Banach algebras: An introduction (Pure and Applied Mathematics: A Series of Monographs and Textbooks 24)

PUR E AND A Series of I\ I'PI. I ED McJ ll o~rap " .I' MATHEMAT I CS and Textbooks BANACH ALGEBRAS an introductio...

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PUR E

AND

A Series of

I\ I'PI. I ED

McJ ll o~rap " .I'

MATHEMAT I CS

and Textbooks

BANACH ALGEBRAS an introduction

Ronald Larsen

PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks COORDINATOR OF THE EDITORIAL BOARD

S. Kobayashi UNIVERSITY OF CAI.IFORNIA AT BERKELEY

1. 2. 3. 4.

S.

YANO. Integral Form u l as in Riemannian Geometry (1970) H y pe rbol i c Manifolds and Holomorphic Mappings (1970) V. S. VLADIMIROV. Eq u ations of Mathematical Physics (A. Jeffrey, ed i tor : A. Littlewood. translator) (1970) B. N. PSHL:NICIINYI. Necessarv Conditions for an Extremum (L. Neustadt, trans lation editor; K. Makowski, transiator) (1971) L. NARICI, E. BECKI.NSI t-IN, and G. 8o\CHMAN. Functional Ana l ysis and Valua K.

S. KOBAYASHI.

tion Theory (1971)

6. 7.

8. 9. 10. 11.

12. 13. 14. IS.

16.

D. S. PASS'tAN. Infinite Group Rings (1971) L. DORNIIOFF. Group Representation Theory (in two parts). Part A: O rd i nary Representation Theory. Part B: M o d ular Representation Theory (1971. 1972) W. BOOTHBY and G. L. WEISS (ed�. ). S }, m m et ri c Space�: Short Courses Presented at Was h i ngto n University (1972) Y. MA'I SlISHIMA. DifT�rentiabJe Manifolds (E. T. Kobaya�hi. t ranslat or ) (1972) L. E. WARD, JR. Topology: An Outline for a First Course (1972) A. BAB·\KIf.\NIAN. Cohomological M � t hod s in G ro u p Th eor y (1972) R. GILM[R. Multiplicative Ideal Theory ( 1972) J. YEH. St ochast ic Processes and t he Wiener Integra) (1973) J. B,\KROS-NETO. Introduction to the Theory of Distrihutions (1973) R. LARSI=.N. F u nction al Analysis: An Introduction (1973) K. Y,\NO and S. ISIIIIHRA. Ta n ge nt and Cotangent H und les : Differential Geometry (1973 )

17. 18.

19. 20.

21. 22. 23. 24.

with Pol y nomial Idenlitiee; ( 1973) R. HI:RMANN. Geome t r y , Physics. and Systems (1973)

C. PROCI�SI. Rings

N. R. W-\LI.�CH. Harmonic Analysis on Homogeneous Spaces (1973) J. DU.I.JI)()l"IiNE. Introduction to the Theor) of Formal Groupe; (1973) I. VAISM.-\N. Cohomolo�y and D iffe ren t i al Forms ( 1 97 3) B.-Y. CHEN. Geometry of Submanifolds (1973) M. MARCUS. Finite Dimensional Multilinear Aigehra (in tWO paris, (1973) R. LARS[N. Banach Algebras: An Introduction ( 1 973 )

In Preparatioll: K. B.

STOLARSKY. Algebraic Number� and Diophantine Approximation

BANACH ALGEBRAS an introduction RONALD LARSEN DEPAR rME~T OF M4TIII M·\TJ('S WI SLEY.\N UNIVERSITY M IOIlI FTO\\ N. CONN I C I ICl T

MARCEL DEKKER, INC.

New York

1973

COPYRIGHT © 1973 by MARCEL DEKKER, INC. ALL RIGHTS RESERVED

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writin~ from the publisher.

MARCEL DEKKER, INC.

95 Madison Avenue, New York, New York LIBRARY OF CONGRESS CATALOG CARD NUMBER: ISBN:

0-8247-6078-6

PRINTED IN THE UNITED STATES OF AMERICA

10016

73-84868

To

Joan and

Nils Erik

PREFACE The exposition in the following pages is an elaboration and expansion of lectures I gave to second year mathematics graduate students at Wesleyan University during the academic years 1970-71 and 1971-72. The aim of the exposition is to provide a compact introduction to the theory of Banach algebras that not only acquaints the reader with fundamental portions of the abstract theory but also illustