (see
_ i
5.1.7) by r0 p, where
'\xn\\n
r0 = lim —T " = lim ,
(n!)n
1 .i
I/(a:)L1i™(^T)B)
p
n—»oo re;
Now
lim ( — j = lim (- tt-J—A =lim = 0.
"—°° Vn-V «—0° \(n +1)!'n!/ n + 1

232
Some Analysis
So ro = 0,r0 p = —j- = oo, whence the series EX(X) converges for
Op
all A and (iii) follows.
5.2.11. Remark. Proposition 5.2.10 can be extended to uni-
tal pseudo-Michael algebra A = (A, {pa}) in the following form:
For every x in A with va(e — x) < 1, for all a , the series for
Log x (defined in 5.2.10) converges absolutely and for all such x
the identity (ii) of 5.2.10 holds. Further, the function Ex = Ex(\)
is analytic for all A G K, for each a; G A.
5.2.12. PROPOSITION. Let A be a unital p-Banach
algebra. The range Re of the expoential function E generates
[algebraically) the principal component G,e [i.e. G,e is the smallest
subgroup of G, containing Re)-
PROOF. Since A is connected and E continuous, Re is
connected. By 5.2.3 (iii), RE C G{. Since e G RE (by 5.2.3 (i))
and Re is connected, Re Q G,e. Write
U ~ {x G A : \\e - x\\ < 1}.
Then, by 5.2.10 ((i), (ii)), U C RE C Gie. Since C/ is an open
neighbourhood of e in G,e and G,e is connected, U and hence
£,E generates G,e (see [20, p.37]).
5.2.13. COROLLARY. RE = Gie iff RE is a subgroup. In
particular, if A is commutative, Re = G,e.
PROOF. The first statement is immediate. For the second we
observe that by 5.2.3 ((i), (iii)), Re is a subgroup and hence the
result.
5.2.14. COROLLARY. (Theorem of Nagumo [11']). If
y G Re then y belongs to a connected abelian subgroup of G,-.
Conversely, every connected abelian subgroup H of G,- is
contained in Re- Hence y G Re (i.e. y has a logarithm) iff y
belongs to a connected abelian subgroup H of G,-.
PROOF. If y e RE then y = E(x), and H = {E(Xx) : -oo <
A < oo} is clearly a connected abelian subgroup of G,-. Next, by
1.1.9, there is a commutative subalgebra Bq with H C B0. Then

§ 2. Exponential and Logarithmic Functions 233
B = Bo is a commutative unital ^-Banach algebra with H C B.
By 5.2.13, the range E(B) = G,e(S) (the principal component in
B). Since e £ H and If is connected we have
H c G,-e(fl) = £(5) c £(A) = HE-
The above two results clearly imply the final assertion in the
corollary.
5.2.15. THEOREM (Gleason-Kahane-Zelazkot). Let A be a
complex unital p -Banach algeba. A linear functional x on A is
a character iff
(i) X(e) = 1.
(ii) ker% comprises non-invertible elements.
PROOF. Suppose that x is a character. Then, by 1.3.10,
condition (i) holds. Further, if x is invertible, x{x)x{xl) =
x(e) = 1, so that x{x) 7^ 0 and (ii) also holds.
It remains to prove the "if" part of the theorem. So assume
that x is a linear functional satisfying (i), (ii). Write M = kerx-
Then x being a linear functional 7^ 0, M is a subspace of A
of codim 1, so that we have A = Ce + M. We shall now show
that x is bounded with ||x|| < 1- If 2; 6 M, x(x) = 0, so that
0 = 1^(^)1 ^ 11 ar 11 _ Next consider an element y G A\M. Then y
has the form
y = \e-x (A gC,A^0,xgM).
Since X~1x G M, by hypothesis (ii) it is not invertible and so, by
3.3.20 (ii), ||e-A_1a;|| tt ^ l. Therefore
\x{y)\ = |x(Ae- x)\ = |A| < |A|||e - A_1x|p = ||Ae - x|p
= \\y\\p (using Ae — x = y).
It follows that ||x|| < 1, in particular that x is continuous.
To prove that X is a character it is enough by 1.3.19, to show
that x G M => x2 G M.
Consider now an x G M. We have to show that x2 G M. We
may clearly limit ourselves to the case that ||i|| = 1. Set
' These authors have obtained the result for Banach algebras.
'' We assume (without loss of generality) that || • || is sm..

234
Some Analysis
fW = E^r^ (AeC)" (1)
ra=0
Since
|X(in)| < Hxll ||zl£<Hp =1 (2)
and
°° \\\n
n=o re-
it follows that the series for /(A) is absolutely convergent and
hence /(A) is an ordinary entire function. Further, since
°° IA I"
l/(A)KE4r = EdAl)>
n=0 "•
/ has order at most 1. By continuity of \ we have
/(A) = x(E^)=xW*)) (4)
where E is the exponential function in A. Since E(Xx) is in-
vertible, /(A) 7^ 0, by (4) and (ii). We now apply Hadamard's
factorization theorem to conclude that
/(A) = eAa+" (a,^C) (5)
where e denotes the classical exponential function. From (l) we
find /(0) = 1 and /'(0) = x{x) = 0 (since x e M). These
imply that in (5) a = /3 = 0, whence /(A) = 1. Therefore
x(z2) = /" = 0, whence x2 G M, completing the proof.
§ 3. Square Roots and Quasi-square Roots
5.3.1. DEFINITION. Let A be an algebra and x G A. An
element y such that y2 = x is called a square root or sg.r. of
x; it may or may not exit. (Since ( — y)2 = y2 = x a sq.r. even

§ 3. Square Roots and Quasi-square Roots 235
when it exists is not unique). Similarly, an element y such that
y o y — x is called a quasi-square root or q.sq.r. of x.
5.3.2. LEMMA. If A has a unity e then an element y is a
q.sq.r. of an element x iff y + e is a sq.r. of x + e.
PROOF. This follows from the identity (j/ + e)2 = e + y o y.
5.3.3. LEMMA, (cf. [4, p.44, Lemma 12]). Let A = (A, P =
{pa}) be a pseudo-Michael algebra, x, y G A,x <-> y, and x o x =
y o y. If va{x+ y) < 2Pa [pa = homogenity index of pa) for all
a then x = y.
PROOF. Write a = \{x + y),b = x - y. Then ab = \(x2 ~
j/2) = —b (using x o x = j/°j/). This implies a o b = a. Since
^a(a) = ^"^(^"'"J') < 1> ^ 4.4.14, a is q. invertible, so that by
1.1.25 b = 0,x = y.
5.3.4. PROPOSITION, (cf. [4, p.44, Proposition 13]). In
a pseudo-Michael algebra A — [A,{pa}) every element a with
va(a) < 2Pa - 1 has a unique q.sq.r.b. with va(b) < 2Pa - 1.
Further, b <-> a and va{b) < va{a)- In particular, every t.nilpotent
element a has a unique t.nilpotent q.sq.r.b.
PROOF. Following Bonsall-Duncan [4, p.44], we use the
method based on the existence of fixed point for obtaining the
results. For each a, fix an rja such that va(a) < r)a < 2Pa — 1.
Then, by 3.5.10(i), we can choose a sm. pa -seminorm such that
9a ~ Pa,
va{a) < qa(a) < r,a < 2»" - 1 < t 2"-"1. (1)
Denote by Ba the closed (commutative) subalgebra of A
generated by a; Ba is a pseudo-Michael algebra. Set 5 = {x e Ba :
9a{x) < r)a for all a}, where r)a satisfies (i) . Then 5 is a closed
subset of A. Define a mapping T by
™ a - x2 ,
Tx = ~T-(xeS).
Then we have
2" - 1 = 2 • 2"-1 - 1 = 2'"1 + (2'-1 - 1) < 2'-1 (since p < 1).

236
Some Analysis
< --2^^ = ^,
so that T is a mapping 5 —> 5. Further, if x, y G 5,
x2 - y2 1
qa{Tx-Ty) = qa( ) < — qa{x + y)qa{x - y)
where fia = 21-^¾ < 21-^ -2^1 = 1. Thus T is a contraction
mapping and so by 4.3.15, T has a fixed point b : Tb = b. It
follows that 26 = a — b2 or b o b = a. Since 6 G 5 C Ba,6 <-> a.
Also,
va(b) < ga(6) < fja < 2"« - 1 < 2""-1.
To prove the uniqueness of b (under the condition va(b) < 2Pa -
1) we suppose that there is a &i with &10&1 = a,va{b\) < 2Pa-l <
2"«_1 (for all a). Then
a&i = (26i + 6j)6i = 6i(26i + 61) = 61a,
i.e. a <-> 61. Since 6 G Ba it is the limit of a sequence of
polynomials in a. Hence 6 <-> 61. Therefore
M& + 6i) < Mb) + MM < 2"""1 + 2""-1 = 2"«.
By 5.3.3, 6 = 61. Further, the uniqueness assertion implies
that 6 is independent of the particular choice of r)a satisfying
va{a) < r)a < 2Pa — 1. Hence, since ua(b) < r)a we conclude
that va(b) < va{d). Finally, the inequalities va(b) < va(a) imply
that ua(b) = 0 whenever va{a) = 0. This means that when a is
t.nilpotent its q.sq.r.b. is also t.nilpotent.
5.3.5. COROLLARY. In a unital pseudo-Michael algebra A =
(A,{pa}) with unity e, every element ai with ua{e — a{) < 2Pa —
1, for all a, has a unique square root 61 with
va{e - 61) < 2Pa ~ l,6i <-> ai; then va{e ~ b\) < va(e ~ a\).

§ 3. Square Roots and Quasi-square Roots 237
PROOF. Write a\ - e = a. Then, by 5.3.4, a has a q.sq.r. 6.
with b <-> a, i/(6) < i^(a). Writing &i = a + 6, by 5.3.2, &i is
a square root of a\. Also, the uniqueness of b\ is an immediate
consequence of the uniqueness of 6.
5.3.6. COROLLARY. In a p-Banach algebra A = (A,\\ • ||)
with v = v\\.\\, every element a with u{a) < 2P ~ 1 is q.invertible
and has a unique q.sq.r. b. with i/(b) < 2P ~ 1, b <-> a, v{b) < v{a);
b is also q. invertible. If a is t. nilpotent so is b.
PROOF. Since a ^-Banach algebra is a Michael algebra the
existence of 6 and all the properties stated in the corollary except
q. invertibility of a,b are immediate consequences of 5.3.4. The
q. invertibility of a or 6 follows from 3.3.19, since v (a),1/(6) <
2" - 1,< 1.
5.3.7. Remark. An element y of an algebra A such that
yn = x (x G A) is called an n th root. Similarly if we write
y°n — y o y o ■ ■ ■ o j/(n factors ) and y°n = x,
then y is called a quasi n th root or q.n th root of x. If A has a
unity e we have the relation e + y°n ~ (e + y)n, so that if y is a
q.n th root of x then e + y is n th root of e + x.
Proposition 5.3.4 can be generalized for q.n th roots as follows:
Each element a of a pseudo-Michael algebra A with
ua{a) < 2"- - 1/(- - 1)
n
has a unique q. nth root b with b <-> a,ua(b) < va{a). In
particular every t.nilpotent element a has a unique t.nilpotent q. nth
root 6. The proof is again by the fixed point theorem method.
Here we take r)a < 2^ - 1/(^ - 1) and define T by
(observe: (e + y)n-e=( i )» + ■■■+( £-1 )yn~1 + yn(ye
A).)
5.3.8. PROPOSITION. Let A be a complex or strictly real,

238
Some Analysis
commutative unital, pseudo-Michael - in particular p -Banach-
Algebra. For every element 6 G A with a(b) = {0,1} there is
a unique element d G \/A such that u = 6 + d is idempotent.
PROOF. Since, by 7.2.21, A is t. spectrally Gelfand we have
{0,1} = <7(6) = {X(6):XGAC}.
It follows that for every x,x(6) = 0 or 1, so that, %(6 ) =
x(6), 62 — 6 G ker %. Therefore
c = 62-6Gv/A = V/A (see 7.2.12).
Now
<j(26 - e) = 2<j(6) - 1 = {-1,1} ^ 0,
hence (26 - e)_1 exists. It follows that -4c(26 - e)~2 G \f~A and
so it is a t.nilpotent element (see 7.4.10). This element has, by
5.3.4 a t.nilpotent q.sq.r. /0 G A. By 7.4.10, /0 G \f~A. It follows
that if f ~ e + fo then
Sett
Then
Also
f2 = e-
<=-»
2 yj
b + d =
- 4c(2/ -
-e 26
2 + :
; 2
e + (26 -
2
e)-2-
P'-
"e/oe
«)/
'A.
and a simple computation shows that (6 + d)2 = b + d. Thus
u = b+d, with d G \/A, is an idempotent. If v = 6+</i(</i G \/a)
is also an idempotent then u — v G \/A, so that, by 7.2.12, 7.2.21,
<t'(u - v) = {0}. If follows from (vii) of 7.2.22, and 7.2.23 that
u — v, d — di, proving the uniqueness of the choice of d G \f~A.
For a motivation of the definition of d see [30, p.48).

§ 4. Complex Vector-valued Line Integrals 239
5.3.9, COROLLARY. If the algebra A is s.s. then every
element b G A, with a(b) = {0,1}, is an idempotent.
PROOF. Since \[A = {0}, d = 0, and b = b + d is
idempotent.
§ 4, Complex Vector-valued Line Integrals and
Cauchy's Theorems
5.4.1. Let X be a complex TLS and T a rectifiable
(continuous) path in the plane C, represented by
z = z(t) (a < t ^ P;a,/3 <E R).
Let / : r —> X be a (vector-valued) function on T. If n = {to =
a < t\ < £2 < • • • < tn = /3} be a partition of [a, /3} we set
SOO = S(* : 4) = £ /(4)(¾. - *,-_!) = £ /(*}) A*y>
where Zj = z(tj),z'j — z(t'j) for some t'- with ij_i < t'- < £y. We
denote max\tj - ij_i| by |7r|. If limS(7r), as \ir\ —> 0, exists we
j
say that the line integral; / fdz exists and write
f fdz = f f(z)dz = lim S(n).
Jt Jt M-o v '
It is clear that if / /(/2, / 17(/2: exist so does / (a/ +
(3g)dz (a,/3 G C) and we have
J (a/ + /fy)</2 = a I fdz + /3 f gdz.
5.4.2. LEMMA. Lei X,F 6e complex TLS's, T C C a pa*A,
f = f(z) : T —* X a function such that / /dz exists. Let T :

240 Some Analysis
X —> Y be a continuous linear transformation. Then J T fdz
T f fdz= f Tfdz
exists and
™ f " tz.
Proof.
T f fdz = TlimJ2f(z'k)A2k
= lim Y(Tf(z'k))AZk = f Tfdz.
|a|->o^--' Jr
5.4.3. COROLLARY. If / fdz exists and x* eX*, t then
x* (f fdz) = f x*f(z)dz = f x*fdz.
PROOF. This follows from 5.4.2 by taking Y = X* and T =
x .
5.4.4. COROLLARY. If X = A is a TA, T and f as in
5.4.2, and x G A then
(i) x^fd^=j{xf)dz.
(ii) (f fdz\x= I (fx)dz.
PROOF. These follow from 5.4.2, by taking Y = A and T =
lx or rx.
5.4.5. PROPOSITION. Let X = (X, ||-||) be a complex Banach
space, T a rectifiable path in C, and f : T —> X a continuous
function. Then / fdz exists and we have
X* denotes the (continuous) dual of X.

§ 4. Complex Vector-valued Line Integrals 241
|| f fdz\\ ^ M\T\. (*)
where M = sup{||/(A)|| : A e T} and \T\ denotes the length of
r.
PROOF (following [14, p.62]). Let T have parametric
representation z = z{t)(a. < t ^ /?). Since / is continuous on T, the
function f(t) = f(z(t)) is continuous on I = [ot,/3]. Since I is
compact, /* is uniformly continuous on I. Hence, given e > 0,
there is a 8 > 0 such that
\\r(f)~r(t")\\<-^ita^t'<t"^i3,
\t' - t"\ < 8, where |T| denotes the length of T. Let 7Ti,7T2 be
two partitions of I with |jti|,|jt2| < f- ^et ^3 ^e a cornrnon
refinement of 7Ti,7T2. Then it is easy to see that
\\S(^)--S(n3)\\<^-\T\=e- (J = 1,2).
Therefore we have
115(^)-5(^)11 < 115(^)-5(^)11 + 115(^)-5(^)11 < E-+€- = e.
It follows from the completeness of X that fTfdz = lim S[n)
M—°
exists. Further, since clearly ||5(7r)|| < M|r|, for each tt, the
inequality (*) follows.
5.4.6. The general existence result for line integrals proved
above (5.4.5) is no longer valid if X is a complex />-Banach space
with 0 < p < 1. However, even for such spaces the line integral
exists for a special class of functions (see 5.4.9).
5.4.7. DEFINITION. Let X be a complex ^-Banach space
and T a rectifiable path in C. A function / : T —> X is called
p-admissible or p-admissible on T if / has a representation
oo
/w = EwW (1)

242
Some Analysis
where Xj G X,(pj = <Pj(z) are bounded complex functions on T
such that the ordinary line integrals /r <pjdz exists and further
$Zll*y||W<°° (2)
where \\<pj\\ = \\<Pj\\r = sup|^j(^)|-
The condition (2) ensures, by the vector analogue of Weier-
strass M-test, that the series in (1) converges uniformly on T. It
follows that the function / will be continuous whenever all the
<Pj are continuous.
oo oo
5.4.8. LEMMA, (a). If f = z__,xj(Pj^ 9 = /Z^/^/ are
/=1 /=1
p - admissible on T so is f +g.
oo
(b) If f = 2_JxiiPi *'s p -admissible and tp a continuous func-
/=i
oo
tion on T then /^^ji^jip) *'s p-admissible on T.
/=i
PROOF, (a) Since
l^(A) + ^(A)|"<|W(A)|" + |^(A)|"
we have
oo oo oo
IZ ll^/ll 11^/ + V'/llr < IZ IM b/llr + £ ll^ill HV'/llr < °°-
/=i /=i /=i
Hence / + # is ^-admissible,
(b)
ll^/V'llr =sup|<py(A) V/(A)|f < ll^/llrllV'lln
so that
IZ ll^/lll^/V'llr < HV'llrdZ ll^/IH^/llr) < °°»
/=i /=i

§ 4. Complex Vector-valued Line Integrals 243
which proves (b).
5.4.9. THEOREM.([31, p.57]). If f is p-admissible on T
then the line integral /r fdz exists and we have
/ fdz = /X; / ipjdz.
h .=1 h
(*)
PROOF. The series in (*) converges absolutely since
YL^xi\ <Pidz\\ < D Wxi\W J ^yfel'^Hltall ll^illr lrl"
oo
where |T| denotes the length of the curve Y. Therefore, this series
converges in X (by 3.1.24).
Let z = z(t) (a < t < /3) be the parametric representation of
T. Let 7r = {a — to < t\ < ■ ■ ■ < tn = /3} be a decomposition of
[a,/?], |7r| = max \tj - tj_1\, and
S = S(n) = J2f(4)(zk~zk-i)
jfc=l
with 4 = z(t'k),zk = z(tk) and tk-i < t'k < tk.
We have:
oo .
II5"" J1XJ / ^Jrf2rll =
n oo oo .
k=lj = l j=l JT
J2xj\^2^-(4)(¾-**-i)- / ^,-^[
j-i \k-i jt )
3 = 1
= ||Si + S2
(1)

244
Some Analysis
where
and
jfc=i
Since
Si = J2xiXi> S2 = J2 xiXi
j—l j=m+l
n
Xj = Yl Vi{z'k){zk - Zk-i) - J <Pjdz.
I / <Pjdz\ < \\<pj\\ \T\, (2)
where we have written \\<Pj\\ for ||£>j||r; we obtain
|A,-|< ll^-H |r| + Ita-H |r| = 2||^-|| |r|.
It follows that
is2ii = ii Yl xixi\\ ^ J2 \\xj\\\x
j—rn+l j=m+l
P
< 1P\V\P Y^ llr •llllwll''
j=m+l
oo
<2|iy £ 11^1111^11" • (3)
j=m+l
From (3), by using (2) of 5.4.7, we can find, for a given e > 0 an
integer N such that for m> JV we have
l|S2||<|. (4)
Next, for any fixed m > TV we can choose a 6 > 0 such that for
a decomposition n with |7r| < 6 we have
\xj\" = I / <Pj<L* ~ J2 ^(4)(¾ - zk-i)Y
jfc=:l
< -sr^ (j = 1,---,m). (5)
2£||z*|
jfc=l
It follows that
e
|Si||<E INI |A,f < -, (6)
3 = 1

§ 4. Complex Vector-valued Line Integrals 245
so that by the choice of N we get, using (1),(4),(6),
oo .
S ~ J2xi I ^idz
IS1 + S2II < ||Si|| + ||S2|| < f,
for any 5 = S(n) with \w\ < 8. Therefore
. 00
/ fdz= lim S = y^Xj / ipjdz
completing the proof.
5.4.10. Corollary. \\fTfdz\\ < Ell^ll II^IKirK-
PROOF. This follows from (*) of 5.4.9, using the inequalities
(2) therein.
5.4.11. DEFINITION. Let X be a complex ^-Banach space,
G C C, an open set, and / : G —> X a function. The function /
is called p- admissible holomorphic on G if / has a representation
00
/ = 1135^ (*)
with £y G X, £>y an ordinary holomorphic function on G, and
further for every compact set K C G, we have
00
HIM Ita-ll* < °° (**)
3 = 1
where ||v?j||jf = sup|^(A)|.
The condition (**) implies that the series /J^jVj(A) con-
j
verges absolutely for every A G if. For any A G G, by taking
K = {A} we conclude that the series )> £j£>j(A) converges abso-
3
00
lutely and we have /(A) = V] Xj<pj(X).
3=1
5.4.12. DEFINITION. A function / •. G -> X is said to
be locally ^-admissible holomorphic at Aq G G if there is an

246
Some Analysis
open set Go such that Ao G Go G G and /|Go —> -X is ^-
admissible holomorphic. If / is locally ^> -admissible holomorphic
at every point A G G we call / locally p -admissible holomorphic
on G. Clearly " p -admissible holomorphic on G " => "locally p-
admissible holomorphic on G", but not vice-versa (see 6.2.9).
oo oo
5.4.13. LEMMA, (a) If f = E1/*3/, g — "^Xjtpj are locally
/= i /= i
p -admissible [respy. p -admissible) holomorphic on G then so is
f + 9-
oo
(b) If f = /_,£/£>/ is locally p-admissible [respy. p-
/=i
admissible) holomorphic on G and tp a holomorphic function on
G then
oo
/=1
is locally p -admissible [respy. p -admissible) holomorphic on G.
PROOF. Similar to that of 5.4.8 .
5.4.14. PROPOSITION. A p -admissible holomorphic function
f : G —> X is strongly holomorphic and its strong derivative f is
p-admissible holomorphic. Moreover, we have
oo
/'(A) = 5>^(A) (*)
/=i
and in general,
oo
/W(A) = E^/")W (»= 1,2,--) (**)
/=i
where f(n> is the nth strong derivative of f and <pj' the nth
ordinary derivative of cpj.
PROOF. We have, for A,A + //GG,

§ 4. Complex Vector-valued Line Integrals 247
/(A+ //)-/(A) ^ lm||
A4
Let D be a closed disc centre A and radius r such that
D C G, and write C = 3D = {// e C : |//| = r}. Using the
classical Cauchy integral representation theorem we have
<Pj(\ + li) -<P,-(A) ,m_
l r i ii i
hh^
dz.
_// {z ~ \ ~ (A z~X) (z — A)2.
Now choose // such that |//| < |, so that for z G C we have
|z - A - //| ^ |z - A| - |//| = r - |//| > ~. Then
<p,-(A + //) -<pj (A)
^-(A)
M
ImIm I, 11 2||^-||c| i
(2)
From (1),(2) we get
/(A+ //)-/(A) ^
^ /= i
9 9 oo
< V llr-IK—-V\\(n-\\p \ii\f — ( —VI nK V llr-II IL,r IK C\\
^ Z^Wxi\\\r2> II^jIIcIMI — \r2) I A*-1 2^ IfjIIII^jIIc- lrJ
3 3 = 1
Since / is p-admissible the last sum Y^ in (3) is finite. By
making // —+ 0 in (3) the R.H.S —> 0 and we get (*).
It remains to prove that /' is also p -admissible. Let K C G
be a compact set. Since G is Hausdorff locally compact we can
find an open set H with
K C H C Kx = H C G and A"i compact .
Choose an open covering {Da} of ii" by open discs Da C IT. Let
6 be the Lebegue number of this covering. Then for any A £ K,

248
Some Analysis
we have (the disc) D = D{\, f) C some Da C H C KY. By
Cauchy's estimates we have:
I^A Jl -,5/3- #
so that 11(p'A\k ^ g - It follows that
OO o oo
Ellr-IIIU' IK < f-VV llr-llll</3-IK < no
llXjllll^jllif ^ \{j) Z^ ll^llll^jllif! < °°>
y=i j=i
whence /' is ^-admissible. Finally, the representation (**) is
obtained by iteration (using (*)).
5.4.15. We proceed to obtain a version of the Cauchy integral
formula for vector-valued functions. First we recall some classical
concepts. We call a closed rectifiable path in the plane C a
contour. The index of a point A G C\X with respect to a contour T
is the integer I given by the integral
K ' ' 2m Jr z-X
If T is contained in an open set G C C, we call T homologous
to zero (in G ), in symbols, T ~ 0, if
I(A,T) = 0 for all A G C\G.
Again, we call T homotopic to 0 (in G), in symbols, T « 0, if T
is homotopic to a constant curve (i.e. a point) in G in the usual
topological sense.
It is known < that:
r«o=>r~o.
Let G C C be an open set and 5 C G a subset. Following [24,
p.241] we say that a contour T C G surrounds S in G if the
following conditions are satisfied: 5 f]T = 0,
' See, for example [8, p.93, 6.10).

§4. Complex Vector-valued Line Integrals 249
T(A'r)- { o if a!c\g.
5.4.16. THEOREM (Cauchy integral theorems). Let X be a
complex p -Banach space (0 < p < 1) and f = /(A) a X -valued
(complex) strongly differentiate function on an open set G C C.
If 0 < p < 1 we also assume that f is locally p -admissible
holomorphic on G. Let T be contour in G with T ~ 0. Then we
have:
(a) Integral theorem: /r f[z)dz = 0.
(b) Integral formulae: For every A G G\T with I(A, T) = 1,
and more generally,
/W(A) = — /"-J^L-T^ (« = 1.2,---) (**)
where /("' denotes the nth strong derivative of f.
PROOF, (a) First assume that X is a Banach space {p — 1).
Then by 5.1.2, 5.4.5 /r f{z)dz exists. If x* G X then by 5.4.3
*{^f{z)dz) = j x*f{z)dz = 0,
by the classical Cauchy theorem since x* f is holomorphic. Since
the above conclusion holds for all x* e X* and X* separates
points (X being ample by 4.7.6) we obtain /r f[z)dz = 0,
completing the proof of (a) for a Banach space X. Now assume that
X is a ^-Banach space and / is ^-admissible holomorphic on
G. Then we have
/(A)^W(A)(AGG)
3
when <pj are holomorphic. By 5.4.9,
J f{z)dz = J2xi J <Pj(z)dz = 0,

250
Some Analysis
since each /r <pjdz = 0, by the classical Cauchy theorem. Next
consider the case where / is only locally ^-admissible holomor-
phic on G. Since Y is compact we find open sets
Gj(j=l,---,n)cG
such that fj = f\Gj —> X is ^-admissible holomorphic on Gj.
It is not hard ' to see that we can find contours Tj such that
Tj C Gj and
r~r! + ■■■ + rn.
It follows that
jvf{z)dz=J2JT f(z)dz = 0
3 = 1---1
(b) In the case X is a Banach space the integral formula (*)
is obtained by applying x* G X*, exactly as in the above proof of
(a). Thus
for every x*, whence (*) follows. The proof of (**) (for this
case) is similar.
Next consider the case where X is a ^-Banach space and /
^-admissible holomorphic. Write
f(z) = 2_^xi(Piiz) (fi holomorphic ) (1)
Then
g{z) = Y.*M*)- (2)
' See [31, pp.62-63 (diagrams)j.

§4. Complex Vector-valued Line Integrals 251
First assume that / is ^-admissible on G. Then we see that each
tpj is holomorphic on Go = G\{A}. Let K C Go, be compact.
By compactness of K there is an r > 0 such that
\z - X\> r for all z <E K.
Therefore
n , n IM*)I „ \\<Pj\\k
Mill* = SUP 71—jn^fl ^ T^-'
so that
< _J Vllx-llllwll''
r(n+l)p
3 = 1
< OO.
It follows that g(z) is ^-admissible holomorphic on Gj, so that
by (*) of 5.4.9 we obtain from (2)
-^[g(z)dz = f>— /\ ^ ydz
2m 7r f-J 22*ih (z-X)n+1
2 — 1 x
^X] n\
3
= -J[n)W (by (**) of 5.4.14). (3)
re!
Again from (1) we get
—. [ g(z)dz = — f , HZ] -dz. (4)
2xiJry ' 2niJr {z - A)"+* K '
From (3),(4) we conclude that
^/rl^IF1^-7 (A)-

252
Some Analysis
The extension of the proof to the case where / is locally p-
holomorphic on G is carried out as in the proof of (a).
5.4.17. Corollary. If /(A) = ^2xj<pj(X) then
3
3
/W(A) = 5>yPH(A).
dz
PROOF. We have
/Wm - n! f /(*) a, -Vr n! / ^
7 l J " 2jti 7r (* - X)n+1 y 3 2ni Jr {z - A)»+*
3
(using the classical Cauchy representation theorem for <pj).
5.4.18. COROLLARY(Cauchy's estimates). Suppose that X G
G C C and To a circ/e centre A, radius r, such that To C G.
TAen:
(i) If X is a complex Banach space and f : G —> X strongly
differentiate (strongly holomorphic) then
ll/"»WII < ^
where M = sup |/(z)|-
|*-A| = r
(ii) If X is a complex p -Banach space and f : G —+ X
admissible p -holomorphic, then
I oo
PROOF, (i) We have from the integral formula
iifWmn - ii n! f fW ./-ii
ll; (A)I1 ~ fe/rFAp "
n! M
2
Mn\
C — ■ r ■ 27rr( using (*) of 5.4.5)

§5. Power Series Operating in TA's
253
(ii) Here we have
W v^„ ..,n!.
using the classical Cauchy estimates. Thus we get (*).
5.4.19. COROLLARY. A (strongly) entire (respy. p -admissible
entire) function on a complex, Banach space (respy. p -Banach
space) is a constant.
PROOF. This follows, as in the classical case, from the Cauchy
estimate for n = 1, by making r —> oo.
§ 5. Power Series Operating in TA's
oo
5.5.1. Let /(A) = 2_]7nA" (7„ e K) be a power series. Let
n=0
A be a unital TA (with unity e) and x G A. We say that /
oo
operates on x if the series Y^7„a;" (recall x° — e ) converges in
n=0
A. Even when A has no unity it is meaningful to consider the
operation of a power series / with /(0) = 70 = 0 on an element
00
x : \ 7„x". (A power series / with /(0) is called a power series
n=l
vanishing at 0.)
Let now A = (A, {pa}) be a locally pseudo-convex algebra.
Then we can speak of / = 2_,7n£" operating absolutely on x
n
if }^pa(xn) converges for each pa. If A is complete (or even
sequentially complete) and / operates absolutely on x then /
also operates on x (because of 3.1.24).
An entire t function / with /(0) = 0 is said to operate on
1 entire in the real case means real analytic everywhere, i.e. is represented
by a power series which converges everywhere in R .

254
Some Analysis
an element x of a TA A if its power series expansion /_,7nA"
n
operates on x; if A is unital we can consider the operation of any
entire function on x. If a power series / operates on every x G A
we say that f operates on A.
5.5.2. Remark. Every entire function / with /(0) =
0 (respy. every entire function /) clearly operates on
any nilpotent element a of a TA A (respy.unital TA A).
Every such / operates also on an idempotent element u
( E ^"u" = 7oe + ( ^ 7« ] « J •
5.5.3. PROPOSITION. Let A = (A, {pa}) be a complete (or
even sequentially complete) locally pseudo-convex TA. Then all
entire functions vanishing at 0 operate on an element x iff
va(x) = \\mpa(xn)n < co,
n
or equivalently
K{x) = snppa(xn)n < CO,
n
for every pa. In particular, entire functions vanishing at 0 operate
on any t. nilpotent element.
PROOF. First suppose that v*a(x) < oo for all a. Choose a
C = Ca > 0 such that i/* < C. Then pa(xn) < Cn (n = 1, 2, • • •)
oo
Let /(A) = z2^n^n be an entire function (vanishing at 0). Then
n = l
lim I7J" = 0. It follows that, given an e with 0 < e < 1, there is
ra—»00
an integer N = Na > 0 such that for n > TV, |7„|« < (eC-1)''" .
Therefore
00
E M^*") = E \ln\PaPa(xn) < E(eC_1)"C"
00
< E f" < oo-

§5. Power Series Operating in TA's
255
Thus }2lnXn
converges absolutely and so also in A. This means
n
that / operates on x, as required .
Next suppose that va(x) = oo for some a. Let p = pa be the
homogenity index of pa. Since va{x) = oo, there is a sequence
0 < ni < 7i2 < • • • of integers such that
Pa{xnk)^ > k (k= 1,2,---).
But then the entire function
which vanishes at 0, does not operate on x, since pa(xnk)/knk > 1
xnk
(so that) —wjt ~h 0 and the series for f(x) does not converge (by
k p
2.1.32).
For the particular case statement in the proposition it is
enough to observe that if x is t. nilpotent then, by definition,
^a(z) = 0 for all a.
5.5.4. COROLLARY. All entire functions vanishing at 0
operate on A iff
v*a{x) < oo for all x and all a.
5.5.5. COROLLARY. All entire functions vanishing at 0
operate on any complete locally sm. pseudo-convex algebra A.
PROOF. If A = (A,{pa}) then since pa is sm., by 3.3.6,
^(^)(^ Pa{x)) < oo, and consequently v*a(x) < oo. The corollary
now follows from 5.5.4.
5.5.6. PROPOSITION. Let A = (A, {pj}) be a pseudo-$-
algebra with p\ < p2 < •• • ■ If there are constants Cj<n (j, n =
1, 2, ■ ■ ■) such that
(i) Pj{x1x2---xn)^cjinp3-+i(x1)---p3-+i(xn) for xu---,xn£
A
(ii) sup tfcj~ ~ Cj < oo,

256
Some Analysis
then there exists a family {qj : j > 2} of sm. pj -seminorms with
{aj '■ 3 ^ 2} ~ {pj : j > 1}, so that A is a locally sm. pseudo- $ -
algebra.
PROOF. Write Bj(r) = {x G A : Pj(x) < r}. By conditions
(i),(ii) we have
Pj(xix2 ■••£„)< cfpj(xi) ■ ■ • Pj{xn). (1)
It follows that the Cartesian n th power
Bf^cj1) = (Bj^cj1))" C 5,(1). (2)
Set
oo
^•+i = ^y+1((J ^+1^71)) Cj"=i,2,-0 (3)
where Cp.+l denotes the py+i-convex hull. Since Pj+i < Pj,
Bj{\) - which is absolutely pj -convex - is also, by 4.1.7,
absolutely Pj+i -convex, so that (2) implies that Uj+i C Bj[l). Also,
from the definition of Uj+\ we have Bj+i^cJ1) C [/,-+1. Thus
B]+1(cji) C [/,.+1 C B,+ 1(l). (4)
By 4.1.10, Uj+i determines a Pj+\ -gauge qj+i which is sm.
(since Uj+i is a subsemigroup as can be easily seen). Taking the
Pj+i -gauges of the sets in (4) and using the relations (*), (* * *)
of 4.1.8 and Remark 4.1.9 we get
Pj"' ^H+i^c^Pj+i- (5)
It follows from (5) that {qn : n > 2} ~ {pn : n > 1}, completing
the proof.
5.5.7. LEMMA. Let A be a commutative algebra. For
X\, ■ ■ ■ ,xn G A set
n

§5. Power Series Operating in TA's 257
where the summation is over all i\,- ■ ■ ,ik with 1 < ii < t2 • --ik ^
n. Then
^2-^ = ^1)(-1)^^1,--,¾). (*)
n! *=1
PROOF. Using the multinomial theorem the coefficient of
the typical term x"1 ■ ■ ■ x"r (1 < r < k) in the development
of <" is (.,1).1.(.,1) (*-')■ since (*-) otthesub-
sets of {1,---,71} which have k elements contain ii,---,ir and
each of these subsets yield the typical term with the coeffi-
n!
cient —'—, ^. Therefore the coefficient of x"1 ■ ■ ■ x"r in
(ni!)---(nr!)
n
H(-1)*w'i',)» is equal to
jfc=l
(^^£^¢-^=1^^^-^=0
when r ^ n, and is equal to n!(—1)" when r — n This completes
the proof.
5.5.8. THEOREM. Let A = (A, {pj}) be a commutative 3-
algebra, where we may assume that {pj} satisfy the condition
pi <P2 < ■■■; (*)
PjM < Pi+i(x)pj+1(y) (**)
(see 4.6.1). If there exist constants c*- such that
(i) Pj(xn) < C*„ py+iW (n,i='l,2---)
(") SUP« ^/¾ = c* < oo (j = 1,2,---)
iAen A is a locally sm. 3 -algebra.
PROOF. Suppose that xi,X2,- • • ,xn G A. Assume first that
Pj + i(xk) ^ 0 (k = 1, ••-,«). Write j/* = a^/p^^a;*); then
Py+i(j/)fc) = 1 (A; = 1,-•• ,n). It follows that

258
Some Analysis
Pj(xix2 ■■■Xn) =
= Py+ifci) • ••Pi+i{xn)pj(yiy2 •••yn)
= P] + l{Xl)---Pj+l(xn)Pj{—7-J2Wicn){yi>---,yn))
^ k
< —,PiMxi)---Pi+i(x*)'52pi(wkn\yi>'--,yn)) (1)
n!
where we have used indentity (*) of 5.5.7 and the subadditvity of
Pj. Using the definition of Wj^' we obtain the inequality.
< E Cj*,r.[Pj + l(»l) + --- + ^^ + 1(^)]0
»1 > — ,»*
< Es>*"<Ec>"
< «;,» (* ) »"■ (2)
From (1),(2) we get
^-(2^2---2^) < (^jEcJ>( k )nn)Pj+iixi)---Pj+i{xn)
k
< cy.nPj+i^O-'-Py+i^n) (3)
where
k
If for some £,^-(2^) = 0, then by condition (**) above we have
Pj{xix2---xn) = pj(xix2---xk_1 xk+1---xnxk)
< Pj + l{xiX2 ■ ■ • £jfc_i xk+i ■ ■ ■ xn)pj(xk) = 0
so that (3) is satisfied trivially. Thus (3) holds without any
restriction on xk. This means that condition (i) of 5.5.6 holds. We

§5. Power Series Operating in TA's
259
shall next show condition (ii) of 5.5.6 also holds. Now
~nT
so that
2n n ,_,
hm —= = 2 hm —== = 2e (5)
"-00 \/n\ "-00 Vn!
where we have used for the last equality the result (1) in the proof
of 4.8.5. It follows from (4),(5) that
Ji™ (CJ>)" = 2e i™ (cj>)" = 2ec*r
The theorem now follows from 5.5.6
5.5.9. Remark. One does not know whether the result in
5.5.8 is valid for p- 3-algebras (0 < p < 1). The above method of
proof, however, breaks down in the p- 3 case, since corresponding
to the limit in (4) we now have lim2n/(n!)" which turns out to
be infinite
( Km p- = lim
n-*°° (n\)n -~'~
n—*oo
(n!);
n1 p —> ep ■ oo = oo).
5.5.10. THEOREM (Zelazko [31, p.140]). Let A = (A, {pa})
be a commutative 3 -algebra, where we assume that the sequence
{pa} satisfy the conditions (*),(**) in 5.5.8. If for every x E A
and every j we have
Vj(x) = suppj(a;")" < oo
then A is a locally sm. 3 algebra.
PROOF. Observe first that v*-{x) = lim„ fn(x), where
fn(x) = maxpj^a;*)^.
Since the py are continuous so are /„. Hence vUx) is the
pointwise limit of continuous functions of x on a complete metric

260
Some Analysis
space. Therefore v* belongs to the first Baire class of functions
and consequently v*- has at least one point of continuity xq in
A. This means that there are constants C, 8 > 0 and an integer
k' such that
pk>(x - x0) < 8 => v*{x) < C.
If k = max(j, k') then we have
pk(x - x0) < 8 => v*(x) < C,
since by hypothesis pj < pJ+1 for every j. So we get from the
definition of i/j that
pk(x -x0)^8=> Pj(xn) < C" (n = 1, 2, • • •)• (1)
In particular, we have Pj(£o) ^ Cn.
Writing y ~ x~ xq, we have for any x satisfying the condition
in (1),
Pi-i(yn) = w-^-ioD^h E±UK*.
A:
i :#: :#:«
n
\jfc=0
< t(nXi('k)pA'rkxt{ k )ckcn~k
k=0 W jfc=0
< 2"C" = (2C)n. (2)
Let now j/ be an element of A with Pk{y) 7^ 0. Writing j/i =
8y/pk{y) we have Pjt(j/i) = 6, so that by (2) we obtain
Pi-M) < (2C)"> which gives
of
Pi-i(yn) < (x)>*(j/)"- (3)
If pjt(j/) = 0 then since k ^ j we have
Pj-i(»n) < Pjivfrjiy'1'1) < M^Mj/"-1) = o.
This means that (3) holds in this case as well.
The integer k figuring above depends on j, so we shall write
k — k(j). Set
m = 1, n2 = k(2), ■■■, nj = k(nj_i + 1).

§5. Power Series Operating in TA's 261
If we set qj = {pn } then {qj} ~ {pj} and by virtue of inequality
(3), the conditions (i),(ii) of 5.5.8 are satisfied and the theorem
follows from 5.5.8.
5,5,11, COROLLARY (Mitagin-Rolewicz-Zelazko [10' p.295]).
A commutative 5 -algebra on which all entire functions vanishing
at 0 operate is a locally sm. 5 -algebra.
PROOF. This follows from 5.5.4, 5.5.10.

CHAPTER VI
SPECTRAL ANALYSIS IN TA's
§ 1. Spectral Properties
6.1.1. PROPOSITION. Let A be a Q algebra (in particular
a p-Banach algebra) and x G A. Then the quasi-resolvent' set
p'(x) is an open subset and the quasi-spectrum* a'(x) a compact
subset of K.
PROOF. If A0 e p'(x) then A0 7^ 0 (by 1.7.6) and -Aq 1x is
q. invertible. Since A is a Q algebra there is an open
neighbourhood U of —Aq x, comprising q. invertible elements. Since the
map A i—> — X~1x is continuous, there is an open neighbourhood
N of Ao such that for A G N, —X~1x G U, so that — X~1x is
q. invertible. Thus N C p'(x) and p'(x) is open, its complement
o'(x) is closed.
Next let U(0) be an open nucleus comprising q. invertible
elements. Since — A_1a; —> 0 as |A| —> oo and U(0) is open there
is a constant C such that — X~1x G U(0) for |A| > C. Hence, for
such A, — X~1x is q. invertible and A G p'(x). This implies that
if A G a'(x) then |A| < C, whence a'(x) is bounded. Being both
bounded and closed (t'[x) is compact, as required.
6.1.2. Corollary. In an I algebra (in particular a unital p-
Banach algebra) A, p(x) is open, o~(x) is compact, and o(x) ^ K.
PROOF. By 3.6.13, A is a Q algebra and so by 6.1.1,
a(x) U{0} = a'(x) is compact. If 0 ^ a(x) then x is
invertible. Since A is an I algebra there is an open neighbourhood
U of x, comprising invertible elements. Since x — Xe —> x as
|A| —+ 0, there is an e > 0 such that for |A| < e, x ~ Xe G U and
hence invertible, so that A G p(x). It follows that 0 ^ a(x) (the
closure of a(x)). On the other hand, since a(x) C a'(x) and
For definition see 1.7.5.

§ 1. Spectral Properties
263
(t'{x) is closed we have a{x) C a'{x) = a{x) U{0}. Hence a(x) =
a(x),a(x) is compact and p(x) is open. Finally, since a(x) is
bounded, a(x) ^ K.
6.1.3. PROPOSITION. 7/ A is a real Q algebra and x G A
then the extended quasi-spectrum o'(x) is compact, in particular
closed, and r(x) < f(x) < oo.
PROOF. Let A = a + i/3 ^ 0, be a complex number. Then
x2 — 2ax x2 — 2ax .,,
2 , a2 = —rvf2 " °» as A -> oo
az + /?z |A|Z
(note |a|/|A|2<4r-^0). It follows that if U is a nucleus of A,
comprising q. invertible elements then there is a C > 0 such that
x2 - 2ax
cxL + pL
and hence y q. invertible, if |A| > C. This implies, by 1.8.5, that
for |A| > C, A ^ &'{x)- This means that cr'(x) is bounded. To
complete the proof of its compactness it remains to show that
cr'(x) is closed, or equivalently, p'(x) is open. Using 1.8.5 and the
fact that
x2 ~ 2ax _ x2 - 2(Re A)a;
iT|2 _ iT|2
is a continuous function of A (for A^O), it can be shown, exactly
as in the proof in 6.1.1 for p'{x) being open, that p'(x) is open,
completing the proof of the proposition.
6.1.4. COROLLARY. If A is a unital Q algebra then v(x)
is compact.
PROOF. It is clearly sufficient to prove that a(x) is closed
in o'{x) = a(x) U{0} and we carry out the proof as in 6.1.2. We
can assume that 0 ¢: a{x), so that 0 G p(x). Then x, and hence
x2, is invertible. Therefore, A being an I algebra we can choose
an open neighbourhood U of i2, comprising invertible elements.
Since
%a,p = (x - ae)2 + /?2e —> z2, asa^-t 0,

264
Spectral Analysis in TA 's
we can find an e > 0 such that for \a\, \/3\ < e, xa$ G U. If
Za,p = x ~ (a + ip)e
then
za,/3Za,/3 ~ Xa,P e U for la|) \P\ < €-
From the choice of U, xa p and hence zaip, is invertible, whence
X~a + i/3e p(x) for \a\, \/3\ < e.
This implies that 0 ¢. g(x), whence &{x) is closed in g'(x), as
desired.
§ 2. The Resolvent Function
6.2.1. DEFINITION. Let A be a unital algebra (over a field
F), x G A and p(x) the resolvent set of x. We assume that
p(x) ^ 0. The vector-valued function
Rx : A i—> x^ = x(\) — (x — \e)~
on p(x) is called the resolvent function of x; x\ is called a
resolvent of x.
6.2.2. PROPOSITION. Let A be a unital algebra and x e A.
Then:
xx — %n = (A - //)2^2^ (A,// G ^(z)) (Hilbert relation); (*)
(e - Ax)-1 - (e - fJ.x)^1 = (A - //)(e - Ai)_1a;(e - //x)_1
(A = 0 or A-1 G p(x),[A - 0 or //-1 e p(a:)). (**)
Proof.
(x-Xe)(x-fiey1 = [(z-//e) + (//-A)e] (x-//e)_1 = e + (//-A)xM.
Multiplying both sides of the above equation, on the left, by x^
we get
(x - ne)~l = xi + (// - A)zAa;M

§ 2. The Resolvent Function
265
which gives
changing signs on both sides we get relation (*). The relation (**)
can be obtained by noting that
(e - Xx)(e - //z)-1 = [(e - fj.x) + (fi-X)x](e - fix)'1
= e + (fj, - X)x(e ~ fix)~ .
6.2.3. COROLLARY. Any two resolvents xx, Xp of x
commute: x\ <-> Xfi.
PROOF. Interchanging A,// in (*) of 6.2.2 we get
x„ - xx = (fj. - X)x„xx (*').
Changing the sign in (*') and comparing it with (*) we get
(A - n)xxx„ = (A - n)x„xx.
Since we may assume that A 7^ // we conclude that xx <-> x^.
6.2.4. PROPOSITION. Let A be a unital Hausdorff TA and
x G A. Assume that
(i) p(x) is open;
(ii) A 1—> xx (A G p{x)) is continuous.
Then the resolvent function xx is a strongly infinitely
differentiate function with
and in general,
dxx 2 1 s
dL^ = „.*?+!
d\n
n\x^+L. (**)
Further, if
(iii) a(x) ± 0 and compact and p*(x) = (p(a;))\{0})_1 |J{0} t
t (p(2)\{0})-1-{A^:AGp(2)\{0}}

266
Spectral Analysis in TA 's
then p^(x) is open and on it (e —Ax) 1 is infinitely differentiate
with
dn-[{e - Ax)"1] = n\xn[{e ~ Ax)"1]" (A € p*{x)). (* * *)
dX
PROOF. If A e p(x) then A + // e p(x) for sufficiently small
// since p[x) is open. Using the Hilbert relation we get
xx+ij, ~~ xx _
Making // —> 0, and using condition (ii) above we get (*). The
formulae (**) is obtained by induction. Assume thus
dn~1x\
Then ^^ = (n - l)!-£-(a:?). Now
d\n y ' d\K x'
0<j+fc=n-l
:^- nxr^nx^i using (*))
where we have written L for lim. Therefore -^^- = (n!)x"+1.
(The factorization of £", „ — x™ used in the above calculation is
based on the property x\+ll <-> x\ which is assured by 6.2.3.)
It remains to prove the assertions on p^(x). Since p(x) is
open, (/)(2)/(0})-1 is also open. To prove that p^(x) is open it
is enough to show that 0 is an interior point of />#(x). Since <r(x)
is compact we have 0 < r(x) < 00. If r(x) = 0, then a(x) — {0}
and so p^(x) — K is open. Next assume that 0 < r(x) < 00. If
0 < |A| < ^K then |A-1| > r(x), so that A-1 e p(x),X e p*(x).
Therefore 0 is an interior point of p^(x) and p^(x) is open as
required.
Finally, the relation (* * *) is established by using (**) of
6.2.2 and induction in exactly the same way as for the proof of

§ 2. The Resolvent Function
267
(**) above.
6.2.5. COROLLARY. Let A* be the continuous dual of A and
x* G A*. Then
Fx.{X)^x*(x)t)
is a K -valued infinitely differentiate function on p[x) with
^-O (•"■»->•
When K = C,FX* is a holomorphic function and x^ a weakly
holomorphic function on p(x).
PROOF. This follows form 5.1.2.
6.2.6. COROLLARY. The conclusions and formulae
(*)>(**)>(* * *) °f 6.2.4 hold in any Hausdorff CI algebra [in
particular, a unital p -Banach algebra or more generally a unital
sm. (F) algebra).
PROOF. It is enough to prove that the hypotheses (i),(ii) of
6.2.4 are satisfied. By 3.6.22, 3.6.23 a unital (F) algebra A is a CI
algebra and so, by 6.1.2, hypothesis (i) of 6.2.4 is satisfied. Again
the continuity of the maps A i—> x — Xe, y >—> J/-1 (A being a C
algebra) implies that their composite map A i—> x> is continuous
(which is hypotheses (ii) of 6.2.4).
6.2.7. LEMMA. In a CI - in particular a p-Banach - algebra
A, xx —> 0 as |A| —> oo (A G p(x))-
PROOF. As |A| —> oo,|A_1| —> 0, and A being a CI algebra
we have
X* = -A-1(e - A-1*)-1 -♦ 0 • e = 0.
6.2.8. PROPOSITION. Let A = (A, ||-||) be a unital p -Banach
algebra and x G A. Then we have
(i) If \X\P > v{x) = lim„||z"||" then X e p(x) and the
resolvent x^ is given by
oo n
Xx = (x-Xe)-1 = ~J2^[, (*)
n=0A
with the series converging absolutely.

268
Spectral Analysis in TA 's
(ii) If A G p(x) and \X — fi\ < \\x^\\ p then fi €E /)(x) and
oo
^ = E(M"A)X+1. (**)
oo
1=0
with the series converging absolutely.
PROOF. Write A = A-1. Since v(Ax) = |A|*V(:e) < 1 we
have, by 3.3.20 (i),
oo ,
X
x^ix-Xey^ ~A(e - Ax)'1 = - £ A"+V = ~ E T^I
n=0 n=0
which is (*). To prove (**), we first note that we can write
x - fie = (x - Xe)[e + (X - n)xx]. (1)
Since
||(a-m)sa|| = |a-mNM< ^-11^11 = 1,
by 3.3.20, e + (A — //)x> is invertible and so by (1)
Xn = [e + (A - fi)x)]~lxi = [e ~ (fi - A)^]-1^
oo oo
proving (**).
6.2.9. THEOREM. Let A be a complex unital p-Banach
algebra and x G A. Then the resolvent function x^ is locally p -
admissible holomorphic on p(x) |J{oo}.
PROOF. Let A=(A, ||-||) where we may assume that || • || is
sm.. By 6.2.8(ii), we have for A0 £ p(x) and |A —Ao|< II^AoH"',
oo
n=0

§ 2. The Resolvent Function
269
Write <pn(\) = (A - X0)n,xn = x^1 and A) = {A G C :
|A - Ao| < ||x>0||_^}. Clearly <pn is holomorphic on D0. li K C
Do and K is compact it is clear that we can enclose * K in a
_ _ i
closed disc D(Ao,ro) with tq < \\xx0\\ p ■ So
H^ll* < sup{|A - A0|",A G Do} < r%.
A
It follows that
oo
ra=0 n n
oo
= II^Anll E ll^r^oir < °° ( since ll^o^ll < 1)
proving that x^ is locally ^-admissible at p0. Write Gx = {A G
C : |A| > v{x)1?}.
By 6.2.8 (i), we have
°° xn~l
xx = {x- Ae)_1 = -J2~^' (*)
ra=l
l
the series on the right converging absolutely for |A| > v(x)i>.
Writing xn = xn~1,cpn(\) = —1/A", we get for x^ the representation
xx = J2xn<pn.
n
Each <pn is clearly holomorphic on Gx- Any compact set in Gj
can be enclosed in an annulus
K = {A G K : ri < |A| < r2}
where v(x)^ < r\ < r2. Now we have for A G K,
' If C — sup{|A - A0| : A G K), then by compactness of K there is a
AiGif with C = |Aj - A0| < ||a;Ao IP^• We can choose for the radius of D
any r0 will C < rn < \\x\a \\~e.

270
Spectral Analysis in TA 's
I'PnWl < -^, SO that \\<pn\\K < "^
It follows that
ri ri
|X«-i|| »,«-i
E ii^mi^ < ^- = E ii V» < °°'
r»=l rl n=l rl
where for the last inequality we have used the absolute
convergence property of the series in (*). This proves the ^-admissible
holomorphy of x^ on Gx. By interpreting Gx as an open
neighbourhood of oo we see that what we have just proved amounts to
x^ being locally p -admissible holomorphic at oo. This completes
the proof of the theorem.
6.2.10. PROPOSITION. Let A be a complex unital p-Banach
algebra and x G A. Let T be a contour in C surrounding
a(x)\J{0} (in C). Then:
f XmxxdX = -2nixm (m = 0,1,2,--- ;x° = e). (*)
In particular, if r > v(x)^ and Cr is the circle |A| = r (A G C)
we have
f \mxxd\= ~2nixm. (**)
JCr
PROOF. Since Tf](p(x) = 0, T C p(x) and so by 5.4.9, we
have
f oo . >m
X™X,dX =-^xn dX. (1)
We can choose r sufficiently large that Cr C p(x) and Cr ~ T
in p(x). Then we have
J_ /" J^?_dA =—f —d\ = i ° (ra ^ ml (21
2jti 7r A"+! 2tt2 7c, A"+* 1 1 (n = m) *■ '

§ 2. The Resolvent Function
271
by the classical results. The formulae (*),(**) now ready follow
from (1), (2).
6.2.11. THEOREM (Beurling-Gelfand spectral radius
formula). In a complex Banach algebra A we have for any x G A,
r{x) = v{x) — lim \\xn\\ « . (*)
If the norm || • || is sm. then we have also
r(x) = v{x) < ||i||. (**)
PROOF. Our proof makes use of the formula (*) of 6.2.10
and is essentially on the lines in the exposition in [24, p.236].
We assume (as we may) that A is unital. By 4.8.11, r[x) ^
v{x) (since here p = 1). So to prove (*) it is enough to show
that i/(x) < r(x). If r > r(x), then by 6.2.10 (**), we have
—. f \mxid\. (1)
7T« JCr
2ni Jcr
Since x\ is a continuous function of A and Cr is compact we
have
M{r) = sup{||x>|| : |A| = r} < oo (2)
using (*) of 5.4.5 we obtain from (1),(2)
||xm|| < —rmM(r) ■ 2nr = rm+1M(r).
Therefore
||zm||m ^ r(rM(r))sr
whence
v{x) < r.
It follows that v(x) < r{x), completing the proof.
6.2.12. COROLLARY. If A is a real Banach algebra then

272
Spectral Analysis in TA 's
f(x) = v{x), where f[x) is the extended spectral radius. In
particular, if A is strictly real then r(x) = v{x).
PROOF. Apply 6.2.11 to the complexification A.
6.2.13. Remark. We shall obtain later (see 7.4.6) extensions
of the above results for p -Banach algebras.
6.2.14. Proposition. Let A=(A,|| • ||), with || • || sm., be
a unital p -Banach algebra and x G A. Then:
(i) o{x) is an upper semi-continuous function of x, i.e., given
an open set GCK with o(x) C G, there is a 6 > 0 such
that a(x + y) C G for every y G A with \\y\\ < S.
(ii) For A G p(x) if d[X) denotes the distance of A from o~(x)
then \\x),\\ > l/d(X)p; whence \\x\\\ -too os d(X) —> 0.
PROOF. By 6.2.6, x^ is a continuous function of A; by 6.2.7,
xx —> 0 as |A| —> oo. It follows that there are numbers Ci,r > 0
such that ||iA|| < Ci for all A with |A| > r, Write L\ = {X <E
K : |A| < r}. Then K = (K\G)f]Dr is compact, so that there
is a C2 > 0 such that \\xx\\ < C2 for A G K. It follows that
if A G K\G then ||a:A|| < C = max{d,C2}. If y in A with
||2/|| < S = C-1 and A G K\G then
x + j/ - Ae = (a; - Ae) [e + a;* j/]
is invertible since A € ^(x) and
\\xxy\\ < ||za||||j/|| < CC_1 = 1 ( see 3.3.20).
This means that A ^ a(x + y), whence a(x + y) C G, proving (i).
If A G p(x) and /x G a(x) then by 6.2.8 (ii) we have |A — fi\ >
II- ^
H^aII p , so that
||x,||>l/|A~//|",||x,||>l/rf(A)^,
proving (ii).
6.2.15. COROLLARY. r(x) -> 0 as x -> 0. TAus r(z) is
continuous at x = 0.
PROOF. Take G = {A G K : |A| < e}. Then G ID <j(0) = 0,

§ 2. The Resolvent Function
273
whence by upper semicontinuity at 0 we have o{x) C G, r(x) < e
for ||i|| < 8. This shows that r(x) —> 0 as x —> 0.
6.2.16. COROLLARY. 7/ A is a /> -Banach algebra and x G A
iAen <r'(a;) is upper semi-continuous.
PROOF. It suffices to observe that v'{x) = 0^(^) where Ai
is the unitization of A.
6.2.17. PROPOSITION (Rickartt [23, p.36]). Let x be an
element of a unital p -Banach algebra A = (A, || • ||) which is
either complex or strictly real. Let V be an open neighbourhood
of 0 in K. Then there is a 8 > 0 such that for every y G A with
\\x — y\\ < 8 and xy = yx we have:
a(y)Ca(x)+V; (*)
a(x)Ca(y) + V. (**)
PROOF. Since a(x) +V is an open set containing a(x), the
relation (*) follows from 6.2.14(i). To prove (**) we will assume
that is is false and show that this leads to a contradiction
Without loss of generality we can take V ~ {A G K : |A| < 2e}.
By our assumption we can find a sequence xn —> x with xn <-> x,
such that
a{x) %Gn = a(xn) + V for all n. (1)
Choose A„ G a(x)\Gn. If // G a(xn) then
A„ - /j ¢ ^, so that |A„ - fi\ > 2e. (2)
Since A„ G o{x) and <r(a;) is compact we can, by passing to a
subsequence if necessary, assume that
A„ -* A0 G a(x). (3)
From (2),(3) we get:
|Ao — //| > 2e > e for every // G <r(a;„) (4)
so that Ao (£ o{xn) for every n. For obtaining the contradiction
we have to consider separately the cases Ao = 0, Ao 7^ 0.
' He obtained the result for complex Banach algebras.

274
Spectral Analysis in TA 's
Case 1. Ao = 0. Then 0 ^ a(xn) and xn is invertible. Since
|//| > e for // G ff(£n) we get f^z"1) = sup|//_1| < e"1. So, by
7.4.6, v{x~l) = r(x~1Y < e~", whence, by 3.7.29(i), x = lima:,,
is invertible contradicting 0 = Ao 6 a{x).
Case 2. Ao 7^ 0. Since Ao ¢ c(a;„), J/n = --^1^ is
q. invertible for all n. By 1.7.12
*'(»»)= {"IT} :Ae*(*»)}- (5)
Since A G c(j/n) iff -AAo G ^(^), and by (4) |Ao + AAo| > e for
-AA0 G o{xn), we get
A , , -AAn , . ||z„||' ^ C
—— < T (6)
1 + A Ao + AoA e e
for some C > 0 (since the sequence x„ being convergent is
bounded). From (5),(6) we conclude that r(y'n), and hence by
GB formula (see 7.4.6.) v(y'n) is bounded. So, by 3.7.29(ii),
lim yn — —Aq x is q. invertible, contradicting the fact that Ao G
a'{x).
6.2.18. COROLLARY. If the algebra A 0/6.2.17 isnotunital
then we have inclusion relations analogous to (*)(**) obtained by
replacing a by a' {the quasi-spectrum).
PROOF. We have only to apply 6.2.17 to the unitization A\
of A.
6.2.19. COROLLARY. If A is commutative, r{x) is
continuous everywhere.
PROOF. This follows from (*),(**) of 6.2.17.
6.2.20. PROPOSITION. The completion A of a commutative
strictly real p -normed algebra A is strictly real.
PROOF. Consider an element x G A. If possible let d~^(x)
contain a complex number Ao = &o + i0o, with /¾ 7^ 0. Write
r) = 21/^01 > 0. Choose a sequence (xn) in A with xn —* x. Since
A is strictly real, 0""^(x„) C R. Since A C A we have also A C A,

§ 3. Pseudo-Resolvent Function
275
and so
aA(xn) C aA{xn) C R. (1)
Write
V = {A e C : A = a + i/3 with a G R, |/?| < q}.
By (**) of 6.2.17, there is a 6 > 0 such that
<r(x) C <j(j/) + V whenever ||x - j/|| < S.
Choose n sufficiently large that \\xn ~ x\\ < 6. Then
a~A(x) ^ a~A.(x") +VCR + V = V.
It follows that A0 G V,\/3q\ < r) ~ §|/?o|, whence /?0 = 0 - a
contradiction. Therefore 0""^(x) C R and A is strictly real.
§ 3, Pseudo-Resolvent Function
6.3.1, DEFINITION. Let A be an algebra (over a field F)
and x G A. Set
/(x) = {AeF: (Ax)' exists}; pp(x)
is called the pseudo-resolvent set of x. Since (Ox)' = 0' = 0,
0 G pp(x) and we have always pp(x) ^ 0. Write
x', = (Ax)' (AG/(x))
and call x\ a pseudo-resolvent or a p-resolvent of x; A i—> x'A is
called a pseudo-resolvent function. '
6.3.2. Remark. The map A i—> —A-1 is a bijection of
p'(x) = p'(x)\{0} onto ^p(x)\{0}, where p'(x) denotes the quasi-
resolvent set of x. If F — K the map is also a homeomrohpism.
If xis q.nilpotent then a'(x) = {0}, p'(x) = F\{0}.
Therefore,
' This function has been considered implicity by Kaplansky [10 " , p.400,
Lemma 3.2].

276
Spectral Analysis in TA 's
/(x) = -{fuo}}-1 (j{°> = {nwxjw = F-
6,3,3, Lemma, (a) x\ <-> i'M (A,//e/(x)).
•^u 3?\
^i A
PROOF, (a) Since Ax <-> //x, by applying 1.1.18 twice we get
-> x'
(b) We have
Ax + (Ax)' + (Ax)(Ax)' = 0 (1)
//x + (//x)' + (//x) (//x)' = 0 (2)
Denoting the expressions in the equations (1), (2) above also by
(1),(2) we get
// x (1) - A x (2) = // • 0 - A ■ 0 = 0,
which gives
//(Ax)' - A(//x)' = A//x[(//x)' - (Ax)'].
. , , lux)' , (Ax)' , , .
Again, (1) x ^ ; and v ' x (2) give
A //
x(//x)' + (Ax) V*)' + ^(Aa:)'^)' = 0.
A
x(Ax)' + M'W + x(Ax)'(//x)' ~ 0.
(3)
(4)
(5)
From (4), (5) we obtain by subtraction
x[(//x)'-(Ax)']=Q-i)(Ax)'(//x)' (6)
and from (3), (6) we get
A^i A^i

§ 3. Pseudo-Resolvent Function
277
which reduces, after a change of sign, to the equation in (b).
6.3.4. LEMMA. In a Q algebra A, for each x G A, pp(x)
is an open set containing 0.
PROOF. For A G pp(x), let U be an open neighbourhood
of Ax comprising q. invertible elements. By continuity of scalar
multiplication there is an open neighbourhood TV of A such that
Nx C U. If follows that N C pP(x) and pp(x) is open.
6.3.5. PROPOSITION, (a) Let A be a Hausdorff C algebra,
and x an element of A such that pp(x) is open in K.
Then x\ is a strongly differentiate function of A on pp(x),
with
^ = -.(1 + ^, w
^| = (-1) w(i+^r1. («,)
We have also the relation
(y)=»!(y)"+1 (A*0). (***)
(b) 7n a Hausdorff CQ algebra A, the formulae (*), (**),
(* * *) are valid for every element x in A.
PROOF, (a) From the equation (1) in the proof of 6.3.3 we
obtain, for A G Pq(x)\{0},
x\ (Aa:)'
Making A —> 0 we obtain
dxV
= — x.
0
dX
If A,// G />p(x)\{0}, by 6.3.3 (b) we have
// A Xfj,
(1)
(2)

278
Spectral Analysis in TA 's
Making // —> A we obtain
hich reduces to
hence we get
dx\
dX
here we have used
m~& <3»
xA 1 dxx xx
A2 A dX A2 '
= ^(1 + ^) = -^(1 + ^)2. (4)
(1). Thus we have proved (*) when A^O.
On the other hand (2) shows that the formula in (4) (or (*)) is
valid for A = 0, completing the proof of (*) for all A.
To prove (**), assume by induction that
^^ = (-1)-^-1)1^-1(1 + ^.
Differentiating both sides of the above equation we get
i£ = (-1)-(.-1)^..(1 + ^)-¾
= (-1)^711^(1 + ^)^--2(1 + ^)2 (using (4))
- {-l)nn\xn(l + x'x)n+1
which is (**). Finally, by successive differentiation we obtain
(* * *) from (3).
(b) This follows from (a) and 6.3.4.
6.3.6. PROPOSITION. Let A be a p-Banach algebra and x e
A. Then x\ exists for all X such that v{Xx) < 1, or equivalently,
|A| < v(x) p, and we have
oo
^ = £(-ir(^r w
ri=l
with the series on the right converging absolutely. In particular,
_ i
x'^ is analytic for \X\ < v(x) <>.
PROOF. Consider the unitization A\ of A. Then we have
(ei + Ax)-1 = ei + (Ax)'

§ 3. Pseudo-Resolvent Function
279
where e\ is the unity of A\. By (*) of 3.3.20 we have
oo
(e1 + Ax)-1 = ^:(-ir(Axr,
n=0
the series converging absolutely. It follows that we have
oo
(with the series converging absolutely).
6.3.7. COROLLARY. If A is a complex p-Banach algebra
— A
then x\ is p -admissible holomorphic on {A G C : |A| < v(x) ?}.
PROOF. Set
V>„(A) = (-i)"A"; xn = xn.
Then (*) of 6.3.6 can be rewritten as
oo
x\ = E xnrpn(X).
n=l
For any ro with 0 < tq < i/(x) p, set
K = Kr0={\EC: |A|^r0}.
Then
Un\\K^rZ.
So
OO oo oo
E Kll lliMli- < E \\*nK' = E ll(r°*)l < oo,
n=l ra=l ra=l
since v(rox) = r^v{x) < 1. Hence the corollary.
6.3.8. PROPOSITION. Let A be a complex ample C -algebra.
If x G A\{0} is q. nilpotent then {x1^ : A G C} is unbounded.
PROOF. First note that by 6.3.1 and definition of q.nilpotent,
pp(x) — C. Since A is a Hausdorff C-algebra, by 6.3.5 (a),

280
Spectral Analysis in TA 's
x\ = {^x)' is strongly differentiable, and hence for x* G A*
F'x,{\) = x*{x\) is an entire function. If possible let x\ be a
bounded function of A. Then F'x»{\) is bounded and so by
classical Liouville's theorem F'x*(\) is a constant. Since Fx,(0) = 0
we must have
F'x.(\) = 0 for all A G C.
Putting A = 1, we get x*(x') = 0. Since A is ample we
conclude that x' = 0 which implies x = 0, contradicting the
choice of x. Therefore {x'n} is unbounded as required.
6.3.9. COROLLARY. (Kaplansky). In a complex normed
algebra A, if x ^ 0 is q. nilpotent then {x1^} is unbounded.
PROOF. Since a normed algebra is an ample C-algebra, the
corollary follows from 6.3.8 .
6.3.10. LEMMA. Let A be a normed algebra and x G A be
q. nilpotent. If A„ G K are such that the sequence ((A„x)') is
unbounded then |A„| —> oo.
PROOF. First note that since x is q.nilpotent (Ax)' exists
for all A G K. Assume now that, to the contrary, |A„| < C for all
n. Since ((A„x)') is unbounded there is, by 2.1.23, a nucleus U
and a subsequence ((A„'x)') such that
^^ 4 U for all n'. (*)
n'
Since |A„'| < C, for all n', we can choose a subsequence (A„»)
of (A„') with A„» —> (some) Ao- Since A is a C -algebra we get
(A„»x)' —+ (Xqx)', whence the sequence ((A„»x)') is bounded.
But by (*),
n"
contradicting boundedness of {{^n"x)')- Hence the Lemma.
6.3.11. PROPOSITION. (Kaplansky). In a complex normed
[or more generally, ample p -normed) algebra A every q. nilpotent
element x is s.t.z.d.. In particular, every element of the radical

§ 3. Pseudo-Resolvent Function
281
\f~A is a s.t.z.d. .
PROOF. We may assume that x ^ 0. By 6.3.8 there is a
sequence (A„) in C with
||(A„x)'|| -^ oo.
By 6.3.10, |A„| -> oo. Write
*n = A„x, yn = x'J\\x'n\\~p
(so that \\yn\\ ~ 1)- Then since
we get
XnVn = -xn\\x'n\\~~P ~ Vn- (1)
Since xn = Xnx the equation (1) becomes
xyn= -x\\z'n\\~~' - A"1^. (2)
Since 11a:^11, |A„| —> oo as n —> oo, and \\yn\\ = 1 we get from
(2) xyn —> 0 (as n —> oo). In exactly similar manner we also get
ynx —> 0. Thus, x is a s.t.z.d., as desired.
6.3.12. COROLLARY. In a real normed - or more generally,
ample p -normed - algebra A every ext. q. nilpotent element x is
a s.t.z.d. .
PROOF. By definition x is a q. nilpotent element of the com-
plexification A. By 4.7.4. (b), A is also ample. So, by 6.3.11, x
is a s.t.z.d. of A and hence by 3.7.16 (ii), a s.t.z.d. of A.
6.3.13. DEFINITION. A TA A is called a topological integral
domain or TID if it has no non-zero t.z.d..
6.3.14. PROPOSITION. A complex ample p-normed algebra
A which is a TID is q.s.s., in particular s.s..
PROOF. This is an immediate consequence of 6.3.11 and the
definition of a TID.

282
Spectral Analysis in TA 's
§ 4. Spectral Algebras
6.4.1. DEFINITION. A unital algebra over a field is called
spectral if for every x in A, &{x) ^ 0.
6.4.2. LEMMA. Every subunital algebra B of a spectral
algebra A is spectral.
PROOF. By 1.7.19 (**), aB(x) 2 <ta(x) ^ 0.
6.4.3. THEOREM. Every complex ample Hausdorff CI algebra
is spectral.
PROOF. If possible let a(x) = 0, for some x in A, so that
p(x) = C. By 6.2.5, Fx- (A) = x*(x)t) (x* G A*) is holomorphic on
p(x) = C. Also, Fx* is bounded since by virtue of 6.2.7, x*(x\) —>
0 as |A| —> oo. By Liouville's theorem, Fx> is a constant which
must by 0 (since ^.(A) -> 0 as |A| -> oo). Thus Fx>{\) = 0
for every x" G A*, whence by ampleness of A,x^ = 0. It follows
that e — [x — Xe)x\ = 0 - a contradiction. Thus (f(x) ^ 0, as
required.
6.4.4. COROLLARY. Every complex Hausdorff locally convex
CI algebra A is spectral.
PROOF. By 4.7.6, A is ample and so the result follows from
6.4.3.
6.4.5. COROLLARY. Every complex Hausdorff locally
sm. convex I algebra A is spectral. In particular, every complex
Hausdorff locally sm. convex division algebra D is spectral.
PROOF. By 4.4.15, 3.6.21, A is a CI algebra and hence the
first statement follows from 6.4.4 . The second statement follows
from the first since D being a division algebra is an I algebra
(G,- = D\{0} is open).
6.4.6. COROLLARY. A complex, ample p-Banach algebra -
in particular, Banach algebra - is spectral.
PROOF. This follows from 6.4.3 since every Banach algebra is

§ 4. Spectral Algebras
283
a CI algebra (by 3.6.23 (b)) and is also Hausdorff.
6.4.7. THEOREM (Zelazko). Every complex unital p-Banach
algebra A is spectral.
PROOF. This theorem goes beyond 6.4.6 since it covers also
those algebras which are not ample.
If possible let there be an element x G A with &(x) = 0.
Then p(x) = C and by 6.2.9, x> is a locally ^-admissible entire
function. If T is any circle in C then T ~ 0 and hence by 5.14.16
(a)
x^dX ~ 0.
On the other hand, by taking m ~ 0 in the formula (*) of 6.2.10,
we obtain
x\i\ ~ —2nie,
where e is the unity of A. This contradiction proves that (f(x) ^
0, for every x G A, and A is spectral.
6.4.8. COROLLARY. Every strictly real unital p-Banach
algebra A is spectral.
PROOF. Let A be the complexification of A. Since A is
strictly real we have
aA{x)=aA(x)^0 (by 6.4.7).
6.4.9. COROLLARY. Every complex or strictly real unital
pseudo-Michael algebra A is spectral.
PROOF. By 4.5.3, A has a projective limit decomposition
A = limAa, where each Aa is a unital pa -Banach algebra. Again,
by 4.5.7 (3),
(TA{x) = {joA{xa), x=(xa). (*)
a
First let A be complex. Then Aa is complex and so by 6.4.7,
aA, (Xa) ^ ^- It follows by (*) that oa{x) ^ 0, and A is spectral.
L
L

284
Spectral Analysis in TA 's
Next let A be strictly real. In view of 1.7.26, 1.9.11 we may
assume that A is commutative. By 1.9.8, Aa is commutative
strictly real and so by 6.2.20, Aa is strictly real. Hence by 6.4.8
and (*), A is spectral.
6.4.10. Remark. A complex unital commutative complete
locally convex algebra may fail to be spectral. To overcome this
deficiency Waelbroeck [13' ] has given a modified definition of
spectrum with respect to which these algebras are spectral. His
modified spectrum which we shall denote by sp(x) is the complement
(in K) of the set of all Ao G K such that (x — Ae)_1 exists and
is (t). bounded in a neighbourhood of Ao- Using the properties of
his spectrum sp(x), Waelbroeck has obtained interesting results
concerning locally convex algebras.
§ 5. Gelfand-Mazur and Other Similar Theorems
6.5.1. LEMMA. If a division algebra A over a field F is
spectral then A is of the form A = Fe, where e is the unity of
A. Moreover, if x — Xxe, the map oj : x i—> \x is an isomorphism
of A onto F.
PROOF. If x G A and Xx G a(x) then x — Xxe is non-
invertible and so i- Xxe ~ 0 since A is a division algebra. Thus,
x — Xxe, A—Fe and ui is clearly an isomorphism.
6.5.2. COROLLARY. If a spectral division algebra A over K
is a Hausdorff TA then the map u> : x >—> Xx is a t. isomorphism.
PROOF. This is because of the result that an isomorphism
between two finite-dimensional TLS's is automatically a homeo-
morphism (here dim A = dimK = 1) (see 2.1.12).
6.5.3. THEOREM. A complex ample CI division algebra A
is of the form A ~ Ce, with the map oj : x i—> Xx (x = Xxe) a
t. isomorphism.
PROOF. Since A\{0} = G, is open, A is Hausdorff. Hence,
by 6.4.3, A is spectral. The theorem now follows from 6.5.1, 6.5.2 .

§ 5. Gelfand-Mazur and Other Similar Theorems 285
6.5.4. COROLLARY (Arens). A complex Hausdorff locally
sm. convex division algebra A is of the form A = Ce (with oj
a t. isomorphism).
PROOF. Since A is a Hausdorff division algebra, by 3.6.10,
A is an 7 algebra. Further, by 4.4.15, A is a C algebra, and
by 4.7.6. it is ample. The required conclusion now follows from
6.5.3.
6.5.5. THEOREM (Gelfand-Mazur). A complex normed
division algebra A is of the form A — Ce and oj : x >—> Xx is a
t. isomorphism. If ||e|| = 1 then oj is also an isometry.
PROOF. Without loss of generality we may suppose that the
norm of A is sm.. Then A is locally sm. convex, so that by 6.5.4,
A = Ce and oj at. isomorphism.
If ||e|| = 1, then ||x|| = ||Aze|| = |AZ| and oj is an isometry.
6.5.6. COROLLARY (Zelazko). A complex p -seminormed
division algebra A = (A,p < ) has the form A = Ce. Consequently,
every locally bounded complex TA B which is a division algebra
has the form B = Ce.
PROOF. We may assume that p is sm.. Let a be an element
of A and Am a maximal commutative subalgebra containing a
(see 1.1.9.); by 1.1.19, Am is a division algebra. For x G Am,
write \\x\\ = ^"(x), where v(x) = lim p(x")«. By 4.8.6, 4.8.7,
v v ra—»oo v '
the above defined functional |-| is a norm on Am, so that (Am,||-
||) is a complex normed division algebra. By 6.5.5, Am = Ce, so
that a = Ae (A G C). It follows that A = Ce, proving the first
assertion. For the second assertion it suffices to observe that, by
4.2.4, B is ^-seminormed (so that it follows from the first).
6.5.7. PROPOSITION. A strictly real, ample CI division
algebra A is of the form A = Re (with oj : x —> Xx at. isomorphism).
In particular, such an algebra is commutative.
As always we assume p^O

286
Spectral Analysis in TA 's
PROOF. As in 6.5.6, for any element a / 0 in A, take a
maximal commutative subalgebra Am with a G Am; Am is a
division algebra and so in particular inverse-closed in A. By 4.7.3
(a), 3.6.27, Am is a commutative ample CI algebra. Further, by
1.9.11, Am is strictly real, and consequently, by 1.9.14, it is also
formally real. By 1.6.20, its complexification Am is a division
algebra which is moreover ample (see 4.7.4 (b)). By 6.5.3, Am =
Ce, whence Am = Re, so that a — Xe (A G R). Since a is an
arbitrary element of A we conclude that A = Re, as desired.
6.5.8. COROLLARY. If A is a strictly real, Hausdorff locally
sm. convex division algebra then A = Re.
PROOF. The same argument as in the proof of 6.5.4 shows
that A is an ample CI algebra. The required result is now an
immediate consequence of 6.5.7 .
6.5.9. PROPOSITION. A commutative real ample CI [in
particular, a Hausdorff locally sm. convex) division algebra A is of the
form A = Re or Ce (A ~ R or C) according as A is formally
real or not.
PROOF. If A is formally real then, by 1.9.14, it is strictly real
and so A = Re (by 6.5.7). It remains to consider the case where
A is not formally real. By 1.6.20 (b), A has a complex structure.
Since ix = jx, with j G A, multiplication by i is continuous so
that A is a complex TA which is moreover, ample by 4.7.4 (a).
By 6.5.3 (&; 6.5.4) we conclude that A = A= Ce.
6.5.10. THEOREM. Every real ample CI division algebra A
is t. isomorphism to R, C, or H (the Hamilton quarternions).
PROOF. In view of 6.5.9, we may assume that A is not
commutative. Let Z denote the centre of A. By 1.1.18, Z is a
commutative division algebra. For any x ^ 0 in A the algebra Z(x)
generated by Z, x, x^1 is a commutative division algebra which is
moreover, ample (by 4.7.3 (a)) and CI (by 3.6.24). Applying 6.5.9
to Z(x) we get
Z(x) ~ Re or Ce.
Thus,

§ 5. Gelfand-Mazur and Other Similar Theorems 287
x = (a + i/3)e (a, /? E R, with /3 = 0 if Z(x) = Re).
It follows that
(a: - (a + i/3)c)(i - (a - t'/9)e) = 0
or,
x2 - 2ax + (a2 + /?2)e = 0,
which implies that A is algebraic of degree 2 at most. By a
theorem of Jacobson ' A is finite-dimensional, and so by the classical
Frobenius theorem, A ~ R, C or H; the isomorphism is
topological since A is finite-dimensional Hausdorff.
6.5.11. COROLLARY (Arens). Every real Hausdorff locally
sm. convex division algebra A is t. isomorphic to R, C or H.
PROOF. As in 6.5.4 we see that A is ample and CI, and the
required conclusion follows from 6.5.10.
6.5.12. Theorem (Zelazko). Every real p-normed division
algebra A — (A,p) is t. isomorphic to R, C or H according as
it is formally real, commutative but not formally real, or not
commutative. If (j denotes the t. isomorphism then
\oj(x)\ ~ v(x) = limp(x")« ,
n
where | ■ | denotes the standard norm'' on R, C or H.
PROOF. Since we do not know d priori that A is ample we
cannot deduce this theorem from 6.5.10. However, we can prove
it by directly considering separately the three cases that arise.
Case 1. A is formally real. For any non-zero x is A
consider a maximal commutative subalgebra Am with x E Am- As
in 6.5.7, Am is a formally real division algebra, so that Am is
' Every algebraic division algebra of bounded degree over a perfect field
is finite-dimensional ("Structure theory for algebraic algebras, Annals of
Mathematics 46 (1945), p.70l").
tt \t\ = (ti)i (teR,C or H and t the conjugate of t).

288
Spectral Analysis in TA 's
now a commutative formally real ^-normed division algebra. Its
complexification Am is a ^-normed division algebra, whence by
6.5.6, Am ~ Ce, Am = Re, x ~ Xe, A = Re.
Case 2. A is commutative but not formally real. Then A
has a complex structure and as in the proof of 6.5.9, we get A =
A = Ce.
Case 3. A is not commutative. Let Z be the centre of A.
For x ^ 0 in A we can form, as in the proof of 6.5.10, the algebra
Z(x) which is now a real commutative ^-normed division algebra.
By the conclusions in cases 1, 2 we have Z(x) = Re or Ce. This
implies, as in 6.5.10, that A is finite-dimensional, and hence by
Frobenius, A ~ R, C or H.
It remain to prove (*). Set ||x|| = |cl»(x)|. Then || ■ || is a
norm on A with \\xy\\ = \\x\\ \\y\\. Since A is finite-dimensional,
|| ■ || ~p. By 4.8.2
v(x) = vp(x) — 1/11.11(1) = lim ||z"||" = lim ||x|| = ||i||.
6.5.13. Remark. If A is a formally real normed division
algebra whose norm || ■ || is normalized, then the t. isomorphism
oj : A —+ R is an isometry: if x = Xxe, \\x\\ = \XX\ = \oj[x)\.
6.5.14. LEMMA. Let A be a unital p -normed algebra which
is a TID. Let A be the completion of A. Then every non-zero
element x of A has an inverse in A.
PROOF. Suppose that x G A has no inverse in A. Then
0 G a^(x), so that 0^(2) 7^ 0. Also, by 6.1.2, a^ix) *s cl°sed and
aA(x) ^ K. It follows that daA(x) ^ 0. If A G aA(x) then, by
3.7.28, x — Xe is a s.t.z.d. of A and so by 3.7.14, it is a t.z.d. of
A. By the hypothesis on A, x = Xe and since x is not invertible,
x = 0.
6.5.15. COROLLARY. If a unital p -Banach algebra A is a
TID then it is a division algebra.
6.5.16. Remark, (cf. Zelazko [31, p.112]). The result in
6.5.15 does not hold for a locally sm. 3 algebra. For example,
consider the algebra £ of entire functions (see 4.6.8 (iii)). £ is

§ 5. Gelfand-Mazur and Other Similar Theorems 289
not a division algebra, since for instance, z G £ has no inverse.
But £ is a TID. To see this, suppose that in £ we have g ^ 0,
and
fk9 ~~> 0 as k —> oo. (*)
Since the zeros of ^ are isolated we can find a sequence rn of
reals with 0 < rn —> oo and such that
inf |ff(*)| >0(n =1,2,---). (**)
If we write
ll/nll* = SUp |/(2r)|,
the family {|| • ||*} of norms is easily seen to be equivalent to the
family {|| ■ ||„} defining the topology in £ (see 4.6.8 (iii)). The
condition (*) implies that ||/jtff|^ —> 0 as k —> 00, for all n. But
this result along with (**) gives: ||/jt||* —> 0. This means that g
is not a t.z.d., proving £ is a TID.
6.5.17. THEOREM, (cf. [31, pp.30-32.]) Let A be a unital
p -normed algebra which is a TID. Then:
A = Ce, A ~ C, if A is a complex algebra
and
A ~ R, C or H if A is a real algebra.
PROOF. Let A be the completion of A. By 6.5.14, every x 7^
0 in A has an inverse x"1 in A. Consider the unital subalgebras
A(x) of A, consisting of all rational functions of x, over K, i.e.
all elements of the form f{x)/g{x) = f(x)g(x)~1 {g{x) 7^ 0),
where f,g are polynomials over K. A(x) is a division algebra
containing x, which being a subalgebra of A is p -normed. If
A is complex, it follows from 6.5.6 that A(x) = Ce, x = Xxe,
A = Ce.
It remains to consider the case where A is real. By 6.5.12,
A(x) ~ R, C or H. It follows that x satisfies a relation of the
form
ax2 + /3x + 7e = 0, where a, (3,7 G R, 7 7^ 0.
a /3
It follows that x_1 = x e £ A, showing that A is a divi-
7 7
sion algebra. Now we apply 6.5.12 to A and conclude that A ~ R,

290
Spectral Analysis in TA 's
C or H.
6.5.18. Remark. Theorem 6.5.17 has been extended by
Zelazko to locally sm. convex algebras in the following form: If
a unital locally sm. convex algebra A has no nonzero g.t.z.d. then
A ~ C if A is complex, and ~ R, C or H if A is real (see [31,
pp.112-14]).
6.5.19. PROPOSITION. Suppose that the norm of a unital
p -normed algebra A satisfies the condition
CNI l|y||<ll*y|| (*)
for some C > 0 and all x,y G A. Then A is t. isomorphic to C
if A is complex and to R, C or H if A is real.
PROOF. In view of 6.5.17 it is sufficient to show that A is a
TID. Suppose that x, yn G A, \\yn\\ = 1, xyn ~~> 0- Then, by (*)
we have
Clllll = Clllll lll/JI < 11It/rail ~~> 0-
II II II II llv'^ll I v'vM
This means that ||x|| = 0, x = 0, so that A has no non-zero
l.t.z.d.; similarly it has no non-zero r.t.z.d.. Thus A is a TID,
completing the proof.
6.5.20. COROLLARY (Arens-Shilov). A unital normed algebra
A satisfying condition (*) above is t. isomorphic to C if A is
complex and to R, C or H if A is real.
6.5.21. Remark. A special case of 6.5.20 was proved
earlier by Lorch and Mazur (see [14, p.127]) in the form: A
complex unital Banach algebra A whose norm satisfies the condition
\\xy\\ = \\x\\ \\y\\ for all x,y in A is (t.) isomorphic to C.
§ 6. Turpin's Theorem on Locally Convex Algebras
6.6.1. Let A be a Hausdorff complete locally convex algebra.
Denote by Pq the set of all continuous semi-norms on A and by
A* (the continuous) dual of A.

§ 6. Turpin's Theorem on Locally Convex Algebras 291
Set
ri(x) = sup lim pa(xn)n .
r2{x) = sup lim |/(x")|«,
oo
rs(x) ~ inf{r : (an)f, an e K, the series y^anXn
n=l
has radius of convergence > r
oo
=> 2_\0inx"' converges in A}.
n=l
We have clearly 0 < fj[x) ^ oo (j = 1,2,3).
6.6.2. LEMMA. r2(z) < ri(z).
PROOF. If / G Av, by 4.3.13 (i) there is a pa such that / is
pa -continuous, and so pa -bounded. Therefore we have |/(x")| <
||/||pa(x") so that
lim \f{xn)\n < lim {\\f\\«pa{xn)n) < 1. \impa{xn)n < n(x).
ra—»oo' v " ra—»oov" " v ' ra—»00 v v
By taking the sup over / G A* we get ^(z) < ^i(^)-
6.6.3. LEMMA. If rs(x) < 00 and |//| > rs(x), then:
(i) the series 2__, ( ~ ) converges absolutely;
(ii) // G p'(x);
(iii) r(x) < ^3(2), where r[x) denotes the spectral radius of x.
PROOF. By definition of rs(x) there is an ro with |//| >
00
ro > rs(x) such that, if Y^ OLnXn is a series over K with radius
ra=l
of convergence > ro, the series V^a„x" converges absolutely.
ra=l

292
Spectral Analysis in TA 's
oo / \\n
Now the numerical series Y( — ) has radius of convergence
oo / \ n
= l/lim|-^|« = |//| > ro, so that by choice of ro, 2_^ I — ) con-
n=l ^'
verges absolutely. This implies, by 3.1.24, that 2^, (~) con"
r»=l ^M'
verges in A, whence, by 2.2.17, ( —-) exists, so that fiGp'(x).
Therefore, if A G o'(x) then A ¢. p'(x) and so we must have
|A| < ^3(2)- It follows that r(x) < ^3(^), which completes the
proof of the lemma.
6.6.4. PROPOSITION. Let A be a complex Hausdorff
complete locally convex algebra which is CQ.
Then we have
ri(x) = r2(x) = r3(x) = r(x). (*)
PROOF. By 6.1.1, a'{x) is compact. So 0 < r{x) < 00.
Suppose that
0 < IAI < -^-.
r{x)
Then r(x) < |A|_1 = | - A_1|, so that -A-1 G p'(x) and
consequently A G ^(x) (see 6.3.2). Since A is a CQ algebra, by
6.3.5, x\ is a strongly holomorphic function with
^ = (-1)^(1 + ^+1.
If / G A*, the F(X) = /(x'i) is holomorphic on pp{x) and in
particular, on |A| < -7—r. Since
r{x)
F(r'(0) = f((~l)nn\xn) = (-l)nn\f(xn)
the Taylor expansion for F is given by
00
n*) = £(-^/(1^8 (note F(°) = /(°) = °)
n=l

§6. Turpin's Theorem on Locally Convex Algebras 293
with the series converging for |A| < \/r[x).
It follows that
(-l)nf(xn)Xn = f((-l)n\nxn) -» 0, as n -» oo
and this is true for every f £ A*. This implies that the sequence
((— l)nXnxn) is weakly bounded. Since A is locally convex it is
also strongly bounded ' , i.e. we have a constant Ma > 0
(depending on A) such that
pa((-l)n\nxn) = pa(Xnxn) < Ma for all n > 1.
It follows that
~hm{/pa(xn) < Tim"(Ma)n|A|-1 = |A|-1.
This being true for every pa we obtain, by taking sup over pai
ri(x) < |A|_1 (for any A with |A|_1 > r(x)).
By taking the infimum over A we get
ri(x) < r(x).
Combining the above inequality with that in 6.6.2 we get
r2{x) < ri(i) < r(x) (< oo). (1)
Next choose r, r' with
r2(x) < r' < r.
By definition of r2, we have for any f E A*.
\/\f(xn)\ < r' < r for sufficiently large n. It follows that
|/ (fr~)\ < 1 (« > some N). Therefore (*£■) is weakly bounded
and consequently strongly bounded. This means that there is an
Ma > 0 such that
We have therefore pa (^-J < Ma lH , ~ < 1, so that by the
t See [24, p.68l.

294
Spectral Analysis in TA 's
00 xn
Weierstrass M-test V^ — converges absolutely in A. Let
ra=l
00
y^ anXn be a power series with radius convergence > r. Then
»=i
an\ ^ r~n (for sufficiently large n) so that
pa{anxn) = \an\pa{xn) < ^^ < Ma (^) .
n=l
It follows that the series V]a„a;" converges absolutely in A. This
n=l
means that r > ^3(2) whence
r2(*) > r3(x). (2)
From (1), (2) we get
^3(2) < r2(x) < ri(x) < r(z). (3)
But by 6.6.3 (iii), r(x) < ^3(2). This together with (3) gives the
relation (*).
6.6.5. THEOREM (Turpin). A commutative complex 3
algebra which is a Q algebra is locally sm..
PROOF. By 3.6.18, 4.6.7, A is CQ. Since A is a locally
convex Q algebra we can find an open absolutely convex nucleus
U comprising q. in vertible elements. Since U is balanced, by 3.6.8
(i), we have r(x) < 1 for x G U. Set V = U/2. Then V is open
and also absolutely convex (see 4.1.2 (c)); since U is balanced,
V C U. If y G V then y = x/2 (x G U), so that
r(v) = r - = -r(x) < - ■ 1 = - < 1.
By 6.6.4, f3(j/) = r(j/) < 1. So, by 6.6.3 (i), y. J/" converges
n=l
absolutely in A.
00
Let <p(X) = V^a„A" be an entire function vanishing at 0.
n=l
Then
\anXn\ < 1 for n > n(A). (*)

§ 6. Turpin's Theorem on Locally Convex Algebras 295
If x G A, then using the fact that U as a nucleus is absorbing we
find Ao 7^ 0 such that a; = Aoj/ (y <E U). Then, using (*),
pa{anxn) = pa(ctn\%yn) = \an\%\pa(yn) < pa{yn)
for n ^ no = re(Ao). Since 52 yn converges absolutely we have
n
oo oo
52 pa{anxn) < 52 p«(yn) < °°>
re—no re—reo
oo
whence )> a„z" converges absolutely in A. Thus, all entire
funereal
tions vanishing at 0 operate on A and the theorem follows from
5.5.11.

CHAPTER VII
GELFAND REPRESENTATION
THEORY
§ 1. Ideals of Topological Algebras
7.1.1. LEMMA, (cf. [20, p.70]). Let A be dense subalgebra of
a TA A. Then:
(i) The closure I in A of an (/. or r.) ideal I of A is an ideal
of A of the same type.
(ii) If the ideal I is regular with a (/. or r.) relative unity u then
I is also regular with u as a relative unity.
(iii) If I ^ A is a closed regular ideal of A then I ^ A and I
is a closed regular ideal of A.
PROOF. It is enough to prove the results when 7 is a 1.ideal.
Clearly 7 is a subspace of A . Further if x G A, a G 7, xa € A,
a/j G 7 and xa —+ x, ap —> a then xaap —> x a. Since each
xaap G 7 it follows that x a G 7, proving 7 is a 1. ideal. Hence
(0-
Next let 7 be a regular 1. ideal with relative r. unity u. If
x G A, xa G A and xa —> x then
ii/-S= lim(xau — xa) G 7 (since xau — xa G 7)
and 7 is regular with u as a relative r.unity, proving (ii).
Finally, if u is a relative unity for closed ideal 7 7^ A, then
u ^ 7 and hence u ^ 7 (since Af|7 = 7) and therefore 7 / A,
proving (iii).
7.1.2. COROLLARY. 7ei A be a TA. Then the closure I of
a I. ideal, a r. ideal, or a bi-ideal I of A is an ideal of A of the
same type.
PROOF. Apply 7.1.1 with A = A.

§ 1. Ideals of Topological Algebras
297
7.1.3. COROLLARY. Every maximal ideal M of A is either
closed or dense.
PROOF. By maximality of M, the closure M = M or A.
7.1.4. DEFINITION. Following Michael [20] we call a TA A
normal if every closed regular 1. (respy. r.) ideal I ^ A is contained
in some closed maximal regular 1. (respy. r.) ideal of A. Further, if
every such I is contained in some closed hypermaximal ideal then
A will be called hypernormal. Trivially, every hypernormal ideal
is normal. Finally, we call a TA A hyponormal if every maximal
regular 1. or r. ideal is closed.
7.1.5. Remark. Since a radical algebra A has no regular
ideal ^ A, such an algebra is vacuously hypernormal and
hyponormal.
7.1.6. LEMMA, (a) Every closed maximal regular I. or r. ideal
M of a hypernormal TA A is hypermaximal.
(b) Every hyponormal TA A is normal.
(c) Every hyponormal TA A is functionally < continuous.
PROOF, (a) By hypernormality there is a closed
hypermaximal ideal Mi with M C M\. The maximality of M => M = M\.
(b) By Krull (1.2.10) if 7 is a closed regular ideal ^ A there
is maximal regular ideal M with I C M. Since A is hyponormal,
M is closed and so A is normal.
(c) Since A is hyponormal every hypermaximal ideal is closed
and so by 1.3.9, 2.1.30 A is functionally continuous.
7.1.7. We give below examples to show that neither of the
properties hypernormality, hyponormality need imply the other.
Example 1. Let A be any radical algebra (A = \/A) and A\
its unitization. Then Ai is a TA under the indiscrete topology.
The ideal A is hypermaximal in A\. Since A is not closed in Ai,
Ai is not hyponormal. On the other hand, since no ideal I ^ A\
is closed it is vacuously hypernormal.
Example 2. The Williamson algebra W. We have seen in
For definition see 2.2.19.

298
Gelfand Representation Theory
3.6.33 that *W is a commutative Hausdorff division algebra over
C. Since {0} is the only maximal ideal of *W and it is closed,
"W is hyponormal. But since {0} is not hypermaximal, "W is not
hypernormal.
7.1.8. PROPOSITION. In a Q algebra A if I ^ A is a
regular I or r. ideal then its closure I ^ A.
PROOF. Since A is a Q algebra, Gq is open. If u is a
relative (r. or 1.) unity of 7 the same is true of u + a for any
ael (see 1.2.8(a)). Therefore, by 1.2.9 (ii), ~u-a ^Gq, so that
(-« + /)flGg =0 and hence also -u + If]Gq = 0 (since Gq is
open). It follows that
-u + 7=-u + 7^A, so that 7 ^ u + A = A.
7.1.9. COROLLARY. Every maximal regular (I. r. or bi-) ideal
M of A is closed. Hence, every Q algebra - in particular [see
3.6.23) a pseudo-Banach or [more generally) a sm. (71) algebra -
is hypernormal.
7.1.10. LEMMA. Let A — (A,\ ■ |) be a unital sm. (F)
algebra with unity e. Then for any ideal I ^ A we have:
1 <<f(e,7) t < |e|; (*)
in particular when \e\ = 1, d(e,I) = 1.
PROOF. Since 0 G 7, d(e,I) < d(e,0) = |e|.
If d[e,x) — \e — x\ < 1 then, by 3.3.18 (ii), x is invertible and
hence x ¢. I (since I ^ A). It follows that, for x G 7, we have
d{e,x) > 1. Hence the inequality (*).
7.1.11. LEMMA. In a TA A, if I\ [respy. Ir) is a closed
l.(respy. r.) ideal of A then (7; : A) (respy. (7r : A)) is closed.
Hence the primitive ideal P associated with a closed maximal
regular I. (respy. r.) ideal Mi (respy. Mr) is closed.
PROOF. If xa £ (7; : A) and xa —> x G A, then, for any
y G A, xay G 7; and consequently xy G 7; (since xay —> xy and
7; is closed). It follows that x G (7; : A), proving that (7; : A)
t d(e, 7) = inf{|e- o| : a el}.

§ 1. Ideals of Topological Algebras
299
is closed. Similarly, (7r : A) is closed. Finally, since P = (M; : A)
(respy. (Mr : A)), P is closed.
7.1.12. PROPOSITION. A primitive ideal P as well as the
radicals \f~A~, \/A, vA of a hyponormal algebra - in particular of
a Q or of a pseudo-Banach algebra - A, are closed.
PROOF. By hyponormality of A and 7.1.11 every primitive
ideal is closed. Also, every maximal regular bi-ideal being
primitive (by 1.5.10) is also closed. Therefore vA ' (respy. \/A) as the
intersection of primitive ideals (respy. maximal regular bi-ideals)
is closed. Finally, since each hypermaximal ideal of A is closed
(see 7.1.6. (c)) their intersection \/A is also closed.
7.1.13. PROPOSITION. Every pseudo-Michael algebra A is
normal.
PROOF. Let A = (A,P), where P is saturated, and
A = limAa the projective limit decomposition, with Aa pseudo-
Banach algebras.
Let / / A be a closed regular (1. or r.) ideal with relative
(r.orl.) unity u. Since u ^ I and I is closed there is a pa €E P
and an e > 0 such that B€ = {x G A : pa(x - u) < e} is disjoint
with I. It follows that
pa(a ~ u) > e (a G I). (1)
Using the notations in (the proof of) 4.5.3, 3.4.15 we can write
a# = a + Na = aa; <pa : x i—> xa = x + Na = z# (7Va = kerpa).
Then we have
pf{aa - «a) = p*{a* - u*) = pa(a - u) > e(a e 7). (2)
Clearly, /a = ^a(7) is a regular ideal of Aa, with relaive
unity ua; ua ¢. Ia because of (2). By 7.1.1, Ja = Ia is a closed
regular ideal (with relative unity ua) ^ Aa. By Krull Ja is
T That V A is closed can also be deduced from the fact that it is the
intersection of all maximal regular 1. ideals.

300
Gelfand Representation Theory
contained in some regular maximal ideal M. Since Aa is pseudo-
Banach, by 7.1.9, Ma is closed. By continuity of ipa and 1.2.17
(iii) M = ^~1(Ma) is a closed regular ideal of A which is
maximal, and clearly I C M. This proves that A is normal.
7.1.14. COROLLARY. A complex or strictly real commutative
pseudo-Michael algebra A is hypernormal.
PROOF. The ideal Ma in the proof above (of 7.1.13) is now
closed hypermaximal since Aa is Gelfand (by 7.2.17, 7.2.19) > .
Let xn be the character determined by Ma\ xa is continuous.
Then, if we set x = Xn° <Pa, *(«) = *„(«<*) = 1, so that x is
a continuous character of A. If M = ker% then M is a closed
hypermaximal ideal. Also, if x G I then (pa(x) G Ja C Ma,
x(x) = 0, so that I C kerx = M. This proves A is hypernormal.
7.1.15. Remark. There are commutative Michael algebras
which are not functionally continuous (and so not hyponormal)
(see [20, p.49]).
7.1.16. Remark. There are hypernormal TA's having dense
regular maximal ideals. For an example, consider the algebra C =
C(R,K) of all K-valued continuous functions on R topologized by
the family {|| ■ ||„} of sm. seminorms, where
11/11,, = sup |/(i)| (/ G C) (cf. Example of 4.6.8)
\t\<n
C is a unital commutative locally sm. 5 algebra. Note that C is
strictly real when K = R. By 7.1.14, C is hypernormal. C has,
however, dense maximal ideals. For instance, if
I = {/ G C : f(t) = 0 for t > (some) t0(f)}
I is clearly an ideal. We claim that I is dense in A. To see this
define for an / G C,
f f(t) if t < n
/»,*(*)={ /(0^7^ if n^t^n + e
{ 0 if t > n + e.
' That An ^ V An holds since Aa admits the regular ideal Ja.

§ 1. Ideals of Topological Algebras
301
Then /„,e G I, and ||/„|£ - /||„ = 0 < e, proving that I is dense
in C. By Krull's lemma there is a maximal ideal M 2 I and then
M is a dense maximal ideal.
7.1.17. PROPOSITION (Rickartt ). Let A be a hyponormal
algebra-in particular a p -Banach algebra or [more generally) a Q
algebra. Let A* be an arbitrary algebra and ip : A —> A* an
epimorphism. Then (p(ker <p) C y/A' where bar denotes closure.
In particular, when A" is s.s., ker <p is closed and \f~A C ker^>.
PROOF. Let Ml be a maximal regular 1.ideal of A*. By
1.2.17 (iii), Mi ~ (p~1(Mf) is a maximal regular 1. ideal of A and
ker <p C Mi. Since A is hyponormal M; is closed. So ker (p C M;.
It follows that
£>(ke7^) C p| Mi = VA*.
When \/A* = {0}, we obtain £>(ker ^>) = {0}, so that ker <p =
ker^. Further, By 1.2.26, <p(VA) C a/a17 = {0}, whence \/A C
ker ^?, completing the proof.
7.1.18. DEFINITION. A bi-ideal I of an algebra A is called
primary or a primary ideal if it is contained in a unique maximal
regular bi-ideal M. An algebra A is called primary if the ideal
{0} is primary - which is the same as A having a unique maximal
regular ideal.
7.1.19. Remark. Every maximal regular bi-ideal is primary.
Moreover, in a Q algebra - in particular in a pseudo-Banach
algebra - the closure 7 of a primary ideal I is primary (if I C M,
where M is a maximal regular bi-ideal, then since M is closed,
I C M and I is primary).
7.1.20. Examples of primary ideals
(i) Denote by d1) = C^)(R,K) the algebra of K-valued
continuously differentiable functions on R. CW is a locally sm. 3
algebra under the family {|| ■ \\n } of seminorm, where
\\f\i1] = sup{|/(0| : |t| < n} + sup{|/W(i)| : \t\ < n}
He has proved this result for Banach algebras [23, p.74).

302
Gelfand Representation Theory
where /(0 denotes the derivative of /. Define in C(0 the
ideals: M0 = {/ G C(0 : /(0) = 0}, h = {f E C(0 : /(0) =
/(0(0) =0}. We have 7i C Mo, and Mo is maximal (since
for / G O1), / - /(0) G Mo). We claim 7i is primary.
To see this, suppose that 7 is an ideal with I Z> I\, I <2
M0. Then there is an / e 7 with /(0) = a / 0. Then
if /3 = /(^(0), <7(i) = f(t) -a-fit, then g E h G I.
Therefore a + fit E I, whence at + /3t2 = (a + /3t)t E 7.
Since clearly /?£2 G 7i c 7 we get at E I, t G 7. Therefore
a = (a + /?i) — j3t E I and I = A. It follows that 7i is not
contained in any other maximal ideal, so that 7i is primary.
(ii) The algebra 0^((0,1]^) is a Banach algebra under the
norm
I /1(1) — II /|| _i_ || /(Oil where II ■ II
|/ | — ||y ||oo i ||y l|oo; wiieie || Hqq
denotes the sup norm. If I\ and Mo are defined analo-
gouly as above then I\ is a primary ideal with Mo as the
containing maximal ideal.
(iii) Denote by CW = CW ([0,1], K) the algebra of all K-valued
functions / whose n th derivative exists and is continuous.
O"' is a Banach algebra with respect to the norm:
II f|H-V In/(-)11 t
\\j\ ~ 2^, r\\\J H°° •
r = 0 '
Set Ir = {/ G CW : /(0) = /(0(0) = ... = /(-)(0) = 0}.
Then it can be shown that 7r (1 < r < n) are primary
ideals with 7„ C 7„_i c ■ ■ ■ C I\ c Md- Also, 7„ is the
smallest primary ideal C Mq (see [10, p.205]).
(iv) In the (F) algebra £ of entire functions (defined in 3.3.14
(iii)) if we set
Ir = {/ G £ : /(')(0) = 0 (0 < j < r)}
then 7r are primary ideals such that
7„c7„_! (n= 1,2,---)
The weighted sum has been taken to make the norm sm..

§2. G elf and Algebras
303
and I0 = M0 = {f <E £ : f(0) = 0} is maximal regular.
Clearly f| /„ = {0}. Since {0} is obviously not primary,
there does not exist a smallest primary ideal contained in
M0.
7.1.21. Remark. For information regarding primary ideals
and other related matters in Banach algebras, see [23, p.92] and
references cited therein, as also [22, pp.238-99].
§2. Gelfand Algebras
7.2.1. DEFINITION. A TA A is called Gelfand algebra if (i)
A 7^ \f~A (ii) every maximal regular 1. or r. ideal is hypermaximal
and closed.
Note that when A is unital, condition (i) is automatically
satisfied (by 1.2.24 (d), e ¢ y/A).
7.2.2 PROPOSITION. Let A be a Gelfand algebra. Then:
(i) The radical \f~A is the intersection of all hypermaximal
ideals of A, i.e. vA = \/A.
(ii) Every character of A is continuous, i.e. A is functionally
continuous; A = Ac 7^ 0.
(iii) A is both hypernormal and hyponormal.
(iv) If x G A then <r'(x) = {X{x) : X e A} |J{0}.
(v) If A is unital and x G A then a(x) = (x(x) : \ e A}.
PROOF, (i) This follows from 7.2.1 (ii), 1.2.22, 1.2.24 (b).
(ii) The ideal ker% is hypermaximal by 1.3.9, so closed by
7.2.1 (ii) and hence \ is continuous by 2.1.30. The condition
7.2.1 (i) ensures the existence of a maximal regular 1. ideal M;
M is closed hypermaximal (by 7.2.1 (ii)) and so determines a
continuous character, so that Ac 7^ 0.
(iii) The condition 7.2.1 (ii) implies that A is hyponormal.
Also, the same condition together with Krull's lemma (1.2.10)
shows that A is hypernormal.

304
Gelfand Representation Theory
(iv) Since always 0 G a'{x) it is enough to consider non-zero
elements on both sides of the equation. If A 7^ 0 in a'(x) then
-A-1£ is not q. invertible and hence, by 1.2.21 (b), 7.2.21 (ii) there
is a hypermaximal ideal M = Mx for which A_1x is a relative
unity. So, by 1.3.10, x(A-1a:) = 1, x(x) = A. Thus a'(x) C
R.H.S. of the equation in (iv). The reverse inclusion follows from
1.7.11, completing the proof.
(v) In view of 1.7.24, it is enough to prove that if A G a(x) then
A = x{x) f°r some \- Since x ~ Ae is not invertible, by 1.2.14,
7.2.1 (ii) there is a hypermaximal ideal M with x — Ae G M. If \
is the character determined by M then x{x ~ ^e) = 0> x(x) = ^-
7.2.3. PROPOSITION. The unitization Ai of an algebra A^
\/A is a Gelfand algebra iff A is a Gelfand algebra.
PROOF. By 1.4.9 (b), there is a bijection between the
hypermaximal ideals Mi (7^ A) of A\ and the hypermaximal ideals M
of A, given by:
MiH->M=Ap|Mi (*)
Mi = M = {zi G Ai : ziti G M} = {zi G Ai : uxi e M} (**)
where u is the relative unity of M.
It follows from (*),(**) that Mi is closed iff M is closed.
These results along with those in 1.4.8 (connecting the maximal
Lor r,ideals of Ai and maximal regular Lor r.ideals of A) prove
the proposition.
7.2.4. Remark. The restriction A 7^ \[A in the statement
of 7.2.3 cannot be omitted. For example, any radical TA A is not
Gelfand (since A = \/A) though its unitization Ai is Gelfand
(by 1.4.11, its sole maximal ideals A is hypermaximal (see 1.1.12)
and closed (see 2.2.9)).
7.2.5. DEFINITION. Let A be an algebra without unity.
Then A is called spectrally Gelfand if it satisfies the condition
<7'(x) = {X(x):XeA}|J{0}(xGA) (*)
where A denotes the set of characters of A.

§ 2. Gelfand Algebras
305
If A is TA then A is called t. (=topologically) spectrally
Gelfand if
a'(x) = {X(x) : X e M |J{0} (x G A) (**)
where Ac denotes the set of continuous characters of A.
7.2.6. DEFINITION. A unital algebra A is called spectrally
Gelfand if
<r(x) = {x(x):Xe*}(xeA). (*1)
Similarly, unital TA A is called t. spectrally Gelfand if
a(x) = {x{x) : X £ Ac} (x G A). (* * 1)
7.2.7. Remark. Since a'(x) = a(x)\J{0}, (*l) => (*) and
(* * 1) => (**).
7.2.8. Proposition, (a) A t. spectrally Gelfand algebra is
also spectrally Gelfand; the two concepts coincide when A is
functionally continuous - in particular when A is p -Banach or even
hyponormal.
(b) Every Gelfand algebra is t. spectrally Gelfand.
(c) The unitizsation A\ of an algebra (respy. TA) A is
spectrally Gelfand (respy. t. spectrally Gelfand). iff A is spectrally
Gelfand (respy. t. spectrally Gelfand).
PROOF, (a) By 1.7.11 (respy. 1.7.24)
X(z) G o'(x) (respy. \(x) £ a(x)) f°r aU X e A-
When A is functionally continuous, A = Ac. Also, ^-Banach =>
hyponormal => functionally continuous. Hence (a).
(b) This follows from 7.2.2 ((ii), (iv), (v)).
(c) If x\ G Ai, xi — \ie\ + x (x G A) then AAl(x\) =
H + a'A(x). Also, Ai = AU{Xo}, Aic = AjU{x0}» where Ai
(respy. Aic denotes the set of characters (respy. continuous
characters) of Ai, A1 (respy. Aj) the set of extensions of characters
(respy. continuous characters) of A to A1; and x0 tne
distinguished character of A\ (kerx0 = A). The assertions (c) of the
proposition are easy consequences of the above relations.
7.2.9. PROPOSITION. Let A be a spectrally Gelfand algebra
and x G A. Then x is q.invertible iff x(x) 7^ ~ 1 for every

306
Gelfand Representation Theory
X G A. If A is a t. spectrally Gelfand algebra and x{x) 7^ ~~1 for
every x ^= Ac then x is q. invertible.
PROOF. If x is q.invertible then by 1.1.24, x{x) is
q. invertible, so that by 1.1.26 (ii), x{x) 7^ -1. If £ is not
q. invertible then -1 £ ff'(i). Since A is spectrally Gelfand there
is a x £ A with x{x) = ~ 1- This completes the proof of the first
assertion.
For the second, assume now that A is t.spectrally Gelfand,
x G A and x{x) 7^ 1 f°r an X e Ac. If possible let x be not
q. invertible; then —1 £ &'{x) and so by hypothesis on A there is
a x e Ac with x(x) = ~ 1, which contradicts our assumption on
X- This contradiction shows that x is q. invertible as required.
7.2.10. COROLLARY. Every unital spectrally Gelfand [re-
spy, t. spectrally Gelfand) algebra A has the Wiener property:
x G A is invertible iff x{x) 7^ 0 for every x ^= A (respy.
every x £ Ac).
PROOF. An element iGA is invertible iff x—e is q. invertible
iff (by 7.2.9) X{x ~ e) ^ -1, i.e. x{x) ^ 1 - 1 = 0, for every
XG A (respy. Ac).
7.2.11. Remark. The commutative unital p-Banach algebra
W considered in 3.4.10 has the Wiener property by 7.2.10, since
it is spectrally Gelfand by 7.2.17, 7.2.8 (b).
7.2.12. PROPOSITION. In a spectrally Gelfand algebra we
have:
\/A=v^t = A«"t = -(/At. (*)
Further, in a t. spectrally Gelfand algebra we have also
Va= cVa^ (**)
so that \/A is closed.
PROOF. By 1.5.12,
VacJacVa. (1)
Also, by 1.7.15 (iii),
t For definitions see 1.5.11, 1.7.13, 1.3.7, 2.2.18.

§ 2. Gelfand Algebras
307
y/A C Aqn C ^A. (2)
If x G \/A it is clear that x G Mx, x{x) = 0^-1 for every x £
A, whence by 7.2.9, x is q.invertible. Thus \/A is a q.invertible
ideal, whence by 1.2.24 (b), \/A C x/A. This together with (1),
(2) proves (*).
Now let A be t. spectrally Gelfand. Then, by (*) of 2.2.18,
y/AC tyAC *tyA.
Therefore, to prove (**) it is enough to show that VA C \JA
and this is done exactly as above for yA (with Ac replacing A).
7.2.13. COROLLARY. A t. spectrally Gelfand algebra - in
particular a Gelfand algebra - is s.s. iff it is q.s.s. iff it is h.s.s. .
PROOF. This is an immediate consequence of (*) of 7.2.12.
7.2.14. PROPOSITION. A t. spectrally Gelfand p-seminormed
s.s. algebra A is p-normed.
PROOF. Let us take A = {A,p). Then, if p(x) = 0 we have:
x, x, ■ ■ ■ —> 0. If x G Ac then x{x) ~~> 0> i-e- x{x) = 0- So
kerpC '\/~A = \fA= {0},
whence p is faithful and p is a /) -norm.
7.2.15. PROPOSITION. Let A^ \/A be either a complex or
formally real commutative TA. Then A is a Gelfand algebra if it
satisfies either of the two conditions:
A is p -seminormed and hyponormal. (*)
A is locally sm. convex and CQ. (**)
PROOF. Let M be a maximal regular ideal with relative unity
u . Then the quotient A* = A/M has u# as unity and A* is a
division algebra (see 1.2.5).
Let now A satisfy (**). Then M is closed in A (by hyponor-
mality) and so A* is ^-normed. By 6.5.6, 6.5.12, A* = Ke and
M is hypermaximal.
Let next A satisfy (*). Then M is closed (by 7.1.9) so that
A# is a Hausdorff division algebra. Further, A* is locally sm.
convex (by 4.4.12 (ii)) and CQ (by 3.6.27). Also, by 1.6.21, A*

308
Gelfand Representation Theory
is formally real whenever A is formally real. It now follows, by
6.5.4, 6.5.8, that A$ = Ke, M is hypermaximal.
Thus we have shown that when either of the conditions
(*)>(**) is satisfied M is closed hypermaximal, so that A is
Gelfand.
7.2.16. COROLLARY. Every complex or formally real unital
commutative algebra satisfying either the condition (*) above or
the condition
A is locally sm. convex and CI (* * l)
is a Gelfand algebra.
PROOF. Since A is unital, A ^ \/~A and further, by 3.6.22
the hypothesis "A is CI" is equivalent to "A is CQ". The
corollary now follows from the proposition.
7.2.17. COROLLARY. Every complex or formally real
commutative p -Banach algebra which is either unital or at least has
a^Va is a Gelfand algebra.
PROOF. By 7.1.9, A is hyponormal. The desired conclusion
now follows from 7.2.15 since A satisfies the condition (*) therein.
7.2.18. PROPOSITION. Let A be a strictly real algebra and
A its complexification. Then A is a Gelfand algebra iff A is a
Gelfand algebra.
PROOF. This follows from 1.9.16 (ii), the result (by virtue of
1.9.18) that M + iM is hypermaximal iff M is hypermaximal,
and the elementary observation that M + iM is closed in A iff
M is closed in A.
7.2.19. COROLLARY. Every commutative strictly real p-
Banach algebra A with A ^ \M is a Gelfand algebra.
PROOF. By 1.9.17 the complexification A of A satisfies the
condition A / V A. Therefore, by 7.2.17, A is Gelfand and so,
by 7.2.18, A is Gelfand.
7.2.20. Remark. C is a 2-dimensional real Banach algebra
C'KI which is a division algebra. cW is not a Gelfand algebra since

§ 2. Gelfand Algebras
309
the ideal {0} which is (regular) maximal (cW being a division
algebra) is not hypermaximal (since codim {0} = 2).
Note that cW is neither formally real (since i2 + i2 — 0) nor
strictly real (since i2 — — 1 is not q. invertible). This example
therefore shows that a commutative real Banach algebra can fail
to be Gelfand if it is neither formally real nor strictly real.
The algebra H]M gives a 4-dim non-commutative real division
algebra which again is not Gelfand (here codim {0} = 4).
7.2.21. THEOREM. Let A be a complex or strictly real
commutative pseudo-Michael algebra with A ^ \/A. Then:
(i) Ac ^ 0 (ii) A is t. spectrally Gelfand.
PROOF, (i) Let A = limAa be the projective limit
decomposition, where Aa are pa -Banach algebras. Since A ^ \f~A~
there is an element x G A which is not q. invertible. By 4.5.6 (i)
there is an a such that xa G Aa is not q. invertible. It follows
that \JAa ^ Aa, and by 7.2.17, 7.2.19, Aa is a Gelfand
algebra. By 7.2.2 (ii), Aa has a continuous character xn (say)- Then
Xn ~ Xo ° <Pa is a continuous character of A, so that Ac ^ 0.
(ii) Suppose that A G a'A{x), A / 0. Then, by 4.5.7 (i),
A G a'- [xa) for some a. But Aa being Gelfand is t. spectrally
-A or
Gelfand and there is a (continuous) character Xa £ Aa = A(Aa),
with X„{x(*) = ^- Then X - Xa ° <Pa ^ &c and x(z) = A.
Further, when A is unital each Aa is also unital (by 4.5.6 (ii)).
In this case if 0 G o{x) then x is not invertible, so that some
xa is not invertible in Aa, whence 0 G <ra(xa)- Since Aa is
Gelfand, by 7.2.2 (v) there is a x„ G Aa with x„(ia) = 0. Then
X ~ X ° Pa e Ac and x{x) = Xa(xa) = 0. On the other hand
if A G <r(x), A / 0 then A G a'(x) and so by what has been
proved above there is a x £ Ac with x(x) = ^- This completes
the proof.
7.2.22. PROPOSITION. Let A be spectrally Gelfand - in
particular Gelfand. Then we have:
(1) For any x, y G A,
(i) a'(x +y)C a'{x) + a'{y) (ii) a'{xy) C a'{x)a'(y);

310
Gelfand Representation Theory
(iii) a(x + y) C <j(z) + a(y) (iv) <t(xj/) C a(x)a(y)
(here we assume A is unital);
(v) r(x + y) < r(x) + r(j/) (vi) r(zj/) < r(z)r(j/).
(2) For any two idempotents u,v(u ^ v) of A such that u <-> v,
(vii) {0} c a'(u -v) = {0, ±1}; r(u - v) = 1.
//" A is unital and «,»^0 or e then
(viii) {0} c<j(u- v) C {0,±1}.
Proof. If xeA then
X(x + j/) = x(z) + X(j/) and x(zj/) = x(z)x(j/)-
From these relations and the definition of spectrally Gelfand the
relations (i)-(iv) follow. Also, results (v), (vi) follow from (i), (ii)
respectively.
It remains to consider (vii), (viii). By 1.7.9 we have:
a'(u),a'(v) C {0,1}. Therefore, by (i),
a'(u -v)C a'(u) + a'(~v) = a'(u) - a'(v) C {0, ±1}.
If we now show that a'(u ~ v) ^ {0} then (vii) will follow.
Suppose to the contrary a'(u — v) = {0}. Then
a'(u - uv) = a'(u(u - v)) C a'(u)a'(u ~ v) C a'(u){0} = {0}.
Therefore u ~ uv <E Aqn = \f~A (see 7.2.12). Since u <-> v, u — uv
is easily checked to be an idempotent. Since it is in \/A, by
1.2.24 (d), u - uv = 0. Similarly v — uv = 0. This, u ~ v - a
contradiction, proving (vii). The result (viii) is proved by similar
arguments.
7.2.23. COROLLARY. (Michael t). Let A be a complex or
strictly real pseudo-Michael algebra. Then for x,y G A with x <->
y, results (i) - (vi) of 7.2.22 hold. Further, results (vii), (viii) of
7.2.22 also hold for A.
PROOF. We may assume that A ^ \[A (if A ~ \J~A the
results hold either trivially or vacuously). Let Am be a maximal
' He considered and proved the results only for complex Michael algebras.

§ 3. The Gelfand Representation
311
commutative subalgebra containing x, y. Then we know by 1.7.26
that for a G Am
CTA„» = cta(«); ffAm(a) = oa{o)
(when A is unital). So we may assume that A is commutative.
But then, by 7.2.21, A is spectrally Gelfand and so the desired
results follow from 7.2.22 .
§ 3. The Gelfand Representation
7.3.1 Let A be a TA and A (respy. Ac) the set of all (respy.
all continuous) characters of A. As remarked earlier A or Ac
may be empty. In the sequel we assume that they are non-empty.
We topologize A by equipping it with the weak topology (or
topology of simple convergence): thus a net xa ~~> X iff Xa{x) ~~* x{x)
for every x G A (xa,X G A). Ac as a subset of A inherits the
relative topology. Let M (respy. Mc) denote the set of hypermax-
imal (respy. closed hypermxaximal) ideals of A. Since there is a
bijection between A (respy. Ac) and M (respy. Mc) (see 1.3.9,
2.1.30) the weak topology on A (respy. Ac) can be transferred to
M (respy. Mc)\ Mc is a subspace of M.
We refer to A or Ac as Gelfand space or spectrum of A; we
refer to M or Mc as the maximal ideal spectrum of A.
If A\ is the unitization of A we denote the corresponding
spectra by Ai, Aic, Mi,Mu.
7.3.2. LEMMA, (a) A ~ Ai\{x0},Ae ~ Alc\{Xo}, where
XH is the distinguished character of Ai and ~ denotes homeo-
morphism.
(b) A is homeomorphic to a subspace of the Cartesian product
K ~ Y[KX [x G A), where each Kx = K. In particular, A,AC
are Hausdorff completely regular spaces.
PROOF, (a) The homeomorphims are given by x ^~* Xi> where
Xl denotes the unique extension of x (see 1.3.17, 1.3.9, 1.4.9).
(b) The map
X~(---,x(aO(e Kx),---)

312
Gelfand Representation Theory
of A into K is clearly bijective. Moreover, the weak topology
of A can be indentified with the relative topology induced by the
product topology of K on the image of A under the above map.
This means that the above map is a homeomorphism. Since each
Kx = R or C is Hausdorff completely regular it follows that A
has also the same properties (since these properties are preserved
by products and subspaces).
7.3.3. PROPOSITION. Let A be a strictly real TA and A its
complexification. Then the map
A : x £ A(A) h-> x ^ A(A) (as defined in 1.9.17)
is a homeomorphism. Similarly, if Ac — A|AC (restriction of A
to the subspace Ac) then
Ac : AC(A) -> AC(A)
is a homeomorphism.
PROOF. We have already seen that in 1.9.18 that A is a
bijection. Since
x{z) = X(z) + *x{y) iz = x + iy, zE A;x,y E A)
X is continuous iff x is continuous. So Ac is also a bijection.
Finally, it is clear that
Xa -+ x iff Xa ~* X-
Thus, A and Ac are homeomorphisms.
7.3.4. Denote by C(A) the algebra of K-valued continuous
functions on A; these functions can also be regarded as functions
on M (since X can be identified with A). For each x E A, define
x on A(= X) by
Hx) = X(x) = x(M) = x(M) (M = kerx).
It is clear from the definition of the weak topology that x E
C(A) = C(X); x is called the Gelfand transform of x. We write
A — {x : x E A} and call A the transform algebra of A. We

§3. The G elf and Representation
313
can consider x operating on A or X according to our
contextual needs. We can also clearly make each x operate on Ac or
Mc- The algebra C(A) is a TA under the weak toplogy. This
means that if fa,f G C"(A) then fa —> f iff fa{x) —> fix) f°r
each x G A. Similarly the algebra C(AC) is a TA under its weak
topology.
7.3.5. PROPOSITION, (i) The map Q : x i—> x is a homomor-
phism of A into C(A) and the map Qc : x \—> x is a continuous
homomorphism of A into C'(A).
(ii) ker$ = \/A (= \J~A if A is sepctrally Gelfand).
(iii) Q is infective iff A is h.s.s. (=s.s. if A is spectrally
Gelfand).
(iv) If A is functionally continuous then Q (= Qc) is a
continuous homomorphism.
PROOF, (i) That Q, Qc are homomorphisms follow from the
fact that each x G A or Ac is a homomorphism of A. Further,
if xa —> x in A and x £ Ac then
*<*{x) = X(xa) -+ x{x) = x(x).
Hence Qc is continuous.
(ii) ker Q — f| Mx(x G A) = \/A = \f~A (when A is spectrally
Gelfand, see 7.2.12).
(iii) This is immediate consequence of (ii).
(iv) When A is funtionally continuous we have A = Ac, so
that Q = Qc is continuous.
7.3.6. COROLLARY. Every real h.s.s. TA (or s.s.t. spectrally
Gelfand algebra) A is formally real.
PROOF. By 7.3.5 ((i|, (iii)), A is (algebraically) isomorphic
to A. Since elements of A are real-valued functions, by 1.6.18, A
and hence A is formally real.
7.3.7. COROLLARY. A commutative strictly real s.s., pseudo-
Michael algebra (in particular, p -Banach algebra) A is formally
real.
PROOF. By 7.2.21, A is t.spectrally Gelfand and so the result

314
Gelfand Representation Theory
follows from 7.3.6.
7.3.8. PROPOSITION. If A = (A,p) is a p-seminormed
algebra, with p sm., then
\x[x)\ < i/(a:)p < p(z)? {x G A,X e Ae). (*)
Hence \\x\\ ^ 1; z/ A is unital with p(e) = 1, then \\x\\ = 1.
PROOF. If M = kerx then Ax = A/M ~ K. Since x is
continuous, M is closed and consequently the identity coset E =
u + M (where u is an element of A such that x(u) = 1) is closed.
Since E2 = E we have for any y £ E, yn <E E (n = 1,2, • • •).
Since E is disjoint with M, 0 ¢. E. It follows that yn /» 0
(since yn £ E and i? is closed). Therefore, by 3.3.4, we have:
p(j/") > 1. If x G A and x(z) = A^0, then A_1x e £. So
p(A-V) = ^-^)^1,
which gives
p(x")>|Ar" = |x(x)|^,
so that we have
\X{x)\^{p(xn)^.
i i
Making n —> oo, we get \x{x)\ ^ v(x)i> ^ p{x)p, which is (*).
Further, Definition 3.5.1. and (*) imply that ||x|| < 1, and ||x|| =
1 when A is unital with p(e) = 1 (since x(e)|p(e)'' = 1)-
7.3.9. COROLLARY. Ac is an equicontinuous family of K -
valued functions on A.
PROOF. For x £ Ac we have
\x{x) - x{y)\ = \x{x - y)\ < p(x ~ y)~? (x,ye A). (*)
The equicontinuity of Ac is therefore a consequence of the
continuity of p.
7.3.10. COROLLARY. If N - kerp, A* = A/N then AC(A)
is canonically homeomorphic to AC(A^).
PROOF. For x £ A(A) define x# £ A(A#) by x#(z + -W) =

§3. The G elf and Representation
315
x{x)- This is well-defined since x vanishes on N (see 3.1.21 (v)).
The map x ^ X^ is clearly a homeomorphism.
7.3.11. PROPOSITION. Let A= (A,p) be a p-seminormed
algebra and A — (A,p) be its completion. Then each x ^= AC(A)
has a unique extension x ^= AC(A) and the map X ^ X *s a
homeomorphism of AC(A) onto AC(A).
PROOF. The existence of the unique extension x °f X follows
from 3.5.11 (b). Again, if x £ AC(A) then kerx 7^ ^ and being
closed we have A 2 kerx, whence x|^ e AC(A). Thus, the map
X ^ X (x — extension of x) is a bijection. It remains to show
that the map is bi-continuous. Suppose that Xa ~^> X- It x € A
then x = lim xn, xn G A. Given e > 0,0 < e < 1, we can find
ra—»oo
an n such that
p(x~xn) < -. (1)
Since Xn{xn) —* x{xn), there is an olq such that for a. > oto,
\Xa{*») ~ X(xn)\ < ^- (2)
Using the inequality (*) of 7.3.9 for p, we have
\Xa{x) - Xa{xn)\ < p(x - xn)~> < ^£j " < £, (3)
e
lxW"X(i)| < 3. (4)
From (2), (3), (4) we obtain, for a > cto,
IXa(*)- *(*)| < ixa(^)"Xa(^)| + |xQ(^) -X(^)| +
Ix(^) - x(i)l
e e e
< 3+3+3^
This implies that Xa -+ X- Thus we have shown that xa —> X =>
Xa -+ X- On the other hand, the reverse implication is trivial since
Xa — Xa\A~, X — x\A- Thus, the map x l—* X is bicontinuous, as
we wished to show.

316
Gelfand Representation Theory
7.3.12. PROPOSITION. The spectrum Ac of a p-seminormed
algebra A is locally compact Hausdorff. If A is unital, or more
generally, if VA ' is regular then Ac is compact Hausdorff.
PROOF. By 7.3.2, we can identify Ac with a subspace of the
cartesian product K = Y\KX (x E A). Write Sx = {A G Kx :
i
|A| < p{x)p} and S = FJ ^ (x E A). Then 5 is a compact
subset of K. By the inequality (*) of 7.3.8, we have: Ac C 5 C
K. Denote by Ac the closure of Ac in S(otK). Take a point
A = (Az) G Ac and write A(x) ~ Xx. Then A(x) = limxa(x) for
some net (x„) m Ac. Since each x„ is a homomophism so is A.
Also, since
i
\Xa(x)\ Kp{x)p
we get
i
|A(l)| ^ p(x)e.
Thus, A is bounded and so, by 3.5.5, continuous. By 1.3.3, A is
either the zero homomorphism ho or A G Ac. Therefore
Ac = Ac or Ac = Ac\J{h0}.
Since 5 is compact and Ac is closed in 5, Ac is compact. Thus
Ac is either compact or locally compact (with Ac as its 1-point
compactification). Also, by 7.3.2, Ac is Hausdorff.
Finally, if VA is regular, with relative unity u, then xu—xE
'\JA C kerx (x E Ac), whence
X{x)x{u) =x(i),x(u) = 1.
It follows that if x^ —+ A then A(u) = limxa(«) = 1, so that
A G Ac and Ac = Ac is compact.
7.3.13. COROLLARY. Any subset Ae of Ac of the form
Ae = {x E Ac : IxC^o)! ^ f}, /or some fixed xq E A and e > 0,
is compact.
PROOF. Clearly Ae is closed in A. Let A^ = A(J{/i0}
be the 1- point compactification of A. Since /10(2¾) = 0, while
t For defintion see 2.2.18.

§3. The Gelfand Representation
317
|x(zo)| > £ for x G Ae it follows that /io is not a limit point of
Ae. Therefore Ae is closed in Aqq and hence is compact.
7.3.14. LEMMA. Let a ^-seminormed algebra A be a direct
sum A= Ai©A2, with Aj(j = 1,2) closed bi-ideals of A. Then
the spectrum Ac = AC(A) has a decomposition
Ae = ASe(jA§e
where A*? are (disjoint) closed subspaces such that
{X\Aj : X e A^} = Ay,e = Ae(Ay) (j = 1,2).
PROOF. The above decomposition into subspaces follows from
1.3.18. It remains to show that they are closed. But this is quite
easy. For, if x £ Ac then x e A?e iff X\M = 0. If xjA2 = 0
and x„, ~~* X then for y G A2
X(j/) = limx„(y) =0, so that
X G A°c, proving A°c is closed. Similarly, A^ is closed.
7.3.15. DEFINITION. A subset 5 of a TA A which has
no unity is said to topologically generate or t. generate A if A
is the smallest closed subslgebra containing 5; 5 is called a
t. generating set of A. If A is unital, with unity e, then the
condition is modified by requiring that A is the smallest closed
subalgebra containing 5 and e.
A TA is said to be finitely or countably t. generated according
as it has a finite or a countable t. generating set of elements.
7.3.16. PROPOSITION. Let Ac be the spectrum of a p-
seminormed algebra A = (A,p) and S a t. generating set of A.
V (xJ>X«, ^ Ac then for xn ~* X0 in Ac it is sufficient if
X„{y) -* Xn{y) for each y G 5. (*)
PROOF. Suppose that (*) holds. Then given x G A and 0 <
e < 1, by definition of t. generating set, there is a polynomial P in,
say, n (possibly) non-commuting variables and elements j/i, • • •, yn
in 5 such that

318
Gelfand Representation Theory
p[x-P(yi,•••,»„)) < -.
Further we have
\xn(x) " X0(x)\ < \xn(x) - xffl(P(yi, • • •, J/„))| + \xa(P(yi,
-x,XP[yi,'",yn))\ + \x0{P(yi,~',yn)) ~ xn{x)\ ■
Now
\x.Ax) -Xa(p(yu---,yn))I < |x«(x-P(yir--,yn))I
< p(x- P(j/i,••-,!/«))"
( by (*) of 7.3.8)
< UJ < "(since 1/p >
Similarly
\x0(P(yi,---,yn))-Xo(x)\ < g-
By our hypothesis
x(j/fc) -> X0(j/fc) ik= !»■•-,«)
so that, since P is a polynomial, we have
x„ (p(j/i , • • •, »n)) = p(x„ (»i). • • •. xa (yn))
-* P{x0{yi),---,Xo(yn)) = x0(P(.yi,--
So we can choose an such that for a > an we have
\xa{p{yi,---,yn)) -xAp{yu---,yn))\ < g-
From the inequalities (2)-(5) we obtain, for a > an,
\xAx) - Xn(x)\ <3 + 3 + 3=e'
whence Xa{x) ~^ x{x), completing the proof.

§3. The G elf and Representation
319
7.3.17. COROLLARY. The spectrum Ac of a separable p-
seminormed algebra A is metrizable.
PROOF. Let {xn} be a countable dense subset of A. Set, for
Xi,X2 e Ac
d(y y] = fl IXi(^)~XaK)l
(*)
Then d is a metric. For, if d(x1,X2) — 0 tnen Xi{xn) = Xi{xn),
for all x„, so that by density of {xn} and continuity of XnX? it
follows that Xi = X2- The symmetry property of d is immediate
from the definition of d. Finally, for the triangle inequality
property of d it is clearly enough to prove that each of the summands
in (*) satisfies the triangle inequality. But this can be established
on the same lines as in the proof of 3.1.10.
Now it is clear that a net xa ~~* X> under d, iff
XM~+X{xn) (n =1,2,---). (*)
The set {xn}, being a dense subset of A, is clearly a t. generating
set of A. So, by 7.3.16, condition (*) is equivalent to the
convergence of the net xn "~> X under the topology of Ac. Thus, the
metric topology coalesces with the topology of Ac, completing
the proof.
7.3.18. COROLLARY. The spectrum Ac of a countably
t. generated p -seminormed algebra A is separably metrizable.
PROOF. If 5 is a countable set of t. generators of A then
the set Si of all finite products of elements of 5, is countable.
Denote by 52 the set of all rational linear combination of elements
from Si (where a rational linear combination in the complex case
means the coefficients of the combination have rational real and
imaginary parts). 52 is clearly countable and dense in A, whence
the required result follows from 7.3.17.
7.3.19. THEOREM. Let A be a pseudo-Michael algebra with
projective limit decomposition
A= limAa (Aa pseudo — Banach algebras).

320
Gelfand Representation Theory
Let Ac,Aa denote the sets of continuous characters of A,Aa
respy. . Let Aa denote the subset of Ac comprising characters
which are pa -continuous. > Then:
(i) Ac = |jAa;
a
(ii) Aa homeomorphic to Aa;
(iii) If we define ipa/3 : Aa -+ A^ by \p = *l>ap{Xa) with
Xp{xp) = Xaifiapixp)), where <pap are the maps
associated with the projective decomposition: xa G Aa,xp G
Ap,<pap{xp) = xa, then {Aa;{<pap}} is an inductive >>
system of topological spaces and continuous maps. Write
Ac = ( the direct limit )limAa. Then there is a bijective
continuous map 0 :? Ac —> Ac given by 0(x)(x) = Xa{x) =
Xa(xa) (for any a) where x = lirnXa-
PROOF. The relation (i) is an immediate consequence of
4.3.13. By 7.3.10, 7.3.11 we have Aa ~ A(Aa) ~ Aa which
is (ii).
If 9(xi(z)) = Xia{xa) = @{x2{x)) = x2oi{xa) for all a then
Xi = X2, so that 0 is injective. It is also surjective. For, if
X G Ac,x £ A-a for some a and X — Xa°<Pa for some Xa £ A-a■
It follows that x — 9(x0)' Finally, the continuity of 0 is clear
from its definition.
7.3.20. COROLLARY. If A is unital, each Aa is compact.
PROOF. By 4.5.6(ii) each Aa is unital and consequently, by
7.3.12, Aa is compact, whence Aa ~ Aa is also compact.
7.3.21. DEFINITION. A TA A which is t. generated by a
single element is called monogenic.
Evidently a monogenic Hausdorff TA is commutative.
T i.e. is a continuous character of the algebra (A,pa).
TT For definition see [12, pp.184-5].
* In general © is not a homeomorphism (see [12,p.161, Remark]).

§3. The Gelfand Representation
321
7.3.22. Examples of monogenic TA's.
(i) The Banach algebra C[0,l]. This has f0(t) = t (t G [0,1])
as t. generator. That /o is a t. generator is a consequence of
the Weirstrass approximation theorem.
(ii) (cf. [10, p.33]). The algebra 21 of complex-valued
continuous functions on unit disc |z| < 1 which are holomorphic on
|z| < 1 is a Banach algebra under the sup norm. This
algebra has fo(z) — z as a t. generator. To see this observe
that if f€(z) = f(z/l + e) (e > 0) then / can be
uniformly approximated by /e in |z| < 1. Also, each /e being
holomorphic on |z| < 1 +e can be uniformly approximated
by polynomials in z in |z| ^ 1, whence /o is a t. generator.
(iii) The Banach algebra L1 = 1^(0, l] (see 3.4.6 (vi)). If /i
denotes the constant function l,/i(i) = 1 (( £ [0,1]), then
it is easy to see that if /" = fi*,,,*fi (n factors),
fi(s) = sn~1/(n -1)- It follows from Weierstrass
approximation theorem and the fact that C[0, l] is dense in L1 that
/i is a t. generator of l_i.
(iv) The subalgebra W^_ of Ws (defined in 3.4.10) consisting of
all elements / = ^f{n)eint with f(n) = 0 for n < 0,
WEI
is closed and so a ^-Banach algebra. It has elt as a
t. generator as can be easily seen.
7.3.23. PROPOSITION. Let A be a monogenic unital p-
Banach algebra which is either complex, or when real is strictly
real or formally real. Let A = Ac be the spectrum of A and a a
t. generator of A. Then the map A : x i—> x(a) *s a homeomor-
phism of A onto the spectrum a(a).
PROOF. By 7.2.16, 7.2.19,7.2.17 A is Gelfand and
consequently, by 7.2.8(b), A is surjective. A is also injective. For,
suppose that Xi(a) = X2(a)- ^ x € A then x = lim„ Pn(a) (Pn e
K[X]), so that
Xl(z) = limx1(Pr,(a)) = limPr,(x1(a)) = UmPr,(x2(a))
= limX2(^(a)) = X2(z)>

322
Gelfand Representation Theory
whence Xi = X2- That A is continuous follows from the
definition of the topology of A (the weak topology). By 7.3.12, A is
compact, and a(a) is clearly Hausdorff. It follows that A is a
homeomorphism, completing the proof.
7.3.24. PROPOSITION (Shilov). If A is a complex monogenic
unital p-Banach algebra, with a t. generator a,p(a) = C\<j(a) is
connected.
PROOF. Since <r(a) is compact we can enclose it in a closed
disc D. Then C\D is connected unbounded and C\D C p(a). If
p(a) is not connected it has a bounded component Go which is
open (since p(a) is open and A is locally connected). It follows
that 8Gq C a[a). If Ao G Go then by the maximum modulus
principle applied to Go we obtain, for any complex polynomial
P,
|P(A0)| < sup |P(A)|< sup |P(A)| = sup|P(X(a))|
A69G0 X6"(") X6A
= sup|X(P(a))|<||P(a)||p.
X6A
Let Ao be the subalgebra of A, generated by a; Ao is dense
in A. If x G Ao then x = P(a) for some P G C[X]. The map
X„ '■ P(«) >—► P(Ao) is a bounded homomorphism. By 3.5.11 (b)
this can be extended to all of A, yielding a (continuous) character
X„ of A. Then Ao = X,,(a) e a(a) (by 1-7.24) while by choice of
Aq,Ao G Go G p(a). This contradiction proves the desired result.
7.3.25. Remark. If K is a compact subset of C such that
C\K is connected then it has been shown by Shilov that there is
a monogenic unital Banach algebra A such that K is homeomor-
phic with c(a), where a is a t. generator of A (see [10, p.72]).
7.3.26. PROPOSITION. Let A be a complex monogenic unital
p-Banach algebra with a t. generator a. Then the transform
algebra A can be identified with the algebra of all continuous complex
functions on o{a) which are holomorphic on its interior a(a)°.
PROOF(cf.[28, p.412]). Since A is homeomorphic to a(a) we
can regard x as a function on a(a) : if A G a(a),X = x(a) then

§3. The Gelfand Representation
323
x(X) = x(x)(x G A). Moreover, x is continuous. Thus, A can
be identified with a subalgebra of the algebra C — C(<r(a)) of
all complex continuous functions on a(a). If x G A and x =
lim Pn(a) then
re—»00
\\x - P„(o)||00 < ||a; - Pra(o)||p
so that x is a uniform limit on <r(a) of polynomials on <r(a). Hence
x = x(X) is holomorphic on a(a)°. On the other hand, if / G C is
holomorphic on (f(a)°, then since, by 7.3.24, p(a) is connected we
can apply a theorem^ of Mergelyan to conclude that / is a uniform
limit on &(a) of polynomials, so that f £ A. This completes the
proof.
7.3.27. PROPOSITION. Let A = (A,p) be a p-seminormed
algebra, with p.sm., andCo(Ac) the algebra of continuous K-valued
functions on the [locally compact) spectrum Ac vanishing at 00.
Then:
(i) Qc : x 1—> x is a continuous homomorphism of A into Co(Ac)
such that
1 1
P||oo < "(x)" ^ p(x)p. (*)
(ii) If A is also t. spectrally Gelfand (in particular, Gelfand)
then we have
r(x) = Halloo < 00, (**)
which implies in particular that o'(x) is bounded.
PROOF. We first observe that x G C0(AC) since 1(/10) =
/io(z) = 0 and Ac |J{/io} is the 1-point compactification of Ac. In
view of 7.3.5(i), to prove statement (i) we have only to verify the
inequality (*). But this is an immediate consequence of (*) of 7.3.8
(since ||z||oo — supx |x(z)|).
When A is t. spectrally Gelfand we have o'(x) = {x{x) '■ X e
Ac} U{°}> whence
p||oo = sup \x{x)\ — r(x), proving (**).
x
7.3.28. COROLLARY. If Qc is a t. isomorphism then there is
a positive constant C such that
t See [25, p.385].

324
Gelfand Representation Theory
p{x)2 < Cp{x2) for all x e A. (*)
PROOF. Since Q~l is continuous there is a constant Cq such that
p(x)i> < ColliHoo- Therefore
p(a;)7 < Co 11 ^ I f 2o = Co||^2||oo < Cop{x2)~p,
where in obtaining the last inequality we have used (*) of 7.3.27.
Thus, we obtain p(x)2 < Cp(x2) (C = Cp0).
7.3.29. PROPOSITION. Let A be a pseudo-Michael algebra
with projective limit decomposition A = limAa. Then we have
i
r(x) ^ sup Va(x) I"* . (*)
a
PROOF. By 4.5.5 (2), we have
r(x) = rA(x) = suprA (xa). (1)
a a
Since Aa is a pseudo-Banach algebra the inequalities (*),(**) of
7.3.27 give
r^{x) < i>a{xa)™ = i>a(x)~p^ . (2)
The desired inequality (*) now follows from (1),(2).
7.3.30. COROLLARY. A t.nilpotent' element x of a pseudo-
Michael algebra A is q. nilpotent.
PROOF. Since all va(x) = 0 we conclude from inequality (*)
of 7.3.29 that r(x) = 0 and x is q. nilpotent, as required.
§ 4. GB Algebras
7.4.1. DEFINITION. A p-seminormed algebra A = (A,p) with
p sm., is called a Gelfand-Beurling algebra or a GB algebra if it
satisfies the condition
' For definition see 4.8.8.

§ 4. GB Algebras
325
r(i)' = i/(i)=Hmp(iB)» (<P(*)) (*)
for any x G A. A ^-Banach algebra A satisfying (*) is called a GB
^-Banach algebra.
7.4.2. Remark. By 6.2.11, 6.2.12 every complex or strictly
real Banach algebra is a GB algebra.
7.4.3. LEMMA. Let A be a p-seminormed GB algebra. Then
we have:
(i) For any x G A, v'{x) is bounded.
(ii) If x,y G A and x <-> j/ £/iere
r(a: + ff)<(r(a:)',+r(y)',)p (*)
r(xj/) ^ r(x)r(j/). (**)
PROOF, (i) Since r(x) = u(x)i' < oo, <r'(a;) is bounded,
(ii) By 4.8.3, we have v(x + y) < v(x)+v(y), v(xy) < v(x)v(y).
It follows that
r(x + y) = v{x + y)~r ^(v(x)+v(y)y? = (r(x)p+r(y)py?
i ill
r(xy) = ^(zj/)'' ^ (i/(x)i/(j/))" = v{x)fv{y)f - r(x)r(y).
7.4.4. PROPOSITION. A unital p-seminormed GB algebra A
is spectral.
PROOF. We have r(x)p = v{x). If v{x) > 0 then r(x) > 0,
so that there is a A ^ 0 in a'(x). It follows that A G a(x) and
a{x) ^ 0. If v{x) = 0, then by 3.3.7(v), x is not invertible. So
0 G a(x) and <r(z) 7^ 0.
7.4.5. PROPOSITION. 7n a spectrally Gelfand GB p-
seminormed algebra we have
\\x\\oo = r(x) = i/(x)p (x£A). (*)
Hence, the concepts "essentially nilpotent", "q. nilpotent" and
"t. nilpotent" coalesce and we have
i/A = {xe A: v{x) = 0} = {x G A : r(x) = 0}. (**)

326
Gelfand Representation Theory
PROOF. The relation (*) above results by combining
(**) of 7.3.27 and condition (*) of 7.4.1. The equivalence of
"q. nilpotent" and "t.nilpotent" follows readily from (*) above;
that of "q. nilpotent" "essentially nilpotent" from (*) of 7.2.12.
7.4.6. THEOREM (Beurling-Gelfand-Zelazko). A complex or
strictly real p-Banach algebra A is a GB algebra.
PROOF. We have to prove (see (*) of 7.4.1.) that
r(x) = v(x)i> (x G A). (*)
Since both sides of (*) remain the same when x is considered an
element of the unitization of A, we may assume that A is unital.
Consider first the case where A is commutative. By
7.2.17,7.2.19, A is a Gelfand algebra. So, by 7.3.27, we have
i
r(x) = H^Hoo ^ u(x)f. (1)
Write
||a;||* = v(x)p. (2)
By 4.8.6(ii), || • ||, is a sm. semi-norm. Write I = ker || • ||*. Then
A# = A/1 is a normed algebra with
||z#||# = ||x + 7||f = Hxll* (see 3.4.15). (3)
The completion B of A^ is a commutative complex or strictly real
(see 6.2.20) Banach algebra. Since B,A are Gelfand algebras, by
7.2.2(v) and (1) (above) we have
r(x*) = sup |x(x#)| = sup \x{x)\ = ||z||oo = r(x). (4)
xeA(B) xeA(A)
By 7.4.2, B is a GB algebra, so that we have
r(a:#)=i/||.||.(a:#). (5)
Now
iw(x#) = lim(||x#lf)^= lim(||x"||*)"
INI* n—»oo ra—»oo
= lim i/(xn)^ (where we have used (3),(2))
n—»oo
l l
= lim v{x)e =v(x)p (using 3.3.7(h)(6)).

§ 4. GB Algebras
327
Thus
i/j|.j|jji(a:#) = i/(a:)*. (6)
From (4),(5),(6) we conclude that r{x) = v(x)e, completing the
proof in the commutative case.
When A is not commutative, consider a maximal commutative
subalgebra Aq with iGAq. Then Aq is a p-Banach algebra which
is again complex or strictly real (see 3.7.32). By virtue of (* * *)
of 1.7.25 we have
rA„{x) = rA{x) = r(x).
Since Aq is commutative, by what we have just proved, rA0(x) =
i
v(x)i>. Therefore we obtain
i
r(x) = rA,Xx) = v{x)p-
completing the proof in the general case.
7.4.7. COROLLARY. In a p-normed algebra A we have for
If A is strictly real we have v(x) < r(x)p.
PROOF. First assume that A is complex and let A be its
completion. Then by applying 7.4.6 to A we get
Hx) = rA(x)P < rA{xY ( by virtue of 1.7.20). (*')
Next, if A is real, let A be its complexification. By considering x
as an element of A and applying (*') we get v(x) ^ r(x)p.
If A is strictly real then a(x) = a{x) and so we then have
v{x) < ?(x)" = r(x)p.
7.4.8. THEOREM (Michaelt). Let A be a pseudo-Michael
algebra which is either complex or strictly real and A=limAa
He considered only the case of complex locally convex algebras.

328
Gelfand Representation Theory
its projective limit decomposition. Then for x G A, x = {xa)(xa G
Aa) we have the GB formula
^a(^) = supya(i)^, where va{x) = lim pa(xn)n .
v ' a n—»00
PROOF. By 4.5.7(2)
rA(x) =supr^ (xa).
a a
Since by 7.4.6, Aa is a GB algebra we have also
TaSX°^ = "aK)*" = M1)7" (since PaC1) = p£{xa))-
Combining these two results we get the above GB formula.
7.4.9. PROPOSITION. Let A be a p-normed algebra. If A is
complex, every q. nilpotent element x of A is t. nilpotent; if A is
real every ext. q. nilpotent element is t. nilpotent; if A is strictly
real every q. nilpotent element is t. nilpotent. The radical yA of
every p-normed algebra A is a topologically nil) (bi-) ideal.
PROOF. The first three assertions are immediate consequences
of (*) of 7.4.7 and the fact that when A is strictly real, r(x) = r(x).
For the final assertion we have to show that every element of \f~A
is t. nilpotent. If x G \/A, then by 1.7.15(H), x is q. nilpotent.
Therefore x is t. nilpotent when A is complex (as just seen above).
To obtain this conclusion when A is real it is enough to show that x
is ext. q. nilpotent. If A ^ 0 in C then '^p" £ \T^ (<* = Re A)
and so this element is q. invertible. It follows from 1.8.5 that A ^
or'(x), so that o\x) = {0} and x is ext.q. nilpotent as required.
7.4.10. PROPOSITION. Let A 7^ \/A be a commutative
pseudo-Michael algebra which is either complex or strictly real.
Then all the three concepts-essentially nilpotent, q. nilpotent and
t. nilpotent-coincide.
' i.e. every element of the ideal is t. nilpotent.

§ 4. GB Algebras
329
PROOF. By 7.2.21 (ii), A is t. spectrally Gelfand. So, by
7.2.12, "essentially nilpotent" is the same as "q.nilpotent". Also,
by virtue of GB formula of 7.4.8, "q.nilpotent" is the same as
"t. nilpotent" . Hence the proposition.
7.4.11. LEMMA. Let A = (A,p) be a GB algebra. Then the
following two conditions are equivalent:
(i) There is a constant C\ > 0 such that
Cip(x)2 < p(x2) for all x G A. (*)
(ii) There is a constant C2 > 0 such that
C2p{x) < r{x)p for all x G A. (**)
PROOF. Assume that (*) holds and apply the (*) inequality
successively to the elements x2, x4, • • • ,xn, where n is of the form
n = 2k, to obtain
crV(*r<p(*"),
from which follows that
C^p(x)^p(xn)±.
Making, n —+ 00, we get
Cip(x) < vp{x) = rp(x) ( since A is GB)
which is (**) (with C2 = Cx). Next assume that (**) holds. Then
C2p{x)2 < (r(*))2' = ^(x)2 = ^(x2) < p(x2),
which gives (*) (with C\ = C|). This completes the proof.
7.4.12. PROPOSITION. lei A = (A, || • ||) be a Gelfand p-
normed algebra which is also a GB algebra. Then the map Q :
x *—> x is a t. isomorphism iff there is a constant C > 0 such that
\\x\\2 <: C\\x2\\ for all x G A. (*)

330
Gelfand Representation Theory
In particular, the above conclusion regarding Q holds when A is a
complex or strictly real commutative p-Banach algebra with A 7^
Va.
PROOF. In view of 7.3.28, we have only to prove the "if part.
It follows from (*) that
., ,, 1,, 0..1 1 1,, ,1,,1 |i 1 ij Li,, on,, 1
||Z|| SC C5||£2||2 SC C2C4||£4||4... ^C(2+22+ 2n)||a;2 ||5T
< CV(x) = Cr(xY = C\\x\\^,
so that we have ||x|| < CHiH^. It follows that x = 0 => x = 0,
which means Q is 1 — 1. Again it follows from the above inequality
that Q~l is continuous. Since A is Gelfand, A = Ac and Q = Qc
is continuous (see 7.3.5). Thus, Q is a t. isomorphism, completing
the proof.
7.4.13. COROLLARY. Q is an isometry iff p— 1 and
||z||2 = ||z2|| for all x G A. (*)
PROOF. If Q is an isometry then ||x|| = ||i||oo ^ llxll^ (by (*)
of 7.3.27). This implies that p= 1 (for, if 0 < p < 1 and ||z|| < 1
1 _ ,
then Hill'' < ||x|| since - > 1).
Further we have,
•^ ^ rwn
= ll^2lloo = Ik II which is relation (*).
II IIl-"-' 11 11 \ /
Conversely, suppose (*) holds and p= 1. Then by 7.4.11 (with
Ci = 1) we get (see proof therein)
||x|| ^ r(x)p = r(x) = \\x\loo ( see (**) of 7.3.27).
On the other hand, by (*) of 7.3.27, HzH^ < ||z||. Thus ||z|| =
||i||oo an(l 9 *s an isometry.
7.4.14. LEMMA. Let A = (A, II • II) be an ample complex p-
Banach algebra with || • || sm., and x,y G A. If there is a constant
C such that
\\E(\x)yE(-\x)\\ < C for all A G C,

§ 4. GB Algebras
331
then xy = yx.
PROOF. Write F{\) = E{Xx)yE{-\x). Then we have
F(A) = [e1+A«+^ + Hy[ei-A« + ^ + H
where ei denotes the unity element of the unitization Ai of A. By
continuity of multiplication in A\ we get
F(\)=[e1+\x+^ + ...}ly-\yx+y-^ +•••].
The first series on the right above is absolutely convergent (by
5.2.2) and the second series also is absolutely convergent (as can
be checked by ratio test). So we can multiply (as in the classical
case) the two series term by term to obtain
F(X) = y + X(xy-yx)+X2(^--xyx+^-j+--- . (1)
Since the series in (1) converges absolutely for all A G C, F(X) is a
strongly entire function. If <p G A*t then <p o F(X) = (p(F(X)) is
an ordinary entire function. Further, it is bounded since
\<p{F(X))\ < |M|||F(A)||p <|M|c,
for some constant C. By Liouville's theorem >p o F is constant, so
that
<p(F(X)) = <p(F(0)) = ip(y).
Since A is ample, F(X) = y for all A. Differentiating the series in
A in (1) we get
0 = F'(0) = xy - yx
so that xy — yx = 0, as desired.
7.4.15. THEOREM (La Page-Hirschfeld-Zelazkott). An ample
' A* — continuous dual of A.
'' These authors obtained the result for Banach algebras (i.e. p — 1).

332
Gelfand Representation Theory
complex p-Banach algebra A — (A, || • ||), with \\ • || sm., which
satisfies the condition
Ci||x||2 < ||z2|| for all x G A (*)
is necessarily commutative.
PROOF. Let A\ be the unitization of A, with unity t\. For
x, y G A, A G C set
z= F(X) = E(\x)yE(-\x). (1)
Clearly
fiei = £(Ax)//ei£(-Ax) (2)
(where we have used 5.2.3 (iii)). It follows that
z - fiei = E(Xx)(y - fiei)E(-Xx). (3)
The equation (3) shows that z — //ei is invertible iff y — //ei is
invertible. Therefore we have
a'(z) = a'(y), r(z) = r(y).
It follows from (*) and 7.4.11 that there is a constant C2 > 0 with
C2\\z\\ ^ r(z)" = r(y)".
Therefore
\\nm = \\zH^rr(yy.
Applying 7.4.14, we conclude that xy = yx, completing the proof.
7.4.16. LEMMA. A complex or strictly real p-Banach algebra
A — (A, || ■ ||), with || • || sm., satisfying the condition
CxII^H2 < ||z2|| for all x G A (*)
is s.s. .
PROOF. The condition (*) together with 7.4.11 and the fact
A is GB (see 7.4.6) gives
Ci||x|| ^ v{x) = r(x)p.

§ 4. GB Algebras
333
Therefore, r(x) ~ 0 => x = 0, whence A is q.s.s. and so s.s.
(see 1.7.18).
7.4.17. THEOREM (Kaplansky). Every strictly real s.s. p-
normed algebra A is commutative.
PROOF. First suppose that A is primitive. By 1.9.15, A is a
division algebra and so by 6.5.12, A is isomorphic to R,C, or H.
The strict reality of A rules out C or H. Thus A ~ R. Next let A
be s.s. and P be a primitive ideal. The quotient Ap = A/P is
primitive and by 1.9.8, Ap is strictly real and so Ap ~ R. Since
A is s.s., f]P — {0}, so that A is isomorphic to a subalgebra
of the direct product fj Ap (of isomorphic copies of R) and so
commutative.
7.4.18. COROLLARY. Every strictly real primitive p-normed
algebra is isomorphic to R.
7.4.19. COROLLARY. Every strictly real p-Banach algebra
satisfying condition (*) of 7.4.16 is commutative.
PROOF. By 7.4.16, A is s.s. and so by 7.4.17, A is
commutative.
7.4.20. Proposition (Yoodt) Let A = (A, || • ||) be a p-
normed algebra with \\ ■ \\ sm. Then the following statements are
equivalent:
(i) A is a Q algebra.
(ii) f(xY = v{x) = lim^oo ||x"||" (x e A).
(iii) f(x)p < ||x|| (x e A).
(Here f denotes the spectral radius in A, where A denotes A itself
when A is complex and the complexification of A when it is real.)
Yood proved the proposition for a normed algebra (i.e. p = 1).

334
Gelfand Representation Theory
PROOF. Assume (i). Then there is an r) > 0 such that for any
x with ||x|| < r), x is q.invertible. Set e = r)~x if A is complex,
and = [(1 + 77)2- l]_1 if A is real. For a given x G A take a
A = a. + i/3 7^ 0 in C with |A|P > e||z||. When A is real we have:
|||A|-2(x2 - 2ax)\\ < |A|-2'(||x||2 + 2"|q|"||i||)
< |A|-2"(||x||2+2|A|"||x||)
(since p < 1, \a\ < |A|)
= (||A-1X|| + 1)2-1< (6^ + 1)2-1
<l + »j — l = »j (by the choice of rj). (1)
When A is complex we have
|| -A_1i|| =|A|-"||i|| < f). (1')
Using 1.8.5, 1.7.8 we conclude from (1), (l') that A ^ gt(z), whence
f(l)"<6||l||. (2)
It follows that
f(Xy = [f{xnY]n < e«||zl"
< v{x). (3)
Now the completion f of A is a GB algebra (see 7.4.6). So we have
v(x)=r(x)p ^r(x)p (sinceACr). (4)
From (3),(4) we obtain i/(x) = f(x)p, proving (i) => (ii). Again,
since || ■ || is sm., (ii) => (iii). Finally, to prove (iii) => (i), assume
to the contrary. Thus, suppose A satisfies (iii) but not (i). Then
we can find an x G A such that ||x|| < 1 and x is not q.invertible.
It follows from 1.7.8 that -1 G <r'(x), so that | - 1| = 1 ^ r(x)p ^
||x|| < 1 a contradiction whence (iii) => (i), completing the proof.

§ 5. Holomorphic Functional Calclus
335
§ 5. Holomorphic Functional Calculus for a
Single Algebra Element t
7.5.1. DEFINITION. Let A be a complex unital ^-Banach
algebra (with unity e) and G C C, a nonempty open set. Write
AG = {x G A:a(x) C G}.
If A G G then a(Xe) = {A}, Ae G Ag, so that Aq is nonempty.
Moreover, by 6.2,14, Aq is an open set. Denote by H(G) the set
of (ordinary) holomorphic functions on G.
7.5.2. LEMMA. With respect to pointwise operations, H(G) is
a commutative unital algebra over C. H(G) is,moreover, a locally
sm. 7 algebra, the metric convergence in it being the same as
uniform convergence on compacta.
PROOF. It suffices to prove the second statement. Since G
is an open subset of C it is locally compact Hausdorff as well as
a -compact. It follows '' that we can find an increasing sequence
Kn of compact sets such that (i) |J Kn = G (ii) if K C G and
K is compact then K C some Kn- For / G H(G), set
p„(/) = sup |/(A)|; |/ = 2^ 5^1 ,n rfV
AeiC „=1 ^ 1+PnUJ
Then p„ are sm. seminorms, | ■ | an F-metric, and (H(G),{p„})
a sm. 7 algebra. Further, it is clear from the definition of | • |
that metric convergence is the same as uniform convergence over
compacta.
7.5.3. Definition. For /gH(G) set
f(x)= f{Xe-~x)-1f{X)d\ {xeAG) (*)
where T is any contour surrounding a(x) in G. If we write
x^ = —x^ = —[x — Ae)^1 then we can rewrite (*) as
' The exposition follows closely that of Rudin [24, pp.240-45].
>> See for example [9, p.241].

336
Gelfand Representation Theory
f{x) = J ixf(\)d\. (**)
7.5.4. LEMMA. The A-valued integral in (**) exists and is
independent of the particular contour Y surrounding o(x) in G.
PROOF. By 6.2.9, 5.4.13(b) the integrand in (**) is locally
^-admissible holomorphic on G\a(x). The integral exists by 5.4.9
and the independence of the integral on the contour T is a
consequence of the Cauchy integral theorem (5.4.16 (a)).
7.5.5. LEMMA. Let A be a complex unital p-Banach algebra
and suppose that x G A and a G p{%)- Let Y be a contour in
G = C\{a} such that Y surrounds &{x) in G. Then
— [ (\-a)n(\e-x)-1d\ = (x-~ae)n (n = 0, ±1,±2, ■ • •)■ (*)
m Jr
2
Proof. Set
Z'Kl JT
JX.
By Hilbert relation (6.2.2, (*)) we have x> = xa + (A — a)x^xa,
from which it follows that
7« = ^ / (A " oc)nxad\ + -L /" (A - a)n+lx^xadX.
Zki Jr Zki Jy
For »/ -1, the evaluation of the first indefinite integral gives
*—~fj—xa so that the corresponding definite integral vanishes
since Y is closed. On the other hand, for n = — 1, the first
integral, vanishes since the index I(Y,a) = 0, by hypothesis. Thus
we obtain the recurrence relation
In+i = Inx~l = In{x -ae). (1)
If In denotes the integral in (*) we have In — -In, so that (1)
gives
In+i = In{x-oce) (n = 0,±l,±2,---).

§ 5. Holomorphic Functional Calclus
337
To complete the proof it is clearly enough to show that Iq = e.
By 6.2.10 (*), with m = 0, we obtain Iq = -e, so that Iq = e
as required.
7.5.6. PROPOSITION. Let A be a complex unital p-Banach
algebra and suppose that x G A and R = R(X) a rational function
over C whose poles do not lie in &(x). Further, let
R(X) = P(X) + £>rs(A - ar)~s (1 < r < m)(l < s < kr)
r,s
be the representation of R (obtained by division algorithm and
partial fraction development) where P is a polynomial (which may
be 0), ar the poles of R, and kr the multiplicities of ar. Set
R(x) = P(x) + ]Tcrs(x - are)-s.
r,s
Then
R(x) = R(x) = J-. [ R(X)(Xe - x)-*dX. (*)
PROOF. Write i?i(A) = J2crs(*-ar)-s. By (*) of 7.5.5 and
r,s
linearity we obtain
Mx) = ^~. I i?i(A)(Ae-x)-1rfA. (1)
zm Jy
Again, by (*) of 6.2.10 and linearity we get
P(x)=±,! P(\)(\e-xyld\. (2)
zm /r
Adding (1),(2) we get the representation (*).
7.5.7. LEMMA. Suppose that a G G and f G H(G). Then
f(ae) = f(a)e.
Proof.
f(ae) = -^ f f(X)(X-ay1ed\
= [^-.(f(X)(X-a)-U\]e
— f(ot)e (by classical Cauchy theorem).

338
Gelfand Representation Theory
7.5.8. Set H = H(AG) = {/: / e H(G)}, where each / is a
A-valued function on Aq. Under pointwise operations H is an
algebra over C.
7.5.9. THEOREM. H(Ag) is a complex Hausdorff TA under
the weak topology:
fa —> f iff fa(x) —> f(x) for each x G Ac.
The map A : / i—» / is a continuous isomorphism of H(G) onto
H(Ag)- In particular, H(Ag) is commutative.
PROOF. It is clear that the map A is linear. If / = 0 then,
by 7.5.7,
/(A)e = /(Ac) = 0 (A G G)
so that / = 0, proving A is 1 - 1 .
Since (Ae — x)"1 = — x^ is locally ^-admissible holomorphic
on p(x), it has a local representation
(Ae — x)" = 2. <Pj{^)xj with <pj
3
1
ordinary holomorphic for A G Gx = {A G C : A > v{x)p} (see
proof of 6.2.9). It follows that, for fa, f G H(G) we have
f~a(x) - f{x) = -L I (/a(A) - /(A))(Ae - x)-1
j
so that
||/a(*)-/(*)|| < 7^Ell^lll/r(/-(A)-/(A))^(A)rfA|P
3
^- \ ■" II II 11/ /IIP II IIPIr>IP
^ (2^2-11^11 ll/--/llrll^llrlrlP
< \\fa-f\\prM,
ipip
where M = -.—— } ||x,-|| ||v7||r < oo. It follows that /<*—*/
(2?rV ^ " J" "^Mr
v / j
in H(G) => fa —> / in H(Ag), whence A is continuous. To

§ 5. Holomorphic Functional Calclus
339
complete the proof of the theorem it remains to show that A is
multiplicative: A(/^) = A(/)A(^). Thus if h = fg we have to
show that
h(x) = f{x)g(x) (x G AG).
If f,g are rational functions without poles in a(x), then by 7.5.6,
h(x) = h(x) = fg(x) = f(x)g(x) = f(x)g(x).
In the general case, we make use of Runge's approximation
theorem to find rational functions /„, gn such that
fn~* f, 9n-* 9 in H(G).
Then fngn —> fg = h. It follows that
h - fg = A(fg) = A( lim fngn) = lim A(fngn)
n—»00 ra—»00
= ^Ijrn A(/„)A(0„) = /£,
completing the proof of the multiplicativity. Finally, the commu-
tativity of H(Ag) is an immediate consequence of that of H(G)
and the isomorphism.
7.5.10. Corollary. x^f(x) (xeAG).
PROOF. Define j G H(G) by j(X) = A. Then, since AG is
commutative, j' <-> / which clearly implies: j(x) = x <-> /(a;).
7.5.11. Corollary, (cf. [I5,p 206]). If f e H(G) Aa« a
oo
power-series representation /(A) = V] a„A" (a„ G C) throughout
n=0
G then
oo
f{x)=^2anxn {xeAG).
N
PROOF. Write PN(X) = ]Ta„A". Then PN -* f in H(G).
So, by construction of map A, we have
oo
f{x) = limPN(x) = HmPN(x) = ^2 otnxn.
n-0

340
Gelfand Representation Theory
7.5.12. LEMMA. If f G H(G) and x G AG iAen /(i) is
invertible in A iff f has no zero in &{x).
PROOF. Suppose that / / 0 on ^(^)- By continuity of /
there is an open set Gi such that
a{x) C Gi C G and / 7^ 0 on Gi.
It follows that g — 1/ f is holomorphic on Gi . Since fg = 1
on Gi, by applying 7.5.9 (with Gi replacing G) we obtain
f(x)g(x) = e (using 7.5.7), so that f(x) is invertible. Next
suppose that f(a) = 0 for some a G a(x). Since / is holomorphic
we can write
f{\) = {\-a)g{\), with</GH(G).
It follows that
/(x) = (a; - ae)flf(i) = g{x)(x - ae).
Since (x — ae) is not invertible (since a G o^z)), by 1.1.30, f(x)
is not invertible.
7.5.13. THEOREM (Spectral Theorem). Suppose that A is a
complex unital p -Banach algebra, G C C an open set. If x G Ag
and f G H(G) then
a(f(x)) = f(a(x)).
PROOF. Now a G a(f(x)) iff /(i) - ae is not invertible.
By 7.5.12, this is equivalent to /(A) - a vanishing in a(x), i.e.
aef(a(x)).
7.5.14. COROLLARY. An n.a.s.c' for a complex unital p-
Banach algebra A to have an idempotent u^0,e is that there is
an element a G A with <r(a) disconnected.
PROOF. The condition is necessary since, if u exists, we have
by 1.7.9, <t(u) = {0,1}, so that <r(u) is disconnected.
To prove that the condition is sufficient assume that there is
n.a.s.c — necessary and sufficient condition.

§ 5. Holomorphic Functional Calclus
341
an element a G A with <r(a) disconnected. Since <r(a) is closed
we have a disjoint decomposition
a{a) = Fi\jF2
where Fj 7^ 0, Fj closed in C (j = 1,2). It follows that we can
find open sets Gj D Fj(j = 1,2) such that G1f]G2 = 0. Set
G = G1\JG2. Then Gjf\a(a)^9. Define / G H(G) by
,m _ f 0 if A G Gi
AAJ - \ 1 if AGG2
Then / is holomorphic and /2 = /■ If u = /(a) then u2 = u.
Also,
ff(«)=ff(/(a)) = /Ka)) = {0,l}.
Since <r(0) = 0, <r(e) = 1, we have u/0,e.
7.5.15. PROPOSITION. Let A be a complex unital p-Banach
algebra and A its transform algebra. Holomorphic functions
operate on A in the sense that if x G A, f — /(A) holomorphic in
an open neighbourhood of &{x) there is a y G A such that y <-> x
and
y = fox, i.e. y(x) = x{y) = /(*(*)) (x £ A)-
The element y is uniquely determined whenever A is h.s.s.
(Va = {o}).
Proof. Set
y=7^ f(Xe-x)-1f(X)d\
Zm Jy
where T is a contour surrounding o{x). By 7.5.10, y <-> x. Since
X is a continuous homomorphism we have
*M = ^■fx((^-x)-lf(X))d\
Zm Jy
= ^f^~x(x)rif(x)dx
Zm Jy
— f(x{x)) (by the classical Cauchy theorem).

342
Gelfand Representation Theory
The uniqueness statement for y follows since, when A is h.s.s.,
X(j/i) = X{y) for all x £ A => j/i = j/.
7.5.16. THEOREM (a) (Wiener- Zelazko). If f = f{t) G W t
and f(t) ^ 0 for any t then 1// e W.
(b) (Wiener-Levi-Zelazko). If f G W and F z's holomorphic in
an open set G containing the range of f then F o / G WA
PROOF, (a) Note first that Xtx : / >-> f{ti),ti G R, is a
character of (V. Let next \ De anY character of W. Since W
is Gelfand (by 7.2.17), \ is continuous. So, if / G W, / =
£~oo /("K"' then
oo
x(/) = £/Wx(e*'T-
— oo
Now |x(e")| ^ r(e") < ||e'*||; = 1. Similarly, |x(e_*"*)| < 1, so
that
Therefore
|x(e«'*)| = l, sothatx(eft)=eftl,
for some t\ G R. It follows that
oo
x(/) = £/(»)x(e*T = /(ti).
~oo
so that % = x<i- The Wiener-Zelazko theorem is now an
immediate consequence of 7.2.10 (since tt x(f) + 0 V/" <* "/(*) 7^ 0,Vi" ).
(b) Since the range of / is <r(/), the holomorphic function
F satisfies the hypothesis (for /) of 7.5.15. Also Wp can be
identified tt with W. Hence, by 7.5.15
Fof = Fof = FofeW" = W.
t See 3.4.10 for Definition of W .
tt This is possible since A = W is s.s. (/ e V^4 => x«(/) = /(*) =
0, VieR=j. / = 0).

§ 5. Holomorphic Functional Calclus
343
7.5.17. THEOREM (Michael), t Let A be a complex unital
pseudo-Michael algebra. Given x G A and a function f
holomorphic in an open neighbourhood of <r{x), then is a y G A such that
y <-> x and
Hx) = f°Hx) (*eAe).
The element y is uniquely determined if A is h.s.s. (in particular
if A is commutative s.s.).
PROOF. Let A = (A,P),P saturated, and A=limAa be
the projective limit decomposition of A. For x = (xa) G A denote
by Ta a contour surrounding aa(xa) = a^ (xa). (Note that by
4.5.7(3)), 0^(^) — IJffaWJ Since Aa is a pa -Banach algebra,
a
by 7.5.3, 7.5.4,
we can set
Va = / (Aea - xa) lf{X)
dX
and the definition of ya is independent of the particular choice
of Ya. In the notation of the proof of 4.5.3, <pap is a continuous
homomorphism of Ap onto Aa, when a -< /?. Then
<Pa^{yp) = <Pap (Ae/s - xp)~1f(\)d\
= I (Xea-xa)-1f(X)dX=[ (Xea-xa)-1f(X)
dX
(since, by 1.7.19, aa(xa) C a/3(x/3)), so that <pap(yp) = ya. It
follows that y = (ya) £ A. If x £ Ac = AC(A), then by 7.3.19,
X G Aa for some a, and x determines x« £ Aa with x„(ia) =
x(z). By 7.5.15 applied to Aa there is a ya G Aa such that
(l/a) = /(Xak))- Hence
x(j/) = /(*(*)) or y= /(£).
He obtained the result for the case of locally sm. convex algebras.

344
Gelfand Representation Theory's
The uniqueness of y, when A is h.s.s. , is clear. Finally, we
have y <-> x, since ya <-> xa for each a.
7.5.18. Remark. The result in 7.5.16(a) can be restated in
the following form: if / = f(t) is a continuous complex-valued
27r -periodic function on R such that
oo
(i) 2Z \f(n)\P < °° (/(n))rae^! denoting the n th Fourier
coefficient of / )
(ii) / does not vanish anywhere on R, then
i 00
<7 = — has the property : 2J lff(n)|P < °°-
J ra= —oo
For p = 1 the above result reduces to a celebrated lemma of
Wiener which he uses for proving his general Tauberian theorem.
§ 6. Automorphisms and Derivations
7.6.1. An isomorphism of an algebra A onto itself is called
an automorphism. If A is unital then every invertible element a
in A determines an automorphism Ja = 4^-^^(^) = o,xa~
(a G Gi,x G A). (Note that the identities IaIa-i = Ia-i^a — ^
guarantee that Ja is bijective; also it is clear that Ia preserves
algebraic operations). The automorphism Ia is called an inner
automorphism of A .
7.6.2. LEMMA. Ia = I iff a e Z (the centre of A). In
particular, if A is commutative every Ia = I.
PROOF. Clear.
7.6.3. In a TA we have automorphisms which are
continuous as well as automorphisms which are also homeomorphisms,
i.e. t. automorphisms. In general, not every automorphism is
continuous nor every continuous automorphism a t. automorphisms.
7.6.4. PROPOSITION. Let A be a TA, with unity e. Then:

§6. Automorphisms and Derivations
345
(i) Every inner automorphism of A is a t. automorphism.
(ii) If A is a (F) algebra - inparticular a $ algebra - then
every continuous automorphism is a t. automorphism.
(iii) In a semi-simple p -Banach algebra every automorphism is
a t. automorphism.
PROOF, (i) This follows since Ia ~ /ara_i,/a,ra_i are
continuous, and I~l = Ia-i.
(ii) This is a consequence of the open mapping theorem (see
3.1.15).
(iii) This is proved in chapter 9 (see 9.4.7).
7.6.5. DEFINITION. Let A be an algebra (over F). A linear
map D : A —> A is called a derivation if D(xy) = (Dx)y + xDy
for all x, y G A.
Let P denote the algebra of polynomials f(x) in a single
variable x over F. For f(x) = 70 + ■ ■ ■ + 7n£" (7.7 G F) define
f'(x) = 71 + 272X + ... + n^nxn~1. Then />—>/' is a derivation
of the algebra P.
7.6.6. Remarks. If P = P(A) denotes the set of all
derivations of A it is clear that it is a linear space (over F ). Moreover,
if we write for Di,D2 G D,[Di,D2] = DiD2 ~ D2Di,t\ien it is
easy to see that [Di,D2] G D;[Di,D2] is called the Lie
product of D\,D2. V is in fact a Lie algebra i.e. a non-associative
algebra (= an algebra without the associative law for
multiplication) whose multiplication satisfies: [D,D] = 0,[Di,[D2, 2¾]] +
[AU£>3,£>i]] + [£>3,[£>i,A!]]=o. {D,DuD2,DseD).
7.6.7. LEMMA. Let D be a derivation of A and u an idem-
potent. Then:
(i) «(Du)« = 0.
(ii) If u <-> Du - in particular if u lies on the centre of A -
then Du = 0.
(iii) If A has a unity e then De = 0 , hence also D[Xe) = 0 for
all A G F.

346
Gelfand Representation Theory's
PROOF, (i) Du = Du2 = (Du)u + uDu (1)
so uDu — u(Du)u + uDu, whence
u(Du)u = 0. (2)
(ii) If u <-> Du then equation (1) gives Du = 2uDu =
2u2Du = 2u{Du)u = 0 (by (2)).
(iii) This follows from (ii).
7.6.8. LEMMA (Leibniz rule).
Dn(xy) = J2C-)Dn-rx{Dry) (n = 1,2,.. . ;x,ye A).
PROOF. By induction, making use of the well-known relation
(W-0-M-
7.6.9. COROLLARY. Suppose that D2x = 0. Then:
(i) Dmx = 0 (m>2),
(ii) D(Dx)m = 0 (m> 1),
(iii) Dn{xn) = n\{Dx)n (n > 1).
PROOF, (i) Dmx = Dm-2(D2x) = 0 (m > 2). (1)
(ii) For m = 1, we have ^(.Dx) = D2x = 0.
Assume now that D(Dx)m-1 -0 (m > 2). Then
£>((£>x)m) = D(Dx{Dx)m-1) = D2x(Dx)m~l + Dx.D{Dx)m~l
= O.iDx)™-1 +Dx.0 = 0.
(iii) Assume that
D"-1^"-1) = (n - l)!(Z?i)n-1. (2)
Then
^"(x"-1) = D{{n - l)!(Z?i)n-1) = (n - l)!^*)""1 = 0 (3)
where we have used (2). By Leibniz rule

§ 6. Automorphisms and Derivations
347
x.
Dn(xn) = Dn(xn~1.x)
= Dnxn~\x + r ) Dn-lxn~l.Dx + J2 \ ) Dn-rxn~l.Dr
Using (1),(2),(3) the above RHS reduces to 0 + n.(n -
\)\{Dx)n~lDx + 0 = n!(Dx)", completing the proof (by
induction).
7.6.10. PROPOSITION. Let D be a derivation of an algebra
A and x an element of A such that Dx <-> x. Then:
(i) Dxn = nxn~lDx (n > 1),
(ii) Dmxn = xn~mam for some element am E A (1 ^ m <
n, n > 2),
(iii) Dnxn = n\{Dx)n + xbn for some element bn E A (n > 1).
PROOF. We prove the results by induction.
(i) Assume that Dxn~l = (n - l)xx~2Dx. Then
Dxn = Dxn-1.x + xn-1Dx = (n-l)xx-2Dx.x + xn-1Dx
= (n-l)xn-1Dx + xn-1Dx = nxn-1Dx.
(ii) Assume that Dm~1xn = xn-m+1am-1. Then
Dmxn = D(Dm~lxn) = £>(x"-m+1am_!)
= Dxn-m+1.am-1 + xn-m+1Dam.1
= (n -m + l)xn-mDx.am-1+xn-m+1Dam-1
_ xn~mam( with am = (n — m + l)Dx + xDam_i).
(iii) Assume that
ZT-V"1 = (n - l)!^)""1 + ¢6,,-1. (1)
Then
Dnxn = Dn-lD{xn)=Dn'1{nxn-lDx) = nDn-1{xn-1Dx)
= nDn-1xn-1.Dx+nJ2[~1)Dn-1-rxn-1DrDx
r=l ^ '
n— 1 / -.
= n[(n - l)!(£>z)"~' + xbn-i]Dx + n^ ( " ~ * J 3ran+r£>r+13.
r=l ^ '

348
Gelfand Representation Theory's
for some element bn, where we have used result (ii) for evaluating
each of summands in the second line.
7.6.11. Let A be an algebra (over F). For a G A, set
Dax = (£a — ra)x = ax — xa.
Then Da is a derivation (see below) called an inner derivation.
Further we have clearly the relations
Da+b = Da + Db; DXa = XDa (AeF).
7.6.12. LEMMA, (i) Da is a derivation
(ii) Da — 0 iff a G Z (the centre of A); Da = 0 for all a if
A is commutative.
(iii) If A is a TA then Da is continuous.
(iv) If A = (A, || • ||) is a p-Banach algebra, with || • ||sm.,
then \\Da\\ < 2||a||.
PROOF, (i) Since Da = la - ra it is linear. Further
Da{xy) = axy — xya = (ax — xa)y + x(ay — ya)
= Dax.y + xDay.
(ii) Clear.
(iii) The continuity of Da follows from that of la,ra.
(iv) ||Da(x)|| = 11oi — io11 ^ 2||a|| ||i||.
7.6.13. Remark. The inner derivations form an ideal' Pi
of the Lie algebra V of all derivations. To see this it is enough to
observe that for any derivation D, [D,Da] = Db, where 6 = Da,
(this can be easily checked).
ra
7.6.14. LEMMA. (Da)n(x) = ^(-l)r(")o"-rxor
r=0
PROOF. The proof is by induction. Assume that
T Since a Lie algebra is anti-commutative ([2, y] — —\y,x\) every ideal
is a bi-ideal.

§6. Automorphisms and Derivations
349
n—l (_ -,^
n - 1
r
(-\)n-lxa
{DaY-\x) = YA-wy r yn-l-rxa-
n-l
Then (Da)n(x) = D*{D*)n~l{x) =Ptt ]T ■' ■
= ---^(-)-((-:^(::^
7.6.15. PROPOSITION. Any derivation D of an algebra A
can be uniquely extended to its unitization A\.
PROOF. Define Di(\e + x) = Dx, where x e A,ei is the
unity of Ai and AgF. It is straightforward to check that D\ is
a derivation of A\. The uniqueness assertion follows from the fact
that for any derivation D' of Ai,D'(Ae) = 0, by 7.6.7 (iii).
7.6.16. PROPOSITION. Let A be a real algebra, D a
derivation of A and A the complexification of A . The (unique) linear
extension D of D to A is a derivation of A .
PROOF. D(x + iy) = Dx + iDy. It is straightforward to verify
that D is a derivation of A.
7.6.17. LEMMA. Let A, A* be algebras and ip : A —> A* an
epimorphism. If D is a derivation of A such that D(ker^) C
ker^> then D induces a derivation D* of A*.
PROOF. If x" e A*,x* = <p(x) set D*x* = ip(Dx). Then
D* is a well-defined map on A* since if <p(x) = (p(\/),ip(x —
y) — 0 and cp(Dx) - <p(Dy) = <p(D(x - y)) = 0 (since x - y e
ker (p, D(ker (p) C ker^>). Further, D* is clearly linear and
D*(x*y*) = (p(Dxy) = (p(Dx.y + x.Dy) = (p(Dx)(p(y) + <p(x)<p(xy)
= u x .y + x .U y .

350 Gelfand Representation Theory's
7.6.18. PROPOSITION. Let A be a p-Banach algebra and
D a continuous derivation of A ■ Then E(D) = ExpD is a
continuous automorphism of A.
PROOF. By 5.2.4, E(D) is an invertible, bounded - hence
continuous - linear operator. So to prove the proposition we have
only to show that E(D) preserves multiplication. Now
OO * OO * oo I \
E(D)(xy) = E>"M = EA£( ^^)^)
ra=0 n—0 r=0 \ /
= f(sH(sH
= (E(D)x)(E(D)y),
completing the proof.
7.6.19. PROPOSITION. In a unital p-Banach algebra A,
E{Da) = IE{a){aeA).
PROOF. Using 7.6.14 we get
oo r~,n oo * ( n ( \ \
oo n 1 r
oo n oo n
ra=0 ra=0
= £(a)x£(-a) = I£(a)(x).
' Note that the expression on the LHS of the equality gives the Cauchy
product of the two series on the RHS.

§ 6. Automorphisms and Derivations
351
7.6.20. THEOREM (Singer-Wermer) t . Let A be commutative
p -Banach algebra which is either complex or strictly real. If D is
any continuous derivation of A then D(A) C \/A.
PROOF (following Sinclair). First assume that A is complex
and also that it is unital. For a G A, x G A = Ac, write
x = a — x(a)e. Then x G ker% and Dx = Da. By 7.6.10 (iii),
Dnxn = n\(Dx)n + xbn.
Therefore x{Dnxn) = n\(x{Dx))n (since x{x) = 0). So x{Dx) =
(n\)~»x(Dnxn)n, so that
(n\)-^\X(Dnxn)^
(n\)-«\\Dn(xn\\^ (by 7.3.8)
|(»!)-*(||01ll*l)£,
where \\Dn\\ denotes the bound of Dn (for definition see 3.5.1).
The above inequality, gives
\ x(Dx) \^ (n\)-;\\D\\H\\x»\\^.
Making n —> oo we conclude (since (n!)~« —► 0) that
| x{Dx) |= 0, Dx G ker%.
Dx G \/A = \J~A (since A is Gelfand).
If A has no unity, consider its unitization A\. By 7.6.15, D
can be extended to D\ on A\. By what we have proved above
we get for x G A,
Dx = Dixe VMf]A= \/A (by 1.4.9(c)).
Now we may assume that A ^ \f~A (if A = \/A, trivially Dx G
y/A). Then by 7.2.17, 7.2.8, and 7.2.12 sfA = y/A. Thus, Dx G
Finally, if A is strictly real consider the complexification A.
By 7.6.16 D has an extension L> to A. Then DACDAC \/a.
Also DA C A. So
' Singer-Wermer proved the result for commutative complex Banach
algebras
x{Dx)
<
<
C

352
Gelfand Representation Theory's
DAC Af]yA= y/A (by 1.9.17).
7.6.21. COROLLARY. Let D be a continuous derivation
of a complex or strictly real, commutative pseudo-Michael
algebra A = (A,{pa}) such that each kerpa is invariant under
D : D(kerpa) C kerpa. Then D{A) C \[A.
PROOF. Let A = limAa be the projective limit
decomposition. The continuous derivation D induces, by 7.6.17, a
derivation Da on Aa which can be extended to a derivation Da on
Aa. Since Aa is a ^-Banach algebra we have by 7.6.20, for any
x G A, Da(xa) C \ Aa. It follows that
<pa(Dx) = Da(xa) C y Aa,
and hence
Dx ef](p-1yAa C y/A (see 4.5.8).
a
7.6.22. Remark. It follows from 7.6.20 that in a complex
or strictly real commutative s.s.^-Banach algebra, 0 is the only
continuous derivation. Johnson has shown that in the case of
a complex commutative s.s. Banach algebra, 0 is also the only
derivation (continuous or not). Further, Johnson and Sinclair have
proved that all derivations on any s.s. Banach algebra are
continuous (see [4, pp.93, 95]).

CHAPTER VIII
COMMUTATIVEf TOPOLOGICAL
ALGEBRAS
§ 1. Function Algebras tt
8.1.1. DEFINITION. An algebra A over K is called a function
algebra if there is a set S and every element of A is a K-valued
funtion / on S and the algebra operations are point-wise; we call
A a function algebra over S and write A=3(S') = 3 = {/-'/£
A}. If 3 is ^-normed (respy. ^-Banach) we call it a ^-normed
(respy. ^-Banach) function algebra. 3 is called a canonically p-
normed (respy. canonically p-Banach) funtion algebra iff all / G 3
are bounded and the p-noTui of A is given by ||/||£o, where
H/lloo = SUps£S |/(s)|.
8.1.2. Examples.
(i) If X is a locally compact Hausdorff space, Cq(X) the
algebra of K -valued continuous functions on X vanishing at
oo, then under the sup norm it is a (canonically) Banach
function algebra.
(ii) If 3 = d(S) is a ^-seminormed Gelfand algebra then 3 is
a canonically normed function algebra.
(iii) The Banach algebra A of holomorphic functions (defined in
7.3.22 (iii)) is a canonically Banach function algebra.
(iv) The algebra C(R) (see 4.6.8 (i)) is a locally sm. 3 function
algebra.
' Some of the theory developed in this chapter, especially in §3, §4 deals
with non-commutative algebras ( = algebras not necessarily commutative).
'' The treatment here is very limited. For more information on the
topic consult the books of A. Browder (Introduction to function algebras,
Benjamin, 1969) and T.W. Gamelin (Uniform algebras, Prentice Hall, 1969).

354
Commutative Topological Algebras
8.1.3. Remark. Every function algebra is evidently
commutative.
8.1.4. DEFINITION. Let 5 = 5(5) be a funtion algebra. For
a subset 5o °f 3 we define
ker So = {s G 5 : /(«) = 0, V / G 5o}-
8.1.5. LEMMA. Let 5 = 3(5) be a function algebra. Then
for an s G 5\ker5, Xs '■ f >—> /(«) *'s a character of 5 ararf
M, = ker x, a (hyper) maximal ideal of 5-
PROOF. By choice of s, Xs 7^ 0 and so x« is surjective and
hence a character.
8.1.6. DEFINITION. A maximal ideal of the form Ms (=
kerxs = {/ G 5 : f(s) = 0}) is called /ized or a fixed maximal
ideal.
8.1.7. LEMMA. The ^-norm ||-|| of a canonically ^-normed
algebra 5 = 5(51) satisfies the condition
ll/2H = 11/112 (/6 5)-
PROOF. ||/2|| = (sup, |/2(s)|)' = ((sup, |/(s)|)')2 = H/ll2.
8.1.8. LEMMA. Let 5 = 5(5) be a function algebra with
ker 5 = 0- Then:
(i) For f e^, the range Z(f) C a'(f).
(ii) // 5 is unital, Z(f) C <r(/).
(iii) 5 is A.s.s. and in particular q.s.s. and s.s. .
PROOF, (i) /(s) = x,(/)G<7'(/), by 1.7.11.
(ii) This follows similarly from 1.7.24.
(iii) If / G tyd then f(s) = Xs{f) = 0 for all s and so / = 0.
Thus ^ = {0}. Since, by 1.7.15,
\/3 = V5C </5 = {0},
5 is also q.s.s. and s.s. as required.
8.1.9. PROPOSITION. Let A = 5(5) be a p -normed
functionally continuous function algebra with its p -norm ||-|| sm. and

§ 1. Function Algebras
355
ker A = 0. Then every f G A is bounded and
H/lloo < H/lloo < ^(/) " < ||/||p-
/n particular, the p-norm \\ • \\ of A dominates the canonical
p-norm || • H^.
PROOF. Since A is functionally continuous and ker A = 0,
for each s G 5,
X3 G A = Ac, where x« is given by Xs{f) = /(«) (/ £ A).
By virtue of (*) of 7.3.8 we have
H/lloo = SUp |/(*)| = SUPX„(/) < ll/lloo < K/)7 < ll/H',
a a
completing the proof.
8.1.10. Let 5 be any set, Ks the set of all K-valued
functions on 5. K is, under pointwise operations, a unital
algebra over K; the unity element of Ks is the constant function
1 : l(s) = s (V s G 5). Ks is a TA under the weak
topology: a net fa —> / in Ks iff /a(s) —> f(s) for each s G S.
The subset B(5) = B(5, K) of Ks comprising all bounded
functions is a subalgebra which inherits the relative topology under
which it is a TA. B(5) is also a Banach algebra under the sup
norm || • H^ : ||/||oo = SUP |/(s)|- Since norm convergence clearly
a
implies weak convergence, weak topology of B(5) is coarser than
its norm topology.
If 5 is a topological space we have besides the algebra B(5)
the algebra C(5) of all continuous functions on 5. Further,
BC(5) = B(5)HC(5)
is a subalgebra of both B(5) and C(S).
Now let 5 be locally compact Hausdorff. A function / G C(5)
is said to vanish at oo if given e > 0 there is a compact set K
in 5 such that \f(s)\ < e for all s G S\K. All functions in C(5)
which vanish at oo form a subalgebra Co(S) of C(5). Note that
Co(S') = C(5) if 5 is compact.
8.1.11. PROPOSITION. C(5) for compact Hausdorff S and

356
Commutative Topological Algebras
Cq(S) for locally compact Hausdorff S are Banach algebras under
the sup norm. Moreover, C(5), Co(S) are s.s..
PROOF. The proof of the first statement being
straightforward is omitted. For the second we note that for 3 = C(5) or
Co(S), ker 5 = 0, so that {Ms : s G 5} are all maximal ideals,
and clearly f]Ms = {0}, whence 7 is s.s..
8.1.12. DEFINITION. Let 3 = 5(5) be a function algebra. A
subset 3o of 3 is said to separate points (of 5) if given any pair
(si,s2) °f distinct points there is an / G 3o with /(«i) ^ /(s2)-
5o is said to strongly separate points if it satisfies in addition the
condition ker 3o = 0- A family 3o of functions separating (re-
spy, strongly separating) points is also referred to as a separating
(respy. strongly separating) family.
8.1.13. Remark. Let 5 be a locally compact Hausdorff
space. Consider the Banach of algebra Co(S).
Any strongly separating subalgebra A of Co(S) which is
s.a. (i.e. closed for conjugates) is dense by the extended Stone-
Weirstrass theorem (see [26, pp.166-7, Theorems A,B]).
Conversely, we have the result that any dense subalgebra A of Co(S)
is strongly separating. To see this, observe first that 5 being
locally compact Hausdorff is completely regular. So for any point
so G S we can choose / G Cq(S) with /(«o) 7^ 0- By
density of A we can choose g G A with \\g - /||oo < l/(so)|-
Then g(so) ^ 0. Again, if si ^ S2 are two points of 5 and
/i G Co(S) is such that /i(si) 7^ /1(^2), we can choose g\ G A
with ||ffi-Alloc < ||A(*i)-A(«2)|- Then gi{s{) ^ gi{s2)- Thus
A is strongly separating.
8.1.14. LEMMA (Rickart). Let 3= 3(51) be a function
algebra which strongly separates points of S. Then, given any finite
set of points sq, si, •••,sn (n > 1) of S, we can find an f G 3
with
f {s0) ^0, f(Sj) = 0{j= 1,2, ---,71).
PROOF. We first assume that n ~ 1 and write sq = s,
sx = t (s 7^ t). We shall show that there is a function fs G 3
with A(«)^0, A(0 = o.

§ 1. Function Algebras
357
By hypothesis there is an /e J with f(s) 7^ f(t). We have
to consider three cases.
Case 1: f(t) = 0. We can take fs = f.
Case 2: f(s) 7^ 0, f(t) 7^ 0. By replacing / by a suitable
multiple we may assume that f{t) = 1 and then /(s) 7^ 1. Set
/, =/-/2; then /.(0 = 0, /,(*)^0.
Case 3: /(s) = 0, f(t) = 1. Choose aj6j with <?(s) 7^ 0
(this is possible since kerS = 0). If g(t) = 0, we can take fs = g,
and if g(t) 7^ 0 we take fs = f — g/g(t).
We next assume that n > 2. By the result for n = 1 just
proved above we can choose fj G 3 such that /j(so) 7^ 0,
/>(*,-) = 0 (j = 1,---,n). Set / = /i---/„ (product). Then
/(so) 7^ 0, while /(sy) = 0 (j = 1, • • ■ n).
8.1.15. LEMMA. Let S = (S,r) be a compact Hausdorff
space. Then the weak topology* tw on S induced by any
separating family 5o of C(S) coincides with the initial topology t.
PROOF. tw is clearly coarser then t. Moreover, tw is
Hausdorff since So is separating 5. It follows that r = tw (since
Tw <= t, tw Hausdorff, t compact).
8.1.16. COROLLARY. If S = (5,t) is locally compact
Hausdorff and Cq(S) the algebra of K -valued continuous functions on
S which vanish at 00, then the weak topology tw on S induced
by any strongly separating family So of C(5) coincides with t.
PROOF. Let 5^ = 5|J{oo} be the 1-point compactification
of 5; Sqc is compact Hausdorff. Extend the functions / in So
to Sqo by defining /(00) = 0. Then the extended functions form
a separating family of C(5oo). The required result now follows
from 8.1.15.
8.1.17 PROPOSITION. Let S be a compact Hausdorff space
and A a s.a.** inverse-closed subunital algebra of C(5). Then
' A net sn —> s in S under tw if f(sa) —> f(s), V f £ So-
TT i.e., f £ A => f G A, where f = /(«), f(s) = /(«), bar denoting
complex conjugate.

358
Commutative Topological Algebras
every maximal ideal M of A is fixed, i.e. of the form
M = Ms = {/ G A : f(s) = 0}, for some sE S.
In particular, every maximal ideal of C(S) is fixed.
PROOF. Suppose that A has a maximal ideal M which is
not fixed. Since 1 e A ker A = 0, and by 8.1.5, for any s E S,
Ms is a (hyper) maximal ideal. So by our supposition M ^ Ms.
By maximality of M, M g M„, whence there is an fs E M
with fs(s) 7^ 0. By the continuity of fs we have /, / 0 on
an open neighbourhood U3 of s. Since 5 is compact there is a
n
finite number of points si ■ ■ ■, sn in 5 with [J Uj = 5, where
3 = 1
Uj = USj. Set
/ = /l/i H fnfn, where /y = fs. (j = 1, • • •, n).
Then f E M. Since
/ = l/il2 + --- + l/„|2>o,
/-1 exists in C(5). Since A is inverse-closed in C(5), /~ E A.
Thus, M contains an invertible element /, which is impossible.
So we must have M = Ms for some s, completing the proof.
8.1.18. COROLLARY. Let S be a locally compact Hausdorff
space and Cq(S) the algebra of K -valued continuous functions
vanishing at oo. Then every maximal ideal of Cq(S) is fixed.
PROOF. Consider the 1-point compactification 5^ of 5.
Then C(£oo) is T the unitization A\ of A = Cq(S). The
maximal ideals of A\ — C(5oo) are, by 8.1.17, all fixed and they are
{Ml : s E Soo}, where Mls = {/i e Ai : /i(s) = 0}. It follows
that the maximal ideals of Cq(5) are
Mlf]A=Ms(sE A),
where Ms = {/ E A : f(s) = 0}, and so are fixed.
8.1.19. Remark. In the proof of 8.1.17, the hypothesis that
T i.e. can be identified with.

§ 1. Function Algebras
359
A is s.a. was explicitty used. However, it is possible, for a sub-
algebra A of C(5) which is not s.a. to have the fixity property
for the maximal ideals. For instance, the Banach algebra ¾ of
example (ii) of 7.3.22 is a subalgebra of C(D), which is not s.a.
(/o ¢ 21; fo{z) = z). % has, nevertheless, the fixity property.
In fact, if M is a maximal ideal of IL and fo(M) = zQ then
M = MZi) (see [10, p.33]).
8.1.20. PROPOSITION. Let the algebra A = C(S), where S
is compact Hausdorff, be a TA under a topology t which is finer
than the weak topology tw. Then A is Gelfand and its spectrum
A = Ac is homeomorphic to S.
PROOF. By 8.1.17, if x e A = A(A), then x = Xs (« e
5). Since Xs[f) = /(s)> by definition of tw,xs is tw -continuous
(/<*-►/ if Xs{fa) = fa{s) -> /(«) = Xs{f))- Since t 2 tw,Xs
is also t -continuous. It follows that the maximal ideals M =
Ms(s G 5) are t -closed, hypermaximal and so A = (A, t) is
Gelfand. Consider now the map A : s \—> Xs of 5 to A. By 8.1.17,
A is surjective. It is also injective since 5 is normal if si ^ S2,
there is a / e C(5) with f(si) ^ /(«2), so that xSl ^ Xs2-
Further, A is continuous, since if sa —* s and / is continuous,
XsM) = f(sa)^f(s) = Xs(f).
Finally, since 5 is compact and A is Hausdorff (see 7.3.2) it
follows that A is a homeomorphism.
8.1.21. COROLLARY. The Banach algebra (C(S), || • H^), or
more generally, the p-Banach algebra (C(5), || • H^) has S for
its spectrum.
PROOF. This follows from 8.1.20 by taking t to be the norm
topology.
8.1.22. THEOREM (Gelfand-Kolmogorov-Stone-Banach). If
Si,S2 are compact Hausdorff spaces such that the Banach algebras
C(5i) and C($2) are isomorphic (as algebras) then Si and S2
are homeomorphic.
PROOF. Since the spectrum A of the Banach algebra A =
C(5) depends only on the algebraic structure of A, we have,

360
Commutative Topological Algebras
by 8.1.21, 5i ~ Ai ~ A2 ^ ^2, where ~ stands for homeomor-
phism.
8.1.23. Remark. In the algebra A = C(R) = C(R,K), not
every maximal ideal is fixed. For, let
I = {/ e A : f(t) = 0 for all t > some tf}.
Clearly I is an ideal and by Krull's lemma we can find a maximal
ideal M containing I. It follows from the definition of I that M
cannot be a fixed ideal.
8.1.24. Remark. Let 5 be a completely regular 7\ -space
and A = BC(5) the algebra of bounded continuous functions on
5. Then A is a unital commutative Banach function algebra. If
A is its spectrum then A is compact Hausdorff. It can be shown
that A is the Stone-Cech compactification /3(S) of S (see [26,
p.331] or [28, p.415]).
8.1.25. Theorem. Let A ~ (A, || • ||) be an ample, complex
or strictly real, p-Banach algebra, with \\ • \\ sm.. Then A is
t. isomorphic to a canonically Banach function algebra iff || • ||
satisfies the condition
C\\x\\2 < ||x2|| for all x, (*)
where C is some positive constant (c/. [28, p.409, Theorem.
4.8.]).
PROOF. Let || • || satisfy (*). Then, by 7.4.15 or 7.4.19,
A is commutative. Also, by 7.4.16, A is s.s.. By 7.4.12, the
map Q : x 1—> x is a t. isomorphism of A onto A - which is a
canonically Banach function algebra.
Conversely, suppose that A is t. isomorphic to a canonically
Banach function algebra 5 = (3, || • ||*)- Then, by 8.1.7, we have
(IIj/ID2 = h2\V (j/es).
It follows from 7.4.11 that there is a constant C2 with
C2||»|r < r(y)" (ye 3).

§ 2. Shilov Boundary
361
Let ip denote the t. isomorphism between A and 3- Then we can
transfer the norm of 3 to A by setting for x G A, j/ = £>(z)
ii iii{t II / ^ 11 ii< II 11 ii<
\\T\\ = K/31TI ^^ \\v\\
\\ \\ II r V / II 11 " 11
then || • ||* ~ || • ||. Since r(y) = r(x), we get
C2\\x\\* < r(x)'.
Since || • ||* ~ || • || there is a constant C\ > 0 such that ||z|| <
Ci||a;||*. Therefore we get
-£r\\x\\ ^r(x)p
whence, by 7.4.11, there is a C > 0 with
ni-rii2 < ii-r2n
as required.
8.1.26. LEMMA. Let A be a p-seminormed algebra. Then
the Gelfand transform A is always a strongly separating subalgebra
of C0(Ae).
PROOF. If x £ Ac and a G A\kerx then a(x) = x(«) 7^ 0.
Again, if xi,X2 £" Ac and Xl 7^ X2 there is an element b G A
with Xi(^) 7^ X2(6), and then b{xi) 7^ &(X2)- Hence the lemma.
§ 2. Shilov Boundary
8.2.1. DEFINITION. Let 5 be a locally compact Hausdorff
space, Cq(5) = 0()(5, K) the algebra of all continuous K-valued
functions on 5 which vanish at oo, and AC Cq(S) a subalgebra.
A closed set F C 5 is called a pre-boundary for A if for each
f & A there is a point Sf G F such that |/(s/)| = ||/||oo-
8.2.2. Remark. The set 5 itself is a pre-boundary for A.
For, if / G A,/ 7^ 0, /(so) 7^ 0 there is a compact set K such

362
Commutative Topological Algebras
that (*)|/(s)| < |/(s0)| for s G S\K. Then s0 G if. Since if is
compact there is a si G if with |/(«i)| = sup|/(s)| = sup|/(s)|
(by virtue of (*)).
8.2.3. DEFINITION. A pre-boundary (for A) is called a Shilov
boundary or boundary if it is contained in every pre-boundary (for
A). The Shilov boundary, whenever it exists, is clearly unique
and is denoted by d^S.
A pre-boundary Fq is called a minimal pre-boundary if there
is no other pre-boundary contained in Fq. Evidently dj^S is a
minimal pre-boundary.
8.2.4. LEMMA. Every pre-boundary F\ contains a minimal
pre-boundary Fq.
PROOF. The family T\ of pre-boundaries contained in F\
is clearly a poset with respect to set-inclusion. By Zorn's lemma
(or rather by the equivalent Hausdorff maximal principle > ) there
is a maximal chain So in Si with F\ G 3o- Denote by Fq the
intersection of all sets F in do', then Fq is closed. We claim Fq
is a pre-boundary. To see this, set for f & A,
*y = {*eS:|/(s)| = ||/||oo}.
If / = 0,Ff = S and F0 C Ff. If / ^ 0 then Ff is compact
(since / vanishes at oo ). Since each F G To is a pre-boundary,
Fff]F 7^ 0 and further it is a closed subset of compact Ff. It
follows that
Fff)F0 = Ff p| F)= 0 (Fff]F)^t
FeT0 FeTn
Since / is an arbitary function in A the above relation shows
that Fq is a pre-boundary. Finally, by its construction Fq is a
minimal pre-boundary.
8.2.5. LEMMA. Let F be a minimal pre-boundary for A.
Then an element sq G 5 belongs to F iff for every open
neighbourhood V of sq there exists an f G A such that
t See [16, pp.31-36;

§ 2. Shilov Boundary
363
SUp |/(s)| < SUp|/(s)| = H/lloo- (*)
«es\v sev'
PROOF. If s0 G F then F\V is not a pre-boundary and
so there is an / satisfying (*). Conversely, suppose for so G S1
and an arbitrary open neighbourhood V of so the inequality (*)
holds for some /. Then since ||/||oo =sup|/(s)|> F 2 S\V, so
seF
that V fl F ^ 0- It follows that s0eF = F.
8.2.6. LEMMA. If S is infinite and A C Co(S) a strongly
separating subalgebra, then every pre-boundary F (for A) is
infinite.
PROOF. If possible let F = {si,'" ,sn} and so any point of
S\F. By 8.1.14, we can choose / G A with
/(s0) 7^0, /(«,-) = 0 (./=1, ---,71).
Then ||/||oo > 0 and / = 0 on F, contradicting that F is a
pre-boundary. So F is infinite as required.
8.2.7. PROPOSITION. The Shilov boundary dAS exists for
every strongly separating subalgebra A of Co(S), where S is a
locally compact Hausdorff space.
PROOF (cf. [23, pp.133-4]). First assume that 5 is finite, say,
S = {si,--,s„}. Then S being compact and discrete, Co(S) =
C(5) = K". By 8.1.14, there is a f, G A(j = 1,---, n) such that
/;(*;) ^0, fj{sk) = 0{k^j).
Since ||/j||oo is attained only at sj, every pre-boundary F must
contain all the Sj which means that F = S. Hence d^S exists
and d^S = S.
Next assume that 5 is infinite; by 8.2.6, every pre-boundary
F is infinite. By 8.2.4, we can find a minimal pre-boundary Fq-
Let F be any pre-boundary. We shall show that Fo C F.
Take any point so G Fo and let N be any open neighbourhood
of so; since Fo is infinite we can certainly select to G Fo with
t0 ^ s0. Set N0 = N\{t0}. Then N0 C N and we have 0 ^
F0\No i- Fo (since t0 G ^¾ s0 G F0, s0 ¢ F0\iVo ).

364
Commutative Topological Algebras
Since No is an open neighbourhood of so, and the topology
of S is the same as the weak topology induced by A (see 8.1.16)
we can find functions fj(j = 1, • • •, n) in A and an e > 0 such
that
N02 N! = {teS :\fj(t)-fj(80)\ <e, j = 1,---, n}. (l)
We have also 0 ^ Fo\Ni ^ Fo. The minimality property of Fo
implies that Fq\Ni is not a pre-boundary. It follows that we can
find an element Si G Fo f] N± and a g G A such that
|ff(*i)| = l!ff||oo = c > sup{|j(t)| : t G F0\Ni}.
Therefore |c_1j(*i)| = 1, \c~1g{t)\ < 1 for t G F0\Ari. By
replacing g by c^1^ we may assume c = 1, so that we have
|ff(*i)| = l, |ff(0l < 1 for all i e FoXiVx. (2)
By taking a sufficiently large integer n and setting g\ = gn we
get
llffllloo = |ffl(sl)| = 1,
lsi(OI < 7,—£Tfiron F°\Nl- W
2SUPJ \\fj\\oo
Set hj = fjgi - fj{sj)gi. Then we have for t G Fo\Ni (using(3))
IMOI = l/i(0-/i(*o)||ffi(OI <e (4)
and for i G iVi.
IMOI < ell»illoo = c- (5)
From (4),(5) we conclude, since Fq is a pre-boundary, that
IIMoo < c- (6)
On the other hand, since F is also a pre-boundary there is a
t\ G F such that
|ffi(*i)l = llffllloo = 1-
It follows that
|/j(*i) - /j(*o)| = |/j(*i) ~ /j(so)||ffi(ti)| = \hj{h)\ < 11/iy11oo < c,

§ 2. Shilov Boundary
365
where we have used (6). Therefore, by (1), t\ G N\, so that
ti G N\ f| ^, whence
7Vf|FD 7V0f|F3 JVif|F^0.
Since TV is an arbitary neighbourhood of so we conclude that
s0 G F = F, F0 C F.
Hence Fq = 3^5.
8.2.8. COROLLARY. If A C C0(5) is on/j/ separating then
also dAS exists.
PROOF. Since A is only separating (and not strongly
separating) all f & A vanish at a point so G 5; by separation hypothesis
there cannot be another point in 5 at which all / in A vanish.
Set
So = S\{s0}, A0 = {f\S0 : / e A}.
Then Ao is clearly a strongly separating subalgebra of Co(So)-
So, by 8.2.7, d^nSo exists. It is easy to see that
dAS0 = dAS.
8.2.9. REMARK. The above corollary may fail if A is not
separating. A counter-example is furnished by the following
algebra A. Write 5 = [—1,1] and denote by C(5) the algebra of
continuous K -valued functions on 5. Let A be the subalgebra
of C(5) consisting of even functions (/(«) = /( — «))■ Then dAS
does not exist since [0,1] [-1,0] are both minimal pre-boundaries.
Note that A does not separate s and — s (0 < s < 1).
8.2.10. PROPOSITION. For every dense subalgebra A of
Cq(S), S itself is the Shilov boundary: dA(S) = S.
PROOF. For sq G 5, choose any compact neighbourhood Vc
of so- By local compactness of 5 there exists a g G Cq(S) such
that g(so) = 1 and g = 0 on S\VC. Since A is dense in Cq(S)
there is a f & A such that
!/(«)-ff(«)l < ^ for all sG 5. (l)

366
Commutative Topological Algebras
Since g(s0) = 1, |/(«o)| > |ff(«o)| - |ff(«o) - /(so)| > 1 - \ = \,
so that
ll/lloo >\- (2)
On the other hand, if s G S\VC, g(s) = 0,
l/WI < £ ■ (3)
From (2),(3) we get
SUp \f(s)\ < ||/||oo-
S\V,;
Since any arbitrary neighbourhood V of so contains some
compact neighbourhood the above inequality implies that the
inequality (*) of 8.2.5 holds for V, whence so G Sm, where Sm is a
minimal pre-boundary for A. Since so is an arbitary point of 5
we get S = Sm = dA(S).
Note that in the above proof we have shown directly that every
minimal pre-boundary is 5, so that we have not made use of a
priori the existence of 3^(5) (which of course is assured by 8.1.3,
8.2.7).
8.2.11. Proposition. Let A = (A,p) with p sm., be a
p -seminormed algebra having Ac 7^ 0. Then:
(i) A is a strongly separating subalgebra of Co(Ac).
(ii) The Shilov boundary d^Ac exists.
PROOF. By 7.3.12, Ac is locally compact Hausdorff and by
virtue of 7.3.27(i), A is a subalgebra. Also, by 8.1.26, A is
strongly separating. Finally, (ii) follows from (i) (see 8.2.7).
8.2.12. COROLLARY. d^A exists if A is a p -seminormed
Gelfand algebra. In particular it exists if A ^ \f~A and A is
a complex, formally real or strictly real, commutative p -Banach
algebra.
PROOF. Since A is Gelfand we have A = Ac 7^ 0 whence the
first assertion follows from 8.2.11. The second statement follows

§2. Shilov Boundary
367
from the first taking into account the results of 7.2.17, 7.2.19.
8.2.13. DEFINITION. A p-seminormed algebra A with Ac ^
0 is called self-adjoint or s.a. if to each x G A there is a y with
y = x, i.e. y(x) = x(x) for every x £ Ac.
Note that if A is real it is automatically s.a. (we can take y = x).
8.2.14. LEMMA. The unitization A\ of a s.a. algebra is a
s.a. algebra.
PROOF. If xi = Aei + x (x G A), Choose y G A such that
y = £ in A. Set j/i = Aei + y. Consider xi G Ai = A(Ai),xi ^
Xo (the distinguished character, ker xo = A). Write x — Xi|^
then x £ A = A(A). We have
yi(xi) = Xi (Aei+ 2/) = A + x(j/) = A + i(x)
= (A + i(x)) = (^W)-
Also, yi(xo) = ^ = ^i(xo)-
Thus, yi = ii, proving A\ is s.a..
8.2.15. PROPOSITION. The transform algebra A of a s.a. p-
seminormed algebra A (with p sm.) is dense in Co(Ac). Hence
dAAc = Ac.
PROOF. By 8.2.11, A is a strongly separating subalgebra of
Co(Ac). Also, by s.a. property of A, A is closed for conjugates
(i.e. f £ A => f £ A). By the extended Stone-Weierstrass
theorem (see [26, pp.166-7]) A is dense in Co(Ac). Finally, by
8.2.10, aiAc = Ac.
8.2.16. Some Examples of Shilov Boundary.
(i) Let D be the closed unit disc in the complex plane and
A = ?)(D) the Banach algebra of all continuous complex functions
on D which are holomorphic on the interior D° of D. By the
maximum modulus principle we have d^D = unit circle S1; thus
here the Shilov boundary coincides with the topological boundary
dD of D.
(ii) Consider the bi-cylinder D2 = {(21,¾) G C2 : |zi|,|z2| ^
1}. Let C(D2) denote the Banach algebra of all complex contin-

368
Commutative Topological Algebras
uous functions on D2. Let A = P(D) be the uniform closure in
C(D2) of the subset consisting of restrictions on D2 of
polynomials in zi,Z2-
We shall now show that
dj^D = distinguished boundary 8qD
= {{zuz2)eD2-. |*i| = |*2| = l}.
Note that 8qD2 is a proper part of the topological boundary
3D = {(^i, z2) G D2 : \zi\ or \z2\ = 1}- It follows from the
maximum modulus principle that 8qD2 is a pre-boundary, so that we
have dAD2 C dQD2. Further, for each point p = (e iBl,eih) G
d0D2 the function f(z1,z2) = (21 + e ^)(^2 + e ^2) on D2.
satisfies, as we shall see, the inequality (*) of 8.2.5 with respect to
any open neighbourhood of p of the form
Vn = {(21,¾) eD2 :\Zl-e i9>\ < r,, \z2 - e ie*\ < r,}.
Since any neighbourhood V of p contains some Vn it means that
the inequality (*) is satisfied for all V, and so by 8.2.5. p G d^D2,
proving 8aD2 = d0D2.
It remains to show that the inequality (*) of 8.2.5 holds for
V = Vn. First observe that |/| attains its maximum value at
zi = e i$l,z2 = e if>2, so that ||/||oo = 4. If (zuz2) G D2\V we
have
\zi-eiBl\>ri or \z2 - e if>2\ > tj. (*)
If \ziI or \z2\ = 1 then we must have by (*) above that either
amp z\ 7^ amp e ,Sl or amp z2 7^ amp e l$2. It follows that either
\z\ + e t6l\ < 1 or 1^2 + 6 t$2\ < 2, so that once again the inequality
(*) of 8.2.5 holds.
(iii) Let C(D) denote the algebra of all continuous K-valued
functions on the closed unit disc D in K. Then C(D) is a Banach
algebra under sup norm. Its subalgebra A comprising restrictions,
of polynomials over K, to D is a normed subalgebra. By Stone-
Weierstrass, A is a dense subalgebra, so that, by 8.2.10, d^D =
D.
8.2.17. PROPOSITION (Shilov). Let A' be either a complex
or a strictly real, unital commutative p -Banach algebra and A a
closed subunital algebra of A'. Let A, A' denote respectively the

§ 2. Shilov Boundary
369
spectra of A, A'. Then every character x ^= ^A /ias an extension
to a character x' °f ^. In particular, whenever A is dense in
C(A) every X G A has an extension to a \' G A'-
PROOF. Consider the restriction map r) : A' —> A given
by v{x') — restriction x'|A, which is clearly continuous. Since
A' is compact, rj(A') is compact and so closed in A. We shall
show that rj(A') is a pre-boundary for A. If x G A C A' and
£' = i'(x') then ||i'||oo = v(x)~p = Halloo- By compactness of A'
there is a Xo G A' such that
P'lloo = |£'(Xo)l = l*(»KXo))li
proving that rj(A') is a pre-boundary. But then 3^A Q r)(A'),
which clearly implies that every x e <^A has an extension x'j
proving the first assertion. When A is dense in C(A), by 8.2.10,
3^A = A, whence the second assertion.
8.2.18. COROLLARY. Every maximal ideal M of A, whose
associated character \ belongs to 3^A, is contained in a maximal
ideal M' of A'.
PROOF. By 8.2.17, x extends to a character x' °f A'; then
M = kerx G kerx' = M1.
8.2.19. Remark. In general not every maximal ideal of A is
contained in some maximal ideal of A'. For example, if we denote
by 51 the unit circle in the plane C and consider the algebra A' =
0(51) = 0(51^) and its subalgebra A comprising the functions
which can be extended to holomorphic functions on the interior
of the circle 51. The function z = e lB G A C A'. Further z is
invertible in A' with z~1 = z. Since z is not holomorphic, z $ A
and z is not invertible in A. It follows that there is a maximal
ideal M of A with z G M. This ideal M is not contained in any
maximal ideal of A' since z is invertible in A'.
8.2.20. PROPOSITION (Shilov). Let A be a complex unital
p-Banach algebra which is monogenic, with t. generator a. Then
3^A can be identified with (the topological boundary) da(a).
PROOF. By 7.3.23, we identify A with a(a) and regard A as

370
Commutative Topological Algebras
a function algebra on a(a). By virtue of the maximum modulus
principle the topological boundary is a pre-boundary for A, so
that d^a(a) C da(a). For proving the reverse inequality take a
point Ao 6 da(a) and an open neighbourhood V of A0 in o{a).
Then
V 5 {A GC : |A- A0| < e}f)a{a), for some e > 0.
Since Ao 6 da (a) there is a Ai G p(a) such that |Ao —
Ai| < §. Set b = (a - Aie)"1. Then 6(A) = 1/(A - Ax),
and |6(Ao)| > f, |&(A)| < £ for A e a(a)\V (since
|A - A0| > e, |A - Ai| > |A - A0| - |Ai - A0| > e - |).
Since 11611«, > f > |S(A)| (A e ff(a)\V), by 8.2.5, A0 e dAa(a).
This completes the proof.
§3. Hull-Kernel Topology
8.3.1. Let A be an algebra (or more generally a ring) with
A 7^ \f~A. Denote by P the set of prime (bi-) ideals P of A with
P 7^ A, by S the set of maximal regular bi-ideals of A, and by
n the set of primitive ideals of A.
8.3.2. LEMMA We have £ C n C />; />, n^0.
PROOF. The inclusion relations follow from 1.5.10, 1.5.9.
Since A ^ y/A, n^0.
8.3.3. For any subset S of P we write
Ks) = is= f]p
Pes
and call k(5) the kernel of S; k(5) is a bi-ideal of A. We have
clearly:
For Si C S2, k(52) C k(5x). (1)
If 7 is a bi-ideal we set h(7) = {P e P : P D 7} and call h(7)
the hull of 7. We have obviously:
For 7C J, h(J) C h(7).
(2)

§ 3. Hull-Kernel Topology
371
The following inclusions are clear from the definitions.
S C hk(5). (3)
7Ckh(7). (4)
By virtue of (1),(3) we obtain
k(5) = khk(S). (5)
Similarly, by (2),(4) we get
h(7) = hkh(7). (6)
(By applying k to (3) we get, by (1), khk(S) C k(S); on the
other hand, khk(S) = kh(k(5)) D k(5). Combining the above
two inclusions we get (5). Similarly, (6) can be proved.)
8.3.4. LEMMA, (i) f]a h(Ia) = h(£a Ia);
(ii)h(/J) = h(i)(jh(J) = h(/|V);
(iii)h(A) = 0;
(iv) h({0}) = />.
PROOF. The properties (i),(iii),(iv) are immediate from the
definition of h. For (ii), we first observe that since IJ C I, J
we have h(7), h(J) C h(7J), so that h(7)|Jh(J) C h(7J). On
the other hand, if P G h(7J) then P D IJ and since P
is prime, P 13 7 or J, so that P G h(7) or h(J), whence
h(7J) C h(7)|Jh(J). Combining this with the reverse inclusion
relation obtained above we get the first equality in (ii). Again,
since h(I)\Jh(J) C h(If)J) C h(7J), the second equality follows
from the first.
8.3.5. PROPOSITION. The set P can be topologized by taking
as its closed sets the family {h(7) : 7 a bi-ideal of A} . The
resulting topology is called the hull-kernel or hk topology. We denote
P with this topology by P^.
PROOF. It follows from 8.3.4 that the family {h(7)} of subsets
of P is closed for arbitrary intersections and finite unions, contains
the empty set and the whole space P. Hence the proposition.
8.3.6. LEMMA. If S is a subset of Phk then S = hk(5),

372
Commutative Topological Algebras
where bar denotes closure in P^.
PROOF. By definition of hk topology, hk(5) is a closed set
and hk(S) 3 S (by(3) of 8.3.3) Further, if h(7) D 5 then h(7) =
hkh(7) D hk(5). Therefore S = hk(S).
8.3.7. COROLLARY. For a closed subset S, among the bi-
ideals I of A with h(7) = 5 the largest one is k(5).
PROOF. First observe that k(5) is a bi-ideal and h(k(5)) =
hk(5) = 5 since 5 is closed. Next, if h(7) = 5, then 7 C kh(7) =
k(5).
8.3.8. PROPOSITION. Phk is a T0-space. The sub space n
is also a Tq -space while the subspace S is T\.
PROOF. If PltP2 e Phk and P2 2 Pi then P2 ¢ h({Pi}) =
{Pi}, proving P^ is To- By a similar argument n is also To-
Finally, if M e £ then {M} = h({M}) = {M} (by maximality
of M, whence £ is 7\.)
8.3.9. DEFINITION, n = n^ is called the structure space of
A, £ = S^ the strong structure space of A.
8.3.10. PROPOSITION ([23, p.79]). Let J be a bi-ideal of an
algebra A. Then:
(i) S^\h(J) is homeomorphic with £j under the map 9 :
M -> J D Af.
(ii) 7/ A* = A/J iAen the hull h(J) o/ J in £^ is
homeomorphic with £^# under the canonical map 9^ : M i—> M/J.
PROOF. By 1.4.3, the map 9 : M ^> M' = Jf)M is a
bijection between S^\/i(J) and £j. To prove that 9 is a home-
omorphism it is enough to prove that
0(hk(S)) = hk(0(S)).
Clearly k(0(S)) = JflK5)- If M e hk(5) then M ^ k(5) and
0(M) 2 Jf)HS), so that 0(M) e h(Jflk(5)). Conversely,
suppose that
jf|MD k(0(S)) = Jf|k(5)-

§ 3. Hull-Kernel Topology
373
Then M 2 Jf)HS)> an^ since M ~£ J, by primality of M,
M 2 k(S), so that M G hk(5). This completes the proof of (i).
To prove (ii), we observe that if 7 is a bi-ideal of A which
contains J, and its image under the canonical homeomorphism
is denoted by 7^, then the correspondence I <—* I# is easily
seen to be a bijection preserving the inclusion relation. Therefore
maximal bi-ideals correspond to maximal bi-ideals. The required
conclusion now readily follows.
8.3.11. PROPOSITION(cf. [23, p.79]). If I is a regular bi-
ideal of A then h(7), h(7) f]R, h(7) f|S are compact under the
hk -topology.
PROOF. Let {Fa} be a family of closed sets in any one of
these spaces such that
n^=0- (*)
Write k(Fa) = Ia and K = ]Tt ia. Since Fa 2 h(7) we have
a
I C hk(7) C k(Fa) = Ia.
So
K 2 Ia 2 I,
and K is regular. Suppose that K ^ A. Then by 1.2.10 there is
an M e S with K C M. But then
M 2 K 2 Ia, so that M e p| h(7a) = p| Fa.
(since h(7a) = hk(71a) = Fa ). The last conclusion contradicts
(*) . So we must have K = V^7a = A. Let u be a relative
a
(bi-)unity for 7. Since
uGA=7T = ^7a
a
there is a finite subfamily of ideals I\, ■ ■ ■, In (Ij = Ia ) such that
u<E J = 7i + --- + 7,,.
' y Ia denotes the smallest bi-ideal containing the Ia 's.

374
Commutative Topological Algebras
But since J D Ij D 7, J is regular. Therefore, since u G J,
J = A. This implies that
n *> = n hk(^) = n w>)=h(j)=0-
i=i y=i /=1
which proves the desired compactness.
8.3.12. Let A be a TA and M (respy. Mc) the set hy-
permaximal (respy. closed hypermaximal) ideals of A. We have
clearly the inclusion relations
Mc c M c s
so that A(,A(C inherit, by relativization, the hk-topology. This
topology on M (respy. Mc can be transferred to A (respy. Ac)
in view of the bijection.
X -> A ( respy. Mc -> Ac) (see 7.3.11).
If 7£ is a subset of A or Ac we define
k(£) = f|kerX (xe£).
If 7 is an ideal of A, the hull h(7) with respect to A (respy. Ac)
is defined by
h(7) = {x G A( respy. Ae) : ker* 2 /}■
Then a subset E C A (respy. Ac) is closed iff 7£ = hk(7£).
8.3.13. LEMMA. The hk -topology on A (or Ac) can be
described in terms of net convergence in the following way: a net
Xa —* X *'n ^e hk -topology if it satisfies the condition: for x G
A and any subnet (xa') °f {Xa)> aXa'{x) = 0 for a^ Xa'" =>
X(z) = 0.
PROOF. It suffices to note that if Mai = kerxa'> M — ker%
then
M 2 p| Ma> iff £(Ma») = 0 for all Ma>, => £(M) = 0 (ieA).

§ 3. Hull-Kernel Topology
375
8.3.14. COROLLARY. If tw denotes the (weak) topology of
the spectrum A (or Ac ) and r^ the hk -topology on it, then r^
is coarser than tw : r^ C T Tvj_
PROOF. Suppose that a net xa —> x in A (or Ac) under tw.
Then Xc*(z) —> x{x) f°r an a; 6 A. In particular, if Xa'{x) = 0
for all Xa' then x{x) = 0> which means that Xa ~y X under r^.
Therefore r^ C rB, as required.
8.3.15. DEFINITION. Let A be a TA with Ac 7^ 0. A
is called quasi-unital if there is an element uo 6 A such that
^o(x) 7^ 0, Vx 6 Ac. If A is unital (with unity e ) it is also quasi-
unital since e(x) = 1 ^ 0 (x £ Ac).
8.3.16. LEMMA. Let A be a s.a. p-seminormed algebra such
that Ac is non-empty and compact. Then A is quasi-unital.
PROOF. For each x ^= Ac choose an element xx of A with
ix(x) = x(xx) 7^ ^- By continuity of xx and compactness of Ac
we can find, as in the proof of 8.1.17, a finite subset {xi, • • ■, Xn}
of Ac, elements Xj = xx. and open neighbourhoods Uj of Xj
n
such that ij(x) 7^ 0 on Uj (j = 1,---,71) and [J Uj = Ac.
3=1
Since A is s.a., we can find yj G A withyy = £j (j = 1,--- ,ra).
Set uo = /.xjyj- Then
j = i
Mx) = Z>/M »,-(*) = EM*)i2 > °
3=1 3=1
(since if x e C/y, x0(x) 7^ 0).
8.3.17. PROPOSITION. lei A be a s.a. complex spectrally-
Gelfand p-Banach algebra with compact spectrum A(=AC). Then
\/A is a regular bi-ideal.
PROOF. If yA = A, yA is trivially regular. So we may
assume that A 7^ \f~A. Let A\ be the unitization of A and let
' i.e. every rui. -open set is rro -open.

376
Commutative Topological Algebras
Ai,A denote respectively the spectra of A\, A. It follows (see
7.3.2(a)) that we have
A1 = {XleA1:x1|AeA}|J{x0}, (*)
where x„ is the distinguished character of A\. Since, by
hypothesis A is compact, so is Ai. By 8.3.16, there is an uo 6 A with
uo(x) 7^ 0 for all x £ A. Since A is spectrally Gelfand we have:
ff'(uo) = «o(A)|J{0}.
It follows that ,".
ffi(«o) = «o(A)(J{x0(«o)},
where o\ denotes the spectrum with respect to A\. Since A
is compact, tio(A) is also compact and hence closed. It follows
from 7.5.14 that there is an idempotent u G A\ with u(xo) = 0,
u(A) = 1. But then xo(«) = 0, u e A. Further, if
x G A and x = Xi|A (xi ^ Xo)
then foljfi - u{x)x{x) = 1 • x(X) = xfa).
This relation holds for every x £ A, whence we get
ux - x e f)kerX = \/A = y/A (by 7.2.12),
whence \f~A~ is regular.
8.3.18. COROLLARY. In a commutative s.a. complex p-
Banach algebra A with compact spectrum, vA is regular.
PROOF. We may assume that A ^ \JA~. Then, by 7.2.17, A
is Gelfand and so spectrally Gelfand. The required conclusion now
follows from 8.3.1 7.
8.3.19. COROLLARY. If A is a strictly real commutative
p -Banach algebra with A compact then vA is regular.
PROOF. The complexification A of A is s.a.:
x+iy (x) = xix) + *x(j/) = x(z) - ix{y) = x~iy (x)-
Since, by 7.3.3, A = A(A) is homeomorphic to A = A(A) and
A is compact, A is compact. It follows from 8.3.17 that V A is

§ 4. Completely Regular Algebras
377
regular. By 1.9.17, V A is self-conjugate and
y/A= Ap|VA.
It follows that
VA= VAp|A + ;(VAp|A) = VA + iy/A.
Therefore (see 1.9.16(1)) \f~A = Rev A is regular.
8.3.20. PROPOSITION. Let A be a commutative s.a.,
complex or strictly real, p-Banach algebra. Then the following two
statements are equivalent:
(i) A/\J~A is unital .
(ii) A = Ac is compact .
PROOF, (i) => (ii) : Since A/\J~A is unital, \/A is regular
whence \/A D V/At is also regular. So, by 7.3.12, (ii) holds.
(ii)=> (i): By 8.3.18 or 8.3.19, \J~A is regular, and so (i) holds.
8.3.21. COROLLARY. If A is s.s., then A is unital iff A is
compact.
8.3.22. Remark, (cf. [12, p.52, RMK 4.7]). The above
corollary may not hold if the hypothesis "A is s.s" is dropped. For
example if A\ is commutative unital Banach algebra and A2(^
{0}) a Banach algebra with trivial multiplication. Then A(Ai) =
0 and compact, while A(A2) ^ 0. If A = Ax x A2 then by 7.3.14,
A(A) ~ A(Ai) is compact but A is not unital.
§ 4. Completely Regular Algebras
8.4.1. DEFINITION. Following Willcox an algebra A is called
completely regular* if it satisfies the two conditions:
t See (*) of 2.2.18
TT Some authors especially Russian use the term regular for completely
regular, following the usage of the term by Shilov who first introduced these
algebras in the commutative case. The nomenclature completely regular is
due to Rickart.

378
Commutative Topological Algebras
(i) The strong structure space S is HausdorfF.
(ii) Each point M G S has an open neighbourhood V such that
the kernel k(V) is a regular bi-ideal.
When A is unital, condition (ii) above can be dropped since it is
automatically satisfied (every ideal then being regular).
8.4.2. Examples of Completely regular algebras.
(i) The Banach algebra Co(-Y) (see 8.4.15).
(ii) The group algebra LX(G) of a locally compact HausdorfF
commutative group G (see [22, p.426]).
(iii) The group algebra LX(G) of any compact HausdorfF group G
(see [23, p.83]).
(iv) A von Neumann algebra (see [23, p.290]).
(v) The Wiener-Zelazko algebra W (see Appendix).
8.4.3. PROPOSITION. Let A be a completely regular algebra.
Then:
(i) S is locally compact Hausdorff.
(ii) S is compact Hausdorff if A is unital.
PROOF, (i) Since k(V) is a regular bi-ideal (by (ii) of 8.4.1)
it follows by 8.3.11 that hk(V) is a compact neighbourhood of M.
(ii) When A is unital, every ideal is regular and so in particular,
{0} is regular. Therefore, by 8.3.11, h({0}) = £ is compact.
8.4.4. PROPOSITION. If A is a completely regular algebra and
J a bi-ideal of A then J and A# — A/J are completely regular.
PROOF. Let Xj denote the strong structure space of J. By
8.3.10 (i),
M is a homeomorphism of S\h(J) on Sj.
By complete regularity of A,S is HausdorfF and hence by the
homeomorphism 6, Sj is also HausdorfF. Further, if MeS has V
as an open neighbourhood with k(V) regular then 0(M) has 0(V)

§ 4. Completely Regular Algebras
379
as an open neighbourhood and k(0(V)) = J f)k(V) is regular (cf.
proof of 1.4.2 (v)). Therefore J is completely regular.
It remains to prove that A# is also completely regular. By
8.3.10 (ii), SA# is HausdorfF. If <p : A —> A* is the canonical
homeomorphism and M = ip~l{M#) (M* G £A#) then M G S^-
Let V be an open neighbourhood of Mo = £>-1(M0 ), M* G Sa#,
such that k(y) is regular. The intersection I, of all M G Sa
with M I) k(y), M 13 J, is also a regular bi-ideal (since I I>
k(V)). It follows that if V* = {M* : M D k(V)} then y# is an
open neighbourhood of M0 with k(V*) = £>(k(y)) regular. This
completes the proof.
8.4.5. PROPOSITION. The unitization Ai of an algebra A is
completely regular iff A is completely regular.
PROOF (cf. [23, p.84]). Assume that A is completely regular.
Since A\ is unital, to prove that it is completely regular we have
only to show that Si = SAl is HausdorfF. Since A is a bi-ideal
of Ai, by 8.3.10 (i), S = S^ is homeomorphic to Si\{A}. By
complete regularity of A, S and consequently Si\{A} is HausdorfF.
To complete the proof that Si is HausdorfF it remains to show that
the point A can be separated by open sets from any other point M°
in Si. Write M° = AflMf. By complete regularity of A there is
an open neighbourhood V of Mo with I = k(V) a regular bi-ideal,
with a relative unity u (say). By 1.4.7. (iv), I\ = I = 7+Ai(ei — u)
is a bi-ideal of A1;- h % A since e\ - u G I\. It follows that
A G Si\hi(7i) = U\ (say) were hi denotes the hull relative to
Ai. For M G V0 consider Mi = M. Then Af|Mi = M. Since
M D I = k(y), Mi D /i. It follows that Vi = d'^V) C hi(7i),
where 0 is the homeomorphism Mi h-> Af|Mi = M (see 8.3.10)
so that Uif]Vi = 0. This completes the proof of the "if part.
For the "only if part assume that Si is HausdorfF. Then S ~
Si\{A}, is also HausdorfF. If M G S then M = Af\Mi (M1 G
Si). Since Si is HausdorfF we can choose an open neighbourhood
Vy of Mi such that A ¢. M^), i.e., k(Vi) = h 2 A. It follows
(see 1.4.7 (ii)) that I = Af] I\ is regular. Since V\ is open we may
assume that
yi = hi(Ji) = Si\hi(Ji).
Then it is easy to see that V = h[(J), where J = Afj^i, is an

380
Commutative Topological Algebras
open neighbourhood of M. Since k(V) = AQk^i) = Af]h = I
is regular, the proof of complete regularity of A is finished.
8.4.6. PROPOSITION. Let A be completely regular and F any
closed subset o/S. Then k(F) is regular iff F is compact.
PROOF. If k(F) is regular then by 3.3.11, h(k(F)) = hk(F) =
F is compact, which proves the "only if" part. For the "if part
assume that F is compact. Then by using complete regularity
of A, we can find a finite open covering {Vj (j = 1,---,71)),
with each k(Vj) regular. By virtue of 1.2.15, k(ViU---U^n) =
k(Vi) 0"' ' k(V„) = I (say) is regular. Since k(F) is also regular,
completing the proof.
8.4.7. THEOREM. Let A be a p-seminormed algebra. If A is
completely regular then the hull-kernel topology r^ on Ac coincides
with the weak topology tw. Conversely, when A is p-seminormed
Gelfand, r^k — Tw on Ac(= A) implies that A is completely regular.
PROOF. Suppose that A is completely regular. We have
always, by 8.3.14, r^ C tw. To prove the reverse inclusion, let F
be a r^-closed set in Ac. Take xo £ AC\F. By complete regularity
of A we can choose a r^-open neighbourhood V of xo such that
k(V) is regular. Let u be a relative (bi-) unity for k(V). Set
F0 = {X e F : X(«) = Vi
Fq is r^-closed.
By 7.3.13, the set
F1 = {x e AC : \x{u)\ > 1}
is tw-compact. Therefore, Fo as a closed subset of F\ is also tw-
compact. Since r^ C tw, Fo is also t^^-compact and so r^-
closed (r^ being Hausdorff by complete regularity of A). Since by
choice xo ¢- Fo and r^ is Hausdorff, each point x £ ^0 has a
reopen neighbourhood U with xo ¢- U = hk(J7). By t^ ^-compactness
there is a finite covering of F by open neighbourhoods U\,- • • ,Un
such that xo ^ hk(t/j) (j = 1, ■••,«). It follows that there is an
element aj G k(Z7j) with Xo(aj) 7^ 0- Also, since AC\V is r^-closed

§4. Completely Regular Algebras
381
and xo ¢- AC\V, there exists ao 6 k(Ac\V) with Xo(ao) 7^ 0. Set
a = aofli■■• an. Then:
Xo(a)^0, (1)
a G ker^U- ■ ■(Jt^lJA^V) " M^U AA^)- (2)
If x £ V then kerx 3 k(y) and so kerx has also u has a relative
unity. It follows that x(u) = 1 whence V f] F C F0, so that we
have
F C F0\J(AC\V).
From (2),(3) we conclude that a 6 kerF. Also, by (1), Xo{a) 7^ 0-
Therefore xo ¢- hk(ir), proving F = hk(F), so that F is r^-closed
and T^ = tw.
For the converse part, assume now that A is Gelfand and tw =
r^. Since A is Gelfand, S = A = Ac, so that S is Hausdorff.
Further, if xo £ A we can choose a 6 G A with Xo(^) = 2. Setting
V = {x £ Ac : |x(^)| > 1}) ^ is an open neighbourhood of xo and
Fc{xeAc:|x(6)|>l}
is compact for tw = r^. Therefore, by 8.4.6, k(V) is a regular
ideal and hence the ideal k(V) I> k(^) is also regular, completing
the proof of the converse (and also of the theorem).
8.4.8. DEFINITION. Let 5 be a topological space and Jo a
family of continuous K-valued functions on 5. Following Naimark
(Neumark) Jo is called completely regular or a completely regular
family if it satisfies the condition:
To each closed set F in 5 and to each point sq ¢. F there is an
/ = /(s) in Jo such that
/(*) = 0 on F and f(s0) 7^ 0. (*)
Jo is called normal if it satisfies the condition:
To each pair of disjoint closed sets F\, F2 in 5 there is an / G Jq
with
f(s) = 0 on Fi and f(s) = 1 on F2. (**)
8.4.9. LEMMA, (a) If S is T\ then every normal family is
completely regular.

382
Commutative Topological Algebras
(b) A space S is completely regular iff the family C(5) is
completely regular.
PROOF. Clear.
8.4.10. Let S = (S,t) be a completely regular Hausdorff
space and C(5) = C(5,K) the algebra of all continuous K-valued
functions on S. Let A be a strongly separating subalgebra of C(5).
Denote by M the maximal ideal spectrum of A. For s G S, set
Ms = M* = {f e A : f(s) = 0}. Since ker A = 0, each Ms G X.
Further, since A separates points of 5, the map f] : s \—> Ms is
injective. Write Mo = {Ma : s G 5}. Since
MoCMcs
we have hk-topology r^ on Mo (got by relativization). Moreover,
since f] is a bijection between 5 and -Moj^hk can be transferred
from Mo to 5 and the transferred topology we denote again by
r^. We also write for an ideal I of C(5)
h0(7) = {M G Mo : M 2 1} = Mo fl K7)-
8.4.11. PROPOSITION (cf. [19, p.57]). rhk C r, and rhk = r
i/f A is a completely regular family of functions.
PROOF. Let F C 5 be a rhk-closed set so that \\QV.{F*) = F",
where F* = Q(F). If / G A then / G k(F*) iff / = 0 on F.
Further, Ms G h0k(F*) iff s G f] ker/, where / ranges in k(F*)
and ker / = {s G 5 : f(s) = 0}. It follows that F = fl ker /. Since
/ is continuous each ker / is r-closed. So F is r-closed, whence
Ilk ^ r-
Suppose now that A is a completely regular family and F(C S)
is r-closed; then I = k[F*) = {/ G A : / = 0 on F}. Clearly
M, G h0(7) = h0k(F*) iff "/ = 0 on F => /(*) = 0".
\i s ^ F there is, by the complete regularity of the family A, an
/ G A with / = 0 on F but /(s) 7^ 0. This means that Ms ^
h0k(F*), whence h0k(F*) = F*, so that F* and therefore F is
r^-closed, proving that r = r^.
Conversely, if r = r^ and F C 5 is closed, then for each point
s ¢ F = h0k(F) there is an / G A with / = 0 on F and f(s) ^ 0.

§ 4. Completely Regular Algebras
383
But this is precisely the condition to be fulfilled for the family A
to be completely regular.
8.4.12. COROLLARY. rhk = r on S for the algebra C(5).
PROOF. Since 5 is a completely regular space, by its
definition C(5) is a completely regular family and hence the result (by
8.4.11).
8.4.13. PROPOSITION. LetS = (S,t) be compact Hausdorff
and A a separating subunital algebra ofC(S) such that every
maximal ideal of A is fixed. Then r^ = r (on S) iff A is completely
regular algebra.
PROOF. If A is completely regular then r^ is Hausdorff.
Since, by 8.4.11, r^ C r, and t is compact Hausdorff (by
hypothesis) it follows from a well-known result in topology that r^ = r
(on 5). Conversely, if r^ = t then rj^ on Mo — £ (every
maximal ideal of A being fixed) is also Hausdorff. Therefore, A being
unital, is completely regular.
8.4.14. COROLLARY. The algebra C(5) is completely regular.
PROOF. This follows from 8.4.12, 8.4.13.
8.4.15. COROLLARY. Let S be a locally compact Hausdorff
space. Then the Banach algebra Co(S) is completely regular.
PROOF. Let 5^ be the 1-point compactification of 5. Then,
by 8.4.14, C(Soo) is completely regular and so, by 8.4.5, Co(S) is
completely regular since C(5oo) is the unitization of Cq[S).
8.4.16. PROPOSITION. Let S ~ (S,t) be a locally
compact Hausdorff space and A be a strongly separating subalgebra
of Cq[S) — Co(5, || ■ ||oo)- Then the following two statements are
equivalent:
(i) A is a completely regular family of functions.
(ii) The hull-kernel topology r^ coincides with t.
If A is also Gelfand then (ii) (or (i)) is equivalent to

384
Commutative Topological Algebras
(iii) A is completely regular algebra.
PROOF. The equivalence of (i) and (ii) has already been
demonstrated in 8.4.11. The equivalence of (ii) and (iii) follows
from 8.4.7.
8.4.17. PROPOSITION. If A is a completely regular p-
seminormed algebra then A is a completely regular family of
functions. Conversely, if A is a p-seminormed Gelfand algebra such
that A is a completely regular family then A is a completely regular
algebra.
PROOF. Assume that A is completely regular. Then, by 8.4.7,
rhk ~ Tvi on Ac - which is locally compact Hausdorff. By 8.1.25,
A is a strongly separating subalgebra of Co(Ac). So by 8.4.16, A
is a completely regular family. For the converse, assume that A is
also Gelfand and that A is a completely regular family. By 8.4.16,
Tw ~ '"hk on Ac(= A = E) so that by 8.4.7., A is completely
regular.
8.4.18. PROPOSITION. Let A be a completely regular p-
seminormed algebra. Then d^Ac ~ Ac.
PROOF. Write d^Ac — F; then F is a closed set in Ac. If
AC\F ^ 0, take a xo £ Ac\ir. Since, by 8.4.17, A is a completely
regular family, there is an a G A with a[F) ~ 0 and a(xo) 7^ 0.
Then II fl11oo -^ 0j but sup|ffl(^)| — 0, contradicting that F is the
xeF
Shilov boundary. Hence Ac = F ~ 3^AC, as required.
8.4.19. PROPOSITION (cf. [20, p.236], [12, p.54]). Let A be a
completely regular p- seminormed algebra. Then:
(i) Any closed set F C Ac = AC(A) is homeomorphic to
Ac(A/k(F)).
(ii) Any open set G C AC(A) is homeomorphic to Ac(k(Ac\G)).
PROOF. By complete regularity of A, r^ = tw on Ac (see
8.4.7). Therefore F = hk(F). It follows that x e F iff x = 0 on
k(F). The homeomorphism in (i) is now clear.

§ 4. Completely Regular Algebras
385
Write AC\G = F. Then xGG iff x<^ iff XT^O on k(F) iff
x|k(ir) G Ac(k(ir)). Hence the homeomorphism in (ii).
8.4.20. PROPOSITION (cf. [12, p.55]). Let A be a completely
regular, complex or strictly real, commutative p-Banach algebra
A(^ \/A). Let I be an ideal of A, Fi a closed subset of A(= Ac),
F2 a compact subset A such that
(hwu^on f*=* w
TAen iAere exists an element ao G If)k(Fi) such that clq = 0 on
F\ and clq ~ 1 on F2.
PROOF. Write I}- = k(Fy) (j = 1,2). Then F}- = h(7y), so that
condition (*) above becomes (h(7) U h(7i)) f| 11(/2) = 0, i-e.,
h(lf)h)f)h(I2) = <t> (1)
which means that no character x of A can vanish at the same
time on both If]Ii and /2- Since /2 is clearly the intersection of
all hypermaximal ideals containing I2 and A is Gelfand, it follows
that A* = A/I2 is s.s.. Since by 8.4.19, A(A#) ~ F2, and F2 is
compact we conclude by 8.6.6 that A* has unity u& = u + I2 (say).
Denote by <p the canonical homomorphism A —> A# = A/I2. We
assert that <p(If)Ii) = A*. At any rate J^ = ^(7P|-^i) is an
ideal of A*. If J# 7^ A* there is a hypermaximal ideal M# with
<p(I f| ^1) ^ M*. Then M = £>-1(M#) is a hypermaximal ideal of
A with I f]I\, I2 C M. But this contradicts relation (1). Therefore
we must have <p(I f]h) = A, as asserted. It follows in particular
that there is an element ao G / f]h such that ao + I2 ~ <p(ao) —
u*. If x G Fi then x = 0 on k(Fi) = 7i so that a0(x) = x(«o) = 0.
On the other hand if x G F2 then x = 0on k(F2) = I2, so that
ao(x) = X(a0) = X#(a0 + /2) = X#(«#) = 1,
where x is the character of A* induced by X- This completes the
proof.
8.4.21. COROLLARY. A being as in 8.4.20, if Fi is a closed
subset and F2 a compact subset of A(A) such that F\f]F2 = 0,
then there exists an element ao G k(Fi) with clq = 0 on Fi and = 1

386
Commutative Topological Algebras
on F2.
PROOF. Suffices to take in 8.4.20, I = A (then h(7) = 0).
8.4.22. COROLLARY. If I is an ideal of A and F a compact
subset of A(A) such that h(I) f] F = 0, there is an element ao G I
such that ao = 1 on F.
PROOF. Suffices to take in 8.4.20, Fi = 0 (then k(Fi) = A)
and F2 = F.
8.4.23. COROLLARY. If A is a completely regular, complex
or strictly real, commutative unital p-Banach algebra then A is a
normal family of functions on A.
PROOF. This follows from 8.4.21, since now A being compact
every closed set in A is compact.
8.4.24. DEFINITION. Let A be a completely regular p-
seminormed algebra and F a closed subset of Mc (the space of
closed hypermaximal ideals). Denote by J = j{F) the set of all
x G A such that x has compact support> disjoint with F. If F is
a single point {M} we write j(M) for j(F). Further, if F = 0 we
write Jo for j(0).
8.4.25. LEMMA. J = j(F) is a bi-ideal of A such that h(J) =
F.
PROOF. Clearly x G j(F) iff there is a compact set Cx such
that x = 0 outside Cx and Cx f| F — 0. From this result it is easy
to see that j(F) is a bi-ideal. Further, if x G F then x{x) ~ 0 for
every x G j(F), whence h(j(F)) 3 F- Now consider a \0 e Ac\ir.
Since Ac is locally compact Hausdorff and F is a closed set there is
an open set U in Ac with x0 G U and U compact and disjoint with
F. By complete regularity of A there is an x in A with i(x0) 7^ 0
and £ = 0 on AC\C/. Then x G i(F) and x0 ¢- HJ(F)), which
proves that h(j(F)) = F.
8.4.26. PROPOSITION. Let A be a commutative completely
* For a function / on a topological space S, by support of f we mean the
closure of the largest subset on which it is not zero.

§4. Completely Regular Algebras
387
regular s.s., complex or strictly real, p-Banach algebra. Let F be
a closed subset of Ac — A. Then J ~ j(F) is the smallest of the
ideals I such that h(7) = F.
PROOF. In view of 8.4.25, it is enough to prove that for any
7 with h(7) = F we have J C 7. If x e J and C is the support of
x then C is compact and Cf]F — 0. By 8.4.22 there is a y e 7
with y — 1 on C. Since x — 0 outside C we have clearly xy = x.
But A being s.s., we get xy = x, whence x e 7, J C 7, completing
the proof.
8.4.27. COROLLARY. TAe closure J is the smallest closed
ideal I of A with h(7) = F.
PROOF. If Me A then M is a closed ideal, so that M 2
J => M 3J. Therefore /i(J) = /i(J) = F. Further, if 7 is a closed
ideal with h(I) = F then J C 7 and so J C 7 = 7, completing the
proof.
8.4.28. COROLLARY. For eacA maximal regular ideal M of
A,j(M) (respy. j(M)) is the smallest primary [respy. closed
primary) ideal contained in M.
PROOF. Apply 8.4.26, 8.4.27 with F = {M}.
8.4.29. THEOREM (Abstract Wiener Tauberian Theorem).
Let A be a commutative s.s. regular, complex or strictly real, p-
Banach algebra and Jq the ideal of elements x such that x has
compact support. If Jq is dense in A then every closed ideal 7(^ A)
is contained in a maximal regular ideal M.
PROOF. If 7 is not contained in any maximal regular ideal
then /i(7) (in M) = 0. Since maximal ideals of A are closed and Jq
is dense in A it is clear that /i(Jo) — 0- By 8.4.26 we get Jo C 7.
But then since 7 is closed and Jo dense we must have 7 = A - a
contradiction.
8.4.30. Remark. For connection between above theorem
and classical Wiener Tauberian theorem see [19, pp. 147-9] or [22,
pp.426-7]. See also [23, p.326] and reference cited therein for other
related results.

388
Commutative Topological Algebras
§ 5. Holomorphic Functional Calculus for Several
Commutative Algebra Elements
8.5.1. We begin by recalling the notion of polynomial
convexity. Let K be a bounded subset of K". Set
h{K) = {X = {X1,---,Xn)eKn:\P(X)\^\\P\\K
for each polynomial P over K"}, where \\P\\k — sup{|-P(m)| : M e
K}. Then clearly h(K) D K and h(K) is called the polynomial
convex hull of K. If h(K) ~ K then K is called polynomially
convex or p-convex.
If K is the unit circle in C, h(K) is the closed unit disc, as can
be seen using the maximum modulus theorem.
8.5.2. LEMMA, (i) h(K) is compact, in particular a p-convex
set K is compact.
(ii) K is p-convex iff for each A0 G Kn\K, there is a polynomial
P0 with |P0(A°)| > \\Po\\k-
(iii) If K is polynomially convex and C > 0, then for each
A0 G Kn\K we can find a polynomial Q with
Q(A°)| > C, \\Q\\K < C.
PROOF, (i) It is clear from its definition that h(K) is closed.
Also, if Ay is the polynomial Ay(Ai, ■ ■ ■, An) = Ay then by taking
P ~ Ay in the definition of h(K) we get for A G h(K), |Ay| <
sup{|//y| : (2 G K} ~ Cj < oo. This means that h(K) is bounded
and consequently it is compact.
(ii) Clear from the definition of p-convexity.
(iii) Given A0, choose first Pq as in (ii) and then set Q(X) =
CP0{\)/\\P0\\K.
8.5.3. Remark. It is known that a compact set K in C is
p-convex iff C\K is connected (see [30, p.37]). Therefore the result
7.3.24 can be restated as: the spectrum <r(a) of a t.generator a of
a complex monogenic ^-Banach algebra is p-convex. In this form
the result is generalized in 8.5.15 (ii).
8.5.4. Recall that a polydisc Pr with poly-radius 6 =

§ 5. Holomorphic Functional Calculus
389
(Si, ■•• ,Sn) is given by
Pj = {A e K" : |Ay | ^ Sj, j = 1, ■ ■ ■, n}.
A subset n of Pg is called a p-polyhedron (in P{-) if there are
polynomials Pi, ■ ■ ■ , Pm such that
n = {A GP,-:11^)1,-..,^(^1}.
8.5.5. LEMMA, (a) P^ itself is a p-polyhedron.
(b) Every p-polyhedron is p-convex.
PROOF, (a) Take m = n, Py = SJ Ay, where Ay is the
polynomial Ay (A) = Ay.
(b) If A0 ¢ K"\n then either some |A^| > Sj or some |Pfc(A°)| >
1. In the first case |Ay(A )| = |A-| > Sj > ||Ay||n and in the second
case we have |Pfc(A°)| > 1 ^ ||Pfc||n- Thus in either case A0 ¢. h(II)
which means that h(II) = II and II is p-convex
8.5.6. LEMMA. Let K be a p-convex set with K C P^ and G
an open set with K C G C K". Then there exists a p-polyhedron
n with k cncc.
PROOF. By 8.5.2 (iii), we can choose for each A0 e Kn\K a
polynomial P?n with P? (A ) > 1, HPr'Hif < 1. By continuity of
Pr I, we have |Pju (A)| > lforA in some (open) neighbourhood .Vc0
of A0. Writing P = Pg and allowing A0 to range in P\G(C P\K)
and using the compactness of V\G we can find a finite family of
neighbourhoods .A/ry, corresponding to polynomials Pry, such that
%U---U%^p\c.
Set n = {A e P : |P^(A)| < \\P^\\k ^ 1}, so that K C II. Suppose
next X¢ G. If also X¢P then of course A ¢. II (since II C P). On
the other hand if A e P then A E P\G and so A e H^ for some j.
Hence \P^3\ > 1, whence A G II. Therefore II C G, as required.
8.5.7. The notion of spectrum of an element can be
generalized to that of joint spectrum of a finite set of elements as
delineated below.

390
Commutative Topological Algebras
Let A be a commutative algebra with unity e and write a =
(oi, ■ • • ,an) (ay e A). If A = (A1;- ■ ■, An) (Ay G K) then we write
I ~ /(A) = /(A, a) = the ideal of A generated by ay — Aye (j =
1, • ■ ■, n). Clearly we have
n
/=/(A) = X>(ay-AyC). (*)
/=1
The joint (or simultaneous) spectrum
a(a) = <r(ai, ■■■, an) is defined by :
a(a) = {AGKn:/(A>A}.
Not that when n = 1, o = o and a(a) = a(a).
8.5.8. LEMMA, (i) A e Kn\a(a) iff there are elements
n
&i, ■ ■ ■, 6„ G A with 2_](ay - Aye)6y = e.
/=i
(ii) A G <r(a) i/f /or every &i, ■■-,&„ G A we Aave
n
2_[{aj ~ Aye)6y ^ G,, where G,- denotes the group of invertible el-
/=i
ements of A.
(iii) A G o'(a) iff there is a maximal ideal M of A, with a}- —
Aye G M (j = 1, ---,71).
PROOF, (i) The stated condition is clearly equivalent to:
/(A) = A. ^
(ii) The condition is clearly necessary and sufficient for /(A) ^
A.
(iii) If A G a(a) then 7(A) ^ A and so by Krull, 7(A) C some
maximal ideal M, whence aj — Aye G 7(A) C M. Conversely, if
ay — Aye. (j = 1, ■ ■ ■, n) are contained in some maximal ideal M
then 7(A) C M ^ A and A G <r(a).
8.5.9. Corollary. A e <r(a) => Ay e <r(ay) (j = l,---,71),
so that
a(a) C <j(ai) x ■ ■ ■ x <r(an).

§ 5. Holomorphic Functional Calculus
391
PROOF. A e a{a) => ay - Aye e M => Ay e <r(ay).
8.5.10. PROPOSITION. Lei A be a unital commutative
Gelfand algebra - in particular, a unital, complex or strictly real,
commutative p-Banach algebra. Then A G o'(a) iff there is a
character X G A with
X{aj) = Ay (j = 1,---, n), where A = (Ai, ■ ■ ■, An).
PROOF. If A e v{a), by 8.5.8 (iii) there is a maximal ideal
M with ay — Aye e M. Since A is Gelfand M is hypermaximal
and let x be the character determined by it. Then x(a/) = ^/>
(A = (Ay)). Conversely, if x £ A satisfies the stated condition then
aj — Aye G M (M = kerx) and A 6 <j(a).
8.5.11. COROLLARY. Lei A = (A, || ■ ||) (|| ■ ||sm.) be a unital,
complex or strictly real, p-Banach algebra. Then v{a) is nonempty
compact with
n
a(a) C J] a(ay) C P£,
where S = (6y), 6y = ||fl/||p-
PROOF. Since A is Gelfand, A = Ac ^ 0. If x £ A and
x(ay) — Ay then by 8.5.10,
A = (Ay) G o(a), so that a{a) ^ 0.
By 7.3.12, A = Ac is compact and since the map x l—*
(x(ai)j"'' )X(an)) is continuous, a{a) is compact. Finally, if
fij G ff(ay) then |//y| ^ r(ay) < ||ay||p (see 7.3.27). This completes
the proof.
8.5.12. COROLLARY. Let A be as in 8.5.10, and oi,---, an e
A. TAen
(i) For any polynomial P over K,
a(P(ai,- ■ ■, an)) = P(a(a1, ■ ■ ■ an)).

392
Commutative Topological Algebras
(ii) If aj G K (j = 1, ■ ■ ■, ra) then
a(ctiai,--- ,anan) = {(c*iAi,-- ■ ,anXn) : A e <r(ai, ■ • ■, an)}.
PROOF. It suffices to observe that each X G A being a homomor-
phism we have:
x{P{ai,---,an)) = P(x(ai),---,x(an))
X(ayay) = ayx(ay).
8.5.13. Remark. Arens has shown that the conclusion in
Proposition 8.5.10 holds also if A is a unital commutative complex
locally sm. 5 algebra (see [31, p.105]).
8.5.14. PROPOSITION. Let A be a unital commutative p-
Banach algebra which is either complex or strictly real, and which
is t. generated by a finite set of elements a\, ■ ■ •, an. Then the
spectrum A(A) is homeomorphic to the joint spectrum <r(a) under the
map
9 -X^ (x(ai),"-,x(an))-
PROOF. By 8.5.10, 6 is surjective. The map 0 is also injective.
To see this, suppose that 0(xj = #^)- Then, for a polynomial
P — P[\\, ■ ■ •, An) over K we have
xAP{aU ■ • ■ , an)) = P(Xi(*l), ■ ■ ' >Xi(an))
= -P(X2(al); ■ ■ ■ jX2(an)) = X2(-P(al, • • ■ J an))-
Since the elements of A of the form P{a\, • • •, an) - where P is
a polynomial over K in n variables - form a dense subalgebra
Aq of A (since ai,---,an t. generate A) and \i ~ X2 on ^0, we
conclude from the continuity of xnx2 that Xi — X2 on ^> proving
injectiveness of 6. Further, the continuity of 0 follows from that of
the x's- By 7.3.12, A = A(A) is compact Hausdorff. Therefore 9
is a homeomorphism.
8.5.15. PROPOSITION. Let A be a unital commutative,
complex or strictly real, p-Banach algebra t. generated by a\,- • • ,an.
Then:

§ 5. Holomorphic Functional Calculus
393
(i) For any C > 0, a G K and A0 G Kn\a(a) there is a
polynomial P (in n variables over K) such that
P(X°) = a, \\P(a)\\<C
where 3= [ai,- ■ ■ ,an). In particular, there is a polynomial
Q withQ(X°) = 1> ||Q(a)||.
(ii) v{a) is polynomially convex.
PROOF. If A0 e K"\<r(a) then
7(A°) = £ A(aj - Aje) = £>y - Aje) A = A,
y=i y=i
where e is the unity of A. It follows that ae G A has a
representation
n
5>y - A^e)6y = ae (1)
y=i
where 6y G A.
Since A is t. generated by a, each 6y can be approximated by
elements of the form P(ai,- • • ,an) where P is a polynomial. It
follows from this and equation (1) that there are polynomials Pj
(j = 1, • ■ • , n) such that
n
\\ae - ^2(aj - \°-e)Pj(a)\\ < C. (2)
3 = 1
Write
P$) = «- E(X3 - Aj)Py(A). (3)
3-1
Clearly P is a polynomial in Ai,---,An. Further, it is clear from
(3), (2) that we have
P(A°) = a
\\P(a)\\<C. (4)
If we take C — 1 = a in (4) then the corresponding polynomial P
which we denote by Q satisfies:
Q(A0) = 1 > \\Q(a)\\.

394
Commutative Topological Algebras
This completes the proof of (i).
To prove (ii) take a A £ a(a). By 8.5.10 we have
A = x(a) = (x(ai),---,x(a„))
for some x G A. It follows that the polynomial Q chosen above
satisfies:
|Q(A°)| = |Q(x(a))| = |x(Q(a))|
< ||Q(a)||' (since Hxll ^ 1)
< 1- (5)
Since K — a(a) is compact, (5) implies that ||Q||/r < 1. On the
other hand, <3(A°) = 1 (by construction). Therefore, by 8.5.2 (ii),
a (a) is p-convex.
8.5.16. Remark. From 8.5.14, 8.5.15 (ii), 8.5.11, it follows
that the spectrum A = A(A), of a finitely t. generated complex
commutative ^-Banach algebra A, is homeomorphic to a polyno-
mially convex compact subset of C". Conversely, it can be shown
that if ii" is a polynomially convex compact subset of C" then there
is a finitely t.generated (with n t. generators) complex
commutative Banach algebra A such that A(A) is homeomorphic with K
(see [10, p.45]).
8.5.17. Remark. The joint spectrum or even the spectrum
of a single element, of a commutative Banach algebra which is not
finitely t. generated, may not be p-convex. For example, in the
Banach algebra B of bounded complex-valued functions on the
unit interval [0, l] (under the sup-norm) the function f(t) = e2vtt
(0 < t < 1) has for its spectrum its range which is the unit circle.
The unit circle, however, is not p-convex (since its p-convex hull
is the closed unit disc).
To establish the main theorem of this section, which asserts
that holomorphic functions of several variables operate on the
space of Gelfand transforms, we need a number of preliminary
results which we proceed to consider.
8.5.18. LEMMA (Arens-Calderon trick [3', p.205]). Let A be a
unital commutative, complex or strictly real, p-Banach algebra. Let

§ 5. Holomorphic Functional Calculus
395
ai; ■ ■ ■ j an £ A and G C K" an open set with <r(ai, ■ ■ ■, an) C G.
Then there exists a finitely t. generated closed subunital algebra B
of A, with aj G B [j = 1, ■ ■ ■, n) and cb(oi, ■ ■ ■, an) C G, where
<tb denotes the spectrum with respect to the algebra B.
PROOF. We may assume that the norm || ■ || of A is sm. Write
K = {AgK" : | Ay) ^ ||oy||p (j = 1,---,71)}.
Then K is compact. By 8.5.8 (i), for each A G K"\<j(ai, ■ ■ ■ ,an)
there are elements &i, ■ ■ ■ bn G A such that
n
5^(ay - Aye)6y = e. (1)
y=i
Denote by B(A) the closed subalgebra of A generated by
e,ai,- ■ • ,an, bi,---,bn. By considering the relation (1) in B[X),
we get, by 8.5.8 (i), that
^$aB(%)(fli>" ">a»») = ffB(X) (s&y)- (2)
Since cB^, is closed there is an open neighbourhood G(A) of A
such that
^)0^) = 0- (3)
It follows, using the compactness of K\G and (3) that there are
points \k G K\G [k = 1, ■ ■ ■, n) such that
(J G(P) D ff\G (4)
fc=i
and
*B(A*)fW) = 0(*= !> ■■■'»)• (5)
Since each 5¾ = B(Afc) is finitely t. generated it is clear that there
is a finitely t. generated subunital algebra B of A with all B^ C B.
By 1.7.19 , \ r ( \ fa\
aB{ai,---,an) CaBk{ai,---,an). (6)
From (4),(5),(6) we obtain
<TB{au---,an)(XK\G)=$.

396
Commutative Topological Algebras
On the other hand, by 8.5.11,
Mai,- ■■,*») CP^ = /JT.
Therefore, cb(oi," • • ,an) C G, as required.
8.5.19. THEOREM (Oka's Extension Theorem.) Let m, r > 0
be integers, ej > 0 (1 < j ^ m + r) 6e rea/ numbers and
P = {A G Cm+r : |Ay| < ey (1 < j < m + r)}.
Lei Pj{j — 1, ■ ■ ■, r) 6e complex polynomials in m variables and 9
the Oka map of Cm -> Cm+r given by:
0:X = (X1,---,Xm)~(X1,---,Xm,P1(X),---, Pr{X)).
Then, for any f holomorphic on an open neighbourhood of9~1{P)
there corresponds an F holomorphic on an open neighbourhood of
P such that
F(9(X)) = f(X)(Xe9-1(P)).
PROOF. See [4, p. 103].
8.5.20. PROPOSITION. Let A ~ (A, \\ ■ ||) be a unital
commutative complex p-Banach algebra, a — {a\,- • • ,an), aj G A (j —
1, ■ ■ ■ n) and G C C" an open neighbourhood of a{a). Then we can
find a finite number of elements an+\, • ■ ■, a^r 6 A such that given
any function f holomorphic on G there is a function F
holomorphic on the polydisc
{XeCN :|Ay|<l+2||ay||i (j = l,--.,JV)}
such that
/(«i(x), ■ ■ ■ a„(*)) - F(h(x), ■ ■ ■, &nM) (X e A).
PROOF. By 8.5.18, we can find a finitely t. generated subunital
algebra B containing a\, ■ ■ ■ an with
aB(a) ~ aB(ai,--- ,an) C G.
(1)

§ 5. Holomorphic Functional Calculus
397
We can take as t. generators (7^ e) of B to be a\,- • • ,am, where
n < m. Fix C > 0 and set a = 2C~" + 2. For any A e C"\<rB(a),
there is, by 8.5.15(i), a polynomial P such that
P(A) = 2C~? +2, ||P(a)|| <C
Then we have
|P(A)| = 2C* +2 > 1 + 2C* > l + 2||P(a)||p. (2)
It follows that there is an open neighbourhood Vp(A) of A such
that
|P(/Z)|>1 + 2||P(o)||p (3)
for all //G Vp(A). Denote by tt the natural projection
(Ai,---,Am) h-> (Ai,---,An) (n ^ m)
of Cm onto C". Write
Pm = {AeCm:|Ay|^ l + 2||ay|| {j = 1, ■ ■ ■ ,m)}. (4)
Then : K = PTO\jr-1(G) is compact. (5)
Also, 7r_1(G) contains (f(b), where 6 = (01, ■ ■ ■ , am) since
tt(<7(6)) = a{a) C G.
It follows from (1),(2),(3) and (5) that there are polynomials
Pi,---,Pr such that Vpl,- ■ ■ ,Vpr cover K. Therefore, for any
A G K, we have _ _ !
||Pt(A)||>l + 2||Pt(6)||? (6)
for at least one k (1 < k < r).
Set
TV = m + r, am+fc = Pfc(6) (A; = 1, ■ ■ ■ ,r),
Pw = {Ae C^ : |Ay| ^ l+2||ay||p {j = 1, ■ ■ ■, N)}.
Let 9 be the Oka map

398
Commutative Topological Algebras
where A = (Ai, ■ ■ •, Am). Since 0(A) € PN => A e Pm and
|Pt(A)|<l + 2||Pt(6)||p (* =1,---,0,
using (4),(6) we get
e-\PN) C jr_1(G).
The function / o 7r is holomorphic on the open neighbourhood
7r_1(G) of 0-1^). By 8.5.19, there is a function F holomorphic
on an open neighbourhood of 9~1{PN) such that
F(0(A-1)) = /o^(A)(Aer1(PiV)). (7)
For Xe A, set Ay = X(«y) (,'= l,-,™).
Then
ft (A) = Pfc(Ai, ■ ■ ■, ATO) = XPfc(6) = x(am+fc) (A = 1, ■ ■ ■, r)
and so
0(A) = (*(«i), ■•■,*(«*)).
Since
|Ay| = |x(«j)l < IHI' (i = 1i---im)
and
\PkW\ = X(«m+t) < ||am+fc||p
we conclude that 0(A) e P^, A e 0_1(PAr). Finally we obtain, by
(7)
f(x{ai),---,x(an)) = f°*fi) = F(0(\)) = F{x{a1),---,x{*n)),
which completes the proof.
The next lemma is needed for proving the main theorem which
follows
8.5.21. LEMMA. If f3n e C, limsup |/?n| = I and 0 < p ^ 1,
n—»oo
then limsup \/3n\p ~ lp.
n—»00
PROOF. Given e > 0, choose r) > 0 such that ?/ = e. Since
limsup|/?n| = / there is a TV such that
n—»oo
|/?n| </ + »? for n > N. (1)

§ 5. Holomorphic Functional Calculus 399
Also, for any m, there is an n > m with
\Pn\ >l-n- (2)
From (1) we get
\Pn\p <{l + r))p <l" + fj" < Z" + 6 (n > N) (3)
and from(2) we obtain
1/^+^=1/^+^(1/^1+^^- (4)
The inequalities (3),(4) show that limsup|^n|p = lp.
n—+oo
8.5.22. THEOREM (Shilov-Arens-Caulderon)t. Let A be a
unital commutative, complex or strictly real, p-Banach algebra.
Let ai,---,an be elements of A, G C K", an open set such that
<r(ai, ■ ■ ■, an) C G and f a function holomorphic'' on G. Then
there exists an element b G A such that
Hx) = xW = /(fii(x), ■ ■ ■, «n(x)) (x e A). (*)
The element b is uniquely determined iff A is s.s. .
PROOF. First assume that A is complex. Then, by 8.5.20,
there are elements an+i, ■ ■ ■, a^r in A and a function F holomorphic
on an open neighbourhood G* of the polydisc
P = PA' = {AGCA': I Ay | < l + 2||ay||p (j = 1,---, N)}
such that
^(x(ai),-",x(ajv)) = /(x(ai),-",x(an))-
Since P is compact and P C G* the distance from P to the disjoint
closed Cn\G* is positive. So we can find an open polydisc PJ with
PCPJC G*.
' These authors considered only the case of complex Banach algebras.
TT holomorphic in the real case means real analytic.

400
Commutative Topological Algebras
Let the Taylor series representation of F on PJ be
it£NN
where N = {0,1,2, ■ ■ ■}. By Abel's lemma, the (multiple) series for
F converges absolutely on P. In particular, the series converges at
MM-- +
fl with Hj = 2||oy|| f (j = 1, ■ ■ ■, N). By root test* for convergence
we have
limsup(|ajE|2fcl||ai||^---2fcjv||aAr||"f )M < 1 (*)
|| fc || —»oo
where ||A;|| = &!+■•■ + k^.
The inequality (*) reduces to
, hi „ m^.-Ih 1
hmsup(|a^|||oi|| p •■■Hojvll p ) 11*11 < -.
II Til ~
K —»O0
By applying 8.5.21, we obtain
limsup(|arH|ai||fcl---||aiV||fc'v)Wr ^ — < 1.
tr VI fcl II -1- II II ^ II I x Oo
i 1
2P
|| A: || —»oo
Therefore, by the root test, the multiple series
E|ry^|P||„1||fcl . . . \\nr.T\\kN
la/fcl llal|| llaJV||
converges, which means that the series
Eatai1,,-a
kN
N
converges absolutely and consequently the series converges to an
element b G A, i.e.,
£^-^=6-
t See [2, p.364'

§ 5. Holomorphic Functional Calculus
401
If X G A, then x being a continuous homomorphism we have
X(b) = Eak^i)kl---x(aN)kN
= F(X{ai),---,x{aN)) = /UK), ■ ■ ■ ,x(ajv)),
so that we have
6(x) = /(fii(x),---,Mx)) (xeA).
This proves the theorem when A is complex.
Next, let A be strictly real and A its complexification. Here
we have
ff(ai,- ■',««) CGCR"
and / holomorphic on G. Then it is knownt that there is a
(complex) holomorphic function F on an open set G C C" with G C G
and F|G = /. Since A is strictly real we have
a{ai,--- ,an) = <r(ai, ■ ■ ■, an) C G C G,
where a denotes the joint spectrum with reference to A. Since A
is a complex algebra, by applying the theorem for this case (just
proved) we can find a b G A such that
X(b) = F(x(ai), ■■■, X(a„)) = F(X(ai)r ■ ■ ,xM)
where x £ A, and x = x|^> by 1.9.18, 1.6.14, x is real-valued.
Since b 6 A, 6 = 6 + z'c (6, c e A).
Therefore we have
X{b)+ix{c) = *(& + *'c) = x(&)
= ^(x(ai),---,x(an)) = /(x(ai),---,x(a„)).
Since the RHS is real we must have
x(6) = /(x(ai)r--,x(an))
which completes the proof of the existence of 6 for the strictly real
case.
t See [3, p.134].

402
Commutative Topological Algebras
For the uniqueness of b result (in both the cases) we note that
if 61 is also an element such that (*) holds when b is replaced by
61 then clearly we have b - 61 G f]kerx = VA~ = vA (since A is
a Gelfand algebra). So 61 = 6 iff \f~A — {0} iff A is s.s..
8.5.23. PROPOSITION. Let Abe a unital commutative,
complex or strictly real, pseudo-Michael algebra such that in its
projective limit decomposition A = limAa, each Aa is s.s. . Let Ac be
the set of continuous characters of A. For a\, ■ ■ ■, an G A write
ct*(01,---,0,,) = {(x(ai),-";x(an)) : XG Ac}.
If f is a holomorphic function on an open set G 2 °'*(ai, ■ ■ ■, an)
then there is a unique element b G A such that
Hx) = /(ai(x),---,an(x)) (xeAe).
PROOF. By 7.3.19 we have Ac = |jAa,Aa ~ Aa, where
Aa = the space of all characters of Aa = the space of all continuous
characters of Aa (by 7.1.6(c), 7.1.9). Since each Aa is a Gelfand
algebra we have (see 8.5.10) the relation
<T*(ai,- ■■ ,an) = (Jo-(aia,--- ,ana),
a
where aja = (pa[aj), and ipa the canonical map A —+ Aa. Since
G I> a(aia,- • ■, ana), by 8.5.22 there is an element ba G Aa such
that
Xa{ba) = f{Xa{aia),--- ,Xa{ana)) {Xa £ Aa).
Since 6^-^ = (pa/3{ajp) we have
Xc*(&a) = /(Xo0^^),"-,^0^^))
Since Aa is s.s. we have (pap{bp) = ba. It follows that lim6a = b
exists and we have clearly
S(x) = /(ai(x)»-"»o»(x)) (xeAc).
To prove the uniqueness of b it is enough to show that VA = {0}.
Suppose that 1 £ A, 1 / 0. Then for some a, xa 7^ 0. Since Aa is

§ 6. Shilov Idempotent Theorem
403
s.s. Gelfand there is \a with xa(xa) 7^ 0. Then x» = Xa°^a £ Ac
and Xa(z) = xa(ia) ^ 0. This shows that C\/A = {0}J which
completes the proof.
8.5.24. Remark. The holomorphic functional calculus
embodied in 8.5.22, for t. finitely generated unital commutative
complex Banach algebras, using A. Weil's integral formula for
holomorphic functions of several variables, was obtained by Shilov.
This was extended by Arens and Calderon for arbitrary unital
commutative complex Banach algebras. An approach to
holomorphic functional calculus based on exterior differential forms had
earlier been given by L. Waelbroeck (see [29, pp.41-50], [12']).
Waelbroeck's method enabled him to show that the
correspondence
/ h-> 6= /(oi,---,0,,) = /(a).
is a homomorphism - a result not obtained by either Shilov or
Arens. Further, Waelbroeck developed the functional calculus for
any complex commutative locally convex complete Hausdorff
algebra. For an introduction to Waelbroeck's method (for
commutative Banach algebras) see [12, pp.41-50].
The above proof of theorem 8.5.22, based on Oka's extension
theorem instead of any Cauchy integral formula is modelled after
the proof found in Bonsall and Duncan's book [4, §20].
§ 6. Shilov Idempotent Theorem
8.6.1. LEMMA. Let R be a ring with unity e, which is a direct
sum R = Ii + I2 of two I. ideals Ij ^ {0} (j ~ 1,2). Then there
are idempotents ey ^ 0 (j — 1,2) in R such that
e = ei + e2, e±e2 = e2ei = 0, Ij ~ Rej (j — 1, 2).
[A similar result for r. ideal decomposition.)
PROOF. Since R = I\ + /2, we have for e the decomposition
' It follows from this that the algebra A is s.s. (since \JA C ''yA = {0}).

404
Commutative Topological Algebras
e = ei + e2 (ei G h,e2 G /2).
Therefore, for x G i?, we have
x = xe = xei + xe2.
If x G /i, then x — xei G /i, xe2 G /2 and since I\ f] /2 = {0}, we
get
x = xei, xe2 = 0.
Therefore, I\ = Re\, e\ = ei, eie2 = 0. Similarly, /2 = Re2,
e2 = e2, e2ei = 0.
8.6.2. PROPOSITION. Let A be a unital t. spectrally Gelfand
algebra - in particular a Gelfand algebra. If A has an idempotent
u/0, e then Ac is disconnected.
PROOF. By 1.7.9, a(u) ~ {0,1}. Since A is t.spectrally
Gelfand there are Xi>X2 £ Ac with Xi(u) = 0j X2{u) = 1- Further,
for any x ^ Ac, x(«) = 0 or 1 (since x(«) = x(«2) = x(u)2)- It
follows that if we set
Fx = {x G Ac : x(u) = 0}; F2 = {X G Ac : x(u) = 1}
then the Fj (j =1,2) are non-empty (since Xj ^= Fj) closed sets
in Ac such that
Fi\jF2 = Ae.
So Ac is disconnected.
8.6.3. COROLLARY. //a unital t. spectrally Gelfand algebra
A is a decomposable I. (or r.) A-module then Ac is disconnected.
PROOF. By 8.6.1, there are idempotents e\ ^ 0, e2 ^ 0 with
ei + e2 — e, when e is the unity of A. Clearly e\ (or e2) 7^ 0, e.
Therefore, by 8.6.2, Ac is disconnected.
8.6.4. THEOREM (Shilovt idempotent theorem). Let A be
a unital commutative, complex or strictly real, p-Banach
algebra whose spectrum A = Ac is disconnected, so that we have a
' He obtained the result for the case of complex Banach algebras.

§6. Shilov Idempotent Theorem
405
decomposition A = AiljAo with Ai, Ao clopen} Then there exists
a unique'' idempotent u in A such that u, ~ 1 on Ai and u, = 0
on Ao-
PROOF. Since Ai P| Ao = 0, for each xi £ &i and xo £ A0
there is an element o,Xixo G A with Xi(axi,xo) 7^ Xo(axi,xo)- ^
follows that we can choose open neighbourhoods ^(xi) of Xi and
W(xo) of xo in ^ such that
XiKxJ ^ x'oK.xJ ^ X', e^(xj,x'0 e ^(Xo)-
The open sets V(Xi) x W(Xn) cover the compact set Ai x Ao and
consequently there is a finite subcovering V(Xj-) x W(x0.) (i =
1, ---,71). Rename the associated elements ax ,x . as ai,---,an
and denote by a the mapping
a :X^ (ai(x),-",an(x))-
Then a(Ai)f|a(A0) = 0, since if xi G Ai, xo £ A0 then (xi,Xo)
belongs to some V(xij) x W(Xoy), so that
Xi(aXi.-Xoy) ^Xo(aXiyXoy)-
Since A is compact Hausdorff there exist disjoint open sets G\,Gq
in K" with a(Ay) C Gy (j = 1,0). Write G = GiUG0 and define
/ on G by setting / = 1 on Gi and / = 0 on Go- Then / is
holomorphic on G, and by 8.5.22, there is a b G A such that
6 = /(«1,- ■ • o„) on A.
Note that 6(x) = 1 or 0 according as x £ Ai or Ao. Hence, since
Ai,Ao 7^ 0, (r(b) ~ {0,1}. By 5.3.8, there is a unique element
d G \M such that u ~ b + d is idempotent. Then u = 6 = 1 on Ai
and u = fi = 0 on Ao, completing the proof.
8.6.5. PROPOSITION. Let A be a unital commutative,
complex or strictly real, pseudo-Michael algebra. If its spectrum Ac is
disconnected, with Ac = AiljAo, Ay (j = 1,0) non-empty clopen
then there is a unique idempotent u in A with
u = 1 on Ai and u, ~ 0 on Aq.
' clopen=closed and open.
'' The uniqueness follows from the monomorphism property of w proved
on pp.397-8.

406
Commutative Topological Algebras
PROOF. By 4.5.1, A has a projective limit decomposition
A = limAa, where Aa are commutative pseudo-Banach algebras.
In the notation of 7.3.19, we have
Ac = |jAa; Aa ~ Aa
a
where Ac, Aa are the spectra of A, Aa respy. and Aa = {x G Ac :
X is pa— continuous }.
Set
Aay = Ayf|Aa (i= 1,0).
If for an a we have Aai, Aao 7^ 0, then by applying 8.6.4. to
Aa we can determine a unique element ua G Aa with ua(xa) ~ 1
or 0 (where Xa = Xo'Pa) according as x £ Aai or Aao- If for an
a, Aai = 0, then choose ua ~ 0a and if Aao = 0, choose ua — ea,
where 0a,ea are the zero and unity elements of Aa.
Suppose now that a -< /3, Xa G Aay then x« ° *Pap £ ^/3/ and
XaiVcpiup)) = X<.° tPapiup) = 1 or 0
according as x« £ A<*1 or Aao- Since <pap(up) is an idempotent
the uniqueness of the choice of the ua implies that we have
<Pap{up) = ua.
But this means that
w = (tia) G lim Aa — A,
and x(u) = 1 or 0 according as x £ Ai or Ao- Finally, by 4.5.6
(iii), u is an idempotent of A and the proof is complete.
8.6.6. COROLLARY. Let A be a commutative, complex or
strictly real, pseudo-Michael [in particular p-Banach algebra) such
that its spectrum Ac is compact and non-empty. Then \JA is a
regular ideal. In particular, if A is s.s., then A is unital (c/.
8.3.17,8.3.18).
PROOF. Let Ai be the unitization of A. Then Ai = A|J{xo}
where xo is the distinguished character of A±. By 7.3.2(c), Aic(=

§6. Shilov Idempotent Theorem
407
(Ai)c) and Ac are Hausdorff. Also, by our hypothesis, Ac is
compact, whence it follows that Ac is clopen in (Ai)c. So, by 8.6.5,
there is an idempotent u G A\ such that
u ~ 1 on Ac and u(xo) — 0.
Since u(xo) — 0 it follows that u G A. Further, for x G A and
X{xu -x) = x(x)x(«) - x(*) = x(z) ■ 1 ~ x(z) = °
(since x(u) = u(x) = 1). It follows that xu — x £ \/A = \/A,
whence \/A is regular with u as relative unity. When A is s.s.,
{0} = \f~A is regular which means that xu — x — 0 [x G A), i.e. u
is unity of A.
8.6.7. A partially ordered set or poset 5, with ordering
relation -<, is called a lattice if any two elements a, 6 in 5 has a least
upper bound or lattice sum a V 6 and a greatest lower bound or
lattice product a A 6. The element a V6 of 5 is characterized by the
properties: a,b -< aV6 and if a,6 -< x G 5 then aV6 -< x. Similarly
a A b has the properties: a A 6 -< a, 6 and z~<a,&=>a;~<aA&.
The least element 0 and the greatest element 1 of 5 - whenever
they exist - are characterized by the properties: 0 -< x -< 1 for
every x G 5.
A lattice L is called distributive if it satisfies the distributive
law
a A (6 V c) = (a A 6) V (a A c) for every a, 6, c G L.
Also, a lattice L with 0,1 is called complemented if each a G L has
a complement a' satisfying
a A a' = 0, a V a' = 1.
A complemented distributive lattice is called a Boolean algebra.
8.6.8. PROPOSITION(Stone.) The set B{ of idempotents of
any unital commutative ring R is a Boolean algebra.
PROOF. Define an order "~< " in B; by: for u,v G B,-, u -<
v if uv = u. It is easily checked that "-< " is a partial order.

408
Commutative Topological Algebras
Moreover, it is straightforward to verify that B; is a lattice with
lattice operations V,A given by.
uVv = u + v — uv; u A v ~ uv.
Bi is further complemented with u' = e — u as complement of
u, where e is the unity of R. Finally, the distributivity of the ring
operations imply the distributivity of the laltice operations. Thus,
Bi is a complemented distributive laltice or a Boolean Algebra.
8.6.9. LEMMA. LetX be a topological space. Then the clopen
sets in X form a Boolean algebra BC0(X), with set-union, set-
intersection and set-complementation as the Boolean operations.
PROOF. Clear.
8.6.10. LEMMA. Let A be aTA and Ac ±§. If u G A is an
idempotent and
Gu = {x e A : X{u) = 1}, Guc = {X E Ac : X(«) = 1}
then Gu,GUc are clopen sets of A,AC respectively.
PROOF. Since u is an idempotent, for any x G A,
x{u)2 = x("2) = *(«)» so that x(«) = 0 or 1.
Write
G°={XeA:xW=0}.
Then clearly A = Gu\JGl. By definition of the (weak) topology
of A it is clear that GU,G°U are closed. So Gu is clopen in A.
Similarly it can be seen that Guc is clopen in Ac.
8.6.11. PROPOSITION. Let A be a unital commutative TA.
Then the map oj : u >—> Gu is a homomorphism between the Boolean
algebras Bi = Bi(A) and SC0(A). Similarly, ojc : u i—> Guc is a
homomorphism between Bi and BC0{AC). If A is also spectrally
Gelfand (respy. t. spectrally Gelfand) then the map oj (respy. ojc)
is a monomorphism.
PROOF. Since
X(u Vc) = x(u) + xW - x(«)x(w) = 1 if x(u) or x(«) = 1, (1)

§ 6. Shilov Idempotent Theorem
409
Again, since x(« A v) = x(u)x(v) = 1 iff x(u) = x(v) = 1, we
obtain
Guf]Gv. (2)
Finally, if u' denotes the complement of u then, x(u') — x{e~~ u) =
1 ~ x{u) = 1 iff x{u) ~ 0) whence
Gu, = A\GU. (3)
It follows from (1),(2),(3) that a; is a (Boolean algebra) homomor-
phism. Similarly u)c is a homomorphism.
Suppose now that A is spectrally Gelfand and GUl = GU2;
then x(ui) = 1 iff x{u2) = 1- Since u\,U2 are idempotents,
X(«i)>x(«2) = 1 or 0. Therefore, x(«i - «2) = 0, for all x^A,
whence
ui - U2 6 V A = v A.
Since ui,U2 are idempotents, ui — u\U2 is also an idempotent (as
easily checked). But
«1 - «i«2 = «i(«i - «2) £ vA,
whence by 1.2.24(d), ui - u\u2 = 0 or ui = U1U2. Similarly,
U2 = U2U1 = U1U2. So ui = ti2- This completes the proof that
w is a monomorphism. Similarly it can be shown that ojc is a
monomorphism (when A is t. spectrally Gelfand).
8.6.12. DEFINITION. A unital commutative TA is called
a Shilov algebra if the map u 1—> Guc of Bi(A) —> B^A,;) is an
isomorphism.
8.6.13. THEOREM. Any unital commutative, complex or
strictly real, pseudo-Michael algebra A is a Shilov algebra. In
particular, a unital commutative, complex or strictly real, pseudo-
Banach algebra is a Shilov algebra.
PROOF. By 7.2.21, 8.6.11, u)c is a monomorphism. Also, by
8.6.5, ojc is an epimorphism. Thus u)c is an isomorphism and A is
a Shilov algebra.
8.6.14. DEFINITION. A unital algebra A (over K) is called

410
Commutative Topological Algebras
spectrally connected if the spectrum &{x) of every element x G A
is a connected subset of K.
8.6.15. THEOREM. A unital commutative, complex or strictly
real, pseudo-Michael (in particular, pseudo-Banach) algebra A is
spectrally connected iff its spectrum Ac is connected.
PROOF. If Ac is disconnected then by 8.6.4, there is an idem-
potent u ^ 0, e in A, so that &{u) ~ {0,1} is disconnected, whence
A is not spectrally connected. On the other hand, if Ac is
connected then x >~~* x(a) being a surjective continuous map of Ac
onto <r(a),<r(a) is connected for every a G A, so that A is
spectrally connected.
8.6.16. COROLLARY. A is spectrally disconnected iff A has
an idempotent ti/0, e.
PROOF. If u ^ 0, e is an idempotent then by 1.7.9, o{u) =
{0,1}, so that <t(u) is disconnected, which proves the "if part.
To prove the "only if part assume that A has an element a
with a(a) disconnected. Since x *~* x{a) is a continuous map of
Ac onto <r(a) it follows that Ac is also disconnected. But then, by
8.6.5, there is an idempotent u ^ 0,1 in A, completing the proof.

CHAPTER IX
NORM UNIQUENESS THEOREMS
§ 1. Norm- uniqueness Theorem of Gelfand
9.1.1. LEMMA. Let \ ■ \j (j = 1,2) be two complete (F)
norms on a LS X [over K) such that
\x\i ^ C\x\2 for all x G X and some C > 0. (*)
Then | ■ |i ~ | • (2-
PROOF. The condition (*) implies that the identity map
J : (Jf, I • |i) — (J\T, I ■ |2)
is continuous. By the open mapping theorem (3.1.15) for (F)
spaces, I is open and consequently J is a homeomorphism and
hence || • ||i ~ || ■ ))2-
9.1.2. LEMMA. Let \-\j (j = 1,2) be complete (F) norms
on a LS X. Then | • |i ~ | ■ (2 iff \ ' \ — \ • |i + | ■ I2 is a complete
(F) norm.
PROOF. First note that | ■ | (as defined above) is always an
(F) norm. If | ■ | is complete then since | ■ |y < | • |, by 9.1.1,
I ■ I/ ~ I ■ I U = i)2)- Hence | ■ |i ~ | ■ |2.
Conversely, assume that \m\i ,~ |*|2- Suppose that \xn — xm\ —>
0. Then \xn - xm\j —> 0 (j = 1,2). By completeness of | ■ |i there
is an element x with \xn - x\i —> 0. Since | ■ |2 ~ | ■ |i we have
also \xn - x\2 —+ 0. So \xn — x\ = \xn — x\i + \xn — x\2 —> 0 and
I ■ I is a complete (F) norm.
9.1.3. COROLLARY. If \ ■ |y (j = 1,2) are two complete (F)
norms and there is a (F) norm |-|o with \x\o < |x|i,|x|2 (x G A)
then I • |i ~ I • I2 •
PROOF. Suppose that | ■ |0 = | ■ |i + | ■ I2 and \xn — xm\ —> 0.
Then \xn - xm\j —+ 0 (j = 1,2). By the completeness of the

412
Norm Uniqueness Theorems
(F) norms | ■ |y there are elements x, y G A with \xn — x\i —>
0, \xn — j/|2 —+ 0. Since | ■ |o ^ | • |i, | ■ I2 we obtain \xn — x\o —>
0, \xn — j/|o —+ 0. By uniqueness of limits property, we get x = y.
Therefore \xn — x\ = \xn - x\i + \xn - a;12 —* 0, proving | ■ | is
complete. Now we can apply 9.1.2 to conclude that | ■ |i ~ | ■ I2,
completing the proof.
9.1.4. THEOREM (Gelfandt). Let A be a commutative
s.s. algebra which is a p -Banach algebra under either the p -norms
II !! ( ' 1 ri\ TiL II II II II
|| ■ ||y (j = 1,2). Then \\ ■ ||i ~ || ■ ||2.
PROOF. First assume that A is complex. We can also assume
that the || ■ ||y are sm.. Then, by 7.4.6 (A, || ■ ||y) is a GB algebra,
so that we have
r(x) = v\{x)p ~ i/2(x)p, where i/y = ^y(a;) (j =1,2)
are topological spectral radii functions with respect to the p-
norms || ■ ||y. Writing
v(x) = vi(x) = 1/2(2:),
by virtue of 4.8.4, 7.4.5 and the hypothesis that A is s.s. we
obtain that v is a /)-norm. Since v{x) ^ ||a;||y {x G A), we can
apply 9.1.3 to conclude that || ■ ||i ~ || ■ H2.
It remains to consider the case where the algebra A is real.
If A is the complexification of A, then by 4.1.16, || ■ ||y can be
extended to || ■ ||y over A such that (A, || ■ ||y) (j = 1,2) are
commutative complex ^-Banach algebras. Furthermore, by 1.6.16, A
is s.s.. So applying the result just proved for complex algebra we
get || ■ ||i ~ || ■ H2, whence (by restriction) || ■ ||i ~ || ■ H2.
§ 2. Rickart Separating Function
9.2.1. DEFINITION. Let A be a LS over K and |-|y (j = 1,2)
be two (F) norms on A. Set
T Gelfand considered only the complex Banach algebra case of the
theorem.

§ 2. Rickart Separating Function
413
A(a) = Ai2(a) = inf{|x|i + \a — x^} (a,x G A).
The function A is called the Rickart separating function, or just
separating function, for | ■ |/.
9.2.2. LEMMA, (i) Ai2(a) = A2i(a) (a <E A).
(ii) A(o) ^ min{|o|i, |o|2}.
(iii) A(0) = 0.
(iv) A(-a) = A(a).
(v) A(a + 6) < A(a)+A(6) [a,b e A).
(vi) If J ■ \j ~ || ■ ||y are p-norms then
A(Ao) = |A|''A(o) (aGA,AGK).
PROOF, (i) This follows by relacing x by a — x in the
definition of Ai2(a).
(ii) By taking x ~ a in the definitions of Ai2(a), A2i(a).
(iii) This follows from (ii) since |0|i = 0 (= |0|2)-
(iv) This follows from the definition of A, using the relations
I - x\i ~ \x\i U = i.2)-
(v) Given e > 0, by definition of A, there are £1,2:2 ^= ^
with
|a:i|i + |a - a:i|2 ^ A(a) + e
\x2\1 + \b - X2I2 ^ A(6) + e.
Therefore
\%i + X2I1 + |a + 6 - xi - X2I2
|zi|i + \x2\1 + \a - xi\2 + |6 - Z2I2
A(a) +A(6)+2e.
From the arbitrariness of e, the inequality (v) follows,
(vi) For A = 0 the equality clearly holds (since both sides are
zero). Assume next that A^O.
Then
A(Aa) ^ ll^lli + ll^a ~~ ^x\\2 ^ W (H^lli + ||a ~ ^1)2}
< |A|''A(a) ( by taking inf over x).
A(a + 6) <

414
Norm Uniqueness Theorems
On the other hand
|A|'A(a) = |A|'A(A_1Aa) < ^{"^-^^(Xa) = A(Aa).
Combining the two inequalities we get the equality (vi).
9.2.3. LEMMA. If A is an algebra and | ■ |y (j = 1,2) sm.
then
Proof.
A(a6) ^ |a& — az|i + |az|2 < |a|i|6 — z|i + |a|2|£|2
< (|o|i + |a|2)(|6- x\i + |z|2)
^ (|a|i + |a|2)A(6).
Similarly
A(a6) < \ab — xb\i + |:r&|2 < \a — i|i|6|i + |re121^»12
< A(a)(|6|i + |6|2),
completing the proof of the lemma.
9.2.4. Let A be the separating function for the (F) norms
(j = 1,2). If A(a) = 0, a is called a separating element for
\i
the I • \j. Write
& = {ae A: A(a) = 0}.
9.2.5. PROPOSITION, (a) aG6 iff there is a sequence (xn)
in A with \xn\i —> 0, \a — xn|2 —> 0.
(b) & is a subspace of A which is closed with respect to both
I-1,-0/=1,2).
(c) If A is an algebra then <B is a bi-ideal.
PROOF, (a) Clear from the definition of A.
(b) That 6 is a subspace follows from 9.2.2((iii)-(v)), result
(a) above, and continuity of scalar multiplication with respect to
either (F) norm. That 6 is closed can be easily shown, using
(a).

§ 2. Rickart Separating Function
415
(c) This follows from (b) and 9.2.3 .
9.2.6. PROPOSITION. Two (F) norms | ■ |y (j = 1,2)
making a LS X into (F) spaces are equivalent iff 6 = {0}.
PROOF. First suppose that 6 = {0}. Then, by 9.2.5.(a), for
any sequence (xn) in A, a G A,
l^nli —> 0,|a — xn\2 —> 0 => a = 0.
But this means that the identity map (A, | • |i) —> (A, | • I2) is
closed and so continuous, by the closed graph theorem (3.1.16)
and consequently it is a homeomorphism by the open mapping
theorem. Thus, 6 = {0} implies that | • |i ~ | • (2- On the other
hand, if | • |i ~ | • I2 and |in|i —+ 0, \a — xn\2 —> 0 then also
\%n\2 —* 0 (since | • |i ~ | • (2)- Thus, xn —> a,0 under | • I2. By
the uniqueness of limit property, a = 0, 6 = {0}.
9.2.7. Let I • |y (j = 1,2) be (F) norms on an algebra A
such that (A, | • |y) are (F) algebras. Let I be a bi-ideal of A
closed with respect to both (F) norm topologies and
n : A —> A/1 = A^ the canonical homomorphism .
Let I • \J be the corresponding quotient (F) norms. Let 6, 6#
be the sets of separating elements for | ■ |y, | ■ \J respectively. Then
we have
9.2.8. LEMMA. w(&) C 6#. Hence 6# = {0} => e C I.
PROOF. If a £ 6 then \xn\i —> 0, \xn — o|2 —> 0 for some
sequence (xn) in A. Since |^(in)|j < |x„|i —> 0 and |7r(xn) —
7r(a)|2 = |7r(a;n ~ a)|2 ^ \xn ~ a\2 —* 0 it follows that 7r(a) G 6#,
whence tt(6) C 6#. If G# = {0} then tt(6) = {0}, so that
& C I.
9.2.9. DEFINITION. An (F) algebra A= (A, I ■ I) is said to
have norm uniqueness property if for any (F) norm | ■ |' on A
making A into a (F) algebra we have | • |' ~ | ■ |. Similarly, a
^-Banach algebra A = (A, || ■ ||) is said to have norm uniqueness

416
Norm Uniqueness Theorems
property if for any /)-norm || ■ ||' making A into a p-Banach
algebra we have || ■ ||' ~ || ■ ||.
9.2.10. THEOREM. Let A = (A, | ■ |x) be a t. spectrally
Gelfand functionally continuous s.s. (F) algebra. If \ ■ {2 is an
(F) norm on A such that (A, | ■ I2) is a t. spectrally Gelfand (F)
algebra then | • |i ~ | • (2 •
PROOF. Let {Ma} be the family of |-^ -closed hypermaximal
ideals of A. Since A is s.s., by 7.2.12,
f]Ma= \/A = ^A={0}.
Since (A, | • \i) is functionally continuous, each Ma is also | ■ |i-
closed. Since A* = A/Ma ~ K it follows that | • |f ~ | ■ \f.
So, by 9.2.6, 6# = {0}, whence by 9.2.8, 6 C Ma. Therefore
&Cf]Ma = {0}, so |-|i~|-|2.
9.2.11. COROLLARY, (cf. Michael [20,p59]). Let A= (A, />)
be a commutative functionally continuous locally sm. pseudo- 7
algebra which is s.s.. If Q = {gy} be a family of locally sm.
pseudo-seminorms on A making it into a locally sm. pseudo- 7
algebra, then Q ~ P.
PROOF. Assume first that A is a complex algebra. Then by
7.2.21, A is t. spectrally Gelfand. The result for complex algebras
now follows from 9.2.10. The result for real algebras A is obtained
by considering the complexification A of A and applying the
result for complex algebras, and deducing from it the result for A
(using the fact that a pseudo-seminorm of A is the restriction of
its extension to A).
§ 3. Topological Modules
9.3.1. Definition. Let A be an algebra (over K) and X a
TLS. X is called a topological A-module or a t. A-module if:
(i) X is an A-module;

§ 3. Topological Modules
417
(ii) The map m# : (a,x) i—> ax (a £ A, x £ X) is continuous.
9.3.2. LEMMA. Let A be a p-normed algebra and X a
p -normed linear space which is an A -module. Then X is a t. A-
module iff there is a constant C > 0 such that
\\ax\\* ^ C||o|| ||i||* for all a e A, x e X (*)
where || ■ ||, || ■ ||* are respectively the p-norms of A,X.
PROOF. If (*) is satisfied then clearly the map m# is
continuous and X is a t. A-module.
Conversely, suppose that X is a t. A -module. Then from the
continuity of the map m# at (0,0) we get for a given e > 0, a
8 > 0 such that
||arc11* < e if ||o||, ||i||* < 8. (1)
For any a^O in A and i^O in X, write
i-i i -i
c P M II P C P /\\ II *\ P
a\ = 8 a\\a\\ , x\ — o x[\\x\\ J
Then ||ai|| = 8, \\xi\\ = 8 so that by (l), we get ||oia;i||* ^ e
which reduces to
||az|| <C||a|| ||x||* {C = e/82). (2)
The inequality (2) is trivially satisfied if a or x is zero. Thus (2)
holds for all a,x, completing the proof.
9.3.3. DEFINITION. If A,X are as in 9.3.2 and condition (*)
is satisfied then we call X a p-normed A-module. If besides, A
and X are complete with respect to their p- norms we call X a
p-Banach A-module.
9.3.4. PROPOSITION. Let A = (A, || ■ ||) be a p-Banach
algebra [with ||-|| sm.) and I a closed regular I. ideal of A. Then
the quotient module A& = A/1 is canonically a cyclic p -Banach
module. Further, A# is irreducible iff I is maximal.
PROOF. In view of 1.5.16 it is enough to prove that A^ is a
p- Banach A-module. Set
||x + /||# = inf{||x + i|| :te I}.
ii ii t in ii >

418
Norm Uniqueness Theorems
It is straightforward to check that || ■ ||* is a ^-norm on A* (II-II
is called the canonical p-norm induced by || ■ || on A#). Further,
A* is complete with respect to ||-||* (cf. proof of 3.4.14). Finally,
we have
||a(x + 7)||# = \\ax + I\\* ^ mi{\\ax + t\\ : t e 1}
< inf{||az + at\\ : t G 7}
< inf{||o|| ||x + i|| : i G 7}
^ \\a\\ ||x + 7||#.
9.3.5. LEMMA. Let A = (A, || ■ ||) be a p -Banach algebra
and X = (X, || ■ ||*) a p -Banach cyclic A-module, with generator
xq. Then I = kerxo is closed in A and $ : a + I >—> ax$ is a
t. isomorphism of A# = A/1 onto X.
PROOF. Since the map a^aio is continuous, 7 = ker xq is
closed. Also, by 1.5.14 (i), 7 is regular. Therefore, by 9.3.4, A*
is a ^-Banach cyclic A-module and <5 is clearly an isomorphism.
Further
11 ■*»/ i r\ 11 * II 11*^" /"*l I 11 11 11 *
11^(0 + 7)11 = ||axo|| ^ C||a|| ||xo|| .
If a\ G a + 7 then axQ = a\XQ, so that
||$(o + /)||* = lloioll* = ||aia;o||* ^ C||oi|| ||zo||*-
By taking the minimum over all a\ G a + 7, we get
|$(a + /)||* <C||a + /||#|
^0
It follows that $ is continuous and consequently by the open
mapping theorem t $ is a t. isomorphism, as required.
9.3.6. PROPOSITION. Let A be a p-Banach algebra, X a
cyclic p -Banach A-module and Dc the set of all continuous A-
endomorphisms of X. Then:
(i) Dc is a p -normed algebra.
(ii) Dc is also a division algebra if X is an irreducible module.
' The open mapping theorem is available since p -Banach spaces are
(F) -spaces.

§4. Norm-uniqueness Theorem for Non-commutative Algebras 419
PROOF. Let 2¾ be a generator of X, 7 = kerzo and <5 the
t. isomorphism A# = A/7 —> X (see 9.3.5). For T G Dc write
j, = $_1Txo, ax0 = $(a + 7).
Then
Q-iT&ta +1) = $~lTaxQ = aQ~lTxQ = ay.
It follows that
||$_1T$(a + 7)11* = \\ay\\* = llau/ll* < ||oi|| \\y\\*
II V / I I II v || || A «7 | | II x I I I I v I I
for any a\ with a\ + 7 = a + 7. By taking the infimum over all
such a\ we get
||$-1T$(a + 7)||* < inf{||ai|| ||y||* : a{\ = \\a + 7||#||j/||*.
Thus, T* = $-17-$ is a bounded linear operator of A#. Set
||T|p = ||T*||. Then || • ||* is clearly a sm. /)-norm on Dc and
Dc is a normed algebra, proving (i).
For proving (ii), consider the algebra D of all A-
endomorphisms of X. When X is irreducible, by 1.5.19, D is
a division algebra. Clearly, Dc is a subalgebra of D. Further,
if T G Dc, T^O then T~l e D (D being a division
algebra). But, by the open mapping theorem applied to X we have
T"1 G Dc, proving (ii).
9.3.7. COROLLARY. For an irreducible p-Banach A-module
X, Dc is t. isomorphic to C if A is complex and t. isomorphic
to R, C or H if A is real.
PROOF. This follows from 9.3.6, 6.5.6, 6.5.12.
§ 4. Norm-uniqueness Theorem for
Non-commutative t Algebras
9.4.1 LEMMA (Bonsall-Duncan). Let A be a p-Banach
algebra and X an irreducible p -Banach A -module. Then, for any
' non-commutative = not necessarily commutative

420
Norm Uniqueness Theorems
non-zero element xq of X, if M = kerzo and I a closed I. ideal
with I 2 M then we can find an r) > 0 such that
r)Bix0C (lf\Bi)x0
where B\ denotes the closed unit ball of A.
PROOF. By 1.5.14, M is a maximal regular 1.ideal which by
9.3.5, is closed. Further, by the choice of I and maximality of M,
we have: I + M = A. It follows that the map
7T0 : a <—* a + M (a G I)
of the p -Banach space I to the p -Banach space A* = A/M is
surjective. By the open mapping theorem, 7To(/n^i) is °Pen;
so that we can find an r) > 0 such that for every a + M with
\\a + M\\ < r)p we have a + M e 7r0(/ H ^l)- This means that
there is an element 6 G 7f| i?! with b + M — a + M, bx0 = ax$.
If a G rjBi then ||a + M\\ < ||a|| < r)p, so that axo = bxQ with
6 G 7f| B\, proving the lemma.
9.4.2. THEOREM (Johnson). Let A be a primitive p-Banach
algebra and it a faithful irreducible representation of A on a p -
normed linear space X such that 7r(a) G B(X) for all a in A.
Then the map
7T : A-^B(J>0
is continuous, where B(X) denotes the algebra of all bounded
linear operators of X.
PROOF. If X is finite-dimensional so is B(X) and then also
A (since n is 1 — 1). Thus, in this case 7r, being a linear map
between two finite-dimensional Hausdorff TLS's, is automatically
continuous. ''
Assume now that X is infinite-dimensional. Let rx denote
the map
a i—> ax = rx(a) (a G A,x G X).
Write
' Bi denotes the open unit ball of A.
tt See [5,14, Cor. 2]

§4. Norm-uniqueness Theorem for Non-commutative Algebras 421
X0 = {xeX:rxeB{A,X)},
where B(A,X) denotes the space of all bounded linear
transformations of A into X. Then Xo is a submodule of X = (X, n),
since
||»"4i(a)|| = ||o6a;|| = ||rz(o6)|| < ||rz||||o6|| < ||rz||||a|||H|-
Since X is irreducible, Xo ~ X or {0}.
If Xo = X, and B\ denotes the closed unit ball of X, for
x £ Bi and a G X we have
||rz(o)|| = \\ax\\ — ||7r(a)xll ^ IKWIIII^II ^ IKWIi-
By Banach-Steinhauss theorem (see 3.2.16) there is a constant
C > 0 with ||rz|| < C (x G Si). It follows that
||7r(a)x|| = ||rz(o)|| < C||a|| {x £ B\,a G A).
whence ||t(o)|| ^ C(||o||), so that by 3.5.5, n is continuous.
To complete the proof it is enough to rule out the other
alternative Xo = {0}. Suppose therefore that Xq = {0}. Then the
definition of Xq implies that for each x ^ 0 in X the operator
rx is not bounded, so that (by 3.5.2):
the set {||oa;|| : ||a|| < 1} is not bounded. (1)
Now by 9.3.7, Dc is a division algebra over K with dimDc < 4.
Since X is infinite-dimensional over K it is also infinite
dimensional over Dc. It follows that X contains an infinite sequence
(xn) of Dc -independent vectors. By replacing xn if necessary by
a suitable scalar multiple we may assume that \\xn\\ = 1 for all
n. Set
Mn = kerxn, /;n = Mif|---f|Mn_i (n = 2,3,---)-
By density theorem (1.5.21) we can find an a G A such that
axj = 0(l<j<n — 1), axn ^ 0 (n > 2).
' We have assumed that the norm || • || of A is sm..

422
Norm Uniqueness Theorems
Then a G Ln,a ¢ Mn, so that Ln g Mn. Since by (l), B\xn is
unbounded, by 9.4.1
(Ln || B\)xn is unbounded. (2)
We now inductively choose a sequence (an) with
ane Ln, \\an\\ < 2~n. (3)
and
||anzn|| > n + ||(oi H h a„_1)a;ri||. (4)
To justify the inductive step, suppose we have chosen a\, ■ ■ ■, an-\.
Then, by using(2), we can select a bn <E Lnf)Bi such that
||6nxn|| > 2n+1(n + ||(ai + ■ ■ ■ + an_i)xn||). (5)
Set an = 2-^n+l^t>bn. Then
||an|| = 2-("+1)||6n|| < 2"("+1) < 2"" ( using bn G Bx),
and using (5),
||anxn|| = 2~"(n+1)||&nxn|| > n + \\(a1 -\ h an_i)xn||.
Next write
oo
k=l k=n+l
These elements are defined since the series representing them
converge absolutely (by virtue of the inequality in (3)) and hence also
in A. Since a^ G Mn [k > n) and Mn is closed, we conclude
that cn G Mn and consequently, cnxn = 0.
It follows that we have
cxn= I ^ ak + cn J xn = I J2 ak I xn,
so that by inequality (4) we get
11 "^fi 11 ^ ll^n^nll
11(0! H h an_i)zn|| ^ n.

§4. Norm-uniqueness Theorem for Non-commutative Algebras 423
This means that 7r(c) = lc is not a bounded operator contradicting
the hypothesis on 7r. Therefore, Xq 7^ {0}, completing the proof.
9.4.3. THEOREM (Johnson). Every primitive ^-Banach
algebra A has p -norm uniqueness property. >
PROOF. Let A = (A, || ■ ||i) and || ■ ||2 a complete p-norm
on A such that (A, || • ||2) is also a ^-Banach algebra. We have
to show that || ■ ((2 ~ || • ||i-
Since A is primitive there is a maximal regular 1. ideal M
such that the left regular representation £# on A# = A/M is
faithful. By 7.1.9, M is closed with respect to either of the norms
II ' llj {J = 1)2)- Let || ■ \yf be the canonical p-norm on A*
induced by || ■ \\j. Then (A^, || ■ ||^ ) are ^-Banach spaces (see
3.1.22, 3.4.16). It follows that n is a faithful representation of
the ^-Banach algebra A = (A, || ■ \\\) on the ^-Banach space
A# = (A#,|| ■ H2 )• By 9.4.2, n is continuous and consequently
we have
11^(0)1111 < C||o||i (aeA)
for some C > 0. This implies that
||7r(a)(x + M)||f < ||7r(a)||f ||x +M||f
< C||o||i \\x + M\\f. (1)
Let u be the relative r. unity for M. Then
7r(a)(u + M) = a(u + M) = a + M,
so that
\\a + M\\* = ||7r(a)(u + M)\\f < C||a||i ||u + M\\*,
where we have used the inequality (i) with x = u. If
a + M = b + M (2),
then
||a + M||f = ||6 +M||f < C||6||i||u + M||^. (3)
' Johnson obtained the theorem for Banach algebras (i.e. for the case
P=l).

424
Norm Uniqueness Theorems
Taking the infimum in (3) over all b satisfying (2) we obtain
\\a + M\\f <C||a + M||f ||u + M||f. (4)
The inequality (4) together with the open mapping theorem im-
plies that || • \\f and || ■ \\J are equivalent. Suppose now
that a in A is a separating element with respect to the norms
|| ■ 11j (j = 1,2), so that there is a sequence (xn) in A with
||xn||i -+ 0, \\a - xn\\2 -+ 0.
Since \\x + M||J < 11ary11 it follows that a + M is a separating
element of A# with respect to || • ||;f. Since || ■ ||f ~ II ■ II, we
" II I IJ 11 11 1 11 11L
must have
a + M = M, a € M, & C M,
where & is the separating set for the || ■ ||j. Since by 9.2.5(a), S
is a bi-ideal we have, by 1.5.3,
6 C (M : A) = {0} ( since C* is faithful ).
So, by 9.2.6, || ■ ||i ~ || ■ ||2, as we wished to show.
9.4.4. THEOREM. Every s.s.(real or complex) p-Banach
algebra A has p-norm uniqueness property.
PROOF. If P is a primitive ideal of A then A/P is a
primitive ^-Banach algebra and so has, by 9.4.3, /)-norm uniqueness
property. Let S be the separating ideal for the p-norms || ■ ||j
on A, where A=(A,||-||i) and || ■ ||2 is a second ^-normon A
making it again into a ^-Banach algebra. Then we have S C P
(the proof is similar to that above showing S C M). It follows
that
S C P| P = \/a = {0} ( since A is s.s. )
whence A has p -norm uniqueness property.
9.4.5. Remark. Theorem 9.4.4 admits a strengthening to
the following form: If || ■ ||j- (j = 1,2) are pj -norms (0 < pj < l)
on a s.s. algebra A such that (A, || ■ ||j) are pj -Banach algebras,
then || ■ ||i ~ || ■ |U.

§4. Norm-uniqueness Theorem for Non-commutative Algebras 425
To obtain the above result we first remark that we can assume
that p\ < pi and || ■ ||j are sm. If we set || • ||J = || • ||j2,
then by 3.2.9. || • ||J is a p\ -norm which from its form is clearly
(topologically) equivalent to || • ||2. Further it is sm.:
£1 £i
ll^slli = (ll^slh)'2 < (ll^lhllslb)"2 = II^IIiIIj/IIi-
It follows that (A,||-||i) is a p\ Banach algebra. By 9.4.4, ||-||i~
|| ■ ||j and so || ■ ||i ~ || ■ ||2 (since || ■ ||J ~ || ■ ||2 ) as claimed.
9.4.6. PROPOSITION. Let A, A* be sm. (F) algebras and
p> : A —> A* an epimorphism. If A* is s.s. and has norm-
uniqueness property then p> is continuous.
PROOF. Write I = ker^>. By 3.6.23(a), 7.1.17, I is closed
and therefore A* = A/1 is a sm. (F) algebra with its (F) norm
| ■ |# satisfying | ■ |# < | ■ | (see 3.1.22, 3.4.15). Since A* is
isomorphic to A" we can shift the (F) -norm | ■ |* to A*. Then
p: A -» A* = (A*,|-|#)
is continuous. By hypothesis A" has norm-uniqueness property.
It follows that | ■ |" (of A*) ~ | • |#, whence
p : A —> A" = (A*, | ■ |*) is continuous .
9.4.7. COROLLARY. If A, A* are p-Banach algebras, A*
s.s. and p : A —+ A* an epimorphism then pi is continuous.
PROOF. By 9.4.4, A* has norm-uniqueness property and so
the corollary follows from 9.4.6 .
9.4.8. COROLLARY. Any automorphism pi of a s.s. sm. (F)
algebra A with norm-uniqueness property is a t. automorphism.
In particular, every automorphism of a s.s. p -Banach algebra is a
t. automorphism.
PROOF. By 9.4.6 applied to p>, p>~1 we conclude that they are
continuous, so that p> is a t. automorphism. The second statement
follows from the first in view of 9.4.4 .

APPENDIX
LEMMA A. The character space A = A(W) is homeomor-
phic to the unit circle S1 in C.
PROOF. We have already seen in 7.5.16 that the characters x
of W are precisely the maps
Xtl:f"f(h) (/eF.tiGR).
Since Xti(e'*) = e'*1 determines x<i it follows that Xtx = Xt2 iff
eit1 =eit2 Tnus
is a bijection of A on S1. Moreover it is continuous (by definition
of the (weak) topology of A). Since A is compact (by 7.3.12)
and 51 is Hausdorff, the map x h~~* x(e'*) is a homeomorphism.
LEMMA B. Let F,C be respectively a closed set and a compact
set in R" such that FC\C = 0. Then there exists a C°° function
g on R" such that g = 0 on F and g = 1 on C .
PROOF. See [H, p.3].
LEMMA C. Let f = f{t) be a C°° 2n -periodic complex
function on R and f"> denote the j th derivative of f. If
f{n)>f'{n) (n e ^) denote the nth Fourier coefficients of f
and f(J> respectively then we have, for n^O,
|/(n)|<|nH|/('-)||L1,
where ||/W)||LI = £ /f \fM(t)\dt.
PROOF. See [K, p.24, 4.4].
THEOREM. W is completely regular.
PROOF. Since by Lemma A, A can be identified with S1, for
proving the theorem it is enough (in view of 8.14.7) to show that
given a closed subset F of 51 and a point elt° E S*\F, there is
a feW' = W' such that / = 0 on F and f{eito) ^ 0. Now

428
Appendix
by Lemma B we can find a C°° function g = g(x, y) on R2 with
g = 0 on F and g(eito) ^ 0. If we set
f(t) = g{elt) = (/(cos i, sin t)
then / is a 27r-periodic C°° function on R with f(t) = 0 if
elt e F and f(t0) ^ 0. Choose integer j such that j > -. Then,
by Lemma C,
El/(»)l'< l/(o)l' + 2E i7^<o°'
n£Z n=lII
where C = H/^H^. It follows that / e W, completing the
proof.
References
[H] H. HELGASON: Differential Geometry and Symmetric
Spaces, Academic Press, New York, 1962.
[K] Y. KATZNELSON: An Introduction to Harmonic Analysis,
Douer Publications, New York, 2nd edition, 1976.

TOPOLOGICAL ALGEBRA
TYPE CONNECTION CHART
O Banach Algebra
yl Locally sm.3 Algebra
O Locally sm. convex Algebra
Locally convex
Algebra
O
O Locally sm. p-Convex
Algebra
q Locally
p-convex Algebra
O C Algebra
Topological Algebra
The arrow indicates that the algebra at the tail of
the arrow has the properties of the algebra at its tip.

BIBLIOGRAPHY
Books Referred
L.V. AHLFORS: Complex Analysis, McGraw-Hill,
International Student Edition, 1979.
T.M. APOSTOL: Mathematical Analysis, Addision Wesley,
World Student Edition, 1963.
S. BOCHNER and W.T. MARTIN: Several Complex
Variables, Princeton Univ. Press, Princeton, 1948.
F.F. BONSALL and J. DUNCAN: Complete Normed
Algebras, Springer Verlag, New York, 1973.
N. BOURBAKI: Topological Vector Spaces, Chapters 1-5,
Springer Verlag, New York, 1987.
N. BOURBAKI: Theories Spectrales, Chapters 1-2,
Hermann, Paris, 1967.
G. CHOQUET: Lectures on Analysis, Vol. 1, W.A. Benja-
men, New York, 1969.
J.B. CONWAY: Functions of One Complex Variable,
Springer International Student Edition, Narosa, New Delhi,
2nd revised edit., 1980.
J. DUGUNDJI: Topology, Printice-Hall of India, New Delhi,
1975.
I.M. GELFAND, D.A. RAIKOV and G.E. Shilov:
Commutative Normed Rings, Chelsea, New York, 1964.
L. GILLMAN and M. GERRISON: Rings of Continuous
Functions, Springer Verlag, New York, 1976.
A. GUICHARDT: Special Topics in Topological Algebras,
Gordon and Breach, New York, 1968.
N. JACOBSON: Lectures in Abstract Algebras V.I., East-
West Student Edition, Affiliated East-West Press, New
Delhi, 1964.
E. HILLE and R.S. PHILLIPS: Functional analysis and
semigroups, Amer. Math. Soc. Coll. Publ. 31, Providence,
R.I., 1957.
R.V. KADISON and J.R. RINGROSE: Fundamentals of the
Theory of Operator Algebras, V.I., Academic Press, New
York, 1983.

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[16] J.L. KELLEY: General Topology, Van Nostrand, Princeton,
1955.
[17] J.L. KELLEY and I. NAMIOKA: Linear Topological Spaces,
East-West Student Edition, Affiliated East-West Press, New
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[18] G. KOTHE: Topologische Lineare Raiime, Springer Verlag,
Berlin, 1960.
[19] L.H. LOOMIS: An Introduction to Abstract Harmonic
Analysis, Van Nostrand, New York, 1953.
[20] E.A. MICHAEL: Locally Multiplicatively-Convex
Topological Algebras, Memoirs Amer. Math. Soc, No. 11,
Providence, R.I., 1952.
[21] D. MONTGOMERY and L. ZIPPIN: Topological
Transformation Groups, Interscience Publishers, New York, 1955.
[22] M.A. NEUMARK: Normierten Algebren, VEB Deutscher
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INDEX
The bold face figures after the subsection numbers below
indicate the corresponding pages in which these numbers occur.
Absolutely convergent series 3.1.23, 109
Absolutely ^-convex 4.1.1, 174
Absorbing set 2.1.15, 78
Almost submultiplicative (a.sm.). function 3.3.3, 118
A-module 1.5.1, 36
Ample algebra 4.7.1, 213
Annihilator (left, right) 1.2.1, 12
Arens algebra 4.6.8 (iv), 211
Arens-Calderon trick 8.5.18, 394
Bi-ideal 1.2.1, 12
Binomial theorem 1.1.7, 3
Bi-primitive ideal 1.5.4, 38
Bi-singular 3.7.1, 158
Boolean algebra 8.6.7, 407
Bounded or n. (=norm) bounded linear transformation
3.5.1, 140
Bound of a linear transformation 3.5.1, 140
C algebra 3.6.4, 148
CI algebra 3.6.20, 3.6.21, 153
CQ algebra 3.6.20, 3.6.21, 153
Canonical character 1.4.10, 36
Canonically ^-normed function algebra 8.1.1, 353
Cauchy net or C-net 2.3.1, 91
Cauchy's estimates 5.4.18, 252
Cauchy's integral theorem 5.4.16, 249
Character 1.3.1, 22
Circle operation 1.1.13, 6
Closed graph theorem 3.1.16, 105
Closed map 3.1.16, 105
Commutant 1.1.7, 3
Complete p -topology 3.1.4, 101
Complete quarter-norm 3.1.13, 105

436
Index
Complete topological linear space (complete TLS) 2.3.1, 91
Completely regular algeba 8.4.1, 377
Completely regular family of functions 8.4.8, 381
Complex structure 1.6.1, 43
Contour 5.4.15, 248
Contour surrounding a subset 5.4.15, 248
Contraction 4.3.14, 194
Contraction mapping principle 4.3.15, 194
Convergence of series and generalized series 2.1.32, 83
Cyclic module 1.5.13, 41
Derivation 7.6.5, 345
V -independent 1.5.20, 43
Directed set 2.1.1, 73
Discrete 2.1.11, 78
Distinguished character 1.4.10, 36
Distributive lattice 8.6.7, 407
Entire functions operating in a topological algebra (TA)
5.5.1, 253
Epimorphism 1.1.10, 4
Equivalence of subadditive (sad.) functionals 3.1.4, 101
Essentially bounded net 2.3.5, 93
Essentially nilpotent element 1.2.25, 22
Examples of:
Completely regular algebras 8.4.2, 378
(F) algebras 3.3.14, 123
Fixed ideals 8.1.17, 8.1.18, 8.1.23, 354, 360
Function algebras 8.1.2, 353
Hypermaximal ideals 1.3.13, 25
Metrizable locally pseudoconvex algebras 4.6.8, 209
Monogenic topological algebras 7.3.22, 321
Primary ideals 7.1.20, 301
^-Banach algebras 3.4.6, 133
^-seminormed algebras 3.4.6, 133
Shilov algebra 8.6.13, 409
Shilov boundary 8.2.16, 367
Exponential function 5.2.1, 227

Index
437
Extended quasi-resolvent 1.8.1, 61
Extended quasi-spectrum 1.8.1, 61
Extended spectrum 1.8.1, 61
Extended spectral radius 1.8.10, 64
(F) algebra 3.3.13, 123
(F) norm 3.1.14, 105
Faithful quarter-norm 3.1.14, 105
Faithful representation 1.5.1, 36
Faithful sad. functional 3.1.1,100
Fixed ideal 8.1.17, 357
Formally real algebra 1.6.17, 50
(F) space 3.1.14, 105
Function algebra 8.1.1, 353
Functionally continuous 2.2.19, 90
Gelfand algebra 7.2.1, 303
Gelfand-Beurling (GB) algebra 7.4.1, 324
Gelfand lemma 3.3.6, 119
Gelfand space 7.3.1, 311
Gelfand transform 7.3.4, 312
Generalized sum 2.1.32, 83
Hilbert relation 6.2.2, 264
Homogenity index of a pseudo-seminorm 3.2.1, 110
Homomorphism 1.1.10, 4
Hull-kernel topology (hk-topology) 8.3.5, 371
Hypermaximal ideal 1.3.6, 24
Hypernormal TA 7.1.4, 297
Hyper-radical 1.3.7, 24
Hyper semi-simple (h.s.s.) 1.3.7, 24
Hyponormal TA 7.1.4, 297
I algebra 3.6.7, 149
Ideal (1., r.) 1.2.1, 12
Index of a point 5.4.15, 248
Inverse (1. r.) 1.1.3, 1
Inverse-closed subalgebra 1.7.27, 61
Irreducible module 1.5.2, 1.5.13, 37, 41

438
Index
Isomorphism 1.1.10, 4
Jacobson density theorem 1.5.21, 43
Joint spectrum 8.5.7, 389
Ker p. 3.1.5, 102
Ker P. 3.1.20, 106
Lattice 8.6.7, 407
Leibniz rule 7.6.8, 346
Lemma (Bonsall-Duncan) 9.4.1, 419
Linear functional 1.3.1, 23
Liouville algebra 1.8.13, 65
Locally bounded algebra 4.2.4 (f.n.), 187
Locally bounded space 2.1.24, 81
Locally pseudo-convex algebra 4.4.1, 195
Locally pseudo-convex space 4.3.1, 189
Locally ^-admissible holomorphic 5.4.12, 245
Locally sm. pseudo-pre-Frechet (#) algebra 4.6.3, 208
Michael algebra 4.5.4, 204
Minimal pre-boundary 8.2.3, 362
Monogenic 7.3.21, 320
Monomorphism 1.1.10, 4
Net 2.1.1, 73
Normal family of functions 8.4.8, 381
Normal TA 7.1.4, 297
Normalized ^-seminorm 3.4.13, 136
Norm bounded (n bounded) 3.2.12, 3.5.1, 114, 140
Nucleus 2.1.9, 77
Operator topology 2.2.4, 85
Open mapping theorem 3.1.15, 105
P -bounded 4.3.9, 192
p-bounded 3.2.12, 114
P -complete 3.1.20, 106
Polynomially convex (p. convex) 8.5.1, 388
Polynomially convex hull 8.5.1, 388

Index
Poset 2.1.1, 73
Power series operating in a TA 5.5.1, 253
Pre-boundary 8.2.1, 361
Pre- (F) algebra 3.3.13, 123
Primary ideal 7.1.18, 301
Prime ideal 1.5.9 (f.n.), 39
(1., r.) Primitive ideal 1.5.4, 38
Principal class 2.3.9, 95
Principal component 5.2.7, 230
Projective limit 4.5.1, 201
Pseudo-convex algebra 4.4.1, 195
Pseudo-convex space 4.3.1, 189
Pseudo Frechet (3) algebra 4.26.2, 208
Pseudo-pre- 3 algebra 4.6.2, 208
Pseudo-resolvent function 6.3.1, 275
Pseudo-resolvent set 6.3.1, 275
Quarter-norm 3.1.8, 103
Quarter-normed algebra 3.3.1, 116
Quasi-inverse, Quasi-invertible 1.1.16, 6,7
Quasi-nilpotent (q.nilpotent) 1.7.13, 56
Quasi-resolvent set 1.7.5, 54
Quasi semi-simple (q.s.s) 1.7.17, 57
Quasi-spectrum 1.7.5, 54
Quasi-square root (q.sq.r.) 5.3.1, 235
Quasi-unital 8.3.15, 375
Radical (Jacobson) 1.2.25, 22
Radical algebra 1.2.27, 22
Radius of convergence 5.1.6,223
Real character 1.6.14, 49
Regular ideal (1. r.) 1.2.6, 14
Regular bi-ideal 1.2.6, 14
Regular representation 1.5.1, 36
Relative unity (1., r.) 1.2.6, 14
Relative bi-unity 1.2.6, 14
Resolvent set 1.7.1, 52
Rickart separating function 9.2.1, 412

440
Index
^-admissible function 5.4.7, 241
^-admissible holomorphic 5.4.11, 245
^-Banach algebra 3.4.4, 132
^-Banach A-module 9.3.3, 417
^-Banach space 3.2.6, 111
^-convex linear combination 4.1.1, 174
^-guage 4.1.8, 178
^-modulus homogenity condition 3.2.1, 110
^-normed LS 3.2.6, 111
^-seminormed LS 3.2.6, 111
Saturated closure 4.3.7, 191
Saturated family of pseudo-seminorms 4.3.7, 191
Self-conjugate (subset, ideal) 1.6.4, 1.6.12, 44, 46
Semi-metric space 3.1.2 (f.n.), 100
Semi-simple (s.s.) 1.2.27, 22
Semi-topological group 2.2.6 (f.n.), 86
Separating element 9.2.4, 414
Separating function (Rickart) 9.2.1, 412
Separating, strongly separating, family of functions 8.1.12,
356
Shilov algebra 8.6.12, 409
Shilov boundary 8.2.3, 362
Simple module 1.5.2 (f.n.), 37
Simultaneous spectrum 8.5.7, 389
Singular (1., r.) 3.7.1, 158
Spectrally connected 8.6.14, 409
Spectrally Gelfand 7.2.5, 304
Spectral radius 1.8.10, 64
Spectrum of an algebra 7.3.1, 311
Spectrum of an element 1.7.1, 52
Strictly real algebra 1.9.1, 67
Strongly analytic function 5.1.3, 222
Strong structure space 8.3.9, 372
Structure space 8.3.9, 372
Submultiplicative (sm.) function 3.3.3, 118
sm. pseudo-convex algebra 4.4.1, 195
Subnet 2.1.1, 73

Index
441
Support of a function 8.4.24 (f.n.), 386
Symmetric subset of a group 2.1.3, 74
Symmetric subset of C 1.8.3 (f.n.), 62
Theorems of:
Arens (Proposition) 3.3.2, 116
Arens-Banach 3.6.16, 152
Arens-Shilov (Cor.) 6.5.20, 290
Banach-Steinhauss 3.2.16, 115
Beurling-Gelfand (spectral radius formula) 6.2.11, 271
Beurling-Gelfand-Zelazko 7.4.6, 326
Birkhoff-Kakutani 2.1.7, 76
Bonsall-Duncan 3.5.9, 144
Gelfand (norm equivalence theorem) 9.1.4, 412
Gelfand-Kolmogorov-Stone-Banach 8.1.22, 359
Gelfand-Mazur 6.5.5, 285
Gleason-Kahane-Zelazko 5.2.15, 233
Jacobson (density theorem) 1.5.21, 43
Johnson 9.4.2, 9.4.3, 420, 423
Kaplansky 1.9.15, 71
La Page-Hirschfield-Zelazko 7.4.15, 331
Michael 4.5.3, 7.4.8, 202, 327
Mitjagin-Rolewicz- Zelazko (Cor) 5.5.11, 261
Nagumo 5.2.14, 232
Oka (extension theorem) 8.5.19, 396
Rickart 4.1.16, 183
Shilov (idempotent theorem) 8.6.4, 404
Shilov-Arens-Calderon 8.5.22, 399
Singer-Wermer 7.6.20, 351
Turpin 6.6.5, 294

442
Index
Tychnoff 2.1.12, 78
Wiener-Levi-Zelazko 7.5.16, (b) 342
Wiener-Zelazko 7.5.16, (a) 342
Yood (Proposition) 7.4.20, 333
Zelazko 5.5.10, 259
Topological A-module 9.3.1,416
Topological group (TG) 2.1.3, 74
Topological linear space (TLS) 2.1.9, 77
Topological integral domain (TID) 6.3.13, 281
Topological nilpotent (t.nilpotent) 4.8.8, 220
Topological semigroup (TSG) 2.2.5, (f.n.) 86
Topological spectral radius 4.8.1, 216
Topological zero divisor (1., r., bi-) 3.7.3, 158
Topological zero divisor (symmetric (s.t.z.d.)) 3.7.11,162
Topological zero divisor (generalized (g.t.z.d.)) 3.7.38,172
Universal mapping properly 4.5.1, 201
Weakly analytic function 5.1.3, 222
Weakly differentiable function 5.1.1, 222
Weakly holomorphic function 5.1.3, 223
Weak topology 7.3.1, 311
Well-behaved family of pseudo-seminorms 4.4.9, 198
Wiener property 7.2.10, 306
Williamson's algebra 3.6.33, 157

List of Special Symbols
The numbers after the symbols indicate the subsections in
which they are introduced
y/A, 1.2.24-25, y/A, 1.3.7, etyA, 2.218,
</A, 1.5.11, Aqn, 1.7.13
C, 1.6.1
A, 1.3.1 , Ac 2.2.18, dAS, 8.23
E{x), 5.2.1; EX(X), 5.2.10; Eq{x), 5.2.6
E \ 5.2.1; e*\ 5.2.1; e, 5.2.1
Gf, 3.6.16
G„ 1.1.4 ; G|, 3.7.1; G\, 3.7.1
G„ 1.1.17; Glq, 3.7.1; GJJ, 3.7.1
G,-e, 5.2.7; Gq0, 5.2.7
K, 1.6.1
la, 2.2.2; /^, 3.6.1
i/p(x), 3.3.5 ; v{x) 3.3.7
'A, 8.3.1; Ea. 8-3-l
R, 1.6.1
ra, 2.2.2; r°, 3.6.1
r(x), 1.8.10; n(x), 6.6.1; r2(x), 6.6.1; r3{x), 6.6.1
p(a:), 1.7.1; ,'(*), 1.7.5; pP{x), 6.3.1
a(x), 1.7.1; a'{x), 1.7.5
S', 3.7.1; Sr, 3.7.1; Sbi, 3.7.1
a;A, 6.2.1; x'A, 6.3.1
3", 3.7.3; 3ri, 3.7.3; 3"', 3.7.3
a <-> b ~ a commutes with b : ab — ba, 1.1.7
0 = empty set

List of Special Abbreviations
GB (algebra)
LS
STG
TA
TG
TID
TLS
TSG
1. ideal
r. ideal
bi-ideal
q.i
l.q.i.
r.q.i
q-
l.q.
r.q.
q.nilpotent
q.sq.r
s.q.r.
ext.
t.
t.z.d.
l.t.z.d.
r.t.z.d.
z.d.
l.z.d.
r.z.d.
Gelfand Beurling
linear space
semi-topological group
topological algebra
topological group
topological integral domain
topological linear space
topological group
left ideal
right ideal
both l.r. ideal
quasi-inverse
left quasi-inverse
right quasi-inverse
quasi in q. invertible
left quasi in l.q. invertible
right quasi in r.q. invertible
quasi-nilpotent
quasi-square root
square root
extended in ext. q. nilpotent
topological in t. automorphism, t. isomorphism
topological zero divisor
left topological zero divisor
right topological zero divisor
zero divisor
left zero divisor,
right zero divisor