Topological Algebras

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

9 Narosa Publishing House, India - 1999 Licenced edition of Elsevier Science - 2000 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, w i t h o u t the prior permission of the copyright owner.

Elsevier Science ISBN for this volume: 0 444 50609 8 Elsevier Science Series ISSN: 0304-0208 Published by: Elsevier Science Sole distributors for Europe, North America and Japan" Elsevier Science

The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

To my grand-children S h o b h a n a and V i v e k

for delaying the completion of the writing of the book

This Page Intentionally Left Blank

CONTENTS

Preface

ix

..............................................

Chapter 1: Algebraic Preliminaries w 1. w 2. w 3. w 4. w 5. w 6. w 7. w 8. w 9.

Some Basic C o n c e p t s and R e s u l t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideals and R a d i c a l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C h a r a c t e r s and H y p e r m a x i m a l Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E x t e n s i o n s of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R e g u l a r R e p r e s e n t a t i o n and P r i m i t i v e Ideal . . . . . . . . . . . . . . . . . . . . . . . Real and C o m p l e x A l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S p e c t r u m and Q u a s i - s p e c t r u m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E x t e n d e d S p e c t r u m and E x t e n d e d Q u a s i - s p e c t r u m . . . . . . . . . . . . . . . . Strictly R e a l A l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 12 22 30 36 43 52 61 67

Chapter 2: Topological Preliminaries w 1. T o p o l o g i c a l G r o u p s and L i n e a r Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . w 2. T o p o l o g i c a l A l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w 3. C o m p l e t i o n s of Topological L i n e a r Spaces and Topological A l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 84 91

Chapter 3" Some Types of Topological Algebras w 1. Q u a r t e r - n o r m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w 2. p - S e m i n o r m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w Q u a r t e r n o r m e d Algebras; ( F ) A l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . w 4. p - S e m i n o r m e d Algebras; p - B a n a c h Algebras . . . . . . . . . . . . . . . . . . . . w 5. B o u n d e d L i n e a r T r a n s f o r m a t i o n s on p - S e m i n o r m e d L i n e a r Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w 6. T o p o l o g i c a l A l g e b r a s with Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w 7. T o p o l o g i c a l Zero Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100 110 116 131 140 148 158

Chapter 4" Locally Pseudo-Convex Spaces and Algebras w 1. w 2. w 3. w 4. w 5. w 6. w 7. w 8.

p-Convexity .................................................... Locally B o u n d e d A l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L o c a l l y P s e u d o - C o n v e x Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Locally Pseudo-Convex Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P r o j e c t i v e Limit D e c o m p o s i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metrizable Locally Pseudo-Convex Algebras . . . . . . . . . . . . . . . . . . . . . Ample Algebras ................................................. Topological Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174 185 189 195 201 207 213 216

viii

Chapter w 1. w 2. w 3. w 4. w 5.

6" S p e c t r a l

Analysis

in Topological

7" G e l f a n d

Representation

262 264 275 282 284 290

Theory

Ideals of Topological Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Gelfand Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 The Gelfand R e p r e s e n t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 GB Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Holomorphic Functional Calculus for a Single Algebra Element . . . 3 3 5 A u t o m o r p h i s m s and Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

Chapter

8: C o m m u t a t i v e

Topological

Algebras

w 1. w 2. w 3. w 4. w 5.

Function Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shilov B o u n d a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hull-Kernel Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Completely R e g u l a r Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic Functional Calculus for Several C o m m u t a t i v e Algebra Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w 6. Shilov I d e m p o t e n t T h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter w 1. w 2. w 3. w 4.

222 227 234 239 253

Algebras

Spectral P r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Resolvent Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Pseudo-Resolvent Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G e l f a n d - M a z u r and O t h e r Similar Theorems . . . . . . . . . . . . . . . . . . . . . Turpin's T h e o r e m on Locally Convex Algebras . . . . . . . . . . . . . . . . . . .

Chapter w 1. w 2. w 3. w 4. w 5. w 6.

Analysis

Vector-valued Differentiability and Analyticity . . . . . . . . . . . . . . . . . . . E x p o n e n t i a l and Logarithmic Vector Functions . . . . . . . . . . . . . . . . . . . Square Roots and Quasi-square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . C o m p l e x Vector-valued Line Integrals and Cauchy's T h e o r e m s . . . Power Series O p e r a t i o n s in Topological Algebras . . . . . . . . . . . . . . . . .

Chapter w w w w w w

5: S o m e

9: N o r m U n i q u e n e s s

353 361 370 377 388 403

Theorems

N o r m - u n i q u e n e s s T h e o r e m of Gelfand . . . . . . . . . . . . . . . . . . . . . . . . . . . Rickart Seperating Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N o r m - U n i q u e n e s s T h e o r e m s for N o n - c o m m u t a t i v e Algebras . . . . . .

411 412 416 419

Appendix

427 429 431 435 443 445

........................................................

Type C h a r t Bibliography

...................................................... .....................................................

Index ............................................................ L i s t of S p e c i a l S y m b o l s

...........................................

List of Special Abbreviation8

......................................

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 Banach 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 A1 a unital algebra. The spectrum with respect to A1 is denoted by hA1 , and set for x in A , a '(x) - a f t ( x ) aAl(X ) ( t h e spectrum of x in A1), and call a '(x) the quasi-spectrum of x. If A is itself unital the spectrum a(x) -= OA(X) make sense and we have the simple relation a '(x) = a(x)I,.J {0}. It is to be noted that always 0 e a '(x) so that a~(x) r 0. For a real algebra A we have, for x in A, besides a ' ( x ) , a ( x ) also the extended quasi-spectrum J '(x) and extended spectrum J(x) (when A is unital)which are defined in the following way. Every real algebra A has a complexification, a complex algebra A. Set ~" '(x) - a ~ ( x ) and ~(x) - a3.(x ) (note ~

I

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.

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, p-seminorm, p-norm. These give rise to different types of topological algebras: quarter-normed algebras, (F) algebras, p-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 p-convexity. This concept leads to some gener-

xi alizations of locally convex spaces and algebras giving rise to; locally p-convex, locally pseudo-convex spaces and algebras; locally sm. p-convex, locally sm. pseudo-convex algebras; pseudo Fr~chet 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 p-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 p-seminorm. In chapter 5 some differential and integral analysis involving vector valued functions is developed. The extensions to p-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 G B algebras i.e. algebras in which the spectral radius formulae of Gelfand and Beurling (in a modified form) hold. The proof of the Beurling-

xii Gelfand-Zelazko theorem that every complex p-Banach algebra is a G B 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 p-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 p-Banach algebras and an extended version for pseudo-Michael algebras. Chapter 8 deals with function algebras, Shilov boundary, hullkernel topology, completely regular algebras, holomorphic functional calculus for several elements of a commutative p-Banach algebra and Shilov idempotent theorem. Finally, in chapter 9 an exposition of the norm uniqueness theorems of Gelfand and Johnson (extended to p-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

w 1.

Some

Basic

PRELIMINARIES

Concepts

and

Results

1 . 1 . 1 . Recall t h a t in a ring R an element el (respy. t er) is called a left (respy. right) unity if eza = a (respy. aer = a) for all a in R. If e is both a 1. ( = l e f t ) u n i t y and a r. ( = r i g h t ) unity then e is called a unity. If R has a unity, R is called unital. We always assume t h a t the unity e # 0. The meaning of a positive power a m of an element a is clear. Also, in a ring R with unity e, we define for any element a, a ~ - e. This is well-defined since unity is unique (see 1.1.2). 1 . 1 . 2 . LEMMA. I f R has a I. unity ez and a r. u n i t y e r then necessarily el - er - e (say) and e is a u n i t y of R which is moreover unique.

PROOF. clear.

el-

(et)e~ -- e l ( e ~ ) -

e~. The uniqueness of e is

1 . 1 . 3 . Let R be unital with unity e. An element a~-i (respy. a~-1) is called a I. (respy. r.) inverse or l.i. (respy. r.i.) of an element a if a [ l a e (respy. a a r 1 - e). An element a -1 is called an inverse of a if it is both a 1.i. and a r.i. If the inverse a -1 of a exits we call a invertible or regular. 1 . 1 . 4 . LEMMA. ( a ) I f a has a l . i . a l I and a r . i . a r 1 then az 1 _ a r l _ a - 1 (say) and a -1 is the unique inverse of a. In particular, the inverse of an element, w h e n e v e r it exists, is unique.

(b) The invertible e l e m e n t s of a unital ring R f o r m a group Gi - G i ( R ) under multiplication. (c) I f a e Gi then - a e Gi a n d ( - a ) -1 - - a -1. PROOF. (a) a / 1 -- a ~ l e -1 ar .

t respy.-- respectively.

a l l ( e a r 1) -- (a~la)a-~ 1 -- ear 1 -

Algebraic Preliminaries

(b) It suffices to observe that if a, b are invertible then b - l a -1 is the inverse of ab and a is the inverse of a -1. (c) This is an immediate consequence of the identity x y =

(x, v

R).

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.

ba-

PROOF. Suppose that a has a unique 1. inverse b, so that e. Then (ab - e + b)a - aba - a + ba - a - a + e - e.

By uniqueness of 1. inverse, a b - e by 1.1.4 (a), a is ivertible.

+ b-

b or a b -

e. Therefore,

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 ~ Akxk, where )~k E F and (each) xk a finite product of elements from S (clearly this set is a subalgebra A ( S ) containing S and every subalgebra containing S contains A ( S ) ) . If A has a unity e then we have also the smallest subalgebra AI(S) containing S and e. We have AI(S) - Fe + A ( S ) - {~e + x " ~ E F , x E A(S)}. If A is only a unital ring then we have the analogous subrings A ( S ) and AI(S). Here A I ( S ) - 7/e + A ( S ) . A subalgebra (respy. subring) A0 of a unital algebra (respy. ring) A is called a subunital algebra (respy. subunital ring) of A if the unity e(of A ) c A0; 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. A0 can have a unity e0 ~= e). For example in the algebra A of 2 • 2 diagonal matrices over a field F the subalgebra A0 comprising diagonal matrices with second entry 0 is unital but not subunital ( e - diag (1, 1),

w 1. Some Basic Concepts and Results

e0-- diag (1,0)). 1.1.7. Let A be an algebra (or a ring). For a,b E A we write a ~ b if a b - b a and say t h a t a and b commute. For two c o m m u t i n g elements a, b we have the Binomial Theorem.

(a + b) '~ --

a'~-kb k, where (~) -- k,(=-k),, n

(.)

k=0

a positive integer. (This can be proved by induction as in the classical case of the theorem, see [13, p.52].) For two subsets $1,$2 of A we write $1 ~ $2 if for every a c $1, b E $2 we have a +-~ b. We also write for an element a E A a n d a s u b s e t S of A, a ~ S if a~-~b, for every b E S . If S ~-~ S we say that the (subset) S is commutative. For a subset S of A we set S ~ = {x C A : x ~ S} and call S I the c o m m u t a n t (or centralizer) of S in A. S" = (S~) ~ is called the double c o m m u t a n t of S. 1.1.8. LEMMA. Let S be a subset of an algebra (or ring) A. Then (i) S' is a subalgebra (subring) of A. (ii) S c_ S" and consequently A ( S ) unital, A I ( S ) C S".

C_ S";

also when A

is

(iii) If S C T C A, where T is a subset, then T I C_ S I. (iv) S ' -

S'".

(v) S is commutative ifft

S C S I iff S" is commutative.

(vi) If S is commutative so are the subalgebras A ( S ) , S " . (vii) If S is a maximal commutative subset then S -

S".

PROOF. (i), (ii), (iii) an clear. By (ii)we have S ' _ (S')" S'". On the other hand, since S _c S" (by (ii)), we get; using (iii), S'" _ S'. Hence S ' - S'" which is (iv). For (v), we observe t h a t t iff-

if and only if.

Algebraic Preliminaries "S is commutative" iff S c_ S'. If S _c S' then by (iii), (iv),

S" C_ S ~ C_ (S") ~, so t h a t S" is commutative. On the other h a n d "S" commutative" trivially implies "S commutative" (since S _ S"). For (vi), we note t h a t by (v), S" is c o m m u t a t i v e and so also

A(S) c_ S" (see (ii)). Finally, for (vii), we have by (vi), S C_ S" and S" is commutative by (vi). The maximality of S implies t h a t S - S". 1.1.9. PROPOSITION. Every commutative subset S of an algebra A is contained in a maximal{ commutative subalgebra Am(S) of A such that

s _c a ( s ) c

s"c_ a = ( 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 c A=(s) PROOF. By 1.1.8 (vi), A ( S ) , S " are c o m m u t a t i v e subalgebras, and S c_ A(S) c S". Since the union of any linearly ordered (with respect to inclusion) family of c o m m u t a t i v e subalgebras is a c o m m u t a t i v e subalgebra we can apply Zorn's lemma to obtain a m a x i m a l c o m m u t a t i v e subalgebra Am(S) D S". If A has unity e then $1 - S O{e} is c o m m u t a t i v e and hence

Am(Sl) ~ Am(S)~ S. By maximality of Am(&) -

Am(S)

we

have

Am(S1)

--

Am(S),

e

E

1 . 1 . 1 0 . Let A,A* be two algebras (over the same field F). A m a p p i n g ~ 9A ~ A* is called a homomorphism if it is linear, and multiplicative, i.e. p(ab) = ~o(a)p(b) for all a,b e A. An injective or 1 - 1 h o m o m o r p h i s m ~ is called a monomorphism and a surjective or onto h o m o m o r p h i s m is called an epimorphism. A h o m o m o r p h i s m ~o which is both 1 - 1 and onto is called an t i.e. if $1 is a commutative subMgebra of A with Am(S) C_$1 then

S,-Am(S).

w 1. Some Basic Concepts and Results 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 A1, A2 (over F) we have the direct product algebra A - A1 x A2" A-

E Al,a2 c A2}.

{(al, a 2 ) ' a l

The algebra operations on A are given by"

(al, a2) + (bl, b2)

-

(al + hi, a2 + b2)

A(al, a2)

-

(Aal, Aa2)

(ai, a2)(bl, 52)

-

(aibi, a252)

where al, 51 E A1; a2, b2 E A2; )~ E F. 1.1.12. An algebra A (over F) can be extended canonically into a unital algebra A1 called the unitization of A. We take for A1 the cartesian product linear space (over F) given by" A 1 -

F x A-

{(,k, a)" )~ E F, a E A}.

Write (1, 0) - el, (0, a) - h. Then" ()~, a) -- )ke1 -~- (2, (12, b) - 12el -[- b. Define multiplication in A1 by" +

+ b) -

+

+ , a + a/,.

It is easy to check that under this multiplication A1 is an algebra and that the map a ~-, a is a monomorphism. We identify h with a and write. ()~, a) = ,~el + a, so that A1 = Fel + A. We observe that A as a linear subspace of A1 has codim t 1. Further it is clear from the definition of multiplication in A1 that A is both a left as well as a right ideal of A1. If B is a subalgebra of a unital A with its unity e ~ B then B1 = Fe + B is unital subalgebra of A called the unitization of t codim- codimension- dim(AliA).

Algebraic Preliminaries BinA. We note that the above construction for A1 can be carried out even when A has a unity e; of course el =fi e (since el r A). Finally, we remark that a ring R has a unitization R1. It is given by RI-~ t xR-7/el+R with (me1 -4- a)(nel + b) - m n e l + mb -4- na + ab (m, n E -~). 1.1.13. Let R be a ring and ' o ' denote the binary operation, called circle operation, on R given by

hob-

a+b+ab

(a, b E R).

We denote R with this binary (multiplication) operation by So So (R); So is a multiplicative system. 1.1.14. LEMMA. Let R be unital, with unity e, and So as defined above. Denote by S - S ( R ) the underlying multiplicative semi-group structure of R. Then the map. r-

r(S)

9x E S ~

x-

e E So

is an isomorphism of S onto So (as multiplicative structures). Hence So is a semi-group with '0 ' as the identity element. PROOF. Clearly r is bijective. Further

7(a) ov(b) - ( a - e) o ( b - e) - a b - e - 7(ab). 1.1.15. COROLLARY. Let R be any ring. Then So - - S o ( R ) is a semi-group with '0' as identity. PROOF. Consider the unitization R1 of R. Then el + R = {el + x : x c R}, under multiplication, is a subsemi-group S* of S(R1). By considering the restriction of the isomorphism (see 1.1.14), r : S(R1) --+ So(R1) to S* we get the desired conclusion for So. 1.1.16. If a o b = 0 then a is called a l.q.i.(=left quasi2 - {0,+1,+2,...} (the ring of integers).

w 1. S o m e Basic Concepts and Results

inverse) of b and b a r.q.i. (=right quasi-inverse) of a; also then b (respy. a ) i s said to be l.q. (respy. r . q . ) i n v e r t i b l e . If a is b o t h 1.q. invertible and r.q. invertible it is called q. invertible or q. regular (the m e a n i n g s of 1.q. regular and r.q. regular are clear). !

1 . 1 . 1 7 . LEMMA. (a) If a E R has a l.q.i, a~l and a r.q.i, a r then a~l- a ir - a I (say) and a I is q.i. of a. Moreover, a ~ a'. (b) The set Gq of q. invertible e l e m e n t s of R is a group under the multiplication 'o '; Gq is a subsemi-group of So. PROOF. (i) T h e proof of the first s t a t e m e n t is similar to t h a t of 1.1.4 (a); for the second we note t h a t a o a ~ - 0 - a ~ o a =2z aa ~ - a~a.

(ii) The proof is similar to t h a t of 1.1.4 (b). 1.1.18. LEMMA. Suppose that a is an invertible (respy. q. invertible) e l e m e n t of a unital ring (respy. ring) R , x E R and x ~ a. T h e n x ~ a -1 (respy. x ~ al). In particular a -1 (respy. a') e {a}" (the double c o m m u t a n t ) .

a -1

PROOF. xa - 1 ( a - l a ) x a -1 -- a - l ( a x ) a - 1 - x. Similarly, x o a I - a I o x.

a-l(xa)a -1-

1.1.19. COROLLARY. Let A m be a m a x i m a l , c o m m u t a tive subalgebra of an algebra A. I f b E A m is invertible (respy. q. invertible) in A then b-1 (respy. b') E A m . PROOF. By 1.1.18, b -1 ~-~ A m and so S A m U{b - 1 } is c o m m u t a t i v e . By 1.1.8 (vi), the s u b a l g e b r a A ( S ) is c o m m u t a t i v e . Since A ( S ) D_ S D A m , b y m a x i m a l i t y o f A m , A m - S - A ( S ) , so t h a t b -1 C S C A m . Similarly b~ c Am. 1 . 1 . 2 0 . LEMMA. Let R be a unital ring with unity e. T h e n an e l e m e n t a of R has a l.q.i. (respy. r.q.i.) b iff e 4- a has e + b 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 I, a I being the q.i. of a. Moreover, 7 -1 " a ~ e § a is an i s o m o r p h i s m of Gq onto Gi. PROOF. The s t a t e m e n t s follow from the identity

Algebraic Preliminaries

(e + a)(e § b) -= e § (a o b).

1.1.21. LEMMA. If A1 is the unitization of an algebra (or ring) A and a E A has a l.q.i. (or a r.q.i.) bl in A1 then bl E A. Hence a is q. invertible in A1 iff its q. invertible in A. PROOF. Write b l - - C ~ e l + b , where b E A and c~E F or 7/ according as A is an algebra over F or a ring. Then a o bl -- O :~ a + o~el + b + o~a + ab = O =~ o~ = O =:~.bl = b.

1.1.22. R e m a r k . Related to o-operation in R is the operation denoted by x and given by a x b = a + b - a b (a, b E R). This operation is again associative and has other similar properties of o. If S• denotes the multiplicative system in R corresponding to x, then the map a H - a is easily seen to be an isomorphism of So onto S• The operations o, x are mutually connected by: aob=

( a x -b); a x b =

(ao-b).

The operation x has been introduced by Hille (following a suggestion of J acobson) 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 . 2 3 . LEMMA. Every np. (=nilpotent) element a of a ring R is q. invertible with

a' - - a § a 2 + . . . - 4 - a k-1 (a k - O ,

ak-17s

(.)

Further, if R is unital with unity e, then e + a is invertible with

(e § a) -1 - a - a 2 + . . . - + - a k-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 § 1.1.24. LEMMA. If ~ is a h o m o m o r p h i s m of a ring R then 99 preserves o-operation and hence also I. (or r.) q. invertibility.

w 1. Some Basic Concepts and Results If R is unital with unity e then ~(e) is the unity of ~(R) preserves I. or r. invertibility.

and

PROOF.

~ ( a o b) - ~(a + b + ab)

--

~9(a) + ~(b) + ~(a)~a(b)

=

p(a) o 99(b) (a. b E R).

F u r t h e r , since p ( 0 ) - 0, a o b -- 0 ::~ ~p(a)o ~p(b) - 0, completing the proof of the first s t a t e m e n t . The second s t a t e m e n t is an i m m e d i a t e consequence of ~p being a h o m o m o r p h i s m . 1 . 1 . 2 5 . LEMMA. In a ring, if a o b is l.q. (respy. r.q.) invertible then b - O.

(respy. b o a ) - a

and a

PROOF. Suppose t h a t a o b -- a. T h e n b-0ob-

(a~zoa) o b - a

lo(aob)-a

loa-0.

Similarly, b o a - a ==~ b - 0. 1 . 1 . 2 6 . LEMMA. Let A be an algebra and u E A an idempotent. Then" (i) For any A # - 1 , A u is q. invertible with (Au)' - - A ( 1 + A ) - l u . In particular u ' - - 8 9 Also, when e ezists, e ' 1 2 e.

(ii) If - u is q. invertible then u is not q. invertible.

O. Hence, in a unital A,

-e

(iii) If A has unity e then e + u is invertible with (e + u) - 1 -_

a, e

_

#

0).

PROOF. (i) A u - A(1 + ) ~ ) - l t l - A2(1 + )~)-ltI -- O, whence (Au)' -- --A(1 + A ) - I u . (ii) ( - u ) o u - - u + u - u 2--u. Hence, by 1.1.25, u - 0 . (iii) (e + u) 1 _ e + u' - e - ~u. Since (Ae)(A-le) - e and Ae +-~ A - l e , (Ae) -1 -- A-le. -

1.1.27. tities:

1

PROPOSITION. (a) In any ring R we have the iden-

(i) ba o ( - b a - bxa) - - b ( a b o x)a

10

Algebraic P r e l i m i n a r i e s

(ii) ( - b a - bxa) o ba - - b ( x o ab)a (iii) - a 2 o b - a o ( - a o b ) (iv) b o - a 2 - ( b o - a ) o a where a, b, x E R.

(b) In a u n i t a l ring R with u n i t y e we have: (v) (bxa + e)(e - ba) -- bx(e - ab)a + e - be (vi) ( e - b a ) ( b x a + e ) - b ( e - a b ) x a + e - b a where a, b, x E R.

PROOF. (i) LHSt - b a + ( - b e - b x a ) + b a ( - b a - b x a ) = - b ( x -4- ab + a b x ) a - - b ( a b o x ) a -

RHS t

(ii) Similar to (i). (iii) a + ( - a o b ) + a ( - a o b) RHS a - a + b - ab + a ( - a -a2

t b-

a2b--a

+ b - ab)

2 o b-

LHS.

(iv) Similar to (iii). (v) L H S - bx(a - aba) + e - ba - b x ( e - ab)a + e - ba (vi) Similar to (v).

RHS.

1.1.28. COROLLARY. (i) ab 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. in vertible. (ii) e - ab is I. (respy. r.) invertible iff e - be is I. (respy. r.) invertible. In particular, e - ab is invertible iff e - ba is invertible. PROOF. ( i ) B y taking x - (ab)~z (respy. (ab)~r) in (ii)(respy. (i)) we conclude that "ab is 1.q. invertible (respy. r.q. invertible)" ::> " ba is 1.q. invertible (respy. r.q. invertible)". This plus symmetry consideration proves (i). (ii) Taking x - ( e - a b ) ~ 1 (respy. ( e - a b ) r I 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.

w 1. Some Basic Concepts and Results

11

1 . 1 . 2 9 . LEMMA. In a unital ring R with unity e, if ab has a r.i. (respy. l.i.) c then be (respy. ca) is a r.i. (respy. l.i.) of a (respy. b). Similarly, in any ring R, if a o b has a r.q.i. (r py. l q.i. ) c b o (r py. o a) a r.q.i. (r py. l.q.i.) o/ a (respy.) b. PROOF. Clear. 1 . 1 . 3 0 . 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, a o b is q. invertible iff a,b are q. invertible. PROOF. If a, b an invertible then as already seen, ab is invertible with b - l a -1 as its inverse. On the other hand if ab, ba are invertible t h a n by applying 1.1.29, 1.1.4 (a) we conclude t h a t a,b are invertible. Hence the s t a t e m e n t s concerning invertible elements. The proof of the s t a t e m e n t s concerning q. invertible elements is similar. 1 . 1 . 3 1 . PROPOSITION. Let R be a ring and x , a E R. If x ~,

(a § x'a)ll and (a § ax')lr exist then (x § a)' exists and (x § a)' - (a § x'a)'l o x' - x ' o (a + ax)'r. PROOF. We have

( a + x a )I l o Ix '

O

(x+a)

-

(a § x'a)ll o ( x ' + x + a + x'x + x'a)

--

(a + x'a)ll o (x' o x § a + x'a)

--

(a § x'a)' l o (a § x'a) - O (since x ' o x -- 0)

Similarly,

(x § a) o x' o (a § ax')'r

--

(x § a § x' § xx' + ax') o (a + ax)'7"

--

(a § ax') o (a § ax)' r

--0

It follows t h a t x § a is b o t h 1.q. invertible and r.q. invertible and so q. invertible. The required conclusions are now clear. 1 . 1 . 3 2 . COROLLARY. With the hypothesis in 1.1.31 we have

(x § a)' - x' - (a § x'a)'l § (a § x'a)'lx' - (a § x'a)'r § x'(a + x'a)'r.

12

Algebraic P r e l i m i n a r i e s

1 . 1 . 3 3 . COROLLARY. I f x +-+ a a n d x ~, (a + axe) ~ exist, t h e n (x -4- a) I exists with (x § a)' - x' o (a + a x ' ) ' - (a + ax')' o x'. PROOF. Since x ~ a, by 1.1.18, x ~ ~ a, so t h a t (a § x'a)~z - (a § ax')~z - (a § ax')~r - (a + x'a)~r. T h e required result now follows by 1.1.31.

w 2.

Ideals

and

Radical

1 . 2 . 1 . Let R be a ring. If I is b o t h a 1.(-left) ideal and a r . ( - r i g h t ) ideal of R it is called a bi-ideal. We will use the t e r m ideal for d e n o t i n g any one of these: a 1. ideal, a r. ideal or a biideal. W h e n R is c o m m u t a t i v e or a n t i - c o m m u t a t i v e , t h e r e is, of course, no distinction between a 1. or a r. ideal and the t e r m ideal has an u n a m b i g u o u s m e a n i n g . If a ring R is an algebra (over a field F ) t h e n 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. T h u s , if the 1-dim real linear space R is given the trivial a l g e b r a s t r u c t u r e (~2 t - {0}) the subset ~ of ~ is a ring ideal which is not an a l g e b r a ideal. This distinction vanishes for a u n i t a l a l g e b r a A since we have the relation )~a = ( ~ e ) a (~ E F; a E A; e - - u n i t y of A ). Let R be a ring (or algebra). Recall t h a t an element a =/: 0 of R is called a l . z . d . ( = l e f t zero divisor) if there is an element b J= 0 with ab = 0. Similarly, a ~= 0 is a r . z . d . ( = r i g h t zero divisor) if t h e r e is a c # 0 with ca - O. Given a subset S of R the left a n n i h i l a t o r a n n i h i l a t o r ~ r ( S ) are defined by: .~l(S) -- { x e R : xa = O,V a e S } ~ r ( S ) = {y E R : ay - 0,V a E S}.

Az(S)

and right

t In a ring R, for two subsets S1,Se we write $1S2 - {xy 9x E S I , y E $2}; also for a subset S , S ~ - SS.

13

w2. Ideals and Radical

{a} then we write ~t(a),.4r(a) for the corresponding annihilator ideals. If S = R we denote the two annihilators by ~l and At. A ring R (respy. algebra A) is called a ring (respy. algebra) with trivial multiplication if R 2 (respy. A 2) - {0}. If s

-

1.2.4. LEMMA. ( i ) ~ l ( S ) ( r e s p y .

~4r(S)is a l.(respy, r.)ideal

of R. (ii) If I is a I. (respy. r.) ideal of R then Az(I) (respy. A~(I))

is a hi-ideal. (iii) Mr, Mr are nilpotent bi-ideals: A 2 _ ~ 2 _ {0}. PROOF. (i) Clear. (ii) Let I b e a l . ideal, x e ~ z ( I ) , y E A a n d a e I . Then x y . a - - x . y a - - O , xy C ~z(I), proving Az(I) is a bi-ideal. Similarly, A~(I) is a bi-ideal. (iii) If a,b e ~z then ab e a A {0}, so that A~ - {0}. Similarly, M ~ - {0}. 1.2.5. PROPOSITION. If a ring R (respy. algebra A ) ~ {0} has no l.-ideals # R,{0} (respy. no 1.(algebra)ideals # A,{0} then R (respy. A) ist 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 1). If R (respy. A) is unital then R (reA) a ivi io ring (r py. algebra) PROOF.

Assume that

R (respy. A) has a 1.z.d. a, so that

ab = 0 with a J: 0, b ~: 0. Then Ii = ~z(b) ~: {0} so that the hypothesis implies that Il = R (respy. A). Thus

nb = {0}

(respy. A b -

(0}).

Since R (respy. A) is a r. ideal its right annihilator I is a biideal with b c I, and since I -7(= {0} we conclude that I = R (respy. A). Therefore R 2 (respy. A 2) - {0}. It follows that the additive subgroup of R (respy. subspace of A) is cyclic of prime order (respy. of dim 1). t The same conclusions hold if a similar condition is put on r. ideals instead of 1. ideals.

14

Algebraic Preliminaries

Next suppose t h a t R (respy. A ) has no 1.z.d., so t h a t R* = R \ { 0 } (respy. A* - A \ { 0 } ) t is closed under multiplication. If a c R* (respy. a e A * ) then R e (respy. A a ) i s a l . ideal ~={0}. Hence, by our hypothesis, Ra-

R ( respy. A a -

A)

(,)

It follows t h a t there is an element e ~= 0 with ea - a. T h e n (e 2 - e)a - a - a 0, so t h a t e 2 - e since R (respy. A ) has no 1.z.d.. For a r b i t r a r y x E R* (respy. A* ), e(ex-x)ex-ex0 and hence e x x, so t h a t e is a 1. unity of R* (respy. A* ). Again, by (.), there is an element b c R* (respy. b C A* ) such t h a t ba = e, so t h a t a has a 1. inverse b. It follows t h a t R* (respy. A*) is a group tt and R (respy. A ) is a division ring (respy. algebra). Finally, if R (respy. A ) is unital then clearly R 2 (respy. A 2) =/= {0} and R (respy. A ) i s 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 m o d u l a r if there is an element u in A such that x u - x (respy. u x - x) E I for all x E A.

T h e condition can be expressed briefly by writing A ( u - 1) (respy. ( u - 1)A)_C I. T h e element u is called a relative r. unity (respy. relative I. unity) for I; u is not unique (see 1.2.8). If u is b o t h a relative 1. unity and a relative r. unity it is called a relative b i - u n i t y or just relative u n i t y . 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. W h e n A is c o m m u t a t i v e , relative r. unity and 1. unity concepts coincide and relative unity has an u n a m b i g u o u s meaning. 1 . 2 . 7 . R e m a r k . In a unital A every ideal I is regular, with unity e as a relative unity (since x e - x = ex - x = 0 E I ). On t 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 . tt A semi-group with a 1. unity in which every element has a 1. inverse is a group.

15

w 2. Ideals a n d R a d i c a l

the other h a n d if A has trivial multiplication then no ideal ~ A is regular (since x u - x, u x - x - - x for all x.) In the ring of even integer 27/, the (principal) ideal 67/is regular (with relative unity u - 4) while the ideal 4?7 is not regular. 1 . 2 . 8 . PROPOSITION. (a) I f u is a relative r. (respy. l . ) u n i t y f o r a regular I. (respy. r.) ideal I a n d a E I then u + a is a relative r. (respy. I.) u n i t y f o r I. (b) Let u, v be relative r. (respy. l.) u n i t i e s f o r I. T h e n u v , a n d hence u ~ (n - 1, 2 , - . . ) , are relative r. (respy. l.) u n i t i e s flor I. F u r t h e r , x ( u - v ) ( r e s p y . (u - v ) x ) C I

(x E A ) .

(c) For a regular bi-ideal I, every relative r. or I. u n i t y f o r I is a relative (hi-} u n i t y . M o r e o v e r , i f u, v are relative u n i t i e s t h e n u-v c I. F i n a l l y , u is a relative u n i t y f o r I i f f u # - u + I is a u n i t y of the q u o t i e n t A # - A / I .

PROOF. (a) Let I be a 1. ideal. T h e n x ( u + a) - x -- ( x u -

x) + x a E I + I - -

I

(x E A ) .

The proof w h e n I is a r. ideal is similar. (b) A s s u m e t h a t I is a 1. ideal. Since x u v - x -- x u . v - x u + x u - x E I +

I--

I

u v is a relative r. unity for I. Taking v - u and using induction we get u ~ is a relative r. unit for I. Further,

The c o r r e s p o n d i n g results when I is a r. ideal are proved similarly. (c) Suppose t h a t u (respy. v ) i s a relative r. (respy. 1.)unity for I. T h e n u - v - ( v u - v) - ( v u - u) E I + I -

I.

Therefore, by (a), u - v + (u - v) is a relative 1. unity.

16

Algebraic Preliminaries

Similarly, xEA, xu-

v is a relative r. unity.

Finally, observe t h a t for

x, u x -

x#

x E I iff x # u

#

-

u#x

#

-

x#

where x ~ x # is the canonical quotient h o m o m o r p h i s m . 1.2.9. LEMMA. Let A be an algebra (or a ring) and I regular I. (respy. r.)ideal with relative unity u. Then: (i) If u C I then I (ii) If I # A then - u

a

A is not l.q. (respy. r.q.) invertible

(iii) A n y I. (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 I is a 1. ideal. (i) Since u E I , xuEI§

for any x C A , x u E I whence I - A .

(ii) Suppose that - u

and hence x - x - x u §

is 1.q. invertible with a as its 1.q.i.. Then

a-u-au - O, so t h a t u I - A - a contradiction.

a-au

E I, and so by (i),

(iii) Obvious. 1 . 2 . 1 0 . LEMMA (Krull). Let I ~ A be a regular ideal. Then there is a m a x i m a l regular ideal M of the same type (l., r. or bi-) as I with I C M. PROOF. If u is a relative unity for I, Apply Zorn's lemma, to the poset t of ideals as I) # A, to obtain M (note t h a t if {I a} in the poset then u r U I s , since u ~ any

by 1.2.9(i), u r I. (all of the same type is any chain of ideals Is).

1 . 2 . 1 1 . COROLLARY. In a unital A contained in a m a x i m a l ideal.

every ideal ~ A

1 . 2 . 1 2 . LEMMA. Let A be unital with unity e. Then:

t For definition see 2.1.1.

is

w 2. Ideals and Radical

17

(i) I f I r A is an ideal then e ~ I. (ii) I f Iz (respy. I~) is a I. (respy. r.) ideal r A then Il (repy. ca, not contain any I. (r py. r.) in rtibl of A.

(iii) A a - A

(respy. a A -

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 c Iz is 1. invertible then e - a z l a E It, so t h a t Iz - A. Similarly I~ -- A if I~ contains a r. invertible element. (iii) If A a = A then there is a b c A with ba = e, whence a is 1. invertible. Conversely, if a is 1. invertible then since a E A a we m u s t have A a = A by (ii). Similarly we can prove a A = A iff a is r. invertible. 1.2.13. LEMMA. In a unital A an e l e m e n t a is l.(respy, r.)invertible iff a is not contained in any I. ideal (respy. r. ideal) 7~ A.

PROOF. This is o b t a i n e d by combining (ii), (iii) of 1.2.12 (noting t h a t A a (respy. a A ) is a 1. (respy. r . ) i d e a l ) . 1 . 2 . 1 4 . COROLLARY. A n e l e m e n t a E A is l.(respy, r.) invertible iff a does not belong to any m a x i m a l 1. (respy. r.) ideal of A. In particular, when A is c o m m u t a t i v e , a is invertible iff it does not belong to any m a x i m a l ideal. PROOF. This follows from 1.2.13, 1.2.10. 1 . 2 . 1 5 . LEMMA. (a) I f I and g a regular bi-ideal of A ideal of A (b) I f I is a hi-ideal of A J is a hi-ideal of A. Further,

is a regular I. (respy. r . ) i d e a l of A then I ~ J is a regular I. (respy. r.) and J a regular hi-ideal of I then J is regular if I is regular.

PROOF. (a) Let I 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 c A, where x w - x = x u - x - x ( v u - v) E I § I - - I.

(1)

18

Algebraic Preliminaries

Also, x w -- x = xv - x - (xv - x ) u e J + J u = J.

(2)

It follows from (1), (2) that I A 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 E A then u x c I and hence, if a E J then u x a E J. Also xa E I, so that u . x a - xa E J. Therefore xa = u x a - ( u x a - xa) C 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)uEJ (since x v - x E I ) . Thus J is regular with relative unity w. 1.2.16. LEMMA. Let ~ " A--~ A* be an e p i m o r p h i s m Then: (i) For any ideal I of A , I * = ~ ( I )

is an ideal of A*

of the

s a m e type (l., r. or hi.) as I.

(ii) I f I is regular with relative unity u then I* is regular with relative unity u* = ~ ( u ) .

(i) Clear. (ii) The regularity of I* follows from the identity ~ ( x ) ~ ( u ) ~(x) = ~(xux) when I is a 1. ideal and from the analogous identity when I is a r. ideal. PROOF.

1.2.17. LEMMA. Let 9a : A ~ necessarily e p i m o r p h i s m ). Then: (i) For an ideal I* of A * , I -

A* be a h o m o m o r p h i s m (not

! p - l ( I * ) i s an ideal of A of the

s a m e type.

(ii) If I* is regular with a relative unity u* belonging to ~ ( A ) , then I is regular with any u E ~ - l ( u * ) as a relative unity. (iii) I f p is an e p i m o r p h i s m and M* is a m a x i m a l regular ideal of A* then M -

~-I(M,) is

a m a x i m a l regular ideal of A.

PROOF. (i), (ii): Clear. (iii) It is enough to observe that the map

I* ~--~ ~9-1(/*)

(I* an ideal of A*)

19

w 2. Ideals and Radical

is a bijection between the ideals of A* and the ideals of A containing ker ~. 1.2.18. R e m a r k . The condition "u* E ~ ( A ) " in ( i i ) o f 1.2.17 cannot be omitted. This is shown by the following counterexample. A is an algebra (over F) with trivial multiplication and A1 its unitization and ~ the natural injection A ~ A1. If a c A and I - - F a then I is an ideal of both A1 and A. As an ideal of A1, I is regular (since A1 is unital) but as an ideal of A it is not regular (see 1.2.7). 1.2.19. Write

LEMMA. Let A be an algebra (or ring) and a E A.

II(a) = {xa q- x : x e A}, It(a) = {ax + x : x e A}. Then Ii(a) (respy. It(a)) is a regular I. (respy. r.) ideal with - a as a relative r. (respy. l.) unity. PROOF. It is easily seen that I t ( a ) i s a 1. ideal and I t ( a ) i s a r. ideal. Further, since x ( - a ) - x = ( - x ) a - x, the regularity for Iz(a) follows. The regularity for I~(a) also follows similarly. 1.2.20. COROLLARY. a E A is 1.q. (respy. r.q.) invertible iff IL(a) (respy. I t ( a ) ) : A. PROOF. If It(a) -- A, then since - a E Iz(a) there is a b E A with

- a - ba § b, whence boa - O, so that Conversely, if a has a 1.q.i.b. then we have -a=ba+bcIt(a), so that, by 1.2.19,

a is 1.q. invertible.

It(a) : A. The proof of the assertion for It(a) 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 Mt (respy. Mr) there is an element b possibly depending on Mt (respy. Mr) such that b o a E Mz (respy. a o b E Mr) (b) a e A is l.q.(respy, r.q.)invertible iff - a is not a relative r. (respy. l.) unity for any maximal regular I. (respy. r.) ideal of A. PROOF.

(a) If a has a 1.q.i. b then b o a : 0 E Mr. Next

20

Algebraic Preliminaries

assume that the stated condition is satisfied for all Ml. If a is not 1.q. invertible then by 1.2.20, Iz(a) ~ A. By 1.2.10, there is an Mz with Iz(a) c__ Ml. By our assumption we can find a b with boa c ML. It follows t h a t

- a - b + b a - (boa) E Il(a) + Mt C_ Mz + Mz - Mz. By 1.2.19, - a is a relative unity for Il(a) and so also for Ml. Hence, by 1.2.9(i), Ml = A - which is impossible. So a is 1.q. invertible, as required The proof of the s t a t e m e n t concerning r.q. invertible is similar (b) The statement to be proved for 1.q. invertibility is clearly equivalent to: a is not 1.q. invertible iff - a is a relative r. unity of some maximal regular left ideal Mz. If - a is not 1.q. invertible then by 1.2.20, Iz(a) r A and so Iz(a) c_ (some) Ml. By 1.2.19, - a is a relative unity for Iz(a) and so also for Mz. Conversely, if - a is a relative r. unity for some Mz then by 1.2.9(ii), a = - ( - a ) is not 1.q.i. The proof of the s t a t e m e n t 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. ~r-~). If there are no maximal regular 1. ideal (respy. r. ideal) in A we define x ~ ( ~ ) - A. 1.2.23. LEMMA. ( a ) I f every element 4 a l.(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 C I, let aIz be a l.q.i. Since a I z o a - 0 we have a I - a - ala E I. By our hypothesis, a I has a l.q.i, b" boa I - O . By 1.1.17 (a) - applied to a~z- we get a - b, which implies t h a t a is q. invertible with a ~ - a~z. (b) If ~ : A ~ A* is an epimorphism and I an ideal then by 1.2.16, I* = ~ ( 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) ~

(respy. ~ / ~ ) is a q. invertible

w 2. Ideals and Radical

21

I. (respy. r.) ideal containing every q. invertible I. (respy. r.) ideal of A. In particular, every e l e m e n t of ~ or ~ / A is q. invertible. (b) x ~ x)~x/r~ (say); x / ~ is a hi-ideal containing every q. invertible I. or r. ideal of A. (c) ~ contains every n i l - in particular nilpotent - I. or r. ideal of A. (d) I f u # O is an i d e m p o t e n t of A then u ~ x / ~ . In particular, when A is unital with unity e, e ~ v/A.

PROOF. (a) Since ~ - ~Mz, if a E x ~ C Mz then a o a a + a + a 2 E Mz, whence by 1.1.21 (a), a is 1.q. invertible. So by 1.2.23 (a) r is q. invertible. Let Iz be any q. invertible 1. ideal of A. If Iz g x ~ then there is an Mz with It g Mz, so that we have" (*)Mz + It - A. If u be the relative r. unity for Ml then by (.) we have - u - a + b (a E M z , b E Iz).

Since b C Iz its q.i. b~ exists and therefore u-

-a-b

=

- a - ( - b ' - b'b) - - a + b' + b ' ( - u - a) - a + (b' - b'u) - b'a E Mz,

which is impossible (since Mz # A). Hence Iz _c Mz, as required. The corresponding result for ~ is proved similarly. (b) For a ~ ~ A and x E A we claim that the principal 1. ideal {ax}z - A a x + 2Zax is q. invertible. If y is any element of this ideal then y - bax + n a x - (ba + n a ) x

(b E A, n @ Z).

Since a E ~/-A, yo -- x ( b a + n a ) E ~ and hence y0 is q. invertible. By 1.1.28(i), y is q. invertible, so that the ideal {ax}z is q. invertible. By (a), {ax}~ C_ r In particular a x e C/--A, whence ~ A is a bi-ideal. Similarly, ~r~ is a bi-ideal. Since r (respy. ~ ) is a q.i.1. (respy. r.)ideal we must have by (a), _c ~ (respy. ~ _c ~ ) . Hence ~ ~/-A. (c) By 1.1.23, every nil ideal is q. invertible and so contained in V ~ (by (b)).

22

Algebraic Preliminaries

(d) If u E ~ then - u E v/A, and - u is q. invertible. But then by 1.1.26(ii), u - 0 - a contradiction. 1.2.25.

The bi-ideal v/A is called the J acobson radical or

radical of A. Following Neumark [22,p.173] an element of ~ 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}z = A a + Za (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}t = {a}r is a nil ideal). 1.2.26. ~(v~)

LEMMA.

If ~9 " A ~

A* is an epimorphism then

c vIA *.

PROOF. By 1.1.24, ~(v/A) is a q. invertible bi-ideal of A*, whence by 1.2.24 (b), ~ ( v / A ) _ v/A *. 1.2.27. An algebra A in which ~ A is called a radical algebra. Note that an algebra is a radical algebra iff every element is q. invertible. Any algebra A0, with trivial multiplication (A02 - {0}) is a radical algebra (since every a e A0 is q. invertible with a' = - a ) . An algebra A 7~ {0} is called s.s. (=semi-simple)if x / ~ -

{0} 1.2.28. LEMMA. In a s.s algebra A we have

PROOF. By 1.2.4 (iii), 1.2.24 (c), Az, Ar C_ X / ~ - {0}.

w3.

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 A; X-1 ( 0 ) - - ker X is called the kernel of X. We denote by A -- A(A) the set of all characters of A; A may be empty.

w 3. Characters and H y p e r m a x i m a l ideals

Let X be a linear space over F. A linear m a p r called a linear functional. 1 . 3 . 2 . LEMMA. field F and to: A ~

X=X*

23

X --. F is

Let A , A * be two algebras over the s a m e A* an epimorphism. If X* E A ( A * ) then

o ~ C A(A).

PROOF. Clear. 1 . 3 . 3 . L EMMA. A h o m o m o r p h i s m i ~ it is non-trivial (i.e. X ~ 0 ~ ).

X" A ~ F is a character

PROOF. If X J= 0 there is an element a0 E A with x(a0) r 0. If fl E F t h e n X ( f l ~ - l a 0 ) - fl, proving X is surjective, whence X is a c h a r a c t e r . The converse is trivial. 1 . 3 . 4 . LEMMA. A subspace B is of codim 1 iff it is the kernel r f u n c t i o n a l d2 on A.

of a L S ( - linear space ) A of a non-trivial linear

PROOF. If r y~ 0 there is an a0 E A w i t h r - A ~= 0. Write ) ~ - l a 0 - - u ; then r 1. For x E A we can write x - r where y x-r e kerr B (say). It follows t h a t A - F u q- B, so t h a t c o d i m B - 1. Conversely, if c o d i m B - 1, there is an ao E A \ B such t h a t A - Fao § B. If x E A and x - Aa0 4-b, define r It is clear t h a t r is a linear functional on A with k e r r 1 . 3 . 5 . LEMMA. A regular I. or r. ideal I, of codim 1, of an algebra A is a hi-ideal, which is moreever, m a x i m a l as a I. or a r. ideal, or also, as a hi-ideal. PROOF. Let I be a c o d i m 1 regular 1. ideal with u as a relative r. unity. Since c o d i m I - 1 we have F u q - I - A. If y c A t h e n y - c~u 4- b, with c~ E F , b E I. Therefore, if a E I,

ay - a ( ~ u § b) - ~ ( a u - a)4- ~a q- ab e I, whence I is a r. ideal so a bi-ideal. T h e p r o o f of this w h e n I is a r. ideal is similar. T h e m a x i m a l i t y assertions are i m m e d i a t e consequences of the c o d i m 1 p r o p e r t y of I. t '0' denotes the zero-functional mapping everything to 0.

Algebraic Preliminaries

24

1.3.6. DEFINITION. A codim 1 regular ideal I of A is called a hypermaximal ideal; by 1.3.5 every hypermaximal ideal is a biideal. 1.3.7. If an algebra A has hypermaximal ideals we denote their intersection by ~/-A. We set ~ f A - A if A has no hyperm a x i m a l ideal. We call ~-A the hyper-radical of A, and we say t h a t A is h.s.s. ( - h y p e r semi-simple)if ~ f A - {0}. 1.3.8. L EMMA. x / A _

~/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 x~-A. 1.3.9. PROPOSITION. The hypermaximal ideals of an algebra A are precisely the kernels M• -- k e r x of characters X of A. Moreover, the correspondence X ~-~ M• is a bijection between the set /~ of characters and the set 91I of hypermaximal ideals. PROOF. Since X is a homomorphism, M• is a bi-ideal, and since X is surjective there is a u E A with X(U) = 1. Further, for any x e A, we have ( . ) X ( x u - x ) = X(x)" 1 - X ( x ) = O, whence M x is regular with u as relative unity. Also, by 1.3.4, codim M x = 1, whence M x is hypermaximal. If M is any h y p e r m a x i m a l 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 # C A # then x # = a ~ u # where a x E F , since dim A # = 1. The map X = XM : X ~ a~ is clearly a character of A with ker X = M = M x. Finally, it is clear from the construction of XM from M t h a t M1 - M2 ::> XM, --XM2" Hence X ~ Mx is a bijection. 1 .3 .1 0 . COROLLARY. (i) The relative unities for M - - M x are precisely the elements u in A satisfying X(u) - 1. In particular, if A has unity e then x(e) - 1 and further x(a) ~ 0 for any invertible a. (ii) The set of all relative unities ( - bi-unities) [or M• is the unity coset E of A # - A I M x.

PROOF.

(i) It is clear from the relation (,) in the proof of

1.3.9 t h a t any u with

X(U)-

i is a relative unity for

M - M•

w3. Characters and Hypermaximal ideals

25

Conversely, let v be any relative unity for M. Choose any element ao c A \ M ; then x(ao) ~ O. Since aov - ao e M we get x(ao)X(V)

-

x(ao),

X ( V ) --

1.

If a is any invertible element then

x(a)x(a-1) --x(e)- 1, whence x(a)

~ O.

(ii) Since M - M• 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 X of A vanishes on x/~.

PROOF. By 1.3.8, V ~ __C_~

__ k e r x , hence the result.

1.3.12. Remark. If A is a radical algebra then A ( A ) ~ ) ( i f A - A ( A ) J: 0, X E A then x / ~ _c k e r x c A, so t h a t A is not a radical algebra). Again, for the m a t r i x algebra A - M,~(F)(n > 1 ) , A ( A ) - 0, since A being simple has no h y p e r m a x i m a l ideals. 1.3.13.

(a)

EXAMPLES

OF

HYPERMAXIMAL

IDEALS"

Let G be a finite group of order n and A FIG] the group algebra of G over a field F of characteristic 0. Write a - ~ g , where the sum in A is over all elements g of G; a E A. Clearly, ag - A. for each g E G, whence a 2 - - ha. 1 then u 2 - u . Writing v - e - u , If we write u - ~a, it is easy to see t h a t M Av is h y p e r m a x i m a l (since it has a 1-dim direct s u m m a n d A u - Fu).

(b) Let L I ( G ) be the group algebra of a locally c o m p a c t Hausdorff group G. Consider the functional h(f)-

fc I(t)dt

where the integral is the (left invariant) Haar integral. Clearly A is linear. Further h(f

9 g)

--

If* g(t)dt--f (/f(s)g(s-lt)d8)dt

Algebraic Preliminaries

26

--

f f(8)d8 f g(8-Xt)d(8-xt) - A ( f ) a ( g ) ,

where we have used the left invariance of the Haar measure. Finally, h ~ 0. For, if U is a nucleus t of G with 0 compact, and to E G \ U , we can, by local compactness of G, choose a positive continuous f with f(to) - O, f ( t ) - 1 on U. Then _

_

A(f)-

fo f(t)dt>

fCt)

t-

> 0,

where #~ denotes the left Haar measure of G. Thus h is a character and M - ker h is a h y p e r m a x i m a l ideal ( M is called the a u g m e n t a t i o n ideal). (c) Let F be a field and F s denote the algebra, under pointwise operations, of all F - v a l u e d of functions f = f ( s ) on a set S. If so c S then the m a p Xso : f ~-~ f(so) is easily seen to be a character and M8 ~ - k e r x % - { f e F S ' f ( s o ) 0)} is a h y p e r m a x i m a l ideal. 1 . 3 . 1 4 . LEMMA. Let J be a bi-ideal of A. If M is a hypermaximal ideal of J then M is a hi-ideal of A. If J is regular in A then M is also regular in A. PROOF. These follow ideal is a regular bi-ideal. The first assertion can way. Let )/ be the character of Take u C J with X ( U ) and

from 1.2.15 (b) since a h y p e r m a x i m a l also be proved directly in the following J determined by M " M - ker X. 1. If x E A , a E M _ J then x a E J

x ( x a ) - X ( u ) x ( x a ) - X(uxa) - X ( u x ) x ( a ) - O. Therefore xa E M, similarly ax E M. Thus M is a bi-ideal of A. t for definition see 2.1.3.

27

w 3. Characters and H y p e r m a x i m a l ideals

1.3.15. PROPOSITION. Let A be an algebra and J a hi-ideal of A.

Then:

(i) E v e r y character X of J x(ax)

(x e A,

satisfies the condition

x(xa)

-

a e J).

(ii) X can be uniquely extended to a character ~ of A. PROOF. 1, then M unity. Take M we have

If M - k e r x , and u an element of J with X ( U ) is a hypermaximal ideal of J with u as a relative x E A. Then ux, x u c J and by hypermaximality of the relations.

UX -- m + ~ t t ; x u

-- m 1 -J- ~ t t

(1)

( m , rtz 1 e M )

Then UXU -- m u -F aU2; UXU -- u m l -F ~U 2

(2)

since m u , u m l E M we obtain from (2) - ~ x ( u ~) - x ( ~ u ~) - x ( u = u ) - x ( Z u ~) - / ~ .

So (1) gives x(u=) - ~-

~-

x(=u).

Now extend X on J to ~ on A by setting ~(=) - x(u=)

- x(=u)

(= ~

A).

Clearly ~ is linear. Further

,t(xy)

:

x(u=y)x(u)-

x(u=.yu)-

x(u=)x(yu)

- ~(=);~(y).

x(ux)x(uy)

This proves that ~ is a character of A which is an extension of X. Further, if X, is any character of A extending X then x , (x) - x , ( x ) x ( u ) - x , (~)x, (',.,) - x, (=,.,) - x(=~)

- :~(=),

which shows that the extension f~ is unique. Finally, x(=a)

-

~(=a)

-- :~(=);~(a)

-

;~(a);~(=)

-- ~(a=)

-- x(a=)

28

Algebraic Preliminaries

where x E A , a E J . 1 . 3 . 1 6 . COROLLARY. Every hypermaximal ideal M satisfies the s y m m e t r y condition for x E A, a 6 J, xa

of J

6 M r ax E M .

PROOF. This follows from 1.3.15 (i) since M = k e r x some character X of J.

for

1 . 3 . 1 7 . COROLLARY. Every character X of an algebra A can be extended uniquely to a character X1 of its unitization A1. PROOF. If X1 C A I , X 1 : )~el~-X ()t E F , x E a) then XI(Xl) -A + X(x) is easily checked to yield, a character of ~i'l of A. The uniqueness of Xi follows from 1.3.15 (ii) (taking J = A ) or can also be seen directly using the fact t h a t for any character X~ of AI we have x l ( e l ) = 1. 1 . 3 . 1 8 . LEMMA. Let A = A1 @ A2 be a direct sum of subalgebras A1,A2. I]" A = A ( A ) , A j = A ( A j ) ( j = 1,2) then A is the disjoint union A - A~ ~ o / s u b s e t s Ajo such that A joI A j -- A j

(i.e. if x c A jo then x I A j c A j

PROOF. Let X E A and X(u) = 1 (u E A). Since u = ttl-~-u 2 with uj C A j , and UlU2 - U2Ul - 0 (since AzA2 - A2A1 A1 A A2 = {0}) we obtain 1 : X(u) : X(tL1) -~- ~'(tL2), X ( t t l ) X ( u 2 ) = O.

From these equations it follows t h a t precisely one of X(tL1), X(tL2) is 1 and the other 0. If X(ui) = 1,X(u2) = 0 then for y 6 A2, s l y = 0, so t h a t 0 - - X(Uly)~-- X ( U l ) X ( y ) - -

X(Y).

This shows t h a t X E Al~ Similarly, if X ( U 2 ) - 1 then X E A ~ 1 . 3 . 1 9 . 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:

w 3. C h a r a c t e r s and H y p e r m a x i m a l ideals

29

(i) X is a character. (ii) M - ker X is closed f o r squares,

i.e. x c M ==~ x 2 E M.

PROOF. (i) ~ (ii), since M is an ideal by 1.3.9. To prove t h a t (ii) => (i), assume t h a t (ii) holds. Since X is a linear functional ~- 0, codim M - 1. For x, y E A we have x -- x ( x ) e + a,

y - x ( y ) e + b (a, b E M ) .

C o m p u t i n g from these relations the p r o d u c t x y and s u b s t i t u t i n g it for x y in X ( x y ) and simplifying we get

X(xy) -- X(x)x(y) + x(ab)

(1)

Taking y - x, b - a in (1) and using hypotheses (ii) we obtain

X(X 2) - X(X) 2 for all x C A

(2)

Replacing in (2), x by x + y we obtain X((x + y)2) _ (X(X) + X ( y ) ) 2 which reduces to

x ( x y + yx) - 2 x ( ~ ) x ( ~ )

(3)

x or y E M ::~ x y + y x E M

(4)

It follows that"

Now we have the identity

( x y - yx)~ + ( x y + yx) ~ - 2((xy) ~+ (~)~) - 2(~y~.y+ y.xy~) (5) where x , y c A. If y E M, then by (4), x y + y x as well as the end expression in (5) belong to M. Therefore, by applying X to b o t h sides of (5) and by using linearity of X and relations (2),(4) we obtain xy - yx E M (6) From (4), (6) we conclude t h a t if y E M then x y c M This means t h a t in (1) we m u s t have x(ab) - 0, so t h a t (1) reduces to

x(~y) - x(~)x(y) proving X is a character, as desired.

Algebraic Preliminaries

30 w4.

Extension

of Ideals

1.4.1. Let J be a bi-ideal of an algebra A. For a 1. ideal Jz of J we set Jl - {a C A" Ja C_ Jz}. (*) Similarly, for a r. ideal Jr of J we set

Jr -- {a E A" aJ c Jr}.

(**)

1.4.2. PROPOSITION. (i)^dt is a 1. ideal of A with JL c_ dz; .]~ is a r. ideal of A with Jr C_ Jr. (ii) If I' is a regular l.(respy, r.)ideal of J with relative r. (respy. l.) unity u, then P is also regular with u as a relative r. (respy. 1.)unity. Further, if I' ~ J then J ~ i', hence in particular ~l ~ A. (iii) If I is a l. (respy. r.) ideal of A and I' - J ~ I then I C I'. (iv) If M ' is a maximal regular I. (respy. r.) ideal of J then 2~I' is a maximal regular I. (respy. r.) ideal of A such that M ' - J 0 ~/I~. Also, if M ~ is a maximal regular hi-ideal of J then Is ~ is a maximal regular hi-ideal of A. (v) If M is a maximal regular I. (respy. r.) ideal of A such that J ~ M then M has a relative r. (respy. l.) unity j belonging to J; further M ' - J ~ M is a maximal regular I. (respy. r.)ideal of J with j as a relative r. (respy. l.) unity and M - I~'. Finally, if M is a maximal regular hi-ideal of A with J ~ M then M I = J ~ M is a maximal regular hi-ideal of J. PROOF. We shall prove only the results for 1. ideals. (i) Clearly Jl i s a l i n e a r s u b s p a c e o f A. For a E J t , x E A we have J . x a J x . a cC_ Ja c Jz,xa E Jz, so that Jz is a 1. ideal of A. (ii) If x C A and a c J then a x c J . By regularity for f , a x axu c I ~. This implies that J ( x - x u ) c f , whence x - x u E P,/~' is regular. Further, by 1.2.8 (b) u 2 is a relative r. unity of I ~, so that if I ~ r J then u 2 ~ I ~. It follows that J u ~ I ' , u ~ I', so that J ~ _T'.

w4. Extension of Ideals

31

(iii) Let I be 1. regular with relative r. unity u. If x E I then J x c A x C I and J x C J (since J is a bi-ideal). Therefore J x c J ['l I - I ' , x c P, so t h a t I __ /~'. (iv) Assume t h a t M ~ is a m a x i m a l regular 1. ideal of J with r. unity u. By (ii), h~/~ is regular with relative r unity u. If I is a 1. ideal of A , I ~ A, with I 2 / ~ _ _ _ M ~ then I is regular with u as relative r. unity and u ~ I (since I ~ A). Further, J N I _ M ' , and since u ~ J N I , J n x = / = J . By maximality of M t , J ~ I - M t. Since I ___ h~/t _ M t we get J ~ / ~ / ' - M ' . For x e I, x u x-(x-xu) e I (using regularity of I ) ; also xu e J (since u e J ) . Thus xu c J A I - M ~. It follows that J x u c_ J M ~ c_ M ~, whence xu c i~I'. Consequently x - x u - ( x u - x ) e 1~I'. This proves I - / ~ / t , h~/t is maximal. (v) Let M be a m a x i m a l regular 1. ideal with relative r. unity u. Since J ~ M, by maximality of M we have J § A. Therefore u has a representation (.) u - j + m (j E J, m E M ) , so t h a t for any x E A x-

~.o

z3 - x -

z(u-

m) - x -

xu + x m ~ M .

Thus j E J is a relative r. unity for M. By restricting x to J we get x - x j E J ['1 M - M ~ showing t h a t M ~ is regular (in J ) w i t h relative r. unity j. Since u ~ M it follows from (.) t h a t j ~ M , j ~ M ' , M ' r J. Consider now a 1. ideal I t of J with I t ~= J , P ~ M t. Then A _ ~ t _ /~/t. Also, by (iii), M C_ h~/'. By maximality of M we obtain I ' - h ~ / ' - M. It follows t h a t M ' JAMJ ~ I ' D I t, so t h a t I t - M t and M t is maximal. Finally, by (iv), M t - J[")/~/' - J A M . 1.4.3. PROPOSITION. Let J be a hi-ideal of an algebra A. Then there is a bijection between the maximal regular 1. (respy. r.) ideals M of A with J ~_ M , and the maximal regular I. (respy. r.) ideals M ' of J given by M'-

JNM,

M-

I~I'.

Algebraic Preliminaries

32

In particular, M' is hypermaximal iff M is hypermaximal. Finally, the map M ~ M' - J A M is a bijection between the maximal regular hi-ideals of A and those of J. PROOF. The first s t a t e m e n t follows from 1.4.2 ((iv),(v)). To prove the particular case s t a t e m e n t assume t h a t M is h y p e r m a x imal and X the character determining it. Let X ~ be the character of J obtained by restriction of X- Then it is clear t h a t M ' = J n M = ker X~ and hence M ~ is a h y p e r m a x i m a l ideal of J. Conversely, if M ~ is h y p e r m a x i m a l ideal of J and X~ the character determining it then X~ can be extended, by 1.3.15, uniquely to a character X - X' of A. If M - k e r x then M is h y p e r m a x imal and clearly we have J A M = M'. Finally, by 1.4.2 ((iv), (v)) M ~ is maximal regular bi-ideal iff M is a m a x i m a l regular bi-ideal. 1.4.4. COROLLARY. V ~ -- J n x/~. PROOF.

N(JNM')

-

n M'-

:

JA(A ~')- J r ] ~

(since in the c o m p u t a t i o n of the right hand end t e r m above the m a x i m a l regular 1. ideals of A which contain J can be clearly dropped). 1.4.5. COROLLARY. If A1 is the unitization of an algebra A then

~-

ANv%.

PROOF. By 1.4.4 (since A is a bi-ideal of A1 ). 1.4.6. Let A Iz (respy. I~) be a r. (respy. 1.)unity u. can form Il (respy.

be an algebra and A1 its unitization. Let regular 1. (respy. r.)ideal of A with relative By taking A1 for A and J - A in 1.4.1, we /~r ). We also set

{Xl e A " u x l e it}. 1.4.r. PRovosITION. (i)i~ (r~py. i~)i~ a I. (~py. ~.)ie~l Il - - { X l E A 1 " X l U e I I } ; Ir --

of A1 with

~ c i, c i~; ~ c i~ c i~

(,)

33

w4. Extension of Ideals Further, if I1 (respy. It) =fi A then il (ii) Given a l.(respy, r.)ideal I1 the ideal I = A ~ I 1 is regular and r. (respy. l.) unity u for I such that el (iii) If I is a regular I. (respy. r.) r. (respy. l.) unity u then -- 1-4- A l ( e l

- it) ( r e s p y .

(respy. it) =fi A1. of A1 with I1 ~ A, we can choose a relative - u E I1 ideal 7s A with relative

(**)

I + (el - u)A1),

I - - A ~ [. Further, if I17 ~ A is a l.(respy, r.) ideal of A1 with I - A A I1 then II C_C_i. (iv) For a regular hi-ideal I with a relative (bi-) unity u we have i -- I -4- A1 ( e l - u ) -- I q- ( e l -

u)A

-

- u),

I + F(el

and [ is bi-ideal. (v) If M is a maximal regular I. (respy. r.)ideal of A then 2~ is a maximal regular I. (respy. r.)ideal of A1 and M - ~/I, M -

AN g4. PROOF. As before we will prove all results only for 1. ideals. (i) It is clear from the definition of it that it is a subspace of A1. Further, if Xl E -ft (so that XlU E It) and yl ael + y (y E A), we have y l X l U -- OLXlU + y ' X l U

E I l + I l -- If.

It follows that ylXl E iz, showing that iz is a 1. ideal of A1. If xl E /~t and y C A then y l x u E Ix. Since y X l E A and Il is regular we get ylZl - yzlu-

(yzlu-

yZl) E Iz,

whence Axl C It, so that Xl E II, and /~L,C /~t. Also, if Iz ~ A then u ~ Ix,el ~ It, and _Tl ~ A1. (ii) Let I1 be a 1. ideal and choose an element. a l -- ,~el

-f- a (~ e F, )~ # 0, a e A) in

I1

\A.

Algebraic Preliminaries

34 Then u -

- A - l a E A, and for any x E A we have XU-

X -- --A-lxal

E

A~

I1 - I,

so t h a t u is a relative r. unity for I1. Also, el - - u -

(iii) If xl

-

Ael

A-lal

E 11.

--k x E A l ( x E A), then

(X 1 -- X l t t ) U -

X l U -- X l t t ' t t

E I-

Il

(since XlU e A).

It follows t h a t X 1 - - X l u E I I , SO that A I ( e 1 - u ) _ II. This inclusion with (first half o f ) ( , ) yields" Iz + A i ( e - u) C_ Iz. To prove the reverse inclusion relation consider an Xl E _TI (so t h a t x l u c Ii ). Then X l - - X l U + X l -- X l u E Il + A l ( e l

- u),

completing the proof of the relation (**). Further, it is clear that It c A['l[l. On the other hand, if a E A , a E it then h U E I~ and so a E Iz(since u is a r e l a t i v e r , unity of Iz). Thus Il - A N-Tl. Finally, consider the ideal I1. By (ii) we can choose for I A N I 1 a relative r. unity u such t h a t el - u E 11. If Yl -- Ael + y E 11 then yl U -- A u + yu E A.

Also, Yl u -

Yl -- Yl (el -- u) E /1 -Jr- /1 -- /1.

So y l u C A ~ I 1 required.

-

h_c

I, whence yl E [. Thus,

~, as

(iv) If xl C A1, Xl - ,~el ~- x (x E A) then X l ( e 1 - it)

--

~ ( e l -- It) Jr X -- XU E F ( e l

-

u)+I

(el - U ) X l

--

A(el - It) + x -

-

it) --[- I

ltx E F(el

Hence I + A l ( e I - tt) - - I + F(el - u) -- I + (e I -- u)A1 i (say). Clearly i is a bi-ideal.

35

w4. Extension of Ideals

(v) By (,) we have / ~ c_/~. Again, by 1.4.2 (iv), M - A["I h4. It now follows, by (iii), that M c h~/. Combining the two inclusions we get / ~ / - / ~ / , and M - A ["1 M - A ~ ]t~/. 1.4.8. COROLLARY. The correspondence I1 ~-~ I - A ~ J 1 is injective over each of the following sets of ideals o] A1 " the set of all hi-ideals I1 ~_ A; the set of maximal I. ideals M1 ~ A; the set of maximal r. ideals M1 ~ A. Also, every maximal regular I. (respy. r.) ideal M of A is of the form M - A ~ I(/I, where h:r

{Xl E A" XlU (respy. uxl) E M } .

PROOF. Suppose I1,J1 ~ A are bi-ideals of A1 with n ~ I 1 - n ~ J1 - I (say). If I A, clearly I1 - J1 - A1 (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 I such that el - u ~ /1, el - v c J1. By 1.2.8 (c), v - u E I __C_ J1. If xl - Ael + y (A E F , y E A) is an element of 11 then we can write: X 1 -- ,~(e 1 -- V)-~- ~(V -- it) -~- )~U -~- y,

)~U @ y - - X l U - - ( y U - - y).

(1) (2)

It follows from (2) that )~U -~- y E XlU @ I C I1 ~- I -- I1( s i n c e XlU E I1)

so that )~u+yEA~Ii-IC__

J1.

(3)

Since e l - v , v - u ~ J1, it follows from (1),(3) that Xl e J1. T h u s , / 1 _ J1, and by symmetry considerations we conclude that I1 = J1. 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 hypermazimal ideal of A1. (b) The correspondence M1 ~-~ M A ~ M 1 is a bijection between the hypermaximal ideals M1 ~ A of A1 and the hypermaximal ideals M of A; moreover M1 - M.

Algebraic Preliminaries

36

(c) ~ F X - A N ~ / A ~ . PROOF. (a) This follows from the construction of A~ (see 1.1.12). (b) By 1.4.7^(v), h : / - h4, M - A NhYr The bijectivity of the map M ~ M - M1 is ensured by 1.4.3. (c) This follows from (b) and the definitions of ~r-~, ~/A1. 1.4.10. R e m a r k . The character of A1 determined by the hypermaximal ideal A is denoted by Xo and called the distinguished or canonical character of A1. It is given by x0

+

-

e A).

It follows from 1.4.9 and 1.3.17 that there is a bijection X ---* X1 between the sets of characters A and AI\{X0}, where X1 is the unique extension of X to A1. 1.4.11. PROPOSITION. Let A be a radical algebra and A1 its unitization. Then: (i) no ideal I ~ A is regular; (ii) v/-X1- A; (iii) A1 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 V ~ _c M ~- A, contradicting A is a radical algebra. (i), (iii): If A1 has a maximal 1. (or r.)ideal M1 r A then M = A A M1 is a regular ideal of A, ~- A, contradicting (i). On the other hand, A is a maximal (1. or r. or bi-ideal) of A1. Thus v/A1 - A and A is the unique maximal ideal of A1. w 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 ~a" A -* E (X) (=the algebra of linear endomorphismst of X) is called a linear representation t i.e. linear maps of X into itself.

37

w 5. Regular Representation and Primitive Ideal

or just a representation of A in X. It is well-known t h a t a representation ~ of A gives rise to an A - m o d u l e structure on X : a.x -- ~ ( a ) x . Conversely, if X is an A - m o d u l e it yields a n a t u r a l representation a ~ la, where la is the linear map on X given by la : x ~ ax (a @ A, x E X ) In particular we have for an algebra A the representation. ~ " a ~ la(a E A), where la is now given by la'xHax

(xEA).

is called the (left) regular representation of A The kernel of the h o m o m o r p h i s m ~ is given by ker ~ = {a E A : a A = 0} --the left annihilator ~qz. ~ is faithful (i.e. is a m o n o m o r p h i s m ) iff ~qz = {0}. In particular, by 1.2.28, ~ is faithful whenever A is s.s.. 1.5.2. If Iz is a regular 1. ideal of A then the regular representation ~ induces a representation ~ # in the quotient space A # - A / I l . ~ # is given by 9 ~ # 9 a ~-. la~, with l ~ ( x § Il) - ax § Iz(x § A ) . ~ # is called the regular representation of A in A #. The representation ~ # is called irreducible if A # is a simple t ( ~ # - ) module. For ~ # to be irreducible it is a necessary and sufficient t h a t I be a maximal regular 1. ideal of A. If ~ # is the regular representation of A in A # - A / I , we write (Iz'A)-ker~#-{aEA'la

~-0}-{aEA'aAC

Il}

Similarly, we write for a regular r. ideal It, (If'A)

- {a E A " Aa C_ It}.

1.5.3. LEMMA. Let I be a regular I. (or r.)ideal of A.

Then

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 I; also J is the largest bi-ideal (of A) contained in I. If I is a regular bi-ideal

th~n ( I A ) - ~. PROOF. T h a t J - ( I ' A ) is a bi-ideal is clear. Assume now t h a t I is a r e g u l a r 1. ideal. Then for x E J, x A C I, so t h a t in particular xu c I, where u is a relative right unity for I.

x-

x - xu § xu E I,

JCI.

Further, if J~ is a bi-ideal of A with j t _ I then J I A C J~ C_C_I, whence J~ CC_(I " A) - J. The proofs of these s t a t e m e n t s when I is a r. ideal are similar. Finally, when I itself is a bi-ideal we have clearly (I" 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 Ml of A with P = (Ml : A) Similarly, if there is a maximal r. ideal Mr of A with P = ( M ~ : A ) then P is called r. primitive. By 1.5.3 a 1. or r. primitive ideal is a bi-ideal In general a 1. primitive ideal of a ring need not be r. primitive, t If an ideal P is b o t h 1. and r. primitive then it is called hi-primitive. If A is c o m m u t a t i v e then all these concepts evidently coincide. In the sequel we shall use the t e r m 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 Mt with {0} = ( M l : A). 1.5.5. LEMMA. Let A, A* be algebras (or rings) and p 9 A ---. A* be an epimorphism. If P is a primitive ideal of A with ker ~ c P, then p ( P ) is primitive. PROOF. If P -- ( M 9A), where M is a maximal regular 1. ideal of A, then ~ ( M ) is a maximal regular 1. ideal of A* and ~(P) -(p(M)'A), whence ~ ( M ) is primitive. 1 . 5 . 6 . COROLLARY. If P is a primitive ideal of A and 8" A ~ AlPA* the canonical homomorphism, then ~ ( P ) - {0'} See [ 4 ' , p p . 4 7 3 - 75]

w5. Regular Representation and Primitive Ideal

39

is primitive. 1.5.7. PROPOSITION.

If A is an algebra (or ring) and A r

V ~ , then n(Mz

9A) - N(M

9A)

where Ml (respy. Mr) runs through all maximal regular I. (respy. r.)ideals of A. In particular A is the intersection of all (1.) 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 ~ c Mz, whence by 1.5.3, ~ c ( M I ' A ) . Also, since Ml is regular we have (M~ 9 A) C Mz. Therefore

_c N ( M , so that x / A - N ( M z ' A ) ,

9A) _c

n

Mz

-

~/A.

Similarly, x / A - n ( M r "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 1. ring ideal of A and write M1 - {x E A; Ax c M}. Clearly, M1 is a 1. algebra ideal and M c M1. Further, if u is a relative r. unity for M then u ~ M 1 . (since U C M l = = a u 2 E M = F M - A , by 1.2.8 (b), 1.2.9 (i)). By ,maximality of M we conclude that M1 - M, whence M is an algebra ideal. Similarly, the proof when M is a r. ideal. Now if P is a p r i m i t i v e ring ideal of A then P (A" M) where M is a maximal regular 1. ideal. If a E P, ~ E F then

c ~ a A - a.c~A C_ aA C_ M, so that aa c P, where P is an algebra ideal. 1.5.9. prime, t

LEMMA.

(Jacobson).

Every primitive ideal P is

t Abi-ideal P in a ring R is called prime or aprimeide~lif P ~ : R ~ndfor ~nybi-ide~ls I,J I J C P = : a I C P or J _ P .

Algebraic Preliminaries

40

PROOF. (cf. 21, p.54). Suppose that I J C P - ( M ' R ) and J ~ P. Then J R ~ M and hence, by maximality of M, J R + M - R. It follows that

I R C I ( J R § M) C_ I J § M C P + M so that I C (M" R ) -

M

P.

1.5.10. PROPOSITION. (a) Every maximal regular bi-ideal M is hi-primitive, in particular prime. Furthermore, the quotient A # - A I 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 Mz be a maximal regular 1. ideal with M __ Mz. Then M C (Mz " A) C_ Mt. Since ( M z ' A ) is a bi-ideal, by maximality of M, M - (Mr" 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 biideal, 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 ~r-~; if A has no maximal regular bi-ideal we define A to be ~/A" ~ - A. 1.5.12. PROPOSITION. In any algebra A we have

PROOF. The first inclusion relation follows since every maximal regular hi-ideal is (by 1.5.10 (a)) primitive; the second is a

w 5. Regular R e p r e s e n t a t i o n and P r i m i t i v e Ideal

41

consequence of every h y p e r m a x i m a l ideal being (by 1.3.5) a maximal regular bi-ideal. 1 . 5 . 1 3 . Let A be an algebra. An A - m o d u l e X is said to be cyclic, with generator xo, if x0 E X and Axo = X . Also, an A - m o d u l e X is said to be irreducible if (i) A x ~ {0} and (ii) the only submodules of X are {0 } and X. If X is an A - m o d u l e , for x0 E X we write kerxo = {a E A : axo-O}. Similarly, for a 0 C A we write k e r a 0 - { x E x ' a 0 x 0}. Clearly kerx0 is a 1. ideal of A and kera0 a submodule of X. 1 . 5 . 1 4 . LEMMA. Let X

be an A - m o d u l e .

Then:

(i) If X is cyclic with generator xo then kerx0 I. ideal of A.

is a regular

(ii) I f X is irreducible then it is cyclic with any non-zero element xo as generator. Further, M -- ker xo is a m a x i m a l regular I. ideal of A. PROOF. (i) Since X = Axo there is an element u c A with xo -- uxo. Then ( a - auo)xo --- axo - axo = 0 , so t h a t a - au E ker x0 whence ker x0 is regular with u as a relative r. unity. (ii) The set X0 = {x e X : Axo = {0}} is clearly a submodule of X. By irreducibility of X we have X0 = {0} or X. But condition (i) in the definition of irreducibility rules out X0 = X and so we must have X0 = {0}. Thus, if xo E X , xo ~ 0 then A x o ~ {0} and so Axo = X, proving X is cyclic with generator x0. Also, by (i), M = kerx0 is a regular 1. ideal. It remains to prove t h a t M is maximal. If L is a l . i d e a l o f A with L D M, we can choose an element b E L \ M , and then bxo ~ O. It follows t h a t Abxo --- X , whence there is an a0 E A with aobxo = xo. But then, for any a E A, a - aaob c ker xo = M c L.

(,)

Since b E L and L is a 1. ideal we conclude from (,) t h a t a E L, which means L - A, proving M is maximal. 1.5.15.

If A is an algebra and Il a 1. ideal of A, then the

42

Algebraic P r e l i m i n a r i e s

quotient A # - A / I z is canonically a left A-module: a ( x § Ii) -- ax + Ii

(a, x E A).

The corresponding representation ~ # is given by, ~ # 9a ~ l~, where l~ is the linear transformations on A # such t h a t l ~ ( x + Il) - ax + Iz (cf.1.5.2). ~ # is faithful (i.e. 1-1) if (It "A) - {0}. 1 . 5 . 1 6 . LEMMA. Let I -- Iz be a regular I. ideal of an algebra. Then:

(i) The quotient module A # - A / I is cyclic. (ii) A # is irreducible iff I is m a x i m a l . PROOF.

(i) If u is a relative r. unity for I then x + I = x u § I - x ( u + I), so that A # is cyclic with u + I as generator. (ii) If 7r - A ~ A # is the canonical module h o m o m o r p h i s m given by x~-~ x + I, then clearly ~r-l(x0), where X0 ranges t h r o u g h all submodules of X - A #, are precisely the 1. ideals of A containing I. Hence the irreducibility of X is equivalent to the m a x i m a l i t y of I. 1 . 5 . 1 7 . LEMMA. Let X xo and I - ker xo.

be a cyclic A - m o d u l e with generator

Then 9 "a

§ I ~

is a module i s o m o r p h i s m of A # - A / I

axo onto X .

PROOF. (I) is well-defined since al + I - - a 2 -~ I ~ a l x o -a2xo. Since every element of X has the form axo, 9 is surjective. Finally, (I) is a module h o m o m o r p h i s m since r

§ I)) -- r

+ I) -- baxo -- b r

+ I)

1 . 5 . 1 8 . Let A be an algebra and X an A - m o d u l e . Write D - / ) (X) - the set of all A - e n d o m o r p h i s m s (i.e. e n d o m o r p h i s m T of the additive group of X satisfying T a x - a T x for all a E A). Clearly /) is a unital algebra, with the identity m a p of X as the unity element. 1,5.19.

LEMMA (Schur). For an irreducible A - m o d u l e

D is

a division algebra.

PROOF.

If T C P , T =fi 0 then T X , k e r X

are submodules

43

w6. Real and Complex Algebras

with T X X, ker T -

~ { 0 } , k e r T r X. By irreducibility of X, {0}, so that T is invertible and T -1 E P.

TX-

1.5.20. Let X be an irreducible A-module and P - D(X). If x 1 , ' " , xn E X these vectors are said to be P - i n d e p e n d e n t if for any T 1 , ' - ' , T n c D,

T1X l

--~- "

" -~- T n

x n

-

0

::~ T1

--

"'"

-

Tn

-

O.

Since /) is a division algebra, X can be regarded as a linear space over P. Then clearly P-independence is the same as linear independence in this linear space. 1.5.21. THEOREM (Jacobson's Density Theorem). Given D independent vectors Xl, " " ,xn and arbitrary vectors Y l , " " , Yn all in irreducible module X , there is an a in A with ax i - yj(1 ~< j 0, over F. Then we have the factorization P ( X ) - ~ - ( ~ ( X - # 1 ) " " ( X - #n), where cz, #j E F and ~j are the zeros of the polynomial P - )~. It follows that P ( # j ) - A ( j - 1 , - . . , n). Further we have the factorization

y - P(x) - )~e- a ( x - # l e ) ' " ( x - #he). If A C a(P(x)), y is not invertible, whence by virtue of 1.1.30, some x - #ke is not invertible. It follows that ttk E a(x) and - P ( # k ) e P(a(x)). Thus, a(P(x)) c_ P(a(x)). Combining this with (.) we get (**). It remains to deal with the case where deg P - 0. In this case, if P(X) - c~o (say), (**) holds since as noted in the proof of the first assertion in (v), each side of (**) reduces to {c~0}. 1.7.4.

LEMMA. Let A be unital, with unity e, and x E A .

Then )k(# O) E if(x) iff -,~-lx i8 not q. invertible PROOF. The identity X - ~e -- - - ) ~ ( e - )~-lx)

(X E A, )~ E F,)~ # 0)

and 1.1.20 show that A E p(x) iff - A - i x 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 A1 its unitization. For x E A we write a' (x) -- alA (X) -- aA~ (x) and call a'(x) the quasi-spectrum of x; its complement p'(x) -

is called the quasi-resolvent set of x. 1.7.6.

LEMMA.

0

e A,

that

always nonempty PROOF. If possible let x have an inverse y + )~el (y E A,)~ E

55

w 7. Spectrum and Quasi-spectrum

F) in A1. T h e n el - x(y + Ael) - xy + Ax C A which is impossible. Hence 0 E aA~ (x) - a'(x). 1.7.7. PROPOSITION. ( i ) o " ( 0 ) - O; ( i i ) a ' ( # x ) #a'(x)(# E F); (iii) If P is a constant-free polynomial over F then

D with the inclusion relation becoming equality when braically closed, so that then

F

is alge-

a'(P(x)) - P(a'(x)). PROOF. These follow from 1.7.3 since for any element a in A we have a ' ( a ) - aA~(a). 1.7.8.

LEMMA (Kaplansky).

Let A

be an algebra x E A.

PROOF. Since X - Ael -- --A(el -- A - i x )

r o)

where el is the unity of the unitization A1 of A, we obtain: x(el

Ael is not invertible in A1 -

)~-lx) is not invertible in A1

- A - i x is not q. invertible in A1 (by 1.1.20) - A - i x is not q. invertible in A (by 1.1.21). 1.7.9. COROLLARY. If u ~ 0 is an idempotent then a'(u) {0,1}. Further, if A is unital, with unity e, and u 7s O,e then a(u) -- {0, 1}. PROOF. The first s t a t e m e n t follows from 1.7.8, 1.1.26 (i). For the second we note t h a t by (**) of 1.7.21, we have a(u) U { o } - a'(u) - {0, 1}. If 0 r a(u) then u is invertible and since u 2 - u we get u - e, contradicting the assumption on u. Thus 0 E a(u), a(u) - g ( u ) U { 0 }

- g'(u) - {0,1}.

56

Algebraic Preliminaries

1.7.10. LEMMA. If u is a relative unity for a regular I. or r. ideal I ~ A then 1 E al(u). PROOF. By 1.2.9 (ii) - u is not q. invertible, whence by 1.7.8, 1 C a'(u). 1.7.11. COROLLARY. If X is any character of A and x E A th n x(x) e PROOF. Since always 0 E a'(x) we may assume that X ( x ) = A -r 0. Then X(A-lx) - 1, so that by 1.3.10, u - A - i x is a relative (bi-) unity of M• whence by 1.7.10, 1 E a ' ( u ) - a ' ( A - l x ) A-la'(x), so that A E a'(x). 1.7.12. LEMMA. q.i. x ~. Then a'(x') -

A s s u m e that x E A is q. invertible with

A a~ } l § A " A E (x) .

o PROOF. First note that 0 E a~(x) and 1+o -- 0 E Since x is q. invertible, by 1.7.8, - 1 r a(x). For A r 0 we have the (easily verifiable) identity:

X' o(--,~-lx)-- (--)~-lx)oX ' : It follows, by 1.7.8 and 1.1.30, that - ~ ~ completing the proof.

~I +XA

I.

E a ~(x) iff A E a ~(x),

1.7.13. D E F I N I T I O N . An element x in A is called quasinilpotent or q. nilpotent if a'(x) -- {0}. The set of q. nilpotent elements of A is denoted by Aqn;O E A q" (by 1.7.7 (i)).

1.7.14. R e m a r k . In the algebra M n ( F ) of all n • 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 Ax is q. invertible for every A c F, in particular, a q. nilpotent element is q. invertible.

57

w 7. Spectrum and Quasi-spectrum

(ii) A nilpotent or essentially nilpotent element is q. nilpotent. (iii) x ~ c A q'~ c ~'-A. PROOF. (i) This follows from 1.7.8, 1.7.13. (ii) If x is nilpotent so is Ax, and hence by 1.1.23, Ax is q. invertible, whence by (i), x is q. nilpotent. Again, if x is essentially nilpotent then by definition x E x/~, and hence also Ax e v/A, Ax 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 e A qn, f f ' ( x ) - {0}. If X e A, X(x) e a'(x) (by 1.7.11) so that X(x) - O, whence x e ~'-A. 1.7.16. R e m a r k . Every element of a radical algebra A (A = v/A) is q. nilpotent, so that A - A an. On the other hand, in the one-dimensional algebra F, 0 is the only q. nilpotent element (since, by 1.7.20, a'(A) = a(A)U{O} = {A, 0}). 1.7.17.

An algebra A is called quasi-semiif 0 is the only q. nilpotent element of A, i.e.

DEFINITION.

simple or q.s.s, A q n : {0}.

1.7.18. R e m a r k . In view of 1.7.15 (iii), every q.s.s, algebra is s.s. Note that the matrix algebra M , ( F ) is s.s. (actually simple) but not q.s.s, when n /> 2 (since there are non-zero nilpotent matrices in M,~(F), n/> 2). 1.7.19. LEMMA. Let A , B be algebras (over the same field), ~ : A--~ B a h o m o m o r p h i s m and x E A. Then we have: C

PA( ) C whenever A , B unity of A.

C

(*)

C

(**)

are unital and B has as unity ~(e),

PROOF. If A E p~A(X) then - A - i x 1.1.24, _

e being the

is q. invertible and so by

58

Algebraic Preliminaries

is q. invertible, so t h a t A E p~B(~(x)). clusion relation in (,) and the second complements of both sides in F. For proving (**), we observe that in (,) of 1.7.21, the first conclusion in as

This proves the first inone follows by taking setby using the first relation (,) above can be written

(,')

p~(=)\{0} c p . ( ~ ( = ) ) \ { 0 }

If 0 E pA(x), x is invertible and then ~p(x) is also invertible, and 0 E PB(P(=)). The inclusions in (**) now readily follow from

(,')

1.7.20. COROLLARY. Let A be a subalgebra of an algebra B and x E A. Then we have:

p~(~) c p~(~); ~(~) c ~ ( ~ )

(,)

p~(x) c p,(x); ~.(x) c_ ~(x)

(**)

whenever B is unital and A is a subunital t algebra of B. PROOF. This follows from 1.7.19 by taking ~ to be the inclusion map A --~ B. 1.7.21. PROPOSITION. If A is a unital algebra over F and x c A, then: p'(x) = p(x)\{0}; (,)

~,(~)

-

(**)

~(~)U{o}.

PROOF. Let A1 be the unitization of A and el,e the unities of A1, A respectively. If A E PAl (x) then for some y E A, # E F we have the equations (X-)lel)(y-~-

~ e l ) -- e I -- (y-~- ~ e l ) ( X -

)lel).

(1)

Multiplying the terms in (1) by e from the left and then by from the right we obtain

( x - ~ ) ( y + ~ ) - ~ - (y + ~ ) ( ~ - ~ ) It follows from (2) that A C pA(x). Thus, i.e. if e is the unity of B

then

eEt.

(2)

w 7. Spectrum and Quasi-spectrum

59

p'(~) - p~, (~) c p~(z) - p(~). On the other hand, if A E pA(X),A :/: 0 then since X- )~e- --)i(e- )i-lx), - - ~ - l x is q. invertible in A and hence also in A1, whence A E p~(x). Further 0 ~ PA~(X), for otherwise either of the equations (1) with A = 0 shows that el c 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

~' (xy) - ~' (y~), ~(xy) U { 0 } - ~ ( y x ) U { 0 }

(,)

(,.1

whenever A is unital. PROOF. First assume that A is unital with unity e. If A E a(xy), A ~ 0 then x y - Ae, and hence also ()~-lx)y -- r -- - - ( r

(~-lx)y)

is not invertible. By 1.1.28 (ii), 1.1.14 (c), - ( e - y(A-lx)) y A - l x - e is not invertible, so that A E a(yx). Thus

r

U{o} c r

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 A1 of A (remember that for any a E A, Oca'(a)). 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 E ker X is not invertible, or equivalently, X(x) # 0 for every invertible element x.

(ii) X(x) c r

Algebraic Preliminaries

60

PROOF. (i) => (ii). If X ( X ) = , ~ t h e n x - ) ~ e E k e r x - J : A, whence it is not invertible, so t h a t )~ E a(x). (ii) ~ (i). If x c k e r x t h e n 0 = X(x)C_ a(x), whence x is not invertible. 1 . 7 . 2 4 . LEMMA. Let A be a unital algebra and X a character

of A. Then X(x) e a(x) (x e A). PROOF. If X(x) -- )~ t h e n x - l e C k e r x = M r therefore x - l e is not invertible, and so I E a(x).

A and

a E A and ~ E F. If there is a non-zero element b E A such that a b - lb, then I E al(a). 1.7.25.

LEMMA. Suppose that A is an algebra over F,

PROOF. In the unitization A1 of A the above condition can be r e w r i t t e n as (a - )~el)b = 0. Since b r 0, a - )~el is not invertible and consequently )~ E al(a).

1.7.26.

Let A be an algebra and a E A.

PROPOSITION.

Then we have: (i)

The double commutant Ao = {a}" is a commutative subalgebra of A containing a such that alAo(a) - alA(a).

(*)

(ii) If Am = Am(a) is a maximal commutative subalgebra of A,

containing a then alAm (a) -- alA (a).

(**)

(iii) If A is unital with unity e then =

(~ ~*)

PROOF. First let A have a unity e. By 1.1.8 (vi), A0 is a c o m m u t a t i v e s u b a l g e b r a and clearly e E A0. If a - l e has an inverse b in A then by 1.1.18 b ~-~ { a -

)~e}' - {a}', whence b E {a}" - A0.

w8. Extended Spectrum and Extended Quasi-spectrum

61

Therefore p A ( a ) C PAo(a), so t h a t by (**) of 1.7.20, p A ( a ) - PAo(a). Also, we have (see 1.1.9) Ao C Am C A. It follows (using 1.7.20) t h a t p A ( a ) = PAre(a)- pA(a). Therefore, by taking setc o m p l e m e n t s in F we obtain (, 9 ,). Next let A be non-unital. By applying, ( , , ,) to the subalzebras (A0)l - Ao + Fei, (Am)l -- Am + Fel of the unitization A1 of A we obtain (.) and (**). 1.7.27. PROPOSITION. Let A q. inverse closed t subalgebra of A. Then

~,;(=) -

,,'~(x)

be an algebra and B

a

(= e B)

Similarly, if A is unital and B a inverse closed subunital algebra of A then

o.(=)-

~,A(x)

(= e .4).

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.

w8.

Extended

Spectrum

and Extended

Quasi-spectrum N

1.8.1. DEFINITION. Let A be a unital real algebra, A its complexification and x c A. We write

~(=)- ~ ( x ) - ~(=) and call 5(x) the extended spectrum of x. Similarly, writing 5'(x) - a(~)l(x), where (A)I denotes the unitization of A, we call 5'(x) the extended quasi-spectrum of x; its set-complement fi'(x) - K\5'(x) is called the extended quasi-resolvent set of x. 1.8.2. LEMMA. If A is a real algebra and x E A then

~'~(~) - ~'~(~) N ~.

(,)

t A subalgebra (respy. subunital algebra) B of A is called q. inverse (respy. inverse) closed if for ~ny x 6 B, the q. inverse x' (respy. inverse z -1) of x in A exists ==~x' (respy. x -1) 6 B .

62

Algebraic Preliminaries

If A is unital we have also -

(**)

a,,

In particular, a~A(X) C 5~A(X), aA(X) C_ 5A(X). ~ PROOF. We have A(# 0) e a~A(X)["l~a ,I i ( z ) A R

iff - A -

1 x

is not q. invertible in ,a iff - A - i x is not q. invertible in A iff Ac r This prove (,) (since '0' clearly belongs to both sides of (,)). The equality (**) follows by applying (,) to A1 and using 1.7.21. 1.8.3. PROPOSITION. Let A be a real algebra and x E A. Then 5'(x) is a s y m m e t r i c t subset of C; also when A is unital, 5(x) is symmetric. PROOF. We have

~(~ # 0) e a'(~)

i~

- A - i x is not q. invertible in

iff

( A_lx )

~

~

~-1

X

is not q. invertible in .~i (see 1.6.7.) iff

i e a'(~), N

and 0 - 0, where bar denotes the conjugation in A (see 1.6.4). Thus 5 ' ( x ) i s symmetric. The s y m m e t r y of 5 ( x ) i s an i m m e d i a t e consequence of (**) of 1.7.21 and the s y m m e t r y of 5'(x). ~

1.8.4. PROPOSITION. Let A be a real algebra, A its complexification, and x E A. Then: N

(i) If ~ - ~ + iZ # 0 (~, Z c ~), _ ~ - 1 x is q. in vertible in A iff z2-2a* is q. invertible in A I~l N

(ii) If A is unital with unity e, then x - Ae is invertible in A iff (x - ae) 2 + fl2e is invertible in A. PROOF. Write z -- - A~- i x ; then 2. - - A - i x where bar denotes the conjugation in A. By 1.6.7, z is q. invertible iff 2 is q. invertible. Since z +-+ ~,, by 1.1.30, z o 2 is q. invertible iff z (hence 2) is q. invertible. Since } Asubset S of C is said to be symmetric if )~ES :=>~ES, where denotes the complex conjugate of ~.

8. Extended Spectrum and Extended Quasi-spectrum

63

x 2 - 2ax z

o

z

w

I;~12

(i) follows. The result (ii) can be proved similarly by using the identity (~ - . ~ ) ( x - ~ ) - ( ~ - o,~)~ + / ~ and 1.1.30. 1.8.5. COROLLARY. If A - a + ifl(a, fl E ~), A # 0 then c ~'(x) i g ~ - (x ~ - 2 . x ) ( . ~ + Z~)-I i~ ~ot q. i~v~rtibl~ i~ A. Similarly, when A is unital, A E 5(x) iff y is not q. invertible in A, and O E h ( x ) iff x is not invertible in A.

PROOF. The first assertion is an immediate consequence of 1.8.4 (i). Since 5'(x) - 5(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 ((Xl + i y l ) x - e X(Xl -+-iyl) ~ X l X - e - XXl, where X, Xl, Yl E A). 1.8.6. PROPOSITION. (Rickart). Let A be a complex algebra and A [~] denote A as a real algebra. Then, for x E A, we have

where bar denotes complex conjugation.

PROOF. We denote the general element of A[~] by x + j y , where j2 _ - 1 ; x l , y E A[R] - A (as a set). By 1.6.10, if x E A A [~] has a q.i. in A[~], it has also a q.i. in A. So, by 1.7.8, a~A(X) C 5AI~I(x ) and hence by 1.8.3. ~'A(:) C

-' aA[~] (x).

To complete the proof it is enogh to show that

~-~,E.~(x) ___~;.(~)U o-~(~). Suppose that

.,, r 4,(x)U,,-~(:),

# __ _ ~ - 1 .

64

Algebraic P r e l i m i n a r i e s

Then # x , p x

are q. invertible. Write (#X)'

if # - - / + i a

-- y,

(-/,acR),

(/~X)' -- Yl"

then in Ale] we have # x - "Ix - j a x .

# x - .ix + j a x ,

It follows from 1.6.8 that we have (")'X)' -- ( ~ X ) '

-- y -

(~X)'

-- Yl.

Therefore # x - y' - yl1 - p x , whence a - 0, # - ~/. Thus #x - ~/x E A [~] - A,

(#x) ~ - y E A

so t h a t #x is q. invertible in A. It follows t h a t ,~ E ~A[~], completing the proof. 1 . 8 . 7 . DEFINITION. An element x of a real algebra A is called e x t e n d e d q u a s i - n i l p o t e n t or ext. q. n i l p o t e n t if 8 A ( X ) -- {0}, i.e. if x is q. nilpotent in A. N

1.8.8. R e m a r k . By (**) 1.8.2, every ext. q. nilpotent element is q. nilpotent. On the other hand, in the real algebra A - C [a], the element i is q. nilpotent but not ext. q. n i l p o t e n t . For, a~4(i) - aA(i) U{0} -- {0} since aA(i) -- {~ ( i - - ce being invertible in A for every real a}. But, by 1.8.7,

ab,(i) - 4 (i) O 4 (i) -

{o, +i}.

1.8.9. LEMMA. L e t A be a real algebra. T h e n every e l e m e n t of x / A

is ext. q. n i l p o t e n t .

PROOF. Take x E x / ~ and let ) ~ - a + i / ~ # 0 . Then, x/A being a bi-ideal, we have y - (x ~ - 2 a x ) ( a 2 +/~2)-1 e v/-A, so t h a t y is q. invertible. It follows from 1.8.5 t h a t A ~ 5~(x), whence 5'(x) - {0}, as desired. 1 . 8 . 1 0 . DEFINITION. Let A be an algebra over K and x E A. The spectral radius r(x) is defined by

r(x) - r A ( m ) - sup{{AI" A ~ ,r'(m)}.

w8. Extended Spectrum and Extended Quasi-spectrum

65

If A is unital and a(x) :/: 0 then we also have

r(x)

-

-

sup(l~Xl 9 tx c ~ ( x ) } .

Similarly, we have the extended spectral radius ~(*) - s u p { l ~ x l

,x c ~ ' ( ~ ) }

for an element x of a real algebra A. When A is unital and 5(x) ::/: 0 we have ~(x) - sup{],x I 9 A ~ a ( x ) } .

1.8.11. R e m a r k .

Clearly we have the inequalities"

0 1 1). But this contradicts the boundedness of a(yx) since I n # [ - nlit[ --. oo (as n --. c~). Therefore 0 ~ a(yx). By a similar argument (with x,y interchanged and It replaced by - i t ) we also get 0 ~ a(xy). Thus, the equation (2) becomes just

(2') So (1) can be rewritten as (1') Since a(xy)7s 0 there is a A(~= 0) in a(xy). Then, by (1'), # + A E a(xy). Once again by (1'),

and so on. Thus, nit + A E a(xy) for all n >/0. Since

once again we are led to contradict the boundedness of a(xy). Thus the Heisenberg equation has no solutions as asserted. 1.8.17. R e m a r k . The result 1.8.16 is an algebraic generalization of the well-known fact that the Heisenberg commutation relation of quantum mechanics" p Q _ Qp _ 2~ri h ( h denoting Planck's constant) has no bounded operator solution.

w9.

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: ~ ( x ) ___~; by 1.8.2, this condition is equivalent to" ~ ' ( x ) - a'(x).

68

Algebraic Preliminaries

If A is unital the "strictly real" condition can also be stated as" 5(x) c_ R, or, 5(x) - a(x). An algebra A is called strictly real if all its elements are strictly real. 1.9.2. R e m a r k . 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 AI ( x ) (0}). Again, if A is unital with unity e, the elements )~e()~ E ~) are strictly real

1.9.3. PROPOSITION. (a) Every real radical algebra A - in particular real algebra Ao with trivial multiplication - is strictly real. (b) Ao 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 5~A(X)- {0}, whence A is strictly real. (b) This follows from 1.6.19 since every element of A0 is nilpotent. 1.9.4. R e m a r k . In A0 we have an example of a strictly real algebra which is not formally real. 1.9.5. LEMMA. The unitization A1 of a real algebra A is strictly real iff A is strictly real. PROOF. If A is strictly real, and xl E A1, xl -- ~e + x (x E A , ~ E R) then 5(X1) -- (~(Ae + X) -~ A + 5(X) C ~ (since 5(x) C_ ~, A E ~), proving A1 is strictly real. Conversely, if A1 is strictly real and x E A we have _

,

which proves that A is strictly real. 1.9.6. PROPOSITION. A real algebra A is strictly real iff for any x E A, x 2 is q. invertible. PROOF.

Suppose that A is strictly real.

By 1.9.5 A1 is

w9. Strictly Real Algebra

69

strictly real, whence -

It follows t h a t

c

~'(x 2) - - a ( ~ i ) x ( x 2) - - a ( ~ ) x ( x ) 2 > ~ t 0 ,

- 1 ~ ~(x2), so t h a t x 2 is q. (by 1.6.10 (ii)). Conversely, suppose t h a t A A. We have to show t h a t ~'(x) a n u m b e r a + i f l with / 3 ~ - 0

whence

invertible in A and so also in A satisfies t h a t condition for all x in _c ~. If possible let ~'(x) contain ( a , / 3 E ~ ) . Write

Y _ ( a x 2 _ (c2 _ fl3)x)/fl(c 2 + f12); then y E A. By 1.7.7 (iii), k'(y) ~ {c~(a + i/3) 2 - (a2 _ f12)( a + ifl)}/~(o~2 + f12) _ i, so t h a t ~,(y2) ~ i 2 _ - 1 . It follows t h a t y2 is not q. invertible in A and so also not q. invertible in A, contradicting our supposition on A. Thus, ~ ( 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 E A, e + x 2 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 i m m e d i a t e 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 = f ( t ) on [ - 1 , 1] which are holomorphically extendable to the closed unit disc. It is not strictly real since the function 1/(t 2 + 1) does not belong to the algebra, as its analytic extension 1/(z 2 + 1) has poles at ~ i . t i.e. if .~E~'(x 2) then ~>~0. tt The condition "e+x 2 is invertible for all x E 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.

Algebraic Preliminaries

70

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 . 1 1 . LEMMA. Let Am be a maximal commutative subalgebra of a strictly real algebra. Then Am is strictly real. PROOF. If x C Am C A. By 1.1.19, (x2) ~ E A m , whence x 2 is q. invertible in Am, Am is strictly real. 1.9.12. Remark. A real algebra A which has a complex s t r u c t u r e is not strictly real. To see this, we may assume, by 1.9.5, t h a t 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 t h a t the real algebra H of Hamilton quaternions is not strictly real (note here however t h a t H has no complex structure). 1 . 9 . 1 3 . PROPOSITION. Every strictly real q.s.s, algebra A is formally real. PROOF. Then

For, suppose t h a t x 2 + y2 = 0 (x, y E A). !

! -

(,)

By strict reality, a ,i ( x2 ) - aA, ( x ) 2 >/ 0, aA, ( y 2 ) >/ 0. It follows from (.) t h a t t a~(x)

2

t -- a~(y)

2

-- O, whence

I r a~A(X) -- aA(x) -- O,a~A(y) -- aX(y) -- O. Since A is q.s.s, we m u s t have x = y = O, proving t h a t A is formally real.

1 . 9 . 1 4 . 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 + x 2 = e 2 + x 2 r 0, whence it is invertible. So, by 1.9.7, A is strictly real.

71

w9. Strictly Real Algebra

Suppose now t h a t A is a strictly real c o m m u t a t i v e division algebra and x, y E A , x2+y2-0. If x - ~ 0 then we have e + x-2y

2 -

e +

(x-ly) 2 - 0 ,

contradicting strict reality. Therefore x = 0 and similarly y = 0. Thus A is formally real. 1 . 9 . 1 5 . PROPOSITION. (Kaplansky [8', p.405]). A primitive strictly real algebra A is a division algebra. PROOF. Since A is primitive there is a m a x i m a l regular 1. ideal Mz with (Mz: A ) = {0} (see 1.5.4). Then X = A / M z is a faithful A - m o d u l e (see 1.5.15) which is further irreducible (by 1.5.16 (ii)). By 1.5.19 we have the division algebra P = P ( X ) . Denote by dim X the dimension of X as a linear space over P. If dim X > 1 there are two P - i n d e p e n d e n t vectors x, y E X. By density t h e o r e m (1.5.21) there is an a E A with ax = y, ay = - x . Then a2x = ay = - x . Since A is strictly real, a 2 has a q.i. b : a 2 + b + ba 2 = 0. It follows t h a t a2 x + bx + ba2x - O, i.e. - x + bx - bx - O,

i.e. x - 0-impossible. So dim X - 1 and X is a division algebra. Since X is a faithful A - m o d u l e , A is a division algebra. 1 . 9 . 1 6 . PROPOSITION. Let A be a strictly real algebra and its complezification. Then" (i) If i r A is a regular I. (respy. r.) ideal of A then I is a regular I. (respy. r.) ideal of A with I # A.

Re 7

N

N

(ii) i f M

a maximal t. of A and M -- Re M then M is a m a x i m a l regular 1. (respy. r.) ideal of A, and M M § iM, MA ~ M ; in particular M is self-conjugate.

(iii) The correspondence M ~ M - M + i M is a bijection between the set of m a x i m a l 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 t h a t [ =/= A. For this, by 1.6.13, it is enough to show t h a t I does N

Algebraic Preliminaries

72

not have any relative r. (respy. 1.)unity of the form iv (v e A). Suppose that I has a relative r. (respy. 1.)unity of the form iv. Then, by 1.2.8. ( b ) , - v 2 - (iv) 2 is a relative r. (respy. 1._)unity for 7. Since _T :fi A, by 1.2.9, v 2 is not q. invertible in A. On the other hand, since A is strictly real v 2 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 + i M . Let ~ - u+iv (u, v E 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 r 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 + i M 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. ,v

N

N

N

1.9.17. COROLLARY. Every maximal regular (l. orr.) ideal of

ft is self-conjugate.

Hence v / A is self-conjugate.

Also, x / ~ -

ANv PROOF. By 1.9.15 (ii), every maximal regular (1. or r.)ideal of A is of the form l ~ I - M + i M whence hT/ is self-conjugate. Since v//i - nhT/, x / ~ is self-conjugate. Finally,

- N M - NIA N

A NIN

AN N

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 ~ ft, where fC is 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 ~ of A is real. But this readily follows from 1.6.15 (b) and 1.9.17. N

CHAPTER TOPOLOQICAL

1.

Topological

II

PRELIMINARIES

Qroups

and

Linear

Spaces

2 . 1 . 1 . Recall first the notion of a poset. A set ~ t o g e t h e r w i t h an ordering relation 0 and a neighbourhood x + V of x ( V a nucleus; cf.2.1.4(v)) such that for I)~I 0 and an open nucleus V such that if ])~] ~ e and x c V then Ax e U. Write h~ - {A e K'I)~ I ~< e} and W - A~V. Since W U)~V, when )~ ranges in h e \ { 0 } , W is open (using 2.1.10 each AV is open). Further, if )~EAe,]#I ~< 1 and x E V then I#AI - I#IIAI ~< e. # A e he and #Ax e W , proving W is balanced. Also it is clear that 0 E W ___ U. Finally, by 2.1.4 (vi) there is a nucleus W1 with W1 __ U. By what has been just proved there is a balanced nucleus W2 _c W1. Then W2 c W1 _C U and W2 is balanced (by (i)). 2.1.17.

PROPOSITION. Every T L S X has a basis ~l of nuclei with the following properties" (i) Each U E LI is balanced and absorbing. (ii) If U1, U2 E ~I, there is a U3 E ~l with U3 c_ U i N U2. (iii) If U c Ll there is a V E ~l with V + V C V. (iv) If U E ~l and A C N there is a V E ~l with AV C_ U. Conversely, given a n o n e m p t y family Ll of subsets of a L S X such that ~l satisfies (i)-(iii), it determines a unique topology in X making it a T L S having ~l as a basis of nuclei. PROOF. See [28, p.96]. 2.1.18. Let X be a TLS and X0 a subspace of X. Then the quotient LS X # - X / X o {x§ x E X} carries a n a t u r a l

80

Topological Preliminaries

topology, viz., the quotient topology" a subset S # C X # is open iff ~ r - l ( s #) is open in X, where 7r is the canonical homomorphism x ~ x # - x + X 0 . 2.1.19. LEMMA. The quotient space X # is a T L S which is Hausdorff iff Xo is closed in X. The canonical map r is open and continuous. PROOF. By 2.1.17 we can choose a base ~/ of nuclei of X. Write L/# - { all subsets U # of X # such that ~ r - l ( u #) E ~/}. Then it is easy to see that ~/# 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 # - X0 is closed. By definition of the quotient topology, ~r is continuous. It is also open as can be easily seen by using 2.1.4 (iii). 2.1.20. A subset S of a TLS X is called topologically bounded or t. bounded or just bounded if for every nucleus U there is a ) ~ - )~(U)~= 0 in ~: such that S __c )~U. 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 c - e(U) such that S C #U for every # E K with I#I >1 E. In particular, there is a positive integer n such that SCnU. PROOF. (i) Clear (if S G AU, S' C S then S ' G AU). (ii) Given a nucleus U there is a nucleus V with V _ U (see 2.1.4(vi)). Since S is bounded there is a # with S _ #Y. It follows that S _C #V - #V __ #U, whence S is bounded. (iii) Choose a balanced nucleus V _c U. By boundedness of S, there is a A -r 0 with S C_ AV, so that if x E S , x - Av (v E V). Set e - I A I . If I#1/> IAI then I#-IA[ ~< 1 , # - l A y e V since V is balanced), so that x-

$v - # . # - l ~ v E #V __C#U.

Also, S CC_nU for any integer n/> e. 2.1.22. THEOREM (Mazur-Kolmogorov). A subset S of a T L S X is bounded iff for each sequence (xn) in S and sequence

w 1. Topological Groups and Linear Spaces

81

(~,~) in ~ with ~,~ --~ 0, we have A,x,~ ~ O. PROOF. Suppose t h a t S is bounded. By 2.1.21 (iii) there is an e > 0 with S__C#U for I#] > e. Write r / - e -1 and c ~ - / ~ - l . T h e n c~S C U for ]al ~< r/. Since there is an integer N such t h a t IA~] ~ ~7 for n/> N, it follows t h a t Anxn E AnS C_ U for n/> N, whence )~x~ ~ 0. Conversely, if S is not b o u n d e d there is a nucleus U such t h a t S ~ )~U for any )~ ~= 0 in K. Therefore, in particular, S ~ n U ( n - 1 , 2 , . . . ) . It follows t h a t we can choose x~ in S such t h a t W ~" r U. Then clearly -h~" 74 0 (though ~1 0) violating the s t a t e d condition, which completes the proof. 2 . 1 . 2 3 . COROLLARY. For S to be bounded it is sufficient that the following condition is satisfied:

For any xn E S,

Xn

~ O.

n

2 . 1 . 2 4 . DEFINITION. A TLS is said to be locally bounded if it has a b o u n d e d nucleus. Note t h a t if U is a b o u n d e d nucleus so is every nucleus V c_ U (see 2.1.21(i)). 2 . 1 . 2 5 . PROPOSITION. Let X be a locally bounded T L S and U a bounded nucleus of X. If #n E K\{O} and #n --~ 0 then { u , U } is a basis of nuclei. Thus, X is first countable and hence semi-metrizable. In particular, every locally bounded Hausdorff T L S is metrizable. PROOF. Suppose t h a t V is any nucleus of X. Since U is b o u n d e d , there is, by 2.1.21 (iii), an e > 0 such t h a t U c #V for all # E K with I#] >/c. Since #n ~ 0 we can choose a sufficiently large n such t h a t I#,~l-l~> e. The U c # ~ I V or #nU c V. The s t a t e m e n t s regarding semi-metrizability and metrizability follow by applying 2.1.7 to the underlying additive TG of X. 2 . 1 . 2 6 . L EMMA. Let X , X* be T L S ' s and T " X --~ X* a linear transformation. If T is continuous at some point xo then it is continuous everywhere. P r o o f . Assume t h a t T is continuous at x0, and t h a t x~ ~ x. Then x~ - x § x0 --+ x0, so t h a t T x a - T x + Txo ~ Txo, whence

Topological Preliminaries

82

T x a ~ Tx. 2.1.27. LEMMA. If T 9 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 x ~ c S

then ~--' -"~ 0 n

TX~n - T ( ~ )

so t h a t

--+ T ( O ) - O ,

proving (by 2.1.23) t h a t T ( S ) is bounded. 2 . 1 . 2 8 . Let X, X* TLS's and T 9X -~ X* a linear transformation. T is said to be t. bounded or (sometimes) just b o u n d e d if it carries b o u n d e d sets to bounded sets. By 2.1.27, a continuous linear t r a n s f o r m a t i o n is t. bounded. 2 . 1 . 2 9 . PROPOSITION. Let X be a first countable (or equivalently 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 t h a t there is a nucleus U* such t h a t T - I ( u *) is not a nucleus. Since X is first countable it has a countable decreasing sequence of open nuclei as basis: U1 D U2 D - - . . By 2.1.10, u___~, is an open nucleus Since T - I ( u *) is 1~ not a nucleus we have: v___~,g T - I ( u , ) (n - 1 2 . . . ) This means t h a t there are elements xn e Un such t h a t ~ ~ T - l ( V *) (n 1 , 2 , - - - ) . Since {Un} is a decreasing basis and xn E Un, Xn --+ 0 and so x , is b o u n d e d (see 2.3.7. (a)). Since T is b o u n d e d (Txn) is b o u n d e d . This means by 2.1.21 t h a t T (z__~) _ T~, ~ 0, so 1% /% t h a t Txn e U* for n /> N. But then z__. e T - I ( u *) (n /> N ) n n contradicting the choice of x~. Hence the proposition. 2 . 1 . 3 0 . LEMMA. Let X be a TLS and f a linear functional t on X. Then f is continuous iff k e r r is closed. PROOF.

It suffices to prove the "if" part.

t i.e. a function with values in •.

We may assume

w 1. Topological Groups and Linear Spaces

83

t h a t f r 0, k e r r is closed. Then, for any e > 0, the translate Xo- {xEX" f(x)-e} o f k e r jr is closed, so that U - X \ X o is an open nucleus. By 2.1.16 (iii), there is a balanced nucleus V c U. We claim that for any x E V, t]'(x)l < e. If not there is a x0 e V with 5 - If(x0)I >/e. Write yo - (e/5)xo. Then I]'(Y0)I- e, so that f ( y o ) - eO where 0 e N, I O I - 1. Hence f(Oyo) - e, so that zo - Oyo E Xo. On the other hand, since x0 c V and V is balanced, m

0E zo - Oyo - --;-xo c V 0

since I-~] ~< 1. Therefore zo E Xo N V, contradicting Xo N V -- O. Thus, ]f(x)l < E (x c V) proving f is continuous. 2.1.31. LEMMA. Let X be a T L S with {Us} as a basis of nuclei and f a continuous functional on X with f (O) - O . Then, given C > O, there is a U o - U~ o such that ]]'(x)I < C for all

xCUo. PROOF. Set G {x E X " If(x)l < C}. Since ] ' ( 0 ) - 0, 0 c G and so by continuity of f , G is an open nucleus. {U~} being a basis we must have G _D some U~ 0 which implies the lemma. oo

2.1.32.

Let X be a TLS. A series ~ x , ~

in X is said to

rt--1

converge to x if the sequence (sn) of partial sums s,~ - X l + ' . "+ xn converges to x in X ' s n ~ x. More generally, a generalized series ~ E ~ x~, where ~q is any indexing set, is said to converge to x if the net sF, where franges through all finite subsets of .4 and S F - ~ x~, converges to x; x is called the generalized s u m c~EF

of the x~

~s . oo

2.1.33. LEMMA. If ~

x,

converges in X

then x,~ ~ 0.

n=l

PROOF. Given a nucleus U, choose a symmetric nucleus W (DO

with W §

U. Suppose that ~ x n - x . r~=l

Then there is an No

Topological Preliminaries

84 N

x,~ - x E W for N / > No. It follows t h a t

such t h a t n=l

XN+I

_

fN+lxn

-- X

c

W+W

c__ U

--

X n-

X

for N / > No.

It follows t h a t xn ~ 0.

w 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) T h e m a p (2, x) ~ )~x of E x A ~ A is continuous. (TA3) The m a p ( x , y ) ~ - . 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, yz --~ y ~ xay~ --~ xy. 2.2.2. DEFINITION. An algebra A is called a weak topological algebra or a WTA if the underlying linear space of A is a TLS and further we have: ( T A 3 ' ) The maps l a : X ~ a x , r a : x ~ xa ( x , a e A) are continuous for all a. Clearly (TA3) ~-, (TA 3' ), so t h a t a TA is a WTA. Using 2.1.26, it is easy to see t h a t in terms of net convergence the condition (TA 3 I) can be expressed as ( T A 3 " ) If x ~ - - ~ 0 then a x a ~ O , x a a ~ O for each a E A .

2 . 2 . 3 . LEMMA. For a W T A A to be a TA it is necessary and s u ~ c i e n t that (,) the map (x,y)~-~ xy is continuous at (0,0). PROOF. We have only to prove the sufficiency of the condition (,). For this it is enough to show t h a t in an A satisfying (.), (TA3) holds.

85

w 2. Topological Algebras

Assume now that (.) is satisfied, and in A x a ~ x0, yf~ --+ y0. By (TA 3") we have o;

(1)

(xa - xo)Yo --+ O.

(2)

xo(y,

- yo)

Again, by (.) we have -

- y0)

0

(3)

From (1),(2),(3)we obtain (by adding the terms) x a y z - xoYo ~ O, or , x a y z -~ xoYo.

Thus,(TA3) holds and A is TA. 2.2.4. E x a m p l e s . (i) Every algebra over K is a TA under the indiscrete topology. (ii) The algebra A - K S of all ~:- valued functions (algebra operations being point (or coordinate)-wise is a TA under pointwise net convergence : i.e., if f ~ , f E A then f~ ~ f if f~(s) --. f ( s ) Vs e S; this topology is called weak topology or topology of simple convergence. A is a commutative TA. (iii) Let .~ be a Hilbert space and B = B ( ~ ) the algebra of all bounded 1.o. t ' s on ~. B is an algebra which not commutative if dim ~ > 1. Under the weak or strong operator topology B is a W T A but not a TA (since multiplication is not jointly continuous; see [22, p.448]). The weak or strong operator topology is defined vic net convergence as follows. Ta --, T in the weak operator topology if (T,~x,y} --~ ( T x , y} for every x,y e -O, where (.} denotes the inner product of ~. Similarly, Ta --~ T in the strong operator topology if l I T ~ x - Txtl ~ o for every x E -O, where I1" II is the norm induced by the inner product" I l x l l - ( x , x ) 8 9 In the norm topology, B is a TA; Tn --~ T in the norm topology if IIT,~- TII ~ 0 (for definition of IITII(T E B), see 3.5.1). t 1.o.=linear operator.

Topological Preliminaries

86

2.2.5. LEMMA. A WTA or a TA is a TG under addition. Under multiplication a TA A is a TSG. t PROOF. The first s t a t e m e n t 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, semi-topologicaltt groups and the map

the groups Gq,Gi

are

7 -1 " a E G q ~ - ~ e + a E G i is a t. isomorphism. PROOF. By (TA1), (TA3), Gq and Gi are semi-topological groups. Also, by 1.1.20, r -1 is an isomorphism. T h a t r -1 is topological follows from the fact that translation is a homeomorphism (see 2.1.4(i)). 2 . 2 . 7 . LEMMA. closed hi-ideal of A.

In a TA (or even WTA) A, 0 -

{0} is a

B

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 x0 _c x0 - 0,0y _c 0 b

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 (respy. WTA). (b) Any direct product or direct sum of TA's (respy. WTA's) is a TA (respy. WTA). TA

PROOF. (a) Clear.

t TSG- topological semi-group, i.e., a semi-group with a topology under which multiplication is continuous. ~t A semi-topological group is a group with a topology under which multiplication continuous (inversion may not be continuous). :~ t. isomorphism - topological isomorphism, i.e. an (algebraic) isomorphism which is also a homeomorphism.

87

w2. Topological Algebras

(b) Since the operations of the direct p r o d u c t are coordinatewise and a direct sum is a subalgebra of the direct p r o d u c t , the s t a t e m e n t s follow. 2.2.9. PROPOSITION. The unitization A1 of a TA (respy. WTA) A is a TA (respy. WTA) under the product topology of A 1 - K • A. Further, A is a closed bi-ideal of A1 and A1 is Hausdorff whenever A is Hausdorff. PROOF. Clearly A1 is a TLS in either case. t h a t A is a TA and t h a t in A1 we have Xlc~

--

)i~e 1 + x~ --+ Xl -- )lel -~- x

Yl~

----

~f~el -~ y/~ --+ Yl -- ~el + y

Suppose now

where x, y C A and )~, #t~, )~, # C ~:. Then

It follows t h a t

) ~ e l + )~y + ~x + xy -- XlYl, showing A is a TA. Next suppose t h a t

A is a WTA, and yla -

# a e l + Ya --+

#el + y - Yl. Then l ~ ( y l ~ ) - )~#oLel + )tya + # a x + xyo~ --~ l ~ ( g e l + y ) - l ~ (Yl). This shows t h a t l ~ is continuous. Similarly r= 1 is continuous. Hence A1 is a WTA. We have already noted t h a t A is a bi-ideal of A1 (see 1.1.12). T h a t it is closed in A1 is clear from the definition of the topology of A1. Finally, if A is Hausdorff, {0} is closed in A and so also in A1 (since A is closed in A1 ) and consequently A1 is also Hausdorff (by 2.1.5,2.1.6). N

2.2.10.

TA

The complexification A

PROPOSITION.

WTA) A

TA

(r py. WTA)

of a real

topology o/

A• PROOF. The proof is on the same lines as the proof of the first p a r t 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 t h a t multiplication in A # is continuous. Let U # be an open neighbourhood of x # y # - (xy) #. Then U - ~ - I ( u # ) , where r is the canonical m a p A -~ A #, is an open neighbourhood of xy. Choose open neighbourhoods V,W, of x , y respy, such t h a t V W c_ U. Then, since r is open (by virtue of 2.1.19), V # - r ( V ) , W # - r ( W ) are open neighbourhoods of x # , y # respy, with V # W # c_ U #, proving the continuity of multiplication in A#. 2 . 2 . 1 2 . COROLLARY. Let A be a TA. Then A/O is a Hausdorff TA. PROOF. This is an i m m e d i a t e consequence of 2.2.7 and 2.2.11. 2 . 2 . 1 3 . THEOREM. Every WTA has a basis ~l of nuclei satisfying (i)-(iv) of 2.1.17 and (v) Given U E Ll and a E A there are V, W E [I such that aVE_U,

WaC_U.

Conversely, if ~l is a nonempty family of subsets of an algebra A satisfying (i)-(iii)(o]'2.1.17.), and ( v ) ( a b o v e ) t h e n ~l 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 ~/ of nuclei satisfying (i)-(iv). Then L/ also satisfies (v) since l~, ra a r e continuous for each a c A. For the converse we observe that by 2.1.17, A is a TLS. To prove t h a t A is a W T A it remains to show t h a t each la and each ra are continuous. By 2.1.26 it is enough to prove t h a t la, ra are continuous at 0. But this is precisely what condition (v) expresses. 2 . 2 . 1 4 . THEOREM. Every TA A has a basis [l of nuclei satisfying (i)-(iv) of 2.1.17 (v) of 2.2.13, and also (vi) Given a U E ~l there is a V E I l such that V 2 C_ U. Conversely, every algebra A together with a family ~l 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 [l as a basis of nuclei. PROOF. If A is a TA then it is a W T A and so by 2.2.13, has

w2. Topological Algebras

89

a basis ~/ of nuclei satisfying (i)-(v). This also satisfies (vi) since, multiplication in A is continuous. The converse s t a t e m e n t follows from the converse part of 2.2.13, and 2.2.3. 2.2.15. R e m a r k .

In terms of convergence of nets the condi-

tions (v),(vi) above take the forms: x~ -~ 0 then ax~ --~ 0, x~a ~ 0;

If

(v I)

x~ -~ 0, yf~ ~ 0 then x~y~ ~ O.

If

(vi I)

2 . 2 . 1 6 . LEMMA. Let A be a TA. We have" (i) If in A,x,~ ~

0 and (y,)

is bounded then x , y n

~

O,

y,~Xn --~ O. (ii) If S , T C_ A are bounded subsets so is ST. PROOF. (i) Given a nucleus U choose a nucleus V such t h a t V 2 _ U. Since (Yn) is bounded then is a scalar A ~= 0 such t h a t {Yn} C AV. Since xn---*O we can find N such t h a t x n E A - 1 V for n/> N. Therefore XnYn E )i-lv

9) i V -

V 2 C U

for n/>

N,

whence xnyn -~ O. (ii) Any sequence in S T is of the form (xn),(yn) are sequences in S , T respy.. Since bounded ~---"--~ 0 (by 2 1 2 2 ) Since {y~} c by result (i), ~"u" , -~ 0 , whence by 2.1.23, S T is

(xnyn) where {xn} C_ S is T is bounded bounded.

2 . 2 . 1 7 . LEMMA. In a Hausdorff, TA (respy. unital TA) A, if oo

(x)

the series E ( - 1 )

nxn (respy. E ( e

n-----1

- x) n) converges then x is

n=O

q invertible (respy. invertible) and its q inverse x' (respy. inverse x - l ) is given by oo X I

~(-1)nx n-1

oo

n (x - 1 -

E(e--x)n). n--O

90

Topological Preliminaries

oo

PROOF.

Write

y -

~ ( - 1 ) ' ~ x n.

Then

xoy

--

x+

n=l oo

lim ( - 1) Nx N + I -- 0 (by 2.1.33).

N--~oo

n--1

Similarly, y o x - 0. Therefore x ~ - y, proving the assertion concerning q invertibility. Again, if we set e - x - z, by 2.1.33, z ~ -~ 0. Therefore we have oo

( e - z) ~

z~ -lim(e-

z ~+1) - e, whence

rt--0

oo

OO

Z zn~ n=O

(e--Z) -1-

x-1 r~:O

completing the proof. 2.2.18.

In a TA A we denote by Ar -

Ar

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 {NX-I(O)'x~Ae} A

,~z-~ _

ifAr162 if Ac - O.

Since ~ - A is the intersection of all closed h y p e r m a x i m a l ideals (of A ) it is closed and we have clearly the inclusion relation

(,) 2.2.19.

DEFINITION. Following Michael, [20, p.48] we call a TA A f u n c t i o n a l l y c o n t i n u o u s if every character of A is continuous, i.e. if A r

A.

In a functionally continuous algebra we have

(**)

w3. Completions of Topological Linear spaces Completions

of Topological Linear Topological Algebras

spaces

91

and

2.3.1. DEFINITION. Let X be a TLS. Two nets (xa),(y#) in X are said to be equivalent, in symbols, ( x a ) , ~ (yf~) if they satisfy the condition: (.) Given a nucleus U of X there are indices a0,/~0 such t h a t x~ - y/~ c U for a >- a0,/3 >-/30. A net (xa) is called a Cauchy net or a C-net if we have (xa) (xa), i.e. if for each nucleus U there is an a0 such t h a t x a - x ~ E U for a,/3 >- a0. 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 (x~) ~ (yz) then (yz) ,.., (xa). (ii) If (x~)... (y~), (yz) ..~ (z.y) then (x~).~ (z.~). (iii) If ( x ~ ) ~ (y~) and (x~,) is a subset of (x~) then (xa,) ..~ (Y,).

(iv) Let (x~) be a net and (x) a principal net. Then (x~) ,.~ (x) i f f Xc~ ----~ X.

(v) If (xa) ... (y~) and ~ C •

then (~xa) ,'. ()~Yt~), in particular

( - x ~ ) ~ (-y~). (vi) If (x~)..~ (yz), ( z ~ ) ~ (u~) then (x~ - z~).~ (yz - u 6 ) . PROOF. (i) Given a nucleus U select a symmetric nucleus V _ U. Since (x~)..~ (yz) there are ao,/30 such t h a t x ~ - y~ E V (a ~- ao,/3 >--/3o) whence y~ - x~ - - ( x ~

- y~) ~ - v

- v c u (~ ~ ~0, Z ~ Z0).

(ii) Given a nucleus U select a nucleus V with V + V __c U, and choose a0,/30,/31,'/1 such t h a t Xa -- y~ C V(o~ ~ o/0,/~ ~ / ~ 0 ) , Yfl - z~/ E V(/~ ~ / ~ 1 , ~ ~- ~1).

Topological Preliminaries

92

Then, for a >- no,'/>- "/1, we have x a - z~ - x~ - yz, + yZ, - zu ~ V + V _ U( where ~' >- ~o, ~1). (iii) Given U, choose ao,f~o such that x a - y~ E U(a >no,/3 >- f~o). Since (x~,) is a subnet there is an a~ >- no. Then x ~ , - y~ c U(a' ~ a~,fl >-/~o), whence (xa,).-. (yz). (iv) Clear. (v) Given U, by 2.1.17(iv), there is a Y with ,~Y _c U. Choose ao,flo such that xc~ - yz ~ V ( a >- a o , ~ ~ ~o). Then a x ~ - ,~yz - ,~(z~ - y z )

E ,~v _c u.

(vi) Select a symmetric V with V + V _C U. By hypothesis there are indices no,/~o,'/o, ~o such that x~ - yz E v ( ~

> ~ o , Z >- ~o), z~ - ~e e v ( ~

>- ~o, ~ > ~o).

Then ( x ~ - z ~ ) - ( y z - u c ) - x ~ - y ~ - ( z ~ - u c ) e V + V _ C (~ ~ ~o,/~ ~ r >- ~o,6 >- ~o). 2.3.3. COROLLARY. ( i ) I f (x~) is a C-net in X a subnet then it is a C-net such that (x~,)... (xa).

U

and (x~,)

(ii) If (xc~) is a C-net in X so is ()~xc~); in particular ( - x a ) is a C-net. (iii) If (x~), (yz) are C-nets so is ( x ~ - y~). PROOF. (i) From (x~)-~ (x~), by 2.3.2 (iii), (x~,) ,~ (xa) and also by 2.8.2.((i), (iii))we get ( x ~ , ) ~ (x~,). (ii) Follows from (v) of 2.3.2. (iii) Follows from (vi) of 2.3.2. 2.3.4. L EMMA. Let X , X * be TLS's and T " X ~ X* a continuous linear transformation. Let (xa), (y~) be nets in X with (xa) ~ (y~). Then (Txa) ... (Txz). 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* 9 Since ( x ~ ) . . - ( y ~ ) t h e r e

w 3. C o m p l e t i o n s of Topological Linear spaces

are indices a0, fl0 such that xa - y ~

93

E U ( a >- a o , f l >- flo). Then

T x ~ - Ty~ - T ( x ~ - yZ) e T ( U ) C_ U*,

proving ( T x ~ ) ~ (Ty~). 2.3.5. DEFINITION. A net (x~) in a T L S X is called bounded if it is bounded as a set (see 2.1.20). A net (x~) in X is called essentially bounded if given a nucleus U of X there is an index ao=ao(U) andascalar A=A(U)~O) such t h a t

{x~ "a >- ao} _ AU. Trivially, a bounded net is essentially bounded. 2.3.6. gent n e t -

PROPOSITION. Every C - n e t (xa) is essentially bounded.

in particular a conver-

PROOF. Given a nucleus U choose a balanced nucleus V such t h a t V + V _ U. Since (xa) is a C-net there is an a0 such t h a t x ~ - x~ o E V for a >-a0, so that x~ E X~o + y

(a >- a0)

Since Y is absorbing (by 2.1.16(ii)) and balanced (by choice) we can choose A/> 1 such that Xao E AV. Then x~ E ~ v + v _q ~ v + ~ v - ~ ( v + v ) c ~ v

(~ >- ~0),

proving t h a t (x~) 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 (x~) is essentially bounded there is an integer N ~> 1 and a s c a l a r A1 such that {xn " n _> N} _c A1V. Using the absorption and balanced properties of V we can choose a A2 with xj e A2V (j- 1,...,N).

94

Topological Preliminaries

Set , ~ - max (1~11,1~21). Then

{z.} c_

___au,

proving the boundedness of the sequence (xn). (b) It is enough to construct a convergent net in R which is not bounded. Write J~ -- {O~1,~2,''"

;~1,~2,'''}

and define a partial ordering in A by ~ 1 -'~ /~2" "" ;C~m -~ ~ n ( m -

1,2,...;n-re,

m + 1,...).

Then A is a directed set (as can be easily verified). Define a net x~ (~/c ~) in E by setting 1 X a n - - n~Xf~ n =

--. n

Then x, ~ 0, but (x~) is unbounded in N (since nXc~nl -- 1 -/4 0).

2.3.8. PROPOSITION. Let A be a TA. Then we have" (i) Let (xa), (yz), (z~), (u~) be essentially bounded nets in A such that (xa) ~ (z~), (Y,) ~ (u~).

(ii) If (x~), (yz) are C-nets then so is (xay~). (iii) If xa ~ 0 and (yz) is essentially bounded then xayz --~ O, y~xa --~ O. PROOF. (i) Given a nucleus U(of A) we can find, using 2.1.17(iii), 2.2.14(vi), a nucleus V with V z + V 2 C U.

(1)

95

w 3. Completions of Topological Linear spaces

By essential boundedness of the nets we have scalars, A, # ~ - 0 such t h a t

{ y ~ . ~ ~- Z~} c ~v, {z~ .~ ~ ~1} c , v .

(2)

Since (x~),-, (z,), (y~),--(u6) we have x a - z, e ~ V ( a >- ao,'~ >"/2),Yfl- u5 E X1V(fl >- fl2,5 >- 52) Choose flo >- fll,fl2 and "~0 >- ~/1, "~2. Then x a y f l - z,~u5

--

( ~ - ~,)y~ + ~ , ( y , - u~) IV )~

&V + # V

1V - V 2 + V 2 c U # -

(a >- ao, fl ~- flo, "~ ~- "~o, 5 ~- 50)

proving (xayfl),': (z,~u6). (ii) Since by hypothesis (xa) ~ (xa), (yfl),-, (yfi) and by 2.3.6, C-nets are essentially bounded w~ conclude using(i) (xayfi) ,,, (x~y~) i.e. (x~yz) is a C-net. (iii) Given a nucleus U, choose a nucleus V such t h a t V 2 c U. Since (y~) is essentially bounded there is a A ~: 0 and a fl0 with {yfl " fl >- rio} c )~V. Since xa --* 0 we have xa E A-1V for a >- a0. It follows t h a t x~vz c v 2 c u (~ ~- ~0,fl ~- rio).

Hence xayz---, O. Similarly, y~xa ~ O. 2 . 3 . 9 . Let X be a TLS. The relation ,~ between nets in X is not only s y m m e t r i c 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 )~ the set of the resulting equivalence classes [(xa)]. To each subset S of X we associate a subset S of )( given by ;~ - { [(xa)] E ) ( " for some representative (xa) of the class we have xa E S 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.

,~ is a LS and the map j "

x ~-, Ix] is

Topological Preliminaries

96

linear. It is injective iff X is Hausdorff.

PROOF. If ~ - [(xa)], ~ --[(yfl)] E X,)~ E K we define linear operations in X by"

-

[(x~)] + [(y,)] - [ ( x ~

=

~[(~.)]- [(~x~)]

+ y~)]

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"

Ix + y ] - [x] + [y], [ ~ ] - ~[~] Finally, if X is Hausdorff and x, y E X, x ~= y there is a nucleus U such that x - y ~ U, so that (x) 7~ (y), [x] # [y]. On the other hand, if X is not Hausdorff it is not T1 (see 2.1.6). Therefore there is an x 7~ 0 with x E 0. If U is any nucleus then by 2.1.5(i), x e 0 _C U, whence (x) .~ (0), I x ] - [0], so that j is not injective. 2.3.11. LEMMA. (i) If ,S'I _C S2(___ X ) then S1 CC_$2. (ii) AS - AS. (iii) If x E S then [x] ~ S. (iv) If Ix] E ~5 then x E S, where bar denotes closure in X. (v) If (x=) is a C-net such that xa E S for a >- ~1 then

[(x~)] e ~ PROOF. (i)-(iii)" Clear. (iv) If [ x ] e S there i s a n e t ( x ~ ) . ~ ( x ) with x a e S ~). By 2.3.2 (iv), x ~ - ~ x, whence x e S.

(for all

(v) Since (x~ 9c~ >- ~1) is a subnet of (x~) 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 [{yz}] E .~. Then y~ - y~, e V(/3,/3' >- /30). Since V, as a nucleus, is absorbing, Y~o E AV for some A/> 1. It follows

97

w3. Completions of Topological Linear spaces

that for /~ >-/30 y, -

- y o) +

c v +

c

+

-

+ v)

c

Hence, by 2.3.11 (v), [{yz}] E ,~U, proving U is absorbing. Now assume that U is balanced. If [(x~)] E / ) with x~ E U, then since U is balanced, Ax~ e U([AI ~< 1), so that A [ ( x ~ ) ] - [(Axe)] e 0 , proving f) is balanced. 2.3.13. THEOREM. X can be made into a TLS such that: (i) j : x ~ [x] is continuous; if X is Hausdorff j is a homeomorphism. (ii) j ( X )

is dense in 2 .

(iii) X is complete. (iv) 2

is Hausdorff.

( v ) ) ( has the following universal mapping property: given any complete Hausdorff TLS X* and continuous linear map 99 : X --, X*, there is a unique continuous linear map ~ " fc ~ X* such that ~ - ~b o j. PROOF. Choose a basis ~/ of nuclei of X satisfying (i)-(iv) of 2.1.17. By 2.3.12, the family ~ - {0 9 U E L/} of subsets of 2 satisfies (i) of 2.1.17. Further, by monotonicity of the (set) map S ~ S (see 2.3.11(1))it satisfies ( i i ) o f 2.1.17. Finally, given U E ~ we can choose a V E /2 with V + V __ U. Select [(x~)],[(y~)] ~ V. We may assume that (x~),(y~) C_ V. Then [(x~) + ( y / ~ ) ] - [(x~ +y/~)] C U since x~ + y~ e U. Thus 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 Z) as a basis of nuclei. It remains to prove the statements (i)-(v) above. By definition of D we have j ( 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 anucleus W of X with W c_ V. If [x] E then x c W _c V. It follows that j - l ( I ~ ) c_ V, whence j - 1 is continuous. This completes the proof of (i). For proof of (ii) it is enough to observe that j ( x ~ ) - [xa]--~ [(xa)]. To prove (iii), Let x~ -[(x~(u))] be a C-net in ~7. Therefore,

Topological Preliminaries

98

given a nucleus U of )~, for each ~/ there is a ~/u such that ~-~,

E U for ~/,~/~ >- "/v , i.e,. we have x ~~ ( ~ )

-

x (~' ~,)

C

U

for all a(-/), a(~/') with ~/, ~/'>- ~/~. This mean that the net (with index ~/) of principal classes ~

By considering ordering

(X~(v)) as

(.(u)-

a net indexed by pairs (7, U) with the

(~/, U) < ('r', U ' ) i f "~ -< -/', U ' _ U it is clear that it converges to the element [(x (u))] of X -.

[(x

o

So

~(u))], completing the proof of (iii).

For proving (iv) take ~ - [(x~)] ~: [0]. Then x~ ~ 0. So we may assume that there is a nucleus U of X with the representative (x~) of ~ disjoint with it. Choose a symmetric nucleus W with W + W c U. We claim that I/V ~ [(x~)]. For, if [(x~)] E I2r then there is a C-net ( y z ) E W with (xa),-~ (yz). It follows that there are xa,, ya, with x ~ , - yz, E W. Then x~,-x~,-y~,+y~,EW+W

c U

contradicting U 0. Thus, I?V is a nucleus of 2 not containing ~, whence X is T1 and so Hausdorff. Finally, for proving (v), take an element [{x~}] e )(. By 2.3.4, (~p(x~)) is a C-net in X* and so converges uniquely to x* e X*. Set -

This is well-defined, since if (x~) ,~ (Ye) then ( ~ ( x a ) ) , , ~ (~(yz)), so that ~ ( Y e ) + x*. Also, ~(x)-

~b([x])- # o j(x).

Finally, if r is a continuous linear map with r o j - ~b o j then clearly r - ~5 on the dense subspace j(X) and hence everywhere.

w3. Completions of Topological Linear spaces

99

This completes the proof. 2 . 3 . 1 4 . THEOREM. If A is a TA then its completion ~i (as a TLS) is a complete TA. PROOF. By 2.3.13, A is a complete TLS. So we have only to show t h a t A is a TA. If ~ - [(x~)], ~ - [(Yt~)] are two elements of A then, by 2.3.8(ii), (x~y~) is a C-net and so determines an element [(x~yz)] of A. We define multiplication in A by setting ~!) - [(x~yz)] (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 t h a t A is a TA, consider a nucleus U and an element a E A. Since la is continuous there is a nucleus V with aV c_ U. Suppose now t h a t ~n - [(x~(n))] -~ 0 in ft.. Then there is a "/0 such t h a t for ~/>- ~/0, x n~(n) E Y if a(~/) >- some an(~). It follows that

ax~(~) c aV c_ U whence 5x~n ~ 0 .

(1)

Similarly, using continuity of ra we get

x~a -~o

(2)

The conclusions (1),(2) imply t h a t A is a WTA. Further, since A is a TA, given a nucleus U, there is a nucleus V such t h a t v 2 c u.

Assume now t h a t

~

- f(x (~))] -~ o, g~ - [(y~(~))] -+ o.

Then we have for some "/o , ~o, x ~~(u),Y ~ (6) E V 2 _c U, whence

x~}6 --~ 0, completing the proof t h a t A is a TA.

(3)

CHAPTER

SOME

III

TYPES OF TOPOLOGICAL ALGEBRAS

w 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: ( q l ) p(0) = 0;

(Q2) p ( - x ) = p(~)(~ c x); (Q3) p(x + y) 0. We write X = (X, P) to denote X with the P-topology.

106

Some

Types o f Topological A l g e b r a s

Two families Pl, P2 of quarter-norms on X are said to be Pl "~ P2 if the Pl-topology and the P2topology are the same: (X,)~ - (X, P2) (cf. equivalence of 2 sad.'s in 3.1.4). e q u i v a l e n t , in symbols,

3.1.18. X

LEMMA.

If pj(j

-- 1 , . . - , n )

are q u a r t e r - n o r m s

on

so are P -- Pl + "'" + Pn,

Further,

p ~ {Pl,'",Pn}

q -- Pl V . . .

V Pn

'~ q.

PROOF. By 3.1.7 (b), p and q are sad. functionals. The verification that they are also quarter-norms is straightforward. Further, since p(x

- x)

o

p i ( x a - x) ~ q ( x ~ - x) ~

0 (j-

1,...,n)

0

the equivalence statement follows. 3 . 1 . 1 9 . LEMMA. ( a ) I f p , p * are q u a r t e r - n o r m s t h a t there are c o n s t a n t s C 1 , C 2 > 0 s a t i s f y i n g

Cap 0 there is an a0 - a0(c) such that p ( x ~ - xf~) < e for

107

w 1. Quarter-norms

all a, fl >- a0. It is called a C-net with respect to a family P of q u a r t e r - n o r m s if it is a C-net with respect to each p in P. X is called P - c o m p l e t e if every C-net with respect to P converges in X. 3 . 1 . 2 1 . LEMMA. Let X family of quarter-norms on X .

(X,P) Then:

be a TLS, where P is a

(i) B~,v - {x c X " p(x) < e} is a s y m m e t r i c nucleus of X , for each p E 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 T L S iff it is P - c o m p l e t e .

(v) If f is a continuous functional on X ker P ___ ker f.

with f ( O ) - 0

then

PROOF. ( i ) B y definition of P-topology, B~,v is a nucleus of X; further it is s y m m e t r i c since p ( - x ) - p ( x ) ( x e X ) . (ii) Since p(0) - 0, for any p, 0 E ker P. Now ker P is closed by virtue of 3.1.5 (iii). So, 0c_ k e r P . If x E k e r P , p(0-x)-p ( - x ) - p(x) -- 0 (Vp), so that 0 , 0 , . . . ~ x, whence x e 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 c k e r P - 0, we have 0 , 0 , . . . ~ x. By continuity of f, the sequence f ( 0 ) , f ( 0 ) , . - . ~ f(x). Since f(0) - 0, we get 0 , 0 , . . . ~ f ( x ) . By uniqueness of limit property in K, we conclude t h a t f ( x ) - O, completing the proof. 3 . 1 . 2 2 . PROPOSITION. Let X (X,p) be a quarter-normed L S and Xo a subspace of X . Write X # - X / X o and define p# on X # by p # ( x + X0) - i n f { p ( x - 4 - a ) ' a e X0}. The

I't "

(i) The canonical map ~r " x ~ x + Xo - x # - r ( x )

satisfies

Some Types of Topological Algebras

108

(ii) The functional p# is a quarter-norm. (iii) The topology of the quotient space X # is induced by p#. (iv) p# is faithful i g Xo is closed in X. (v) If p is complete so is p#. PROOF. (i) Evident. (ii) We have

p#(x + y+ Xo)

0. If x C A and p(x) ~ r0 then y-

x + x0 E B __ A~o;

in p a r t i c u l a r xo - 0-4- xo c Ano. It follows t h a t zU.

o -

yU.

o -

x 0 U . o __ U +

U.

Given any nucleus V, we can find, using 2.1.14 (vi), 2.1.16 (iii), a closed balanced (hence s y m m e t r i c ) nucleus U with U + U __ V. Therefore

xU,~o C_ U + U C_ V. This means t h a t if

p(x) < r0, p(y) < t See [16, pp.200-1]

1

no

(i.e. y E Uno)

Some Types of Topological Algebras

118

then xy E xUno c_ V, proving t h a t the map (x, y) ~ xy is continuous at (0,0), whence by 2.2.3, A is a TA. 3 . 3 . 3 . DEFINITION. Let S 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) 0 such t h a t

p(x, y)1 1 for all n >>.1, or p(x n) ~ O. PROOF. Then

Suppose t h a t p(x ~~ < 1 for some integer no > O.

p(x .O)

o

k

Write M m a x { p ( x " ) : 1~< n 0, choose No such t h a t for k>lNo, p(x kn~ < ~ . Write N - N o n o . Then for n>~N, n = q n o + r , with q / > N and r < n 0 . Hence

p(X n) ~ p(xqn~

e .Mr) < --~

e,

and p(x '~) --~ O. 3 . 3 . 5 . DEFINITION. Let p be a non-negative real-valued function on the semi-group S. Set up(x) - lim sup p(x '~) -};up (x) - sup p(x n) 88 n----4 O 0

n

w

Quarter-norm Algebra; (F) Algebras

119

Then clearly 0 ~< ~;(x) ~< ~;(x) ~< ~r Moreover, it is evident t h a t v p ( x ) < oo iff vp(x) < oo. 3 . 3 . 6 . LEMMA. (Gelfand t ) . For an a. sm. p we have

Up(X)-

lim p(xn) -} < oo

(x E S).

tl----*oo

If p is sin. then we have also u p ( x ) - i n f . p ( x = ) 8 8 PROOF. Assume first that p is sm. and set c - i n f p(x'~)~ ( 0, there is an integer k > 0 such t h a t 1

p ( z k) -~ < c + ~.

(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) < k. Then

1

qCn)

rCn)

k

n

nk '

so t h a t

q(n) n

---~

1 k

~

as

D,-----~

c~

(3)

(since r ( n ) / n k < 1). Using ( 2 ) a n d sm. property of p we get n

(4) Using (1), we have, for sufficiently large n,

c < p(x ~) .~- O) is also an a.sm. quarter-norm (respy. pseminorm) with tltp I - [ p l , tp ,,~ p. (ii) If p is a sm. quarter-norm (respy. 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 I P is sin. and p*,,~ p. PROOF. (i) Clearly pl _ tp is a quarter-norm (respy. pseminorm). Also,

p'(xy)-

tp(xy) ~ 1} (see

rt--0

[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 ~" - (~,1" la) is a unital (F) algebra. ~' is not locally bounded (ibid, Theorem 2 ) a n d hence, by 3.3.2, the (F) norm I" la is not a.sm.. Finally, ~' is separable since the set of polynomials with complex t rationalt coefficients is dense in ~. (iv) The entire functions form an algebra also with respect to pointwise linear operations and Hadamard multiplication" if oo

f-

co

~ a , ~ z '~, g -

~

n--O

n--O

~nz '~ the Hadamard product f x g

is defined by t i.e. having rational real and imaginary parts.

w

Quarter-norm Algebra; (F) Algebras

125

oo

rt=0 (Note that pointwise multiplication of f and g corresponds to Cauchy multiplication of the associated power series.) We denote the above algebra by $ ( • ~'(• has no unity element (since 1 + z + z 2 + . . . is not entire). Also, we have clearly

If • gig 0 there is a c o m p a c t set g = KE such t h a t f = 0 on S \ K ) . C o ( S ) i s also a Banach algebra under the sup norm; of course, if S is compact, Co(S) -- C ( S ) . If S is locally compact Hausdorff and Soo is its one-point compactification then it is easy to see t h a t C(Soo) is the unitization of Co(S). 3 . 4 . 7 . PROPOSITION (Zelazko). Let G be a discrete TG. Then the set L p = L p ( G ) = Lp(G,K), O < p~< 1, of all K-valued functions x = x ( s ) ( s e G) such that ~ , Ix(s)[P < co, is an algebra (over K) under pointwise linear operations and convolution as multiplication: if x, y E L p then its convolution product is defined by x , y ( s ) - Etx(t)y(t-ls). For s e G, set xs(t) - 1 or 0 according as s - t or s r t Then xs E L p, and the map s ~-~ x8 is 1-1 and x~, xt = x~t (so that G is multiplicatively embedded in

w4.

p - S e m i n o r m e d Algebras; p - B a n a c h Algebras

135

LP(G)). Moreover, L p is a p - B a n a c h algebra under the p - n o r m - E 8

PROOF. T h a t L p is a LS and I1" lip is a p - n o r m are clear (the

subadditivity of []. IJp depends on the inequality (s + t) p 0 , 0 < p ~ < [ix * Yllp

=

x ( t ) y ( t - l s ) [ p 0 such that

on

X* - (X*,p*) be LS's (with p r 0) T is said to be n. bounded if there is

p* (Tx) 1p* (Tx,,) ~ c~, which contradicts t h a t C3 < oc. On the other hand,if there are only a finite number of x , with 0 < p(x,~) ~< 1 then we may assume after omitting these that p(x,) - 0 for all n, p * ( T x , ) co. Since p=/= 0 there is a y with p ( y ) - 1. Set z , ~ - x , ~ - y ; then by 3.1.2 (iii), p ( z n ) - 1. Since

p* (Txn) - p* (Ty)l ~1 and - 1 - n t Since 0 ~< fn ~< 1 and fn(O) - 1 we have I l f n l l - 1. Also 1 Thus f0 is a t.z.d . . f o f , ~ 0 since Ilfof,~ll 0, whence 11/21111/wl(a) < cr Since

[[x~ - Xml] 0 t Gelfand considered only the case p - 1.

168

Some Types of Topological Algebras

that

of A.

llx II

c

II.II

p- o m

PROOF. Since a unital p - B a n a c h algebra is a CI algebra (by 3.6.23(b)) and the boundedness in A is the same as the boundedness with respect to ]]. ]1 (by 3.2.13), the corollary follows from 3.7.24. 3.7.2,6. COROLLARY. In a unital p - B a n a c h 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 OGi t 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 0Gi is disjoint with Gi. 3.7.2',7. COROLLARY. Every element of the radical x / ~ of a unital p - B a n a c h algebra is a s.t.z.d. (cf. 3.7.17). PROOF. If a E v/A then since na E x/~, na is q. invertible and consequently (e § na) -1 exists. It follows t h a t (~ § a) -1 = n(e § na) -1 exists, whence by 3.7.26, a - l i m , ( ~e + a) is either invertible or a s . t . z . d . . But a cannot be invertible since a E x / ~ , so a is a s.t.z.d, as required. 3 . 7 . 2 8 . PROPOSITION. Let A be a unital p - B a n a c h algebra, x c A and a(x) the spectrum of x. If a(x) 7s tt and )~ c Oa(x) then x - Ae is a s.t.z.d. . PROOF. Since A E On(x) and a(x) is closed, there is a sequence )~n e p ( x ) = K \ a ( x ) s u c h t h a t )~, ~ )~. Then x A~eCG~.. x - ~ e ~ Gi. Since x - ) ~ , ~ e ~ x - ) ~ e it follows t h a t x - Ae E ,~Gi and consequently, by 3.7.26, x - ~e is a s . t . z . d . . 3.7.2!9. PROPOSITION (Rickart). Let A - ( A ,

li" II)

be a

,t For a subset S of a topological space X we denote by OS the frontier of S i.e., OS(- OS') - s - A s ' , where S ' = X \ S and bar denotes closure. tt This condition is satisfied for every element x if A is complex or strictly real. Moreover, whenever cr(x) ~ 0 we have also O~(x) ~ O, since by 6.1.2, a(x) ~ [~, is closed but not open (by connectedness of [K).

169

w 7. Topological Zero Divisors

p - B a n a c h t algebra with I111 ~m. and xn --. x in A with x,~ +-+ X (for all n). Let v - Vll. I. Then"

(i) If A is unital, x,~ invertible a n d V(Xn 1) i8 bounded then x is invertible.

(ii)

If xn are q. invertible q. invertible.

and

u(x~)

bounded

then

x

is

PROOF. (i) Suppose that V(Xn 1) ~< C. Then we have

V(XnlXrt -- XnlX)

V(Xnl)V(Xn -- X)

(using 3.3.7. (iii)) ~
0. It follows t h a t 0 E 0a(x2), so that by 3.7.28, x ~ 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(= s z U s ~)- s hi- s t - s ~ - 3 u - 3 "t- 3 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 . 3 4 . R e m a r k s . 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 = f ( z ) on the closed unit Izl ~< 1 which are holomorphic on tzl < 1, the function fo(z) = z (Iz] 0 and ~P +/~P - 1

(.)

ax + fly E S. It is called absolutely p-convex if x, y e S ; c ~ , / ~ E K

and I~1p+I/~t p~ O,x E flS}.

The non-negative realvalued function p is called the p-gauge (gauge if p - 1). The gauge of S is also known as the Minkowski functional of S. It is clear from the definition of the p-gauge t h a t we have"

Ps2 10. (iii) p ( A x ) -

[A["p(x) for all A E K, provided S is balanced.

provided S is

(iv) p ( x + y) ~ p(x) + p(y) for all x , y E X p -convex.

In particular, p is a p-seminorm if S is absolutely p-convex (and also absorbing). 1

PROOF. (i) Since S is absorbing, 0 E S and hence 0 E a ~ S for all a > 0, whence p(0) - 0 1

(ii) x E a ~ S

1

iff Ax E (APa)~S(A > 0). Hence p(Ax) -

~p(x) (iii) Assume first that

I A I - 1; then A - 1 S - S 1

since S is

1

1

balanced. It follows that Ax E a ~ S iff x E a ~ A - 1 S - a ~ S . Therefore p(Ax) - p(x). Next for arbitrary A e K, A ~= 0, write A - [A]# where I # l - 1. Then, by (ii) (since I # ] - 1). Finally, the relation clearly holds if A - 0. 1

(iv) Given e > 0, we can choose a, fl > 0 such that x E a ~ S , 1

y C f3-~S ; a < p(x) + e, fl < p(y) + e. Since, by 4.1.5, 1

it follows that

1

1

180

Locally Pseudo-convex Spaces and Algebras

From tile arbitrariness of c we conclude that

;(x + y) < p ( x ) + ;(y) proving the first result of (iv). For the second result we remark t h a t it follows from this and (iii), since an absolutely p-convex set is both p-convex and balanced. 4 . 1 . 1 1 . LEMMA. Let S j ( j - 1 , . . . ,n) be absorbing balanced sets and S - $ 1 ~ " " ~ Sn; then S is absorbing and balanced. If pj, p are p -guages of Sj, S respectively then p - pl V . . . V pn. t PROOF. T h a t S is absorbing and balanced are easy consequences of the definitions (see 2.1.15). Since S c Sj we have pj ~ 0 such that p(xk) < e < 1 for all k. Then 1

1

N

e p xk C U, so that e p z E U, whence /5(z) 0,,~z C U then by (3), ~x,)~y E U, so that

p()~x),p()~y) < 1, whence AP max(p(x),p(y)) < 1.

(7)

For a given e > 0, we can choose the A > 0 such that 1

Ap
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 {Us} as a basis of pseudo-convex nuclei t h e m the gauges p~ associated with the U~ determine a family P - - { p ~ } of pseudo-seminorms Pa.

~. The element xt~ E Xt~ is identified with the element x - (x~) in X suchth~t x ~ = 0 if c ~ # p , x ~ - x ~ if ~--fl.

Locally Pseudo-convex Spaces and Algebras

190

4 . 3 . 3 . Let G* denote the set of all pseudo-seminorms on a LS X. We introduce an order -< in G* by writing i

I

p -< p' if p~ < p ;

1

I

(i.e. p(x)-; 0. Therefore S is bounded. -

-

Let X - (X, P), Y - (Y, g)) be locally pseudo-convex spaces, with P saturated. Then, a linear transformation T " X --~ Y is continuous iff for each q~ E ~_ there is a pa E P and a constant C - C a ~ > 0 such that 4.3.11.

PROPOSITION.

p~

(,)

where pa,p~ are respectively the homogenity indices of pa,p~. 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 q/~ and E > 0 there is a pa and a fi > 0 such that

(**)

p~(x) O such that I

pl

pl

p (~v) 0 then is a p0 E - P and a 5 > 0 such that (1)

197

w4. Locally Pseudo-convex Algebras 1

For x , y

~ A with po(x),po(y) ~ O, set

1

Xl

--

1

t~PX/po(X)

P---~

1

Yl -- ~ o(---~y/po(y)~o where P0 is the homogenity index of p0. Then po(xl) - - 5 - - PO(Yl), whence by (1), p ' ( x l y l ) < e which reduces to

Cpo(x)

oa

~

opo(y)oo

(2)

where C - E/5 2p'Ip. It remains to consider the case where po(x) or po(Y) - O. Suppose t h a t po(x) - O. Then po(n2x) - 0 for any integer n ~< 1. For arbitrary y c A we can choose n (large enough) t h a t

, y

p'(y)

n

rt pl

"

Then, by (1), p ' ( n x y ) - p ' ( n 2 x . y / n ) 0).

It follows t h a t ( x s ) - S varying a m o n g all the finite subsets of - is a C - n e t . Since A is c o m p l e t e xs ~

(some) x E A.

Since ~ is c o n t i n u o u s we o b t a i n y~ - - ( g 9 ( x s ) ) ~

( S varying)

--, (~p(x))~ -- x~,

i.e. y~ -- x~, so t h a t y - ~p(a), proving t h a t ~p is "onto". We n e x t show t h a t ~ - 1 is continuous. Consider a net (x (~)) in A such t h a t

c A) T h e n by 3.4.16 we have p a ( z (~) - x) -- p#a (x ('x) - x a ) --+ 0 for all a.

Locally Pseudo-convex Spaces and Algebras

204

This means that x (~) -~ x in A, proving the continuity of ~9-1. To complete the proof of the first statement of the theorem it remains to show that A0 - A0. Now any non-empty open set Ifd of A0 contains an open set V of the form

where ( ~ j

N...

N, o,

are open sets in fi.~j and r--1 the natural projection

~rA~-+ A~j ( j - 1 , - - . , n ) choose an (Xrt+l E .~ with an+l >" a j ( j = 1 , . . . ,n). Then we have the homomorphisms

~j,Olnj_l

"

(j= 1,...,n).

l~Olnj_l ~ l~Otj

Set _

,,-1

Gc~,~+l -- ~1,~,~+1 Then

Set

~n,~n+l

~o,j,o,,,+, (G~,,+,) ___r C~rtA-1

n "

(j= 1,...,n).

c~n+l

~

9

Since A~.+I is dense in A~.+~ we can find x E A so that x~.+~ E (~,~+~. It follows that ~(x) E U, so that U n A 0 # 0. Therefore A0 is dense in A0. But A0 is complete, being t. isomorphic to the complete space A and consequently closed. Thus, A0 - fi-0. The second (or converse) statement follows since every pseudoBanach 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. p-convex algebra then it will be referred to as a p-Michael algebra. Finally, if p - 1, A is called

w5. 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 lim/}~ of pseudo-Banach 4--

algebras [~s. PROOF. of A.

It suffices to apply 4.5.3 to the completion

B -

4 . 5 . 6 . PROPOSITION (Michael). Let A -

(A, P) be a pseudoMichael algebra with projective limit decomposition A--limzi.s (A~ a pseudo-Banach algebra). For x E A, x--limxs (i) x -

(xs C fis) we write x -

(xs). Then:

(xs) is q. invertible in A iff xs E fis is q. invertible

(for ach (ii) A is unital iff all ft~

are unital, and then the unity e (es), es being unity of fis. (iii) u = (u~) is idempotent in A iff for each a, u~ is idempotent in fis. (iv) If A is unital, then an element x - - (xs) is invertible iff for each (~, xa is invertible in fia. PROOF. (i) If x is q. invertible then, by 1.1.24, xs is q. invertible in As and so also in As _~ As. Conversely, if for each a, x s , has q.i. xs, ' then ( x ~ ) C A To see this, note t h a t i f / ~ > a , ~9sZ ..3,Z --~ As the element !psf~(x~) is q.i. of ipsz(xf~) - xs, so that ~f~(x~)-

x~. ' It follows t h a t

~(x~) e limfi,~ - A, whence

there is an element x I E A with (xl)s - x s. Since multiplication and addition are component-wise it is clear t h a t x ~ is q.i. of x. (ii), (iii): These follow at once since multiplication in A is component-wise. t Such algebras h~ve ~lso been considered by Areas.

Locally Pseudo-convex Spaces and Algebras

206

(iv) The proof is similar to that of (i). 4.5.7. COROLLARY. For x -

(x(2) E A we have:

(1 / a~(x) -- Ua2~(x(2); c~

(2) rA(x) -- sup rA,(xa); c~

(3) aA(X) -- U a~i~(x~) whenever A is unital. cg

PROOF. By 1.7.8 A(r 0) E a ~ A ( X ) i f f - - A - i x is not q. invertible, i.e. by 4.5.6 (i), iff for some a, - A - l x a is not q. invertible in A~ iff A E a'^ (xa) This proves (1) and (2)readAs " ily follows from (1). For (3), assume now that A is unital; then each .1i(2 is unital. From (1) and 1.7.21 we obtain (x)U(0)

-

9

If 0 ~ aA(X) then x is invertible and consequently each xa is invertible, so that 0 ~ UaA~(x~). On the other hand, if 0 e c~

aA(X) then x is not invertible and consequently, some x(2 is not invertible so that 0 E aA1 (x(2). Thus, the equality in (3) holds in both cases.

4.5.8. COROLLARY. PROOF. bi-ideal. So

N~I(~/Aa)

_ x/~-- N~9~l(v~a).

(2

(2

By 1.2.17 (i), each ~ I ( ~ A ~ ) o r

(2

(fi)~l(~~a) is a

(2

are bi-ideals of A. By 1.2.26, ~ ( V ~ ) C V~(2. So x / ~ _ I. On the other hand, if x E I then x(2 E V~(2 and so x(2 is q. invertible. Consequently, by 4.5.6 (i), x is q. invertible so that I is q.i. bideal, whence I ___ x/~. Thus x / ~ I. Again, if x E J then

w6. Metrizable Locally Pseudo-convex Algebras

207

x~ E V//i~, x~ is q. invertible and consequently x is q. invertible. Thus, J is a q. invertible ideal whence J c_C_x/~. w6.

Metrizable

Locally

Pseudo-convex

Algebras

4.6.1. PROPOSITION. Every first countable locally pseudoconvex algebra A is semi-metrizable and has the form A - (A, P), where P - { p l , p 2 , ' " } is a countable well-behaved family of pseudo-seminorms such that Pl ~ P2 -< "'" t

(,)

PJ p j ( x y ) ~ Pj+I (X)~176

Pj+I

(j1 , 2 , . . . ) , where pj, indices of Pj,Pj+I.

are

Pi (**)

respectively the homogenity

PROOF. That first countability implies semi-metrizability lows from 2.1.7. Since A is first countable its topology is duced by a countable family Q - {q,} of pseudo-seminorms. P ~ n - ql V . . . V qn and P ' - { p ~ } . Then p~ -< p~ -< . . . . PI --P~. By 4.4.4, there an integer i(1) such that

folinSet Set

Pl (xy) -- pl1 (xy) ~ C l p ~ ( 1 ) ( x ) ~o, pi(1) I (y)-~P'

where Pl, p, are homogenity indices of PI,P~(1) and C1 a constant __E_

which we may clearly suppose /> 1 9 Set p2 - C~~ i'( 1 ) "

Then

P2 ~" P'i(1) and satisfies pl

pl

For P2 we can similarly choose P~(2) ~ with i(2) /> i(1) ~ 2 such that C

!

! 1

1"

t Recall that pj -K Py+l means: pjoy ~ pyPiu

and py+l ~ py.

208

Locally Pseudo-convex Spaces and Algebras

where we can assume t h a t C2 ) C1

9

Set p 3 -

C 2"~2 v"~ i(2)

and we

have

Thus proceeding we get ~3 _ { P l , P 2 , " ' } with the desired properties. It is clear from the way we constructed t h a t P is equivalent to a cofinal subset of P~ ~ whence ~3 ... pt ..~ Q 9 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-~ algebra. If A is also complete it is called a pseudoFrechet or a pseudo-~ algebra. 4.6.3.

DEFINITION. A p s e u d o - p r e - ~ algebra A is called a locally sm. p s e u d o - p r e - ~ algebra if its topology can be induced by a countable family Q = {qn} of sm. p n - s e m i n o r m s for some p,~ with 0 < p n ~ < 1, n = l , 2 , . . . . 4 . 6 . 4 . LEMMA. Every pseudo-pre-~ algebra A is of the form

A = (A,~_) where ~_ = {qn} is a countable family of pseudoseminorms with ker Q = {0}. If A is locally sm. pre-pseudoalgebra then we can choose ~_ such that each q E ~_ is sin.. PROOF. This follows from 4.6.1, 3.1.21 (iii). 4.6.5. DEFINITION. If in a pseudo-pre- ~ algebra A = ( A , Q ) , all q E Q are p - s e m i n o r m s for some fixed p ( 0 < p ~ < 1) then A is called a p-pre- ~ algebra. If in addition each q is s m . , then A is called a locally sm. p-pre- ~ algebra. The meaning of a p- ~ algebra or a locally sm. p- ~ algebra is clear. Finally, when p = 1, we say simply ~ algebra or locally sm. ~ algebra as the case may be.

The locally sm. pseudo-~ (respy. locally sm. p - ~ ) algebras are precisely projective limits of sequences of pseudo-Banach (respy. p-Banach) algebras. 4.6.6.

PROPOSITION.

PROOF. This follows from 4.6.4, 4.5.3. 4 . 6 . 7 . PROPOSITION. Every pseudo-pre- ~ (respy. pseudo- ~)

209

w 6. Metrizable Locally P s e u d o - c o n v e x Algebras

algebra A is a p r e - ( F )

(respy. ( F )

PROOF. Let A ker ~ - {0}. Then

(A,Q),

Ixl - ~

1

-2~

algebra.

~- -- {qn " n -

l + q,(x)

1,2,---}

and

)

is a (F)norm and A ( A , Q ) is complete.

(n, I.I). Further, I'l is complete whenever

4.6.8. algebras

of metrizable

Examples

locally pseudo-convex

(i) Let (p,) be a sequence of real numbers such t h a t 0 < pn N2. It follows that for n/> N - max(N1, N2) and arbitrary rn

p(xny m) N,

p((x + y)2n)~ v ( x ) ; then

V(--)~-lx)- I~1-,~(~) < 1, so that by 3.3.19, -A-~x is q. invertible. It follows by 1.7.8 that 1

~'(.) c {~ ~ K I~I-< ~(~)a} so that (,) holds. N

If A is real we apply (,) to A and deduce (**). 4.8.12. COROLLARY. In any p - B a n a c h algebra a t. nilpotent element is q. nilpotent. 1

PROOF. Since r(x) ~< v(x)7, hence the result.

v(x) -

0 ::~ r(x) -- 0 and

CHAPTER V SOME

1.

Vector-valued

ANALYSIS

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 f : G --~ X is called weakly differentiable if there is a function g : G --~ X such t h a t for each x* E X*, the scalar-valued function x ' f (A) - x*(f(~)) has the scalar-valued function x*g(~) as its derivative in the usual sense; we write g - f~(w) and call f~(w) the weak derivative of f. The function f is called strongly differentiable if for each 2 E G, f ( # ) - f(A) lim exists in the topology of X. We denote this limit by /'()~) and call f ' the strong derivative of f. 5.1.2. LEMMA. A strongly differentiable function f is weakly differentiable and continuous, and moreover, the strong derivative fl coincides with the weak derivative fl(w). PROOF. It follows from the definition of the strong derivative that lim ( f ( # ) - f(,~)) - lim (/~ - )~)f'()~) - 0 proving f is continuous. Again, lim

x* f (tt) - x* f (A)

= lim x * ( / ( # ) - / ( ) ~ ) ) ~-~ #-~

x*f'(~),

whence f is weakly differentiable with f l ( w ) _ fl. 5.1.3. DEFINITION. A function f : G - - - , X is called weakly analytic if x*f(A) = x*(f(~)) is an analytic function of ~ on G in the usual sense, for each x* E X. We call f strongly analytic on G

w1. Vector-valued Differentiability and Analyticity

223

if around each point Ao E G _C K, there is a neighbourhood {A E K "l A - ,k0l < r0} in which f has a power series representation t oo

rt--0

where x , E 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 t h a t oo

rt--0

Then, for each x* E X*, we have oo

n=O

so t h a t f 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 f : G --, X, weak holomorphy of f ==~ strong holomorphy of f (see [24, p.79]). 5 . 1 . 6 . Let X -

(X, I1" tl) be a p - B a n a c h space and oo

rt:0

a series in X. P u t t In the representation we find it convenient to write the scalars on the right side (instead of the customary left side)

Some Analysis

224

1

1

r0 -

1

lim I[x,]l~,R0 - ro 1 , R - Rg - ro ~ (lim - lim sup).

Ft--~ OO

R is called the radius of convergence and {A e K ' I A I -- R} the circle t of convergence of the series (,). Next let X - (X, {p~}) be a complete n a u s d o r f f locally pseudo-convex space. Set 1 n---~ (x)

R

-

1 P~

infR~-infr~

R again is called the radius of convergence of the series ( , ) and {A e K ' I A I - R} its circle of convergence. 5 . 1 . 7 . PROPOSITION. (a) The series (,) converges absolutely

uniformly for IAI R the terms of the series are unbounded and consequently the series diverges (i.e. does not converge).

(c)

diiT

got ] om (,)

ti t d oo

rt--1

PROOF. We will first treat the case where X is a p - B a n a c h sp~ce ( x , II-II). The proofs are all modelled after those of the corresponding classical results. oo

ll .ll(l l ) - - which is a

(a) The absolute-valued series n-0

numerical series - converges, by the classical Abel's lemma, for IAI p < R0, or equivalently, for IAI < R. Further, this series t In the real case the circle of convergence is to be interpreted as the end-points of an interval.

w 1. V e c t o r - v a l u e d

Differentiability

converges uniformly for IA[p R, choose r so t h a t IAI > r > 1

rP


~-~, or [[Xn[[ > rp n It follows t h a t ---~ 0 0 .

T h u s the series (,) is not b o u n d e d (for [A] > R ) and so not convergent (by 3.2.15). (c) If we put r ' -

1

lim[l(n+l)xn+l[[~,

then

tl--* oo

1

n-i-1

r' --n-,oolim[{(n + 1) a--4-f}P [[Xn+l II h--4-Y] --K-1

1 _

_

di~rn~ ]]xn+ll['~-I

_ r,

1

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 t h a t if [A[ < R~ (for each a , ) so t h a t oo

the proof above for (a) shows t h a t

~-[~pa(Xn)([A[P") n < oo, for n--0

each pa, where X ( X , { p ~ } ) . Hence ( a ) f o l l o w s for the locally pseudo-convex case. Further, if [A[ > R t h e n there is an a such t h a t IAI > R~, whence it can be shown, as above, t h a t p~(xnA n) --~ c~, so t h a t the series is not p a - b o u n d e d . Therefore the series is once again divergent, proving (b) for the general case. Finally, if we set r ~'

-

pa((n + n----~ o o lim

1

1)Xn+l)"+'

then as in the proof above for p - B a n a c h spaces we o b t a i n r~ - r ~ot~ whence R ~ - R, where R' is the radius of convergence of the

Some Analysis

226 series (**).

5.1.8. PROPOSITION. The function oo

rt--0

is strongly differentiable inside its circle of convergence and we have

E oo

ft()~) _

Xnn~n-l"

(**)

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 x , e X = (X, {p~}) and assume that the radius of convergence of the series (,), R > 0 (if R = 0 there is nothing to prove). Choose a A c N: with IA] < R. Take a real number r with IAI < r < R. Writing O(3

n--1

we have, for ]#1 < r, Oo

#-

A

-g

(1)

--n=l

where ft. - ( ~ " - A " ) / ( # - A ) - n)~ n - 1 . Clearly fll - O , and for n >/2, we get n-1

fin -- ( ~ - )i) E k'~k-lI~n-k-1

(2)

k=l

(as can be easily checked). Since IAI ' I~t] < r, and

I~.1 -< t . - al ( ~ k ) k-1

~ k =n-1 X k ~< n 2 , w e

obtain from (2)

~"-~ -< I.- ~1~=~"-= (- ~>2).

(3)

w 2. Exponential and Logarithmic Functions

227

Therefore oo

co

rt--1

r

n=

1

n:

2

(since fll -- O) oo

n--2

(using (3)) i.e.

(4)

P~ n=l

n--0 oo

For the series

+

", wo h ve

n--O

r~' -

lim p~(x,+2) • (n + 2 ) ~ - -

n - - ~ O 0 an

I1~11 < ~.

(4)

N e x t , for any fixed m / > N we can choose a 5 > 0 such t h a t for a d e c o m p o s i t i o n r w i t h Irl < 5 we have n

I~jl~ - [ f r ~ i d z - ~ ~j(z'~)(zk - Zk_l)l p k=l s

~
0 such that Iz - A

>rforallzEK.

Therefore

IlCj

I~j(z)l

IK - zEK sup ] z - ,~In+l ~

II~jll~ r n+l

so that oo

K

IlxjllllCjll~

)-~ IlxJll r(,~+i)p

j=l

J

1


k (k - 1, 2 , . . . ) . B u t t h e n the entire function oo

~nk

k-1

which vanishes at 0, does not o p e r a t e on x, since pa(x nk)/k nk > 1 xnk (so t h a t ) ~ 74 0 and the series for f ( x ) does not converge (by k~ 2.1.32). For the p a r t i c u l a r case s t a t e m e n t in the p r o p o s i t i o n it is e n o u g h to observe t h a t if x is t. n i l p o t e n t then, by definition, v~(x) = 0 for all a. 5 . 5 . 4 . COROLLARY. All entire functions vanishing at 0 oper-

ate on A iff v~*(x) < c~ 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})

t h e n since pa

is sm., by 3.3.6,

v~(x)(1 2} of sin. pj-seminorms with {qj " j >>2} ,-- {pj " j >1 1}, so that A is a locally sm. pseudo-~algebra.

PROOF. Write B j ( r ) (i),(ii) we have

{x E A ' p j ( x )

< r}. By conditions

pj(XlX2"''Xn) 1 2} ~ { p n ' n >1 1}, completing the proof. 5.5.7. LEMMA. 9 " , x n C A set Xl,

W(,•) k ( xl~

Let A

9,

be a commutative algebra.

x,) -

(x,1 + . . . + x,k)" n

For

257

w5. Power Series Operating in TA's

where the s u m m a t i o n is over all i l , . . . , i k n. Then

with 1 0 such that for IAI < e , x - ~ e E U and hence invertible, so that A e p(x). It follows that 0 ~ a(x) (the closure of a(x)). On the other hand, since a(x) c a'(x) and

t For definition see 1.7.5.

w 1. Spectral Properties

263

at(x) is closed we have a(x) c_ a'(x) - a(x) [,J{O}. Hence a(x) a(x),a(x) is compact and p ( x ) i s open. Finally, since a ( x ) i s b o u n d e d , a(x) :/: K. If A is a real Q algebra and x E A then the extended quasi-spectrum St(x) is compact, in particular 6.1.3.

PROPOSITION.

PROOF. Let A - a § ifl ~ 0, be a complex number. T h e n x 2 - 2ax

x2 - 2ax

(note lal//IAI 2 ~< & I~l ~ 0) It follows t h a t if U is a nucleus of A, comprising q. invertible elements then there is a C > 0 such t h a t x2 - 2ax

E U,

and hence y q. invertible, if IAI > C. This implies, by 1.8.5, t h a t for IA] > C, A r 5'(x). This means t h a t 5'(x) is bounded. To complete the proof of its compactness it remains to show t h a t 5'(x) is closed, or equivalently, ~'(x) is open. Using 1.8.5 and the fact t h a t x 2 - 2ax x 2 - 2(Re A)x

is a continuous function of A (for A ~= 0), it can be shown, exactly as in the proof in 6.1.1 for p'(x) being open, t h a t ~'(x) is open, completing the proof of the proposition. 6.1.4.

COROLLARY. If A is a u nital Q algebra then 5(x)

is compact. PROOF. It is clearly sufficient to prove t h a t 5(x) is closed in 5'(x) - 5(x)I.J{0} and we carry out the proof as in 6.1.2. We can assume t h a t 0 ~ 5(x), so t h a t 0 E ~(x). Then x, and hence x 2, is invertible. Therefore, A being an I algebra we can choose an open n e i g h b o u r h o o d U of x 2, comprising invertible elements. Since _

_

+

0

264

Spectral Analysis in TA's

we can find an e > 0

such t h a t for ]a),ifl[ < e ,

z~,z -

xa,/~EU. If

9 - (~ + i/~)~

then z~,~,~

- x~,~ e U for [a[, I/~[ < e.

From the choice of U, x~,/~ and hence z~,~, is invertible, whence A - cz + i/~ E ~(x) for tat, I~] < e. This implies t h a t 0 ~ 6(x), whence 6 ( x ) i s desired.

w 2.

The

Resolvent

closed in 6'(x), as

Function

6 . 2 . 1 . DEFINITION. Let A be a unital algebra (over a field F ) , z E A and p(x) the resolvent set of x. We assume t h a t p(x) ~ O. The vector-valued function

R,

~ ~ x~ - x ( A ) -

(~- ~)-1

on p(x) is called the resolvent f u n c t i o n of x; x~ is called a resolvent of x.

6.2.2. PROPOSITION. Let A be a unital algebra and x E A. Then: x~ - x~ - (A - # ) x ~ x t, (A,# E p(x))

(~- ~)-1

_ (~- ~)-1

(Hilbert relation);

_ (~ - ~ ) ( ~ - ~ ) - , ~ ( ~ _

(A - 0 or A-I E p(x), # - 0 or ~ - I E p(x)).

(,)

,~)-1 (**)

PROOF.

(x_~)(x_,~) ~ [(~_~)+(~_~)~] (~_,~)-1_~+(~_~)~. Multiplying both sides of the above equation, on the left, by x~ we get ( x _ ~ e ) - I x~ + (~_ A)x~x~

265

w2. The Resolvent Function

which gives changing signs on both sides we get relation (,). The relation (**) can be obtained by noting that

(~ - ~x)(~ - ~ ) - ~

6.2.3.

-

[(~- ,x)+

(~ - ~)~](~ - , x ) - ~

--

e Jr-(~- ~)x(e-

~x) -1.

COROLLARY. Any two resolvents xa,

xu of x com-

mute" x~ ~-~ xu.

PROOF. Interchanging )~,# in (,) of 6.2.2 we get

,'). Changing the sign in (,') and comparing it with (,) we get

Since we may assume that A ~= # we conclude that xa ~ x~. 6.2.4.

PROPOSITION. Let A be a unital Hausdorff TA and

x c A. Assume that

(i) p(x) is open; (ii)

A ~ xx (A C

p(x))

is continuous.

Then the resolvent function xa is a strongly infinitely differentiable function with dxa = ~ d~

(.)

and in general, d"xa = . ! ~ + 1 d~ ~

(**)

Further, if

(iii) a(x) # 0 and compact and p # ( x ) - (p(x))\{0}) -1 U{0} (p(~)\{o}) -~ - { ~ - ~ - ~

c p(~)\{o}}

266

Spectral Analysis in TA's

then p # (x) is open and on it (e-,,~X) -1 with dn dA n [ ( e - )iX) -1]

i8 ~nfin~tely

differentiable

n ! x n [ ( e - )~X)-I] n ()~ E p # ( X ) ) .

--

* * :4:)

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 XA+ # -- X A

xA+#X A .

Making # --~ O, and using condition (iN) above we get (,). The formulae (**) is obtained by induction. Assume thus d n - i xA

dan-1 - - ( n -

Then d

dnx~ dAn

= (n-

1)!x~.

1)! d

~-~(x~). Now

1

dA (x'~)=L,~o-~(x'~+~

-

x~)

-

L.-,o x ~ + g - x ~ ~ #

dx~ n-1 dA 9rLxA

-

-

x~+gx~

O<j+k=n- 1

n x ~ + l ( using (,))

where we have written L for lim. Therefore dd~,~- (n!)x~ +1. (The factorization of x~+~ - x~ used in the above calculation is based on the property x~+u ~-~ xA which is assured by 6.2.3.) It remains to prove the assertions on p#(x). Since p(x) is open, (p(x)/{O}) -1 is also open. To prove that p#(x) is open it is enough to show that 0 is an interior point of p# (x). Since a(x) is compact we have 0 ~ r(x) < co. If r ( x ) - O, then a(x) - {0} and so p # ( x ) - ~ is open. Next assume that 0 < r(x)< co. If 0 < [AI < - ~ then I)~-11 > r(x), so that )~-1 E fl(X),)t E p # ( x ) . Therefore 0 is an interior point of required.

p#(x) and p#(x) is open as

Finally, the relation ( , , , ) i s established by using ( * * ) o f 6.2.2 and induction in exactly the same way as for the proof of

267

w2. The Resolvent Function

(**) above. 6.2.5. COROLLARY. Let A* be the continuous dual of A and x* E A*. Then F~,(A)-x*(x~) is a ~ 0 such that a(x + y)C_ G for every y c A with Ilyll < ~. (ii) For ~ c p(x) if d(1) denotes the distance of ~ from a(x) then IIx~ll ~> l/d()~)P; whence I I ~ l l - ~ ~r as d ( ~ ) ~ O. PROOF. By 6.2.6, x~ is a continuous function of )~; by 6.2.7, xa ~ 0 as I)~l ~ c~. It follows that there are numbers C l , r > 0 such that Ilxall < c1 for all )~ with 1)~1 > r, Write D r - {A e K " I)~l 4 r}. Then g - ( K \ G ) N D ~ is compact, so that there is a C2 > 0 such that Ilxal[ < C2 for ~ E K. It follows that if A E K \ G then Ilx~ll < c max{C1,C2}. If y in A with IlY]I < (~ - - C - 1 and )~ E K \ G then

is invertible since )~ ~ p(x) and

II~yll ~< IIx~llllull


1

IIx:~[[-~, so that

II~ll ~ 1/1 ,x -~l p, IIx~ll ~

lld(~) p

proving (ii). 6.2.15. COROLLARY. continuous at x -- O.

PROOF. Take G -

r(x) -+ 0 as x ~ O. Thus r(x) is

{A E N ' [ A I < e}. Then G ___ a ( O ) - O ,

w2. The Resolvent Function

273

whence by u p p e r semicontinuity at 0 we have a(x) c_ G, r(x) ~< e for IIx]] < 5. This shows t h a t r(x) ~ 0 as x -~ O. 6 . 2 . 1 6 . COROLLARY. If A is a p - B a n a c h algebra and x c A then al(x) is upper semi-continuous. PROOF. It suffices to observe t h a t a'(x) - aAl(X) where A1 is the unitization of A. 6 . 2 . 1 7 . PROPOSITION ( R i c k a r t t element of a unital p - B a n a c h algebra either complex or strictly real. Let V of O in K. Then there is a 5 > 0 such IIx - yll < 5 and xy - y z we have"

o(y) c

[23, p.36]). Let x be an A(A,I I 9II) which is be an open neighbourhood that for every y E A with

+ v;

(,)

c o(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 t h a t is is false and show t h a t this leads to a contradiction W i t h o u t loss of generality we can take Y = {A E K : IAI < 2e}. By our a s s u m p t i o n we can find a sequence xn --~ x with xn ~ x, such t h a t a(x) ~ C,~ = a(xn) + V for all n. (1) Choose An C a ( x ) \ G n .

If # e a(xn) then

An - # ~ V, so t h a t IAn - #1/> 2e.

(2)

Since A, e a(x) and a(x) is compact we can, by passing to a subsequence if necessary, assume t h a t

e

(3)

From ( 2 ) , ( 3 ) w e get: IA0 - #I I> 2e > e for every # E a(xn)

(4)

so t h a t A0 r a(x~) for every n. For obtaining the contradiction we have to consider separately the cases A0 - 0, A0 ~ 0. t He obtained the result for complex B~nach ~lgebras.

274

Spectral Analysis in TA's

C a s e 1. ) ~ 0 - 0. Then 0 ~ a(x,) and xn is invertible. Since Igl > c for # e a(x~) we get r ( x ; 1) - s u p ]#-11 ~< e -1. So, by 7.4.6, u ( x ; 1) - r(x;1)P ~< e-p, whence, by 3.7.29(i), x - limx,~ is invertible contradicting 0 - )~0 E a(x). C a s e 2. ,~0 J= 0. Since ,~0 r a(xn), q. invertible for all n. By 1.7.12 a'(Y n) -

1 +a

Yn

-

9a e

-- )io l xn

is

(5)

Since )~ E a(y,) iff -)~,~0 E a(xn), and by (4) I)~0+ )~)~0l > e for - ~ o E a(xn), we get 1

IIx ll

I

1 + )~1 - I)~~ + ,~o,~ I ~
0 (since the sequence xn being convergent is bounded). From (5),(6) we conclude that r(y~n), and hence by G B formula (see 7.4.6.) u(y~) is bounded. So, by 3.7.29(ii), lim y,~ - -)~o ix is q. invertible, contradicting the fact that )~0 E 6.2.18. COROLLARY. If the algebra A of 6.2.17 is not unital then we have inclusion relations analogous to (,)(**) obtained by replacing a by a I (the quasi-spectrum). PROOF. We have only to apply 6.2.17 to the unitization A1 of A. 6.2.19. COROLLARY.

If A is commutative, r(x) is continu-

ous everywhere. PROOF. This follows from ( , ) , ( * * ) o f 6.2.17. 6.2.20. PROPOSITION. The completion ft of a commutative strictly real p-normed algebra A is strictly real. PROOF. Consider an element x E ti. If possible let a ' i ( x ) contain a complex number )~o - c~o + i~o, with /~o r 0. Write 1 r ] - ~1/~ol > 0. Choose a sequence ( x ~ ) i n A with x~ ~ x. Since

A is strictly real, ~a(x,~) c_ ~. Since A _C Ji we have also Ji _c ti,

275

w3. Pseudo-Resolvent Function and

so

~ ( x . ) c ~(~.) c ~ .

(1)

Write V - {A c C" A - a + i~ with a e R, lilt < ~}. By (**) of 6.2.17, there is a 5 > 0 such that

a(x) c a ( y ) + V whenever I I x - Yll < ~. Choose n sufficiently large t h a t I I x , - ~ll < ~. Then

~5(x) c ~ ( x . ) + It follows t h a t

v c R + v - v. 1 ~1~01, whence / ~ 0 -

A0 e V,l~01 < r l -

0-

a

contradiction. Therefore ~ i ( x ) __ N and A is strictly real.

w 3.

Pseudo-Resolvent

6 . 3 . 1 . DEFINITION. and x E A . Set

Function

Let A be an algebra (over a field F)

pV(x) - {A E F " (Ax)' exists}; pV(x) is called the pseudo-resolvent set of x. Since (0x)' 0 c pP(x) and we have always pV(x) r 0. Write

9'~ - ( ~ ) '

0~ -

0,

(~ c p~(~))

and call x~ a pseudo-resolvent or a p-resolvent of x; A H x~ is called a pseudo-resolvent function, t 6.3.2.

Remark.

The m a p

A ~

_~-1

is a bijection of

p ' ( x ) - p ' ( x ) \ { 0 } onto pP(x)\{O}, where p ' ( x ) d e n o t e s the quasiresolvent set of x. If F - K the map is also a h o m e o m r o h p i s m . If xis q. nilpotent then a'(x) - {0}, p'(x) - F \ { 0 } . Therefore, t This function has been considered implicity by Kaplansky [10 ", p.400, Lemm~ 3.2].

276

Spectral Analysis in TA's

p P ( x ) - - { F \ { 0 } } -1 [..J{0}- {F\{0}} U { 0 } - F. 6.3.3. LEMMA. (a)x~ ~ xu'

(~,~ e p~(~))

I

(b) x . #

4 = ~_ ~ A

(A,~ e p ~ \ { o } )

PROOF. (a) Since Ax ~ #x, by applying 1.1.18 twice we get I Xtt.

(b) We have Ax + (Ax)' + (Ax)(Ax)' - 0

(1)

ux + (u~)' + (u~)(u*)' - 0

(2)

Denoting the expressions in the equations (1), (2) above also by (1), (2)we get u • ( 1 ) - ~ • ( 2 ) - u . 0 - ~. 0 -

0,

which gives ,(~)'

- ~(u~)' - ~ u x [ ( u ~ ) ' - ( ~ ) ' ] .

(3)

Again, (1)x (#x)' and 'Ax" ( ~ x (2) give A tt

x(u~)' + (~)'(u~)' + ~(~)'(u~)' - 0 .

(4)

9( ~ ) , + (~)'(u~)' + ~ ( ~ ) ' (u~), - 0 .

(5)

#

From (4), (5) we obtain by subtraction

1

'(u~)'

and from (3), (6)we get

Ap

A#

(6)

w3. 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 E A, pC(x) is an open set containing O. PROOF. For A E pP(x), let U be an open neighbourhood of $x comprising q. invertible elements. By continuity of scalar multiplication there is an open neighbourhood N of A such that N x c_ U. If follows that N C_ pP(x) and pC(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 ~. Then x~ is a strongly differentiable function of ~ on pP(x), with dx~ dA = - x ( 1 + x~) 2, (,)

dnxk _ (_l)nn!xn(1 + xk) n+l dA,~

(**)

We have also the relation

dn (Xk) d1,~

__ , (_~)n+l

-~-

n.

(A # o).

(b) In a Hausdorff CQ algebra A, the formulae (.), (. **) are valid for every element x in A.

(***) **),

PROOF. (a) From the equation ( 1 ) i n the proof of 6.3.3 we obtain, for ~ E pq(X)\{O}, 9 A

--

A

t = - x ( 1 + xA).

(1)

Making A --+ 0 we obtain

[dx'

(2)

If A, p C pP(x)\{O}, by 6.3.3 (b) we have

!

I

Spectral Analysis in TA's

278

Making #--~ ~ we obtain d,~

(3)

,~2,

which reduces to x~ 1 dx~ ~2 t ~ d~

x~ ~2 '

whence we get dx~ _ x~ (1 + x~) -- - x ( 1 + x~) 2 dA A

(4)

(,)

when )~ ~- 0 . On the other hand (2) shows that the formula in (4) (or (,)) is valid for ~ = 0, completing the proof of (,) for all ~. To prove (**), assume by induction that dn-lx~ _ ( _ l ) n - l ( n 1 ) ! x n - l ( 1 + Xk) n dan_ 1 where we have used

Thus we have proved

Differentiating both sides of the above equation we get dnx~ d/~n

--

( - - 1 ) n - l ( n -- 1)iX n - l " n(1 + xk) n-1 dxk

--

9 d,k (--1)n-ln!xn-l(1 + X~) n - l " - x ( 1 + x~) 2 (using (4))

=

(-1)nn!xn(1 + X~) n+i

which is (**). Finally, by successive differentiation we obtain (, 9 ,) from (3). (b) This follows from ( a ) a n d 6.3.4. 6.3.6. PROPOSITION. Let A be a p-Banach algebra and x E A. Then x~ exists for all ~ such that v(Ax) < 1, or equivalently, 1

lA] < v(x)

p, and we have oo

x~ - ~ (-1)'~(Ax) n

(,)

n=l

with the series on the right converging absolutely. In particular, 1

x~ is analytic for I)~l < v(x)

o.

PROOF. Consider the unitization A1 of A. Then we have (el + ~x) - 1 -

ex + (~x)'

279

w 3. P s e u d o - R e s o l v e n t Function

where el is the unity of A1. By (,) of 3.3.20 we have (3O

~(-1)"(Ax)",

( e I -~- ) ~ X ) - 1 -

rt=0

the series converging absolutely. It follows that we have (x)

x~ - (Ax)' - ~

(-1)n(Ax) n

rt--1

(with the series converging absolutely). 6.3.7.

COROLLARY.

If

A

is a complex p - B a n a c h

then x i is p - a d m i s s i b l e holomorphic on {A e

cl l

algebra I

< ~(~)-~)

PROOF. Set r

(-1)nAn; x . -- x n.

Then (,) of 6.3.6 can be rewritten as oo

n--1 1

For any r0 with 0 ~ r0 < v ( x ) - ~ , set K -- Kr ~ -- {A E C ' l ~ l

r0}.

Then

lie.IlK < So oo

oo

0(3

n= 1

n= 1

n= 1

since v ( r o x ) - r~v(x) < 1. Hence the corollary. 6.3.8. PROPOSITION. Let A be a complex ample C-algebra. I f x e A\{O} is q. nilpotent then {x~ " A e C} is unbounded. PROOF. First note that by 6.3.1 and definition of q. nilpotent, pV(x) - C. Since A is a Hausdorff C-algebra, by 6.3.5 (a),

280

Spectral Analysis in TA 's

z~($z)' is strongly differentiable, and hence for x* E A* F~.($) - x*(x~) is an entire function. If possible let x~ be a bounded function of )~. Then Fzl.()~) is bounded and so by classical Liouville's theorem Fz'.()~ ) is a constant. Since Fz'.(0 ) = 0 we must have F~. (A) - 0

for all A E C.

Putting ) ~ - 1, we get x * ( x ' ) - O. Since A is ample we conclude that x' - 0 which implies x - 0, contradicting the choice of x. Therefore {x~} is unbounded as required. 6.3.9. COROLLARY. (Kaplansky). In a complex normed algebra A, if x ~ 0 is q. nilpotent then {x~} 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 E A be q. nilpotent. If ~ . e K are such that the sequence (()~.x)') is unbounded then I~,~l ~ c~. PROOF. First note that since x is q. nilpotent (~x)' exists for all )~ c ~:. Assume now that, to the contrary, I)~n[ ~< C for all n. Since (()~,x)') is unbounded there is, by 2.1.23, a nucleus U and a subsequence ((An,x)') such that U for all n'.

n t

(.)

Since IAn, I ~< C, for all n', we can choose a subsequence (An,,) of (An') with An,,--+ (some) A0. Since A is a C - a l g e b r a we get (An,,x)'--~ (A0x)', whence the sequence ((An-X)') is bounded. But by (*), a II

r u,

contradicting boundedness of ((~n",)'). 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

281

w 3. P s e u d o - R e s o l v e n t Function

x/~

is a s . t . z . d . .

PROOF. We may assume that sequence ()~,~) in C with

By 6.3.10,

~.

x r

0. By 6.3.8 there is a

Write

x' /llx ll

1

I

Xn -- ~ n X ~

(so t h a t I]Ynll- 1). Then since !

XnX~ ~

--Xn ~ X n

we get !

1

x~y~ - - x , llxnll-~ - y , ~ . Since x n -

(1)

Anx the equation (1) becomes Xyn

-

- - X [ I X n' l I - ~ -- ~ n l y n

.

(2)

I Since Itx,~ll, lAn[--~ co as n - - . co, and I t y n I [ - 1 we get from (2) xyn --~ 0 (as n --. co). In exactly similar m a n n e r we also get y,~x --+ O. Thus, x is a s.t.z.d., as desired.

6 . 3 . 1 2 . COROLLARY. In a real n o r m e d - or more generally, ample p - n o r m e d - 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 comN plexification A. By 4.7.4. (b), A is also ample. So, by 6.3.11, x is a s.t.z.d, of ti and hence by 3.7.16 (ii), a s.t.z.d, of A. 6 . 3 . 1 3 . DEFINITION. A TA A is called a topological integral d o m a i n or TID if it has no non-zero t.z.d.. 6 . 3 . 1 4 . PROPOSITION. A complex ample p - n o r m e d algebra A which is a T I D is q.s.s., in particular s.s.. PROOF. This is an immediate consequence of 6.3.11 and the definition of a TID.

Spectral Analysis in TA's

282 w4.

6.4.1.

Spectral

DEFINITION.

Algebras

A unital algebra over a field is called

spectral if for every x in A, a(x) # 0. 6.4.2. LEMMA. Every subunital algebra B of a spectral alge-

bra A is spectral. PROOF. By 1.7.19 (**), aB(X) D_aA(X) ~: 13. 6.4.3. THEOREM. Every complex ample Hausdorff CI algebra

is spectral. PROOF. If possible let a ( x ) = ~, for some x in A, so t h a t p(x) = C. By 6.2.5, F~.(A) = x*(x~) (x* e A*) is holomorphic on p(x) = C. Also, F~* is bounded since by virtue of 6.2.7, x*(x~) 0 as I~] ~ cr By Liouville's theorem, F~. is a constant which must by 0 (since Fz.(A) ~ 0 as I)~1 ~ o0). Thus Fz.($) = 0 for every x* E A*, whence by ampleness of A, x~ = 0. It follows t h a t e = ( x - Ae)x~ = 0 - a c o n t r a d i c t i o n . Thus a(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.

Every complex Hausdorff locally sm. convex I algebra A is spectral. In particular, every complex Hausdorff locally sm. convex division algebra D is spectral. 6.4.5.

COROLLARY.

PROOF. By 4.4.15, 3.6.21, A is a CI algebra and hence the first s t a t e m e n t follows from 6.4.4. The second statement follows from the first since D being a division algebra is an I algebra (Gi = 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

w4. Spectral Algebras

283

a CI algebra (by 3.6.23 (b)) and is also Hausdorff. 6.4.7. THEOREM (~;elazko). 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 E A with a(x) = O. Then p(x) = C and by 6.2.9, x~ is a locally p-admissible entire function. If F is any circle in C then F ,-~ 0 and hence by 5.14.16

r X~d~ - 0 . On the other hand, by taking m - 0 in the formula (.) of 6.2.10, we obtain

r x~ d)~ -- - 2~ie, where e is the unity of A. This contradiction proves that a(x) 0, for every x E A, and A is spectral. 6.4.8.

COROLLARY. Every strictly real unital p-Banach al-

gebra A is spectral. PROOF. Let A be the complexification of A. Since A is strictly real we have

aA(x) -- aA(x ) -7/:0 (by 6.4.7). 6.4.9. COROLLARY. Every complex or strictly real unital pseudo-Michael algebra A is spectral.

A-

PROOF. By 4.5.3, A has a projective limit decomposition limAa, where each A~ is a unital pa-Banach algebra. Again,

by 4.5.7

(3), OA(X) -

U

x -

(,)

First let A be complex. Then fi.~ is complex and so by 6.4.7, a2~(x~ ) ~ 0. It follows by (,) that ffA(X) r O, and A is spectral.

284

Spectral Analysis in T A ' s

Next let assume that strictly real and (.), A

A be strictly real. In view of 1.7.26, 1.9.11 we may A is commutative. By 1.9.8, As is commutative and so by 6.2.20, Aa is strictly real. Hence by 6.4.8 is spectral.

6.4.10. R e m a r k . A complex unital commutative complete locally convex algebra may fail to be spectral. To overcome this deficiency Waelbroeck [13 t ] has given a modified definition of spect r u m 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 )~0 e K such that ( x - Xe) -1 exists and is (t). bounded in a neighbourhood of )~0. Using the properties of his spectrum sp(x), Waelbroeck has obtained interesting results concerning locally convex algebras.

w 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 f o r m A -= Fe, where e is the unity of A. Moreover, if x = ~ze, the map w : x ~-~ Az is an i s o m o r p h i s m of A onto F. PROOF. If x E A and )~z E a ( x ) then x - ~ze is noninvertible and so x - )~,e = 0 since A is a division algebra. Thus, x - - Axe, A - Fe and w is clearly an isomorphism. 6.5.2. COROLLARY. I f a spectral division algebra A over K is a H a u s d o r f f TA then the map w : x ~ ~, is a t. i s o m o r p h i s m . PROOF. This is because of the result that an isomorphism between two finite-dimensional TLS's is automatically a homeomorphism (here dim A - d i r e r - 1) (see 2.1.12). 6 . 5 . 3 . THEOREM. A complex ample CI division algebra A is of the f o r m A = Ce, with the map ~ : x ~ ~ (x = $~e) a t. i s o m o r p h i s m .

PROOF. Since A \ { 0 } - Gi is open, A is Hausdorff. Hence, by 6.4.3, A is spectral. The theorem now follows from 6.5.1, 6.5.2.

w5. 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 w a t. isomorphism).

PROOF. Since A is a Hausdorff division algebra, by 3.6.10, A is an I 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 - C e and w " x ~ Az is a t. isomorphism. If []e[[- 1 then w is also an isometry. PROOF. W i t h o u t loss of generality we may suppose t h a t the n o r m of A is sm.. Then A is locally sin. convex, so t h a t by 6.5.4, A - Ce and w a t. isomorphism. If I1 11- 1, then [ I x l l - llama I - I A z l and w is an isometry. 6 . 5 . 6 . COROLLARY (Zelazko). A complex p - s e m i n o r m e d division algebra A - (A,p ~ ) has the form A - Ca. Consequently, every locally bounded complex TA B which is a division algebra has the form B - Ca.

PROOF. We may assume t h a t p is sm.. Let a be an element of A and Am a m a x i m a l c o m m u t a t i v e subalgebra containing a (see 1.1.9.); by 1.1.19, Am is a division algebra. For x E Am, 1

write ] ] x [ [ - v~(x)

1

where v ( x ) -

lim p(xn) -~ By 4.8.6 4 8.7

the above defined functional [[-[[ is a norm on Am, so t h a t (Am, ][" ]) is a complex normed division algebra. By 6.5.5, A m - Ca, so t h a t a - Ae (A E C). It follows that A - Ca, proving the first assertion. For the second assertion it suffices to observe t h a t , by 4.2.4, B is p - s e m i n o r m e d (so that it follows from the first). 6.5.7.

PROPOSITION. A strictly real, ample CI division alga-

bra A is of the form A - ~e (with w " x ~ Az a t . isomorphism). In particular, such an algebra is commutative.

t As Mw~ys we ~ssume p :/: 0

286

Spectral Analysis in TA's

PROOF. As in 6.5.6, for any element a r 0 in A, take a maximal commutative subalgebra Am with a E 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, A,~ Ce, whence A m - Be, so that a - A e (A E ~). 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 sin. convex division algebra then A - Re. PROOF. The same argument as in the proof of 6.5.4 shows t h a t 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 sin. convex) division algebra A is of the form A - B e 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 = ~e (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 = j x , with j E A, multiplication by i is continuous so t h a t ft. 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 . 1 0 . 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 r 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 ) ) a n d CI (by 3.6.24). Applying 6.5.9 to Z(x) we get Z(x) = ~e or Ce. Thus,

w5. Gelfand-Mazur and Other Similar Theorems

287

x - (a + iS)e (a, ~ c R, with 5 - 0 if Z(x) - Re). It follows t h a t (~ -

(~ + iZ)~)(~

- (~ - iZ)~) -

0

or, x 2 - 2 ~ x + (~2 + Z 2 ) ~ _ o,

which implies t h a t A is algebraic of degree 2 at most. By a theorem of Jacobson t A is finite-dimensional, and so by the classical Frobenius theorem, A _~ ~, C or H; the isomorphism is topological since A is finite-dimensional Hausdorff. 6 . 5 . 1 1 . COROLLARY (Arens). Every real Hausdorff locally sm. convex division algebra A is t. isomorphic to ~,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 . 1 2 . T h e o r e m (Zelazko). Every real p - n o r m e d division algebra A (A,p) is t. isomorphic to ~, C or H according as it is formally real, commutative but not formally real, or not commutative. If w denotes the t. isomorphism then 1

Ico(x)l- v ( x ) -

limp(x'~)x tl

where I" I denotes the standard norm tt on ~, C or H. PROOF. Since we do not know t~ 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. C a s e 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, A,~ is a formally real division algebra, so t h a t Am is t 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.701" ). tt It I - ( t t ) 8 9 ( t E ~ , C or HI and t the conjugate of t).

Spectral Analysis in TA's

288

now a commutativeN formally real p - n o r m e d division algebra. Its complexification A,~ is a p - n o r m e d division algebra, whence by 6.5.6, Am = Ce, A m = Re, x = At, A = Re. C a s e 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 --

C a s e 3. A is not commutative. Let Z be the centre of A. For x J= 0 in A we can form, as in the proof of 6.5.10, the algebra Z(x) which is now a real commutative p - n o r m e d division algebra. By the conclusions in cases 1, 2 we have Z(x) = ~e or Ce. This implies, as in 6.5.10, t h a t A is finite-dimensional, and hence by Frobenius, A_~ ~, C or kD. It remain to prove (.). Set l]xII = Iw(x)]. Then I1" ]] is a n o r m on A with ]]xy]l--IIxII []yl]. Since A is finite-dimensional, 11" II ~ P- By 4.8.2

v(x)-

Vv(X ) - v[i.l[(X) - l i m IIx'~I[ 88- l i m I I x l l - IIxll.

6.5.13. Remark. If A is a formally real normed division algebra whose norm I1" II is normalized, then the t. isomorphism w : A --~ ~ is an isometry: if x - Aze, l l x l ] - I A z l - Iw(x)]. 6 . 5 . 1 4 . L EMMA. 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 t h a t

x E A has no inverse in r

Then

0 e hA(X), so t h a t aA(x ) 7~ 0. Also, by 6.1.2, aA(x ) is closed and a~(x) r g. It follows t h a t OaA(x ) 7~ O. If A e a2(x ) then, by 3.7.28, x - Ae 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 = Ae and since x is not invertible, X

---

0.

unital p-Banach algebra A is a TID then it is a division algebra. 6.5.15.

COROLLARY.

If a

6.5.16. Remark. (cf. Zelazko [31, p.112]). The result in 6.5.15 does not hold for a locally sm. ~ algebra. For example, consider the algebra E of entire functions (see 4.6.8 (iii)). ~" is

w 5. G e l f a n d - M a z u r and Other Similar Theorems

289

not a division algebra, since for instance, z E $ has no inverse. But ~" is a TID. To see this, suppose t h a t in ~r we have g ~ 0 , and (,) fkg --+ 0 as k ~ oc. Since the zeros of g are isolated we can find a sequence rn of reals with 0 < rn -~ cr and such t h a t inf

Ig(z)] > 0 ( n -

1,2,...).

(**)

If we write iif~ll*=

sup ]f(z)l , i=1=~,,

the family {ll" I1~} of norms is easily seen to be equivalent to the family {ll" II-} defining the topology in ~ (see 4.6.8 (iii)). The condition (,) implies t h a t Ilfkgl~ ~ 0 as k --+ oc, for all n. But this result along with (**) gives: Ilfkll* ~ 0. This means t h a t g is not a t.z.d., proving ~' is a TID. 6 . 5 . 1 7 . THEOREM. (cf. p - n o r m e d algebra which is a A = Ce, A "~ C, if A is and A"~,Cor H if A is a

[31, pp.30-32.]) Let A TID. Then: a complex algebra

be a unital

real algebra.

PROOF. Let .zi be the completion of A. By 6.5.14, every x =/= 0 in A has an inverse x -1 in A. Consider the unital subalgebras A ( x ) of .zi, consisting of all rational functions of x, over ~:, i.e. all elements of the form f ( x ) / g ( x ) -- f ( x ) g ( x ) -1 (g(x) r 0), where f , g are polynomials over K. A ( x ) is a division algebra containing x, which being a subalgebra of A is p - n o r m e d . If A is complex, it follows from 6.5.6 t h a t A ( x ) = Ce, x = Aze, A--Ce. It remains to consider the case where A is real. By 6.5.12, A ( x ) " ~, C or H. It follows t h a t x satisfies a relation of the form a x 2 § ~ x § "/e -- O, where a, ~ , - / c ~, " / ~ O. It follows t h a t x-1 = - - a x - -eft E A, showing t h a t A is a divi.7 -7 sion algebra. Now we apply 6.5.12 to A and conclude t h a t A ___ ~,

Spectral Analysis in TA's

290

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 _ ~ , C or H if A is real (see [31, pp.l12-14]). 6 . 5 . 1 9 . PROPOSITION. Suppose that the norm of a unital p - n o r m e d algebra A satisfies the condition

cIIxlJ Ilyll-< I1~11

(*)

for some C > 0 and all x, y E A. Then A is t. isomorphic to C if A is complex and to ~, C or H i r A is real. PROOF. In view of 6.5.17 it is sufficient to show t h a t A is a TID. Suppose that x , y , e A, llYn]] = 1, xyn ~ O. Then, by (,) we have

cIl~ll = ctl~ll Ify=ll-< II~y.ll-~ o. This means that Ilxll = 0, x = 0, so that A has no non-zero 1.t.z.d.; similarly it has no non-zero r.t.z.d.. Thus A is a TID, completing the proof. 6 . 5 . 2 0 . COROLLARY (Arens-Shilov). A unital normed algebra A satisfying condition (,) above is t. isomorphic to C if A is complex and to ~, C or H if A is real. 6 . 5 . 2 1 . R e m a r k . A special case of 6.5.20 was proved earlier by Lorch and Mazur (see [14, p . 1 2 7 ] ) i n the form: A complex unital Banach algebra A whose norm satisfies the condition ]]xy]] = I]x]] ]]Y]I for all x , y in A is (t.)isomorphic to C.

w6.

Turpin's

Theorem

on Locally

Convex

Algebras

6.6.1. Let A be a Hausdorff complete locally convex algebra. Denote by Pc the set of all continuous semi-norms on A and by A* (the continuous) dual of A.

w6. Turpin's Theorem on Locally Convex Algebras

291

Set

rl( )

1

sup

limpc~(x'~) ~ ,

p~EPc n~oo

1

sup lim If (xn) l -~, lEA*

n--~oo co

inf{r" (a,)~ ~ an E K, the series ~ a , A " n=l

has radius of convergence > r GO

=> ~ a , ~ x n converges in A}. n=l

We have clearly 0 ~ rj(x) ~ co (j - 1, 2, 3). 6.6.2. LEMMA. r2(x) oo

r0 > r3(x) such that, if ~

~ , ~ n is a series over K with radius

n--1 oo

of convergence /> r0, the series ~ c ~ n x n converges absolutely. rt=l

Spectral Analysis in TA's

292

has radius of convergence

Now the numerical series ~ n--1

- 1/lim Ih!~.188- I'1 > r0, so that by choice of r0, ~

con-

n'--1

verges absolutely. This implies, by 3.1.24, that ~ verges in A, whence, by 2.2.17,

(:)' -

x

con-

n--1

exists, so that # E p~(x).

Therefore, if ,~ E a'(x) then )~ ~ p'(x) a n d s o w e m u s t have I,~] ~< r3(x). It follows that r(x) no -- n(A0). Since E

yn converges absolutely we have

rt

oo

n--no

oo

lr/,-~ lq, O

O(3

whence ~ ~ , ~ x n converges absolutely in A. Thus, all entire funcn=l

tions vanishing at 0 operate on A and the theorem follows from 5.5.11.

CHAPTER GELFAND

w 1.

VII

REPRESENTATION THEORY

Ideals of Topological

Algebras

7.1.1. LEMMA. (cf. [20, p.70]). Let A be dense subalgebra of a T A A. Then:

(i)

The closure I in A of a n ( l . o r r . ) i d e a l I of A is an ideal of A of the same type.

(ii) /f the ideal I is regular with a (l. 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 -[ ~ -A and is a closed regular ideal of A. PROOF. It is enough to prove the results when I is a 1. ideal. Clearly I i s a s u b s p a c e o f A. Further if ~ E A , ~ E I , x ~ E A , az C I and xa --~ 5, az --~ ~ then x~a~ ~ x a. Since each x~a~ C I it follows that ~ a C I, proving I is a l . ideal. Hence M

(i). Next let I be a regular 1. ideal with relative r. unity u. If -~ c A, x~ C A and x ~ 5 then ~u-~-lim(x~u-x~)EI

(sincex~u-x~EI)

and I is regular with u as a relative r. unity, proving (ii). Finally, if u is a relative unity for closed ideal I ~- A, then u ~ I and hence u ~ I (since A ~ I I) and therefore I-7(= A, proving (iii). 7.1.2. COROLLARY. Let A be a TA. Then the closure I of a l. 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.

w1. 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. R e m a r k . 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 t continuous. PROOF. (a) By hypernormality there is a closed hypermaxireal ideal M1 with M C M1. The maximality of M ::~ M - M1. (b) By Krull (1.2.10) if I 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. E x a m p l e 1. Let A be any radical algebra (A - x/rA) and A1 its unitization. Then A1 is a T A under the indiscrete topology. The ideal A is hypermaximal in A1. Since A is not closed in A1, A1 is not hyponormal. On the other hand, since no ideal I r A1 is closed it is vacuously hypernormal. Example

2.

The Williamson algebra ~ .

t For definition see 2.2.19.

We have seen in

Gelfand Representation Theory

298

3.6.33 that ~ is a commutative Hausdorff division algebra over C. Since {0} is the only maximal ideal of ~ and it is closed, is hyponormal. But since {0} is not hypermaximal, ~ is not hypernormal.

7.1.8.

PROPOSITION. In a Q algebra A if I # A is a regular l 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 I the same is true of u + a for any a ~ I (see 1.2.8(a)). Therefore, by 1.2.9(ii), - u - a r Gq, so that ( - u + I)(1Gq = 0 and hence also - u + I N G q = 0 (since Gq is open). It follows that -u+I--u+I#A,

so that I ~ - u + A - A .

7.1.9. COROLLARY. Every maximal regular (l. r. or hi-) ideal M of A is closed. Hence, every Q algebra - i n particular (see 3.6.23) a pseudo-Banach or (more generally) a sm. (F) algebra is hypernormal. 7.1.10. LEMMA. Let A = ( A , I . I) be a unital sin. algebra with unity e. Then for any ideal I # A we have: 1 >.r ( h E I).

(1)

Using the notations in (the proof of) 4.5.3, 3.4.15 we can write a#-a4-Ns-as;

99s'x~xs-x§

# (Ns-kerps).

Then we have P#a (as -- us) -- p#a (a # -- u #) -- ps(a -- u) >t e(a E I).

(2)

Clearly, I~ - ~ ( I ) is a regular ideal of A~, with relaive unity u~; u~ ~ I~ because of (2). By 7.1.1, J~ - I~ is a closed regular ideal (with relative unity u~) ~ A~. By Krull J~ is t That ~ is closed can also be deduced from the fact that it is the intersection of all maximal regular 1.ideals.

Gelfand Representation Theory

300

contained in some regular maximal ideal M. Since fi.a is pseudoBanach, by 7.1.9, M~ is closed. By continuity of ~p~ and 1.2.17 (iii) M - ~ I ( M ~ ) is a closed regular ideal of A which is maximal, and clearly I C M. This proves that A is normal. 7 .1 .1 4 . COROLLARY. A complex or strictly real commutative pseudo-Michael algebra A is hypernormal.

PROOF. The ideal M~ in the proof above (of 7.1.13) is now closed hypermaximal since A~ is Gelfand (by 7.2.17, 7.2.19)t. Let X~ be the character determined by Ma; X~ is continuous. Then, if we set X - X~~ X ( U ) - X~(U~)- 1, so that X is a continuous character of A. If M - k e r x then M is a closed hypermaximal ideal. Also, if x E I then ~p~(x) E Ja c_ Ma, X(X) - O, so that I ___ker X - M. This proves A is hypernormal. 7.1.15. R e m a r k . There are commutative Michael algebras which are not functionally continuous (and so not hyponormal) (see [20, p.49]). 7.1.16. R e m a r k . There are hypernormal TA's having dense regular maximal ideals. For an example, consider the algebra C = C(~, ~:) of all ~:-valued continuous functions on ~ topologized by the family {lI" Iin} of sm. seminorms, where

llfll, - sup If(t)l (f c c)

(cf. Example of 4.6.8)

Itl 1. If x E A and X ( x ) - A # 0, then A - i x e E. So

p(,~-nxn) -- p ( ( , ~ - l x ) n ) ~

1,

which gives

p(~)

~

I~Xl"- Ix(x)l "'~,

so that we have 1

1

Making n ~ c~, we get Ix(~)l-< ~(~)~ -< p(~)~, which is (,). Further, Definition 3.5.1. and (,) imply that IIXII 0,0 < e < 1, we can find n---~ o o

an n such t h a t

E

/ 3 ( ~ - xn) < ~.

(1)

Since X,~ (xn) --+ X(Xn), there is an a0 such t h a t for a >- a0, E

E x o ( x . ) - x(~.)l < 5.

(2)

Using the inequality (.) of 7.3.9 for /3, we have 1

I ~ ( ~ ) - ~ ( x ~ ) l < ~ ( ~ - ~,)~
a0,

I ~ ( ~ ) - i(~)]

~< i ~ ( ~ ) - ~ ( ~ ) 1 E

~

+ Ixo (~,) - x(~,)l +

E

< 5+5+5-~. This implies t h a t ~ --+ f:. Thus we have shown t h a t X~ ~ X f;~ --~ f:. On the other hand, the reverse implication is trivial since X~ - ~ [ A , X - f;[A. Thus, the map X ~ ~7 is bicontinuous, as we wished to show.

Gelfand Representation Theory

316

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 ~ / A t is regular then Ac is compact Hausdorff. PROOF. By 7.3.2, we can identify A~ with a subspace of the cartesian product K - 1-[Kz (x c A). Write Sz - {A e g ~ " 1

IAI - a0 we have

Ix~(P(yl,""

,y,~)) - x o ( P ( y l , " ' , y , ~ ) ) l

From the inequalities (2)-(5) we obtain, for a >~ a0, E

E

E

whence X~(X)--* X(X), completing the proof.

E

< ~.

(5)

w3. The Gelfand Representation

319

The spectrum Ac of a separable pseminormed algebra A is metrizable. 7.3.17.

COROLLARY.

PROOF. Let {xn} be a countable dense subset of A. Set, for XI,X2 E As

d(x,, X2) - ~

1

n=l

- x=(

2n 1 + (X~-(-x~-

n)l

(.)

Then d is a metric. For, if d(X1,X2)= 0 then X,(X•) = X2(X,~), for all xn, so t h a t by density of {xn} and continuity of X~,X2 it follows t h a t X~ = X2. The s y m m e t r y 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 s u m m a n d s 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 t h a t a net X~ --+ X, under d, iff

xo

x(x,)

= 1,2,...).

(,)

The set {x=}, 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 X~ ~ 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 A c of a countably t. generated p-seminormed algebra A is separably metrizable. PROOF. If S is a countable set of t. generators of A then the set S1 of all finite products of elements of S, is countable. Denote by $2 the set of all rational linear combination of elements from S1 (where a rational linear combination in the complex case means the coefficients of the combination have rational real and imaginary parts). $2 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-

limA~ (A~ pseudo-Banach algebras).

Gel]and Representation Theory

320

Let A~,~,~ denote the sets of continuous characters of A, fia respy.. Let Aa denote the subset of Ac comprising characters which are p~ -continuous. $ Then: (i)

A~ - U A c ~ ;

(ii) A~ homeomorphic to }X~; (iii) /f we define r

2z(y~)

-

" }X~ ~

2~(~z(~z)).

}XZ by flZ -

wh~r~

~

~,~

ciated with the projective decomposition:

with

r th~

map~

a~o-

]ca E Aa,kZ E

AZ, ~5~(kZ) - ~ , then { / ~ ; { ~ Z } } is an inductivett system of topological spaces and continuous maps. Write s - ( t h e direct limit ) l i m / ~ . Then there is a bijective co.ti.uous m~p o $ s ~ ~ g i ~ . by (for any a) where f t - limfia.

o(2)(~)

-

xo (x) -

fla(xa)

PROOF. The relation (i) is an immediate consequence of 4.3.13. By 7.3.10, 7.3.11 we have Aa _~ A(A~) _~ / ~ which is (ii). If O(f:l(X)) - X ~ ( x ~ ) - O()~2(x)) - X2~(x~) for all ~ then X1 - X2, so that O is injective. It is also surjective. For, if X C A c , X C A~ for some a and X - )~a o p~ for some ) ~ E A~. It follows that X - O(X,). Finally, the continuity of O is clear from its definition. ^

7 . 3 . 2 0 . COROLLARY. If A is unital, each Aa is compact. PROOF. By 4.5.6(ii) each /~a is unital and consequently, by 7.3.12, / ~ is compact, whence Aa _~/~a 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.

i.e. is a continuous character of the algebra (A,p~). tt For definition see [12, pp.184-5]. In general O is not a homeomorphism (see [12,p.161, Remark]).

321

w 3. The Gelfand Representation

7.3.22. E x a m p l e s

of m o n o g e n i c TA~s.

(i) The Banach algebra C[0, 1]. This has fo(t) - t (t e [0, 1]) as t. generator. That f0 is a t. generator is a consequence of the Weirstrass approximation theorem. (ii) (cf. [10, p.33]). The algebra 9.i of complex-valued continuous functions on unit disc [z I ~< 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 rE(z) - f ( z / l + e ) (e > 0) then f can be uniformly approximated by f~ in [z I ~< 1. Also, each s being holomorphic on Izl < 1 + c can be uniformly approximated by polynomials in z in Izl ~< 1, whence f0 is a t. generator. (iii) The Banach algebra k I - kl[0, 1] (see 3.4.6 (vi)). If fl denotes the constant function 1 , f l ( t ) 1 (t e [0,1]), then it is easy to see that if f ? - ]'1 * " ' * fl ( n factors), f~(s) - s"-l/(n1). It follows from Weierstrass approximation theorem and the fact that C[0, 1] is dense in k I that fl is a t. generator of [-1. (iv) The subalgebra W~_ of W ~ (defined in 3.4.10) consisting of all elements f -

~ - ~ ] ( n ) e int with ] ( n ) -

0 for n < 0,

nE~

is closed and so a p-Banach algebra. t. generator as can be easily seen.

It has

e~t as a

7.3.23. PROPOSITION. Let A be a monogenic unital p B a n a c h algebra which is either complex, or when real is strictly real or f o r m a l l y real. Let A - Ac be the spectrum of A and a a t. generator of A. Then the map A " X ~ x ( a ) is a h o m e o m o r p h i s m 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(5), A is surjective. A is also injective. For, suppose that X, (a) - x2(a). If x e A then x - limn Pn(a) (P,~ e K[x]), so that X.(z)

-

limx.(P.(a)) -limPn(x.(a))n

=

lrl

lim X2 (Pn (a)) - X2 (x). rt

limP.(x2(a)) l,l

Gelfand Representation Theory

322

whence X1 - 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 \ a ( a ) is

connected. PROOF. Since a(a) is compact we can enclose it in a closed disc D. Then C \ D is connected unbounded and C \ D _ 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 aG0 C_ a(a). If A0 E Go then by the maximum modulus principle applied to Go we obtain, for any complex polynomial

P, [P(Ao)I

~
0 such that C l p ( x ) 2 ~ p(x 2) for all x E A.

(,)

(ii) There is a constant C2 > 0 such that C2p(x) 0 with

c~llzlj < r(z)~-~(y)~. Therefore

IIF(A)I I -Ilzll < ~r(y) ~. Applying 7.4.14, we conclude that x y -

2

yx, completing the proof.

7.4.16. L EMMA. A complex or strictly real p-Banach algebra

- (A, ll" II), ~ith II" II ~m, sati4yi~g th~ co~ditio~

c111~112 ~ IIx211for

all x E A

(,)

i8 8.8..

PROOF. The condition (,) together with 7.4.11 and the fact A is G B (see 7.4.6) gives ClllZll ~ l](x) - r ( x ) p.

333

w4. GB Algebras

Therefore, r(x) = 0 ~ x = O, 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 ~, 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 A p -- A / P is primitive and by 1.9.8, A p is strictly real and so A p "" ~. Since A is s.s., N P = {0}, so that A is isomorphic to a subalgebra of the direct product I] Ap (of isomorphic copies of ~) and so commutative. 7.4.18.

COROLLARY. Every strictly real primitive p-normed

algebra is isomorphic to ~. Every strictly real p-Banach algebra satisfying condition (,) o/7.4.16 is commutative. 7.4.19.

COROLLARY.

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, II. 11) be a pnormed algebra with II" II sin. Then the following statements are equivalent: (i) A is a Q algebra. (ii) ~(x)P - L,(x) - l i m , ~ o IIx-ll ~

(x e A).

(iii) f(x) p ~< ]lxll (x e A). (Here ~ denotes the spectral radius in A, where A denotes A itself when A is complex and the complexification of A when it is real.)

t Yood proved the proposition for a normed algebra (i.e. p - 1).

Gelfand Representation Theory

334

PROOF. Assume (i). Then there is an rl > 0 such that for any x with Ilxtl < rl, x is q. invertible. Set c - r1-1 if A is complex, and - [ ( l + r l ) 8 9 - 1] -1 if A is real. For a given x e A take a

- a + ifl ~ 0 in C with I~1 p > cllxtl. When A is real we have" II~l-~(x ~ - 2~x)ll ~< I~l-~(llxll ~ + 2~l~l~llxlI)

~< j~l-2~(ll~ll 2 + 21~1~11~11) (since p ~< 1, I~1-< I~1)

-(11~-1~11 + 1) ~ - 1 < (~-1+ 1 ) ~ 1 ~l 1), (iii) D~(x '~) - n!(Dx) '~ (n >i 1). PROOF. (i) D ' ~ x - - D m - 2 ( D 2 x ) - 0 (m > 2). (ii) For m - 1, we have D ( D x ) - D = x - O. Assume now that n ( n x ) m - i - o ( m ) 2). Then D ( ( D x ) m) -

(1)

D ( D x ( D x ) m - l ) - D 2 x ( D x ) m-1 + D x . D ( D x ) m-1

- O.(Dx) m-1 -4- D x . O - O.

(iii) Assume that Dn-l(x n-l) -(n-

(2)

1)!(Dx) n-1.

Then Dn(x n-l) - D((n-

1)!(Dx) n - l ) - ( n -

where we have used (2). By Leibniz rule

1)!D(Dx) n - l -

0

(3)

w 6. A u t o m o r p h i s m s

D n ( x n)

--

347

and Derivations

Dn(xn-l.x)

D n x n - 1 .x +

D'~-lxn-I.Dx

rt D n _ r x n _ l . D r x

+ r=2

Using (1),(2),(3)the 1)!(Dx)'~-IDz

+ 0 -

r

above RHS reduces to 0 + n . ( n completing the proof (by induc-

n ! ( D x ) '~,

tion). 7.6.10. PROPOSITION. A

Let D

be a d e r i v a t i o n o f an algebra

a n d x an e l e m e n t o f A s u c h t h a t D x ~ x . (i) D x ~ - n x ' ~ - l D x (n >1 1),

(ii)

Dmx n -

xn-mam

for some element

Then: am e A

(1 ~< rn
~2),

(iii)

D'~x n - n ! ( D x ) ' ~ + xbn f o r s o m e e l e m e n t b , e A (n >1 1).

PROOF. We prove the results by induction. (i) Assume that D x '~-1 - ( n 1 ) x Z - 2 D x . Then Dx n

-

Dxn-l.x

+ xn-lDx

(n - 1 ) x n - l D x

-

(n - 1 ) x Z - 2 D x . x

+ xn-lDx

(ii) Assume that D m-ix n Dmx n

-

- nxn-lDx.

_ xn-m+lam_l.

--

D(Dm-lx")-D(x"-m+lam_l)

__

Dxn-m+

=

(n - m + 1 ) x n - m D x . a m _ l

-

X'~-mam(

1 . a m _ 1 q-- x n - m +

with

+ xn-lDx

1Dam-

Then 1

+ xn-m+lDam_l

am -- (n - m + 1 ) D x + x D a m - 1 ) .

(iii) Assume that on-ix

n-l-

(n-

1)!(nx)

n-I

q-xbn-i.

(1)

Then Dnx ~

_

D.-~D(x. ) - D.-*(nx.-*Dz)

=

riD"- i x " - t.Dx + n

~l(nr--1

=

n[(n -

1)!(Dx)"-~ +

r

_ nD.-~(x.-XDx )

1)D,-,-1-, x ,-,-1D rDx n - 1 zran+rDr+ 1x.

xbn_~lDx + n r--1

=

n!(Dz)" + zb.,

r

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 E A, set Dax-

(~a - ra)X -- a x -

xa.

Then Da is a derivation (see below) called an inner derivation. Further we have clearly the relations D~+b -- D, + Db; D~, - AD, (A E F). 7.6.12. LEMMA. (i) Da is a derivation (ii) D a = O iff a E Z (the centre of A ) ; D a = O for all a if A is commutative. (iii) If A is a TA then Da is continuous. (iv) If A (A,[[. [[) is a p - B a n a c h algebra, with ][. [[sin., then [[Da[[ 0 there is a compact set K in S such t h a t IS(~)I < ~ for all s E S \ K . All functions in C(S) which vanish at c~ form a subalgebra C0(S) of C(S). Note t h a t C0(S) = C(S) if S is compact. 8.1.11.

PROPOSITION.

C(S) for compact Hausdorff S and

356

Commutative Topological Algebras

Co(S) for locally compact Hausdorff S are Banach algebras under the sup norm. Moreover, C(S), Co(S) are s.s.. PROOF. The proof of the first statement being straightforward is omitted. For the second we note that for ~ - C(S) or Co(S), ker~ - O, so that { M ~ ' s E S} are all maximal ideals, and clearly N M~ - {0}, whence jr is s.s.. 8.1.12. DEFINITION. Let ~ = ~(S) be a function algebra. A subset ~0 of ~ is said to separate points (of S) if given any pair (Sl, s2) of distinct points there is an f E ~0 with f ( s l ) # f(s2). ~0 is said to strongly separate points if it satisfies in addition the condition ker ~0 = 0. A family ~0 of functions separating (respy. strongly separating) points is also referred to as a separating (respy. strongly separating) family. 8.1.13. R e m a r k . Let S be a locally compact Hausdorff space. Consider the Banach of algebra C0(S). Any strongly separating subalgebra A of C0(S) which is s.a. (i.e. closed for conjugates) is dense by the extended StoneWeirstrass 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 S being locally compact Hausdorff is completely regular. So for any point so c S we can choose f e C0(S) with f(so) r O. By density of A we can choose g e A with IIg - fll~ < If(s0)l. Then g(so) # O. Again, if Sl ~ s2 are two points of S and fl c C0(S) is such that fl(Sl) # fl(s2), we can choose gl e A 1 with ]lgl-flllcr < ~[fl(Sl)--fl(82)[. Then g l ( s l ) # gl(s2). Thus A is strongly separating. 8.1.14. LEMMA (Rickart). Let ~ - ~(S) be a function algebra which strongly separates points of S. Then, given any finite set of points so, 8 1 , ' " ,Sn (rt >1 1) of S, We can find an f E with f(so) # O, f(si) - 0 (j - 1 , 2 , - . . , n ) . PROOF. We first assume that n - 1 and write so - s, sl - t (s r t). We shall show that there is a f u n c t i o n f~ E with f ~ ( s ) r f~(t)-O.

w 1. Function Algebras

357

By hypothesis there is an f E ~ with f(s) ~: f(t). We have to consider three cases. C a s e 1: f ( t ) = O . We can take f s = f . C a s e 2: f(s) :/: 0, f(t):/: O. By replacing f by a suitable multiple we may assume that f ( t ) - 1 and then f(s) ~ 1. Set f~ _ f

_

f2;

then

f~(t) - O, fs(s) ~- 0.

C a s e 3: f(s) = 0 , f ( t ) = 1. C h o o s e a g e ~ } with g(s) r (this is possible since ker ~} = 0). If g(t) = 0, we can take f8 - g, and if g(t) ~ 0 we take f~ = f - g/g(t). We next assume t h a t n t> 2. By the result for n - 1 just proved above we can choose fj E ~ such t h a t fj(so) ~ 0, fj(sj)--0 (j1 , . . . , n ) . Set f fl""fn (product). T h e n f ( s 0 ) ~ = 0 , while f ( s j ) = 0 (j=l,...n). 8 . 1 . 1 5 . LEMMA. Let S - (S, 7) be a compact Hausdorff space. Then the weak topology t rw on S induced by any separating family ~o of C ( S ) coincides with the initial topology 7.

PROOF. 7w is clearly coarser then 7. Moreover, rw is Hausdorff since ~0 is separating S. It follows t h a t 7 = r,~ (since 7~ c 7, r~ Hausdorff, v compact). 8 . 1 . 1 6 . COROLLARY. If S -- (S, 7) is locally compact Hausdorff and C0(S) the algebra of K-valued continuous functions on S which vanish at oo, then the weak topology 7~ on S induced by any strongly separating family ~o of C(S) coincides with r. PROOF. Let Soo = S U{cc} be the 1-point compactification of S; S ~ is c o m p a c t Hausdorff. E x t e n d the functions f in ~}0 to S ~ by defining f ( c c ) - 0. Then the extended functions form a separating family of C(Soo). The required result now follows from 8.1.15.

Let S be a compact Hausdorff space and A a s.a. tt inverse-closed subunital algebra of C(S). Then 8 . 1 . 1 7 PROPOSITION.

t Anet so-+ s in S under Tw if f(s~) ~ f(s), V f E ~ o . t~ i.e., f E A ==>] E A, where f - f(s), ](s) - 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 - { f E A " f (s) - O}, for some s E 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 c A k e r A - 0, and by 8.1.5, for any s E S , M~ is a (hyper) maximal ideal. So by our supposition M ~= M~. By maximality of M, M ~ Ms, whence there is an fs E M with f~(s) ~ O. By the continuity of ]'8 we have ]'8 r 0 on an open neighbourhood Us of s. Since S is compact there is a finite number of points S l " - , s ~

in S with L J u j -

s, where

j=l

/_/3" -- Us~. Set f-

f l f l + " " f,~s

where f j -

f~j ( j -

1,... ,n).

Then f E M. Since

f - If~l 2 + . . . + If.I 2 > 0, f-1 exists in C(S). Since Z is inverse-closed in C(S), f - 1 E a . Thus, M contains an invertible element f, 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 C0(S) the algebra of K-valued continuous functions vanishing at oo. Then every maximal ideal of C0(S) is fixed. PROOF. Consider the 1-point compactification So~ of S. Then C ( S ~ ) i s t the unitization A1 of A - C0(S). The maximal ideals of A1 - C(S~) are, by 8.1.17, all fixed and they are {M~ 9s C Soo}, where M~ {fl e A1 " fl(s) 0}. It follows that the maximal ideals of C0(S) are -

-

-

MINA-M~(sEA), where M~ - { f C A ' f ( s ) -

0}, and so are fixed.

8.1.19. R e m a r k . In the proof of 8.1.17, the hypothesis that t i.e. c~n be identified with.

w 1. Function Algebras

359

A is s.a. was explicitty used. However, it is possible, for a subalgebra A of C(S) which is not s.a. to have the fixity property for the maximal ideals. For instance, the Banach algebra 92 of example (ii) of 7.3.22 is a subalgebra of C(D), which is not s.a. (fo ~ ~t; fo(z) - z). ~t has, nevertheless, the fixity property. In fact, if M is a maximal ideal of 92 and fo(M) = zo then M = Mzo (see [10, p.33]). 8.1.20. PROPOSITION. Let the algebra A - C(S), where S is compact Hausdorff, be a T A under a topology r which is finer than the weak topology rw. Then A is Gelfand and its spectrum A - Ac is homeomorphic to S. PROOF. By 8.1.17, if X C A -- A(A), then X - X8 (s E S). Since x ~ ( f ) - f(s), by definition of Tw,X~ is rw-continuous (f~ --~ f if x ~ ( f ~ ) - f ~ ( s ) ~ f ( s ) - x s ( f ) ) . Since r D_ rw,xs is also r-continuous. It follows that the maximal ideals M M~(s ~ S) are r-closed, hypermaximal and so a (a,r)is Gelfand. Consider now the map A 9s ~ X~ of S to A. By 8.1.17, A is surjective. It is also injective since S is normal if Sl # s2, there is a f ~ C ( S ) w i t h f ( s l ) # f(s2), so that X~x # X~2. Further, A is continuous, since if s~ ~ s and f is continuous, X,~(f)

- f(s~) ~

f(s) -

x,(f)-

Finally, since S 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),ll.llc~), or more generally, the p-Banach algebra (C(S),I[-I1~) ha~ S its spectrum. PROOF. This follows from 8.1.20 by taking r to be the norm topology. 8.1.22. THEOREM (Gelfand-Kolmogorov-Stone-Banach). If $1, $2 are compact Hausdorff spaces such that the Banach algebras C(S1) and C($2) are isomorphic (as algebras) then S 1 and $2 are homeomorphic. PROOF. Since the spectrum A of the Banach algebra A C(S) depends only on the algebraic structure of A, we have,

Commutative Topological Algebras

360

by 8.1.21, S1 --~ A1 -'~ A2 -~ S2, where _~ stands for homeomorphism. R e m a r k . In the algebra A - C ( U ) - C ( R , K ) , every maximal ideal is fixed. For, let 8.1.23.

I-- {fcA'f(t)-Ofor

not

a l l t > somety}.

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. R e m a r k . Let S be a completely regular Tl-space and A - BC(S) the algebra of bounded continuous functions on S. 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-(~ech compactification f l ( S ) o f S (see [26, p.331] or [28, p.415]). 8.1.25. Theorem. Let A - (A, II" II)be an ample, complex or strictly real, p-Banach algebra, with I1" II ~ m T h e n a is t. isomorphic to a canonically Banach function algebra iff II-II satisfies the condition

CIIxll = ~< IIx~ll for all where C is some positive constant (cf.

(,)

x,

[28, P.409, Theorem.

4.8.]). PROOF. Let I1" II 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 ff 9x ~ ~ 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 a = (a, [[" II*). Then, by 8.1.7, we have

(liyll*) ~ -Ily~ll *

(y e a).

It follows from 7.4.11 that there is a constant C2 with

c~]]yil*-< r(y)~

( y e a).

w2. Shilov Boundary

361

Let ~9 denote the t. isomorphism between A and ;~. Then we can transfer the norm of a to A by setting for x E A , y--~o(x)

I1~11'- I1~(~)11"- Ilyl{* then I1" II* ~ I}" II, Since r ( y ) - r(x), we get

Since I1" II* ~ I1" tl there is a constant Clllxll*. Therefore we get

C1 >

0 such that Ilxtl

62 flxll ~ r(x)~ C1

whence, by 7.4.11, there is a C > 0 with

cllxll = sup{lg(t)l.t c Fo\N1}. Therefore I c - l g ( s l ) l - 1, [c-lg(t)l < 1 for t e Fo\N1. By replacing g by c-lg we may assume c - 1, so that we have I g ( 8 1 ) l - 1,

]g(t)l < 1 for all t E Fo\N1.

(2)

By taking a sufficiently large integer n and setting gl - gn we get

]]gllloo --[gl(Sl)]- 1, Igl(t)l


3 ~,

[b(A)[ ~< 3 ol-

for

A E

a(a)\V

(since

- Xol

Since [[blloo > ~3 > I/~(A)[ (k e a(a)\V ) , by 8.2.5, A0 E c3Aa(a) . This completes the proof.

w3.

Hull-Kernel

Topology

8.3.1. Let A be an algebra (or more generally a ring) with A =/- x/~. Denote by P the set of prime (bi-) ideals P of A with P ~- A, by ~ the set of maximal regular bi-ideals of A, and by II the set of primitive ideals of A. 8.3.2. LEMMA We have E C_ II C p; p, II=fi O. PROOF. The inclusion relations follow from 1.5.10, 1.5.9. Since A ~= x/~, II J= 0. 8.3.3. For any subset S of P we write

k(S)-Is-

r~ P

PES

and call k(S) the kernel of S; k(S) is a bi-ideal of A. We have clearly" For $1 _C $2, k(S2) _c k(Sx). (1) If I is a b i - i d e a l w e set h ( I ) - { P E P the hull of I. We have obviously:

9P_~ I} and call h(I)

For I __ J, h(J)_ h(1).

(2)

w3. Hull-Kernel Topology

371

The following inclusions are clear from the definitions. S _c hk(S).

(3)

I c_ kh(I).

(4)

By virtue of (1),(3) we obtain k(S) - khk(S).

(5)

Similarly, by (2),(4)we get h ( I ) - hkh(I).

(6)

(By applying k to (3)we get, by (1), khk(S) c k(S); on the other hand, khk(S) - kh(k(S)) _ k(S). Combining the above two inclusions we get (5). Similarly, (6) can be proved.) 8.3.4. LEMMA. (i) ~ a h ( I a ) = h(E~ I~); (ii) h(IJ) = h(I) U h(J) = h(I ~ J); (iii) h(A) = 0; (iv) h ( { 0 ) ) = P. PROOF. The properties (i),(iii),(iv) are immediate from the definition of h. For (ii), we first observe that since I J c_ I, J we have h(I),h(J) c h(IJ), so that h(I) Uh(J)_C h(IJ). On the other hand, if P E h(IJ) then P 2 I J and since P is prime, P _~ I or J, so that P E h(I) or h(J), whence h ( I J ) c h(I) U h(J). Combining this with the reverse inclusion relation obtained above we get the first equality in (ii). Again, since h(I) U h(J) _ h(I A J) _c h(IJ), 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(I) : I a hi-ideal of A } . The resulting topology is called the hull-kernel or hk topology. We denote P with this topology by Phk" PROOF. It follows from 8.3.4 that the family {h(I)} 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(S),

372

Commutative Topological Algebras

where bar denotes closure in Phk"

PROOF. By definition of hk topology, hk(S) is a closed set and hk(S)D S (by(3)of 8.3.3)Further, if h(I)___ S then h ( I ) hkh(I) D hk(S). Therefore S - hk(S). 8 . 3 . 7 . COROLLARY. For a closed subset S, among the hiideals I of A with h ( I ) - S the largest one is k(S).

PROOF. First observe that k(S) is a bi-ideal and h ( k ( S ) ) hk(S) - S since S is closed. Next, if h ( I ) - S, then I _c k h ( I ) -

k(s). 8.3.8. PROPOSITION. Phk is a To-space. is also a To-space while the subspace E is T1.

The subspace H

PROOF. If P1,P2 6 Phk and P= ~ P1 then P2 r h({P1})= {P1}, proving Phk is To. By a similar argument II is also To. Finally, if M 6 E then {M} = h ( { M } ) = {M} (by maximality of M, whence E is T1.)

A,

8.3.9. DEFINITION. H - HA is called the structure space of ~ -- ~ A the strong structure space of A. 8.3.10. PROPOSITION ([23, p.79]). Let J be a hi-ideal of an

algebra A.

Then:

(i) r ~ A \ h ( J ) i s homeomorphic with Ej

under the map e 9

M---~ J ~ M . (ii) If A # - A / J then the hull h(J) of J in EA is homeomorphic with ~A# under the canonical map 8 # 9M ~-~ M / J .

PROOF. By 1.4.3, the map ~ 9 M ~-, M ~ - J ~ M is a bijection between E A \ h ( J ) and Ej. To prove that 0 is a homeomorphism it is enough to prove that 8 ( h k ( S ) ) - hk(8(S)). Clearly k ( 8 ( S ) ) - J ~ k ( S ) . If M 6 hk(S) then M _ k(S) and ~(M) _ J [-1 k(S), so that 8(M) 6 h(J ["lk(S)). Conversely, suppose that

J F"I M

k(O(s)) - J r'l k(S).

w3. Hull-Kernel Topology

373

Then M D J Ak(S), and since M ~ J, by primality of M, M _~ k(S), so that M E hk(S). This completes the proof of (i). To prove (ii), we observe that if I is a bi-ideal of A which contains J, and its image under the canonical homeomorphism is denoted by I#, 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 hi-

ideal of A then h(I), h(/) N II, h(I) f ] E hk -topology.

are compact under the

PROOF. Let {F~} be a family of closed sets in any one of these spaces such that

VIF -0. Write k ( F s ) - I s

and K -

(,)

t Ia. Since F~ _D h(I) we have S

I c_ hk(I) __ k ( F ~ ) - I~. So

K~Is~I, and K is regular. Suppose that K ~ A. Then by 1.2.10 there is an M C E with K c M. But then M D K D Is, so that M E I'] h ( I s ) -

NFs.

(since h(I~) - hk(F~) - F~). The last conclusion contradicts ( . ) . So we must have K - ~-~'~Is - A. Let u be a relative S

(bi-)unity for I. Since

ucA-K-~Is S

there is a finite subfamily of ideals I 1 , ' . . , I,~ (I i - I~j) such that

uCJ-Ii+...+In. t ZI~

denotes the smallest bi-ideal containing the I~ 's.

374

Commutative Topological Algebras

But since J D Ij D I, J is regular. J - A. This implies that n

n

/=1

/=1

N Fj - N

Therefore, since u E J,

n

hk(Fj)

-

N

h( j)

/=1

-

h(J)-

o.

which proves the desired compactness. 8.3.12. Let A be a T A and ~ (respy. ~ c ) the set hypermaximal (respy. closed hypermaximal) ideals of A. We have clearly the inclusion relations ~c_~__E so that ~4, ~r inherit, by relativization, the hk-topology. This topology on ~ (respy. ~c can be transferred to A (respy. Ac) in view of the bijection. -~ A (respy. 34c-* Ac) (see 7.3.11). If E is a subset of A or Ac we define k(E) -- n ker X

(,k" E E).

If I is an ideal of A, the hull h(I) with respect to A (respy. Ac ) is defined by h(I) - {X e A( respy. A+)" k e r x _D I}. Then a subset E C A (respy. Ac ) is closed iff E 8.3.13. described in Xa --+ X in A and any

hk(E).

LEMMA. The hk-topology on A (or Ac) can be terms of net convergence in the following way: a net the hk-topology if it satisfies the condition: for x E subnet (Xa') of (X~), " X a , ( x ) - 0 for all X ~ , " ~

x(x) - o . PROOF. It suffices to note that if Ma, - ker X~', M - ker X then M D_ N M~, iff ~(M~,) - 0 for all M~,, =~ ~(M) - 0

(x E A).

w3. Hull-Kernel Topology

375

8 . 3 . 1 4 . COROLLARY. If 7w denotes the (weak) topology of the spectrum A (or A c ) and rhk the hk-topology on it, then rhk is coarser than 7w "Thk C_ t 7w.

PROOF. Suppose that a net Xa -+ X in A (or Ac ) under rw. Then X~(x) -+ X(x) for all x e A. In particular, if X~,(x) = 0 for all X~' then X ( X ) = 0, which means that Xa -+ X under rhk. Therefore rhk C 7w, as required. 8.3.15. DEFINITION. Let A be a T A with Ac J= 0. A is called quasi-unital if there is an element u0 E A such that t~0(X) =/= 0, VX c A~. If A is unital (with unity e ) it is also quasiunital since e(x) - 1 J= 0 (X e Ac). 8.3.16. LEMMA. Let A be a s.a. p-seminormed algebra such that Ae is non-empty and compact. Then A is quasi-unital. PROOF. For each X E Ac choose an element x x of A with ~cx(X) - X(Xx) ~ O. By continuity of xx and compactness of Ac we can find, as in the proof of 8.1.17, a finite subset { X 1 , ' " , X,~} of A~, elements x j - Xxi and open neighbourhoods Uj of X1 tI

~i(X) =/= 0 on Uj ( j -

such that

1,...,n)

and

U u j - Ac. j=l Since A is s.a., we can find yj E A withgj - ~j ( J - 1 , . - . , n ) . n

Set u o - ~ ~

xiy i. Then

j--1 n

a0(x)-

n

j(x)yj(x)j=l

I j(x)l > 0 j=l

(since if X e Uj, xo(X) r 0). 8.3.17. PROPOSITION. Let A be a s.a. complex spectrallyGel/and p-Banach algebra with compact spectrum A ( - A c ) . Then v/A is a regular hi-ideal. PROOF. If V ~ - A, v/A is trivially regular. So we may assume that A ~= v/A. Let A1 be the unitization of A and let t i.e. every rbk-open set is rw-open.

376

Commutative Topological Algebras

A1,/~

denote respectively the spectra of A1, A. It follows (see 7.3.2(a)) that we have A1 -- {X1 E t I " x , I A

c a} U{xo},

(,)

where X0 is the distinguished character of A1. Since, by hypothesis A is compact, so is A1. By 8.3.16, there is an u0 ~ A with uo(X) ~ 0 for all X E A. Since A is spectrally Gelfand we have: -

It follows that

O-I(UO ) --

~tO(t)LJ{~o(ttO)},

where al denotes the spectrum with respect to A1. Since A is compact, ~0(A) is also compact and hence closed. It follows from 7.5.14 that there is an idempotent u E A1 with fi(X0) = 0, ~(A) = 1. But then Xo(u) = O, u E A. Further, if z E A and X = x~IA

ux(x)

then

(X1 # XO)

-- fi(X)~c(X) - 1. Sc(X ) - 3c(X ).

This relation holds for every X E A, whence we get u x - x e r'l ker X - ~ x/~ (by 7.2.12), whence v/A is regular.

In a commutative s.a. complex pBanach algebra A with compact spectrum, V ~ is regular. 8.3.18.

COROLLARY.

PROOF. We may assume that A =fi x/~. 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 x / ~ is regular. PROOF. The complexification A of A is s.a." A

=+iy

A

(x) - x ( = ) + i x ( y ) - x ( = ) - i x ( v ) - = - i y

(x).

N

Since, by 7.3.3, A - A ( A ) i s homeomorphic to A - A(A) and A is compact, ,~ is compact.

It follows from 8.3.17 that v ~

is

w4. Completely Regular Algebras

377

regular. By 1.9.17, v/A is self-conjugate and

It follows that

Therefore (see 1.9.16(i)) x / ~ - Rev/-~ 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//V~ is unital . (ii) A = A~ is compact . PROOF. (i) ~ (ii) 9Since A / / ~ is unital, v/A is regular whence ~-A 2 x/~ t is also regular. So, by 7.3.12, (ii) holds. (ii)~ (i)" By 8.3.18 or 8.3.19, x/~ is regular, and so (i) holds. 8.3.21. compact.

C OROLLARY. If A is s.s., then A is unital iff A is

8.3.22. R e m a r k . (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 A1 is commutative unital Banach algebra and A2(=fi {0}) a Banach algebra with trivial multiplication. Then A ( A 1 ) 0 and compact, while A ( A 2 ) # 0. If A = A1 • A2 then by 7.3.14, A(A) --~ A(A1) is compact but A is not unital.

w4.

Completely

Regular

Algebras

8 . 4 . 1 . DEFINITION. Following Willcox an algebra A is called completely regulartt 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 E is Hausdorff. (ii) Each point M E E 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. E x a m p l e s of C o m p l e t e l y r e g u l a r algebras. (i) The Banach algebra C0(X)

(see 8.4.15).

(ii) The group algebra IX(G) of a locally compact Hausdorff commutative group G (see [22, p.426]). (iii) The group algebra LI(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 p (see Appendix). 8.4.3. PROPOSITION. Let A be a completely regular algebra. Then" (i) E is locally compact Hausdorff. (ii) E 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})= E is compact. 8.4.4. PROPOSITION. I r A is a completely regular algebra and J a bi-ideal of A then J and A # - A / J are completely regular. PROOF. Let E j denote the strong structure space of J. By 8.3.10 (i), O" M H J ~ M is a homeomorphism of E \ h ( J ) on Ej. By complete regularity of A,E is Hausdorff and hence by the homeomorphism 0, ~J is also Hausdorff. Further, if M E E has V as an open neighbourhood with k(Y) regular then 0(M) has 0(Y)

w4. Completely Regular Algebras

379

as an open neighbourhood and k ( 0 ( V ) ) - J N k(V) is regular (of. 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), I],4# is Hausdorff. If g) 9 A --+ A # is the canonical homeomorphism and M - ~ - I ( M # ) ( M # E l ] a # ) then M E l~a. Let V be an open neighbourhood of M 0 - g~-l(M0# ), M0# E EA#, such that k(V) is regular. The intersection I, of all M C EA with M D k(V), M _D J, is also a regular bi-ideal (since I _D k(V)). It follows that i f V # - { M # "M_D k(V)} t h e n V # is an open neighbourhood of M0# with k(V #) - g~(k(V)) regular. This completes the proof. 8.4.5. PROPOSITION. The unitization A1 of an algebra A is completely regular iff A is completely regular. PROOF (cf. [23, p.84]). Assume that A is completely regular. Since A1 is unital, to prove that it is completely regular we have only to show that ~1 -- ~A1 is Hausdorff. Since A is a bi-ideal of Ai, by 8.3.10 (i), E - EA is homeomorphic to EI\{A}. By complete regularity of A, E and consequently E I \ { A } is Hausdorff. To complete the proof that ~1 is Hausdorff it remains to show that the point A can be separated by open sets from any other point M1~ in El. Write M ~ - A N M1~ By complete regularity of A there is an open neighbourhood V of M0 with I - k(V) a regular bi-ideal, with a relative unity u (say). By 1.4.7. (iv), I1 - / ~ - I ~ - A i ( e i - t t ) is a bi-ideal of A1; I1 ~ A since e l - u @ 11. It follows that A c ~1\h1(I1) - U1 (say)were hi denotes the hull relative to A1. For M C V0 consider M1 - M. Then A N M 1 - M. Since M D I - k(V), M1 ~ I1. It follows that V1 - 8-1(V) C_ h1(I1), where 0 is the homeornorphism M1 ~-~ A N M1 - M (see 8.3.10) so that U1/1 V1 - 0. This completes the proof of the "if" part. For the "only if" part assume that ~]1 is Hausdorff. Then E E i \ { A } , is also Hausdorff. If M E E then i - A N M 1 (M1 E El). Since ~1 is Hausdorff we can choose an open neighbourhood V1 of M1 such that A ~ hik (V1) , i.e., k(V1) - /1 ~ A. It follows (see 1.4.7 (ii)) that I - A ('//1 is regular. Since V1 is open we may assume that V1 -- h ~ ( J i ) - ~ i \ h i ( g i ) . ^

Then it is easy to see that V - h~(g), where J - A N gl, is an

380

Commutative Topological Algebras

open neighbourhood of M. Since k ( V ) - ANk(V1) -- A ~ I 1 -- 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 of E. Then k ( F ) is regular iff F is compact.

PROOF. If kCF) is regular then by 3.3.11, h ( k ( F ) ) = h k ( 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 , - . . , n ) } , with each k(Vj) regular. By virtue of 1.2.15, k(Vl[.J...[,JVn) k(V1) ~ . . . k(Vn) = 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 rhk on Ac coincides with the weak topology rw. Conversely, when A is p-seminormed Gelfand, 7hk - - 7w On A c ( - - A ) implies that A is completely regular.

PROOF. Suppose that A is completely regular. We have always, by 8.3.14, rhk ___ rw. To prove the reverse inclusion, let F be a T~-closed set in A~. Take X0 c A~\F. By complete regularity of A we can choose a rhk-open neighbourhood V of Xo such that k(V) is regular. Let u be a relative (bi-) unity for k(Y). Set Fo - {X E F " X(u) - 1};

F0 is r~-closed. By 7.3.13, the set F1 - {X E AC " IX(u)I >>.1}

is T~-compact. Therefore, F0 as a closed subset of F1 is also T~compact. Since 7hk C Tw, F0 is also 7hk-compact and so 7hkclosed (7hk being Hausdorff by complete regularity of A). Since by choice X0 ~ F0 and Thk is Hausdorff, each point X E F0 has a rhkopen neighbourhood U with X0 ~ U - hk(U). By rhk-compactness there is a finite covering of F by open neighbourhoods U 1 , ' " , Un such that X0 ~ hk(Uj) (j = 1,... ,n). It follows that there is an element aj E k(Uj) with xo(ai) r O. Also, since Ac\Y is ~hk-Closed

w4. Completely Regular Algebras

381

and X0 r A t \ V , there exists a0 6 k(Ac\V) with xo(ao) r O. Set

a - aoal ""an. Then" xo(a) r 0,

(1)

a 6 ker(U1 U " " U UnU Ao\V) _Cker(F0 U At\V)"

(2)

If X 6 V then ker X - k(V) and so ker X has also u has a relative unity. It follows that X(U) - 1 whence V ~ F c F0, so that we have

r c F0 From (2),(3) we conclude that a 6 ker F. Also, by (1), xo(a) # O. Therefore X0 r hk(F), proving F = hk(F), so t h a t F is rhk-Closed and rhk = rw. For the converse part, assume now that A is Gelfand and rw = rhk. Since A is Gelfand, E = A = Ao, so t h a t E is Hausdorff. Further, if X0 6 A we can choose a b 6 A with xo(b) = 2. Setting V = {X 6 Ac: Ix(b)[ > 1}, V is an open neighbourhood of X0 and VC_ { x 6 A o ' I x ( b )

I/> 1}

is compact for Tw - 7hk. Therefore, by 8.4.6, k(V) is a regular ideal and hence the ideal k(Y) _D k ( Y ) i s also regular, completing the proof of the converse (and also of the theorem). 8.4.8. DEFINITION. Let S be a topological space and Y0 a family of continuous K-valued functions on S. Following Naimark (Neumark) :To is called completely regular or a completely regular family if it satisfies the condition: To each closed set F in S and to each point so r F there is an f = f ( s ) i n 70 such that

f (s) - 0 on F and f (so) :/: O.

(,)

70 is called normal if it satisfies the condition: To each pair of disjoint closed sets F1, F2 in S there is an f E ~ro with f(s)-0 8.4.9.

LEMMA.

completely regular.

on F1 and f ( s ) -

1 on F2.

(**)

(a) If S is T1 then every normal family is

Commutative Topological Algebras

382

(b) A space S is completely regular iff the family C(S) is com-

pletely regular. PROOF. Clear. 8 . 4 . 1 0 . Let S -- (S,r) be a completely regular Hausdorff space and C(S) - C(S,~:) the algebra of all continuous K-valued functions on S. Let A be a strongly separating subalgebra of C(S). Denote by N the m a x i m a l ideal s p e c t r u m of A. For s E S, set M~ - M A - {]" C A" f ( s ) - 0}. Since k e r A - 0, each Ms E J~. Further, since A separates points of S, the m a p 12 : s ~-* Ms is injective. Write N0 - {Ms : s E S}. Since

)4o_C N c_ r~ we have hk-topology rhk on J~0 (got by relativization). Moreover, since f~ is a bijection between S and J~0, rhk can be transferred from ~ 0 to S and the transferred topology we denote again by rhk. We also write for an ideal I of C(S)

ho(1)- {M E ~to" M 2 I} - ~to n h(1). 8.4.11.

PROPOSITION (cf. [19, p.57]), rhk C_ r, and rhk -- r

iff A is a completely regular family of functions. PROOF. Let F C S be a rhk-Closed set so t h a t h0k(F*) - F*, where F* - 12(F). If f e A then f e k(F*) iff f - 0 on r . Further, M+ C h0k(F*) iff s e n k e r f, where f ranges in k ( r * ) and k e r f - {s e S " f ( s ) - 0}. It follows t h a t F - n k e r f . Since f is continuous each ker f is r-closed. So F is r-closed, whence rhk C r. Suppose now t h a t A is a completely regular family and F ( C S) is r-closed; then I - k(F*) - { f e A" f - 0 on F}. Clearly M+ c h0(I) - h0k(F*) iff "f - 0 on F ~ f ( s ) - 0". If s ~ F there is, by the complete regularity of the family A, an f C A with f - 0 on F but f ( s ) ~ O. This means t h a t Ms h0k(F*), whence h0k(F*) - F*, so t h a t F* and therefore F is rhk-Closed , proving t h a t r - 7 h k . Conversely, if r - rhk and F C S is closed, then for each point s ~ F - h0k(F) there is an f e A with f - 0 on F and f ( s ) ~ O.

w4. 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. r h k - - 7 on S for the algebra C(S). PROOF. Since S is a completely regular space, by its definition C(S) is a completely regular family and hence the result (by 8.4.11). 8.4.13. PROPOSITION. Let S - (S,r) be compact Hausdorff and A a separating subunital algebra of C(S) such that every maximal ideal of A is fixed. Then rhk -- r (on S) iff A is completely regular algebra. PROOF. If A is completely regular then rhk is Hausdorff. Since, by 8.4.11, rhk _C r, and r is compact Hausdorff (by hypothesis) it follows from a well-known result in topology that rhk = r (on S). Conversely, if rhk = r then rhk on ~0 = E (every maximal ideal of A being fixed) is also Hausdorff. Therefore, A being unital, is completely regular. 8.4.14. COROLLARY. The algebra C(S) 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 C0(S) is completely regular. PROOF. Let S~ be the 1-point compactification of S. Then, by 8.4.14, C(S~) is completely regular and so, by 8.4.5, C0(S) is completely regular since C(Soo) is the unitization of C0(S). 8.4.16. PROPOSITION. Let S - (S,r) be a locally compact Hausdorff space and A be a strongly separating subalgebra of C0(S) - C0(S, I1" I]oo). Then the following two statements are equivalent" (i) A is a completely regular family of functions. (ii) The hull-kernel topology rhk coincides with r. If A is also Gelfand then (ii) (or (i)) is equivalent to

Commutative Topological Algebras

384

(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 pseminormed algebra then fi is a completely regular family of functions. Conversely, if A is a p-seminormed Gelfand algebra such that fi is a completely regular family then A is a completely regular algebra. PROOF. Assume that A is completely regular. Then, by 8.4.7, r]~k - r~ on A~ - w h i c h is locally compact Hausdorff. By 8.1.25, A is a strongly separating subalgebra of C0(Ac). So by 8.4.16, .3. 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, 7~ = 7hk on A~(= A -- E) so that by 8.4.7., A is completely regular. 8.4.18. PROPOSITION. Let A be a completely regular pseminormed algebra. Then OAAc -- Ac. PROOF.

Write O,iAc = F; then F is a closed set in A~. If

A ~ \ F =fi 0, take a X0 E Ac\F. Since, by 8.4.17, A is a completely regular family, there is an a e A with a(F) -- 0 and h(X0) =fi 0. Then ]I5]1oo > 0, but suplh(x)] = 0, contradicting that F is the • Shilov boundary. Hence A~ -- F - c3,iAc , 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

(ii) Any open set G C_ Ar

A~

Ae(A) is homeomorphic to

is homeomorphic to A~(k(A~\G)).

PROOF. By complete regularity of A, 7hk -- 7w 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.

w4. Completely Regular Algebras

Write A ~ \ G - F. T h e n x E G i f f x r Fiffxr x I k ( F ) c A~(k(F)). Hence the h o m e o m o r p h i s m in (ii). 8.4.20.

385

k(F) iff

PROPOSITION (cf. [12, p.55]). Let A be a completely

regular, complex or strictly real, commutative p-Banach algebra A ( ~ v/A). Let I be an ideal of A, F1 a closed subset of A ( - Ac), F2 a compact subset A such that

(h(z)U F1)N -o. Then there exists an element ao E I N k(F1) such that 50 - 0 F1 and a o - 1 on F2.

(,) on

PROOF. Write I s. -- k(Fj) ( j - 1,2). Then Fj - h(Ij), so t h a t condition (,) above becomes (h(I) U h(I1)) N h(I2) - 0, i.e.,

h(I N 11)Rh(I2)

- 0

(1)

which means t h a t no character X of A can vanish at the same time on both I N I1 and /2. Since /2 is clearly the intersection of all h y p e r m a x i m a l ideals c o n t a i n i n g / 2 and A is Gelfand, it follows t h a t A # - A / I 2 is s.s.. Since by 8.4.19, A ( A #) ~_ F2, and F2 is c o m p a c t we conclude by 8.6.6 t h a t A # has unity u # - u + / 2 (say). Denote by 99 the canonical h o m o m o r p h i s m A --, A # - A/I2. We assert t h a t 99(IN I1) - A #. At any rate J # - ~ p ( I ~ I 1 ) i s an ideal of A #. If J # r A # there is a h y p e r m a x i m a l ideal M # with g)(I n I1) c_ M #. Then M -- p - l ( M # ) is a h y p e r m a x i m a l ideal of A with I N I1,/2 _ M. But this contradicts relation (1). Therefore we must have 99(I N I1) - A, as asserted. It follows in particular t h a t there is an element a0 E I N I1 such t h a t a0 § - to(a0) u #. If X e F1 then X - 0 on k(F1) - I1 so t h a t a0(x) - x(ao) - O. On the other hand if X E F2 then X = 0 on k(F2) = / 2 , so t h a t

ao(x) - x(ao) - x# (ao-4- I 2 ) -

X#(U #) -- 1,

where X # is the character of A # induced by X. This completes the proof. 8 . 4 . 2 1 . COROLLARY. A being as in 8.4.20, if F1 is a closed subset and F2 a compact subset of A ( A ) such that F i e F 2 - O, then there exists an element ao E k(F1) with ~o - 0 on F1 a n d - 1

Commutative Topological Algebras

386

on F2. PROOF. Suffices to take in 8.4.20, I -

A (then h(I) - 0).

8 . 4 . 2 2 . COROLLARY. If I is an ideal of A and F a compact subset of A(A) such that h(I) ~ F - 0, there is an element ao e I such that 5 o - 1 on F. PROOF. Suffices to take in 8.4.20, F1 - 0 (then k(F1) - A) and F 2 - F. 8.4.23.

COROLLARY. If A is a completely regular, complex

or strictly real, commutative unital p-Banach algebra then ~i 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 pseminormed algebra and F a closed subset of J~c (the space of closed h y p e r m a x i m a l ideals). Denote by J = j ( F ) the set of all x c A such t h a t ~ has compact s u p p o r t t disjoint with F. If F is a single point {M} we write j ( M ) for j ( F ) . Further, if F - 0 we write J0 for j(0).

8.4.25. LEMMA. J -- j ( F ) is a bi-ideal of A such that h ( J ) F.

PROOF. Clearly x C j ( F ) iff there is a compact set Cz such t h a t ~ - 0 outside Cz and Cz ~ F - 0. From this result it is easy to see t h a t j ( F ) is a bi-ideal. Further, if X e r then X(x) -- 0 for every x e j ( F ) , whence h ( j ( F ) ) 2 F. Now consider a 2:o e A c \ F . Since A~ is locally compact Hausdorff and F is a closed set there is an open set U in Ac with X0 E U and U compact and disjoint with F. By complete regularity of A there is an x in A with X(Xo) # 0 and ~ - 0 on Ac\U. Then x e j ( F ) and Xo ~ h ( j ( F ) ) , which proves t h a t h ( j ( F ) ) - F. 8 . 4 . 2 6 . PROPOSITION. Let A be a commutative completely t For a function f 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 A c - A. Then g - j ( F ) is the smallest of the ideals I such that h(I) - F. PROOF. In view of 8.4.25, it is enough to prove t h a t for any I with h(I) - F we have J C_C_I. If x E J and C is the s u p p o r t of then C is c o m p a c t and C N F - 0. By 8.4.22 there is a y E I with ~) - 1 on C. Since ~ - 0 outside C we have clearly &~ - ~. But A being s.s., we get xy - x, whence x E I, J _c I, completing the proof. 8 . 4 . 2 7 . COROLLARY. ideal I of A with h (I) - F.

The closure J is the smallest closed

PROOF. If M E A then M is a closed ideal, so t h a t M _~ J ==~ M D g. Therefore h(J) - h(J) - r. Further, if I is a closed ideal with h(I) - F then J C_ I and so J C I - I, completing the proof. 8 . 4 . 2 8 . COROLLARY. For each 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 . 2 9 . THEOREM (Abstract Wiener Tauberian T h e o r e m ) . Let A be a commutative s.s. regular, complex or strictly real, pBanach algebra and Jo the ideal of elements x such that ~ has compact support. If Jo is dense in A then every closed ideal I ( r A) is contained in a maximal regular ideal M. PROOF. If I is not contained in any m a x i m a l regular ideal then h(I) (in ~ ) - 0. Since maximal ideals of A are closed and J0 is dense in A it is clear t h a t h(Jo) - 0. By 8.4.26 we get J0 c I. But then since I is closed and J0 dense we must have I - A - a contradiction. 8.4.30. Remark. For connection between above t h e o r e m and classical Wiener Tauberian t h e o r e m 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

w 5. H o l o m o r p h i c 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 N:n. Set

h(K) - {,,~- ( ~ 1 , ' " , ,,~n) E

K n"

fP(A)I IIPIIK

for each polynomial P over N:n}, where IIPIIK - - s u p { l P ( ~ ) l : g c K}. Then clearly h(K) D_ K and h ( K ) i s 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 V-convex set K is compact. (ii) K is p-convex iff for each ~o e K'~\g, there is a polynomial Po with IPo(A~ > liPolIK. (iii) If K is polynomially convex and C > O, then for each ~o c ~ n \ K we can find a polynomial Q with

> c,

IIQlIK c.

PROOF. (i) It is clear from its definition that h(K) is closed. Also, if Aj is the polynomial hi(A1,-.. ,A,~) = Aj then by taking P - Aj in the definition of h ( g ) w e get for A e h ( g ) , IAjl ~< sup{l#jl : fi e K} = Cj < c~. This means that h ( g ) i s bounded and consequently it is compact. (ii) Clear from the definition of p-convexity. (iii) Given ,~0, choose first P0 as in (ii) and then set Q(A) -

CPo( )/IJPo]l 8.5.3. R e m a r k . 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 a(a) of a t.generator a of a complex monogenic p-Banach algebra is p-convex. In this form the result is generalized in 8.5.15 (ii). 8.5.4.

Recall that a polydisc Pd with poly-radius 5 =

w 5. Holomorphic Functional Calculus

389

(51,-'-,5~) is given by

P ~ - {a ~ ~ " lajl-< ~j, J - 1,... ,~}. A subset II of P g is called a p-polyhedron (in P g) if there are polynomials P 1 , ' " , Pm such that H-

{,~ c [?~." IIPI(,~)I,...,

IPm(X)l ~< 1}.

$.5.5. LEMMA. (a) Pg itself is a p-polyhedron. (b) Every p-polyhedron is p-convex. PROOF. (a) Take m -

n, Pj - 5j-IAj, where A1 is the poly-

__+

nomial h i ( A ) - )~j. 0 (b) If ,~o r 0 53. or some IPk(~~

1. In the first case I A j ( ~ ~

>

0 I,kjl > 53./> ]IAjlIH and in the second

case we have IPk(,~~ > 1 i> IIPkllri. Thus in either case ,~o ~ h(H) which means that h(II) - H and II is p-convex 8.5.6. LEMMA. Let K be a p-convex set with K C_ F$ and G an open set with K C_ G C_ K '~. Then there exists a p-polyhedron II with K c II C G. By 8.5.2 (iii), we can choose for each ~0 E R:n\K a polynomial PS0 with P~0(~ 0) > 1, IIPslIg ~ 1. By continuity of PnOOF.

_~

--+

PS ,) we have IPs0(A)I > 1 for A in some (open) neighbourhood A/S0 of ~0. Writing F -- Pd and allowing Ao to range in P\G(_C P\R:) and using the compactness of P\G we can find a finite family of neighbourhoods AfD , corresponding to polynomials PSi, such that

~s,, 9 U" Set II - {X C e "IPD

U-~f,o - P\a.

(X)I .< IIPx; IlK -< 1}, so that

K C II. Suppose

next A r G. If also~ A r F then~of course A r H(since II _ P). On the other hand if A E F then A c P\G and so A E A/~j for some j. _.+

Hence IP~jl > 1, whence A C 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 d ( a i , ' - ' , a n ) (aj C A). If A - ( A I , ' " , A n ) (Aj C K) then we write I - I ( A ) - I ( A , d ) - the ideal of A generated by a j 1 , . . . , n ) . Clearly we have

Aje ( j -

n

I-

I(A) - ~

A(aj-

(,)

Aje).

j=l

The joint (or simultaneous) spectrum --

a(al,...,an)

o(~)

-

{:~ e ~:". I(:~) # A}.

Not that when n -

8.5.8.

is defined by"

a(d)

1, ~ -

LEMMA.

~ ~nd ~ ( ~ ) -

~(a).

(i) A C K n \ a ( d ) i g there are elements

n

(ii) A C

j=l a(d) iff for every b l , . . . , b ,

E

A

we have

n

(aj - Aje)bj ~ Gi, where Gi denotes the group of invertible el-

j=l ements of A.

(iii) A C a(d) iff there is a maximal ideal M of A, with aj Aje C M (j - 1, . . . , n). PROOF.

(i) The stated condition is clearly equivalent to:

I(A)- A. (ii) The condition is clearly necessary and sufficient for I(A) y6 A. (iii) If A C a(d) then I(X) ~ A and so by Krull, I(X) C_ some maximal ideal M, whence a j - Aje E I(A) C_ M. Conversely, if aj - Aje. (j - 1 , . . . ,n) are contained in some maximal ideal M then I(A) _ M y6 A and ,~ E a(d). 8.5.9. COROLLARY. ,k C a(d) =~ Aj C a(a:i ) ( j so that

(r(~) C o ( a l ) x . . .

x o'(an).

1 , . . . , n),

w 5. Holomorphic Functional Calculus

391

PROOF. ~ C a(d) ::~ aj -- ,,kje E M ::~ ~f E a ( a j ) . 8.5.10.

PROPOSITION.

Let A be a unital c o m m u t a t i v e Gelfand algebra - in particular, a unital, complex or strictly real, c o m m u t a t i v e p - B a n a c h algebra. Then A E a(d) iff there is a character X C A with x ( a j ) -- Aj (j -- 1 , . . . , n), where

X --

(,~1,'",)in).

PROOF. If X E a(d), by 8.5.8 (iii) there is a maximal ideal M with a j - Aje C M . Since A is Gelfand M is hypermaximal and let X be the character determined by it. Then x ( a j ) - Aj, (A

-

(Aj)). Conversely, if X E A satisfies the stated condition then

aj - Aje E M ( M - kerx) and ~ E a(d).

s . 5 . 1 1 . COROLLARY. L~t A - (A, II " If) (ll " ll,m.) b~ ~ ~it~l, complex or strictly real, p - B a n a c h algebra. Then a(d) is n o n e m p t y compact with n

~(~) c I I ~(aj) c p~, j----1 --,

1

PROOF. Since A is Gelfand, A x(aj)Aj then by 8.5.10,

Ac #- 0. If X E A and

- (,kj) c a(d), so that a(d) # O. By 7.3.12,

A

(x(ai),...,X(an))

--

Ae ~ is compact and since the map X ~-* is continuous, a ( d ) i s compact. Finally, if 1

#j ~ a ( a j ) then I#il ~< r(ai) ~ llaj[[~ (see 7.3.27). This completes

the proof. 8.5.12. COROLLARY. Let A be as in 8.5.10, and al, A. Then

(i) For any polynomial P over K, a ( P ( a , , . . . , a,~)) - P ( a ( a l , . . . an)).

,an E

C o m m u t a t i v e Topological Algebras

392

(ii) / f a j E ~ :

(j-l,...,n)

a(alal,...

then

,anan) - {(alAl,"',anAn)"

'~ E a ( a l , . . . , a n ) } .

PROOF. It suffices to observe that each X E A being a homomorphism we have: x ( P ( a I , " " , an)) - P ( x ( a l ) , " " , x ( a n ) ) X(ozjaj) - oLjx(aj).

8 . 5 . 1 3 . R e m a r k . Arens has shown that the conclusion in Proposition 8.5.10 holds also if A is a unital commutative complex locally sin. ~ algebra (see [31, p.105]). 8 . 5 . 1 4 . PROPOSITION. Let A be a unital c o m m u t a t i v e pB a n a c h algebra which is either complex or strictly real, and which is t. generated by a finite set of elements a l , ' " , an. Then the spectrum A(A) is h o m e o m o r p h i c to the j o i n t spectrum a(d) under the map t?:

X ~-~

(X(al), "'" , x ( a n ) ) .

PROOF. By 8.5.10, 0 is surjective. The map 8 is also injective. To see this, suppose that t?(Xt) - t~(X2). Then, for a polynomial P - P ( A I , - - ' , An) over K we have X1 ( P ( a l , " " , an)) - P ( X I ( a l ) , ' ' ' , X1 (a,)) = P(x2(al),...,x2(an))-

x2(P(al,...,an)).

Since the elements of A of the form P ( a l , " " ,an) - where P is a polynomial over K in n variables - form a dense subalgebra A0 of A (since a l , ' - - , a n t. generate A) and X1 = X2 on A0, we conclude from the continuity of X~, X2 t h a t X~ = 2:2 on A, proving injectiveness of 0. Further, the continuity of t~ follows from that of the X'S. By 7.3.12, A = A(A) is compact Hausdorff. Therefore 0 is a homeomorphism. 8 . 5 . 1 5 . PROPOSITION. Let A be a unital commutative, complex or strictly real, p - B a n a c h algebra t. generated by al," " ,an. Then"

393

w 5. Holomorphic Functional Calculus

(i) For any C > O, a E ~( and ft ~ ~ Kn\a(ff) there is a polynomial P (in n variables over K) such that p ( ~ 0 ) _ c~, IIP(g)ll < C where d = ( a l , " " ,an). In particular, there is a polynomial Q with Q ( ~ O ) = 1 > IIO(~)l.

(ii) a(d)

is polynomially convex.

PROOF. If ~o C Kn\a(ff) then n

tl

I(~~ -- E A(aj- ,X~ j=l

E ( a j - ,~~

A,

j=l

where e is the unity of A. It follows that ae E A has a representation n

E (aj - AOe)bj - c~e

(1)

j-1

where bj E A. Since A is t. generated by ~, each bj can be approximated by elements of the form P ( a l , ' " , a,~) where P is a polynomial. It follows from this and equation (1) that there are polynomials Pj (j = 1 , - . . , n ) such that l't

II~e- ~ ( a j - ~~

I < c,

(2)

a;)Pj(2)

(a)

j=l

Write tI

P(a)- . - ~(a; --~

w

0

j-1

Clearly P is a polynomial in )~1,''" ,)in. Further, it is clear from (3), ( 2 ) t h a t we have p ( , ~ o ) _ oe

I]P(~)I] < C.

(4)

If we take C - 1 - ~ in (4) then the corresponding polynomial P which we denote by Q satisfies" Q(~o)-

1 > ]lQ(d)ll.

Commutative Topological Algebras

394

This completes the proof of (i). To prove (ii) take a A e a(d). By 8.5.10 we have

A-

X(d)

-

(X(al),...,x(an))

for some X E A. It follows that the polynomial Q chosen above satisfies:

IQ(X~

-

IQ(x(~))l-

Ix(Q(~))[

1

n -+-II(al + . . . + a,-~)x=ll.

Next write

co

c-Ea

co

,

co- E

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 ak E Mn (k > n) and M n is closed, we conclude that cn E M,~ and consequently, c n x n - O. It follows that we have CXn --

E

ak § Cn Xn

k=l

ak

Xn,

k:l

so that by inequality (4) we get

Ilcx~ll >~ tla.x~ll- II(ax +"" + a~-i)x~ll ~> n.

w 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, X0 r {0}, completing the proof. 9.4.3. THEOREM (Johnson). Every primitive p-Banach algebra A has p - n o r m uniqueness property, t PROOF. Let A = (a,I I 9]11) and ]l" {]2 a complete p - n o r m on A such that (A, II" II=) is lso p-B n ch algebra. We h ve to show that ]l" [[2 ~ ]i" [[1. 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 I]" I[i ( J 1,2). Let [l" I[~ be the canonical p - n o r m on A # induced by [[. IIj. Then (A #, II" [ [ ~ ) a r e p-Banach spaces (see 3.1.22, 3.4.16). It follows that 7r is a faithful representation of the p - B a n a c h algebra A = (A, [I" II1)on the p - B a n a c h space

A # - (A #, []. 112#). By 9.4.2, r is continuous and consequently we have

II (a)ll

CIJaJJ1 (a C A)

for some C > 0. This implies that

ttTr(a)(x § M)l]#2

~. by L e m m a C, o~

I/(~)1 ~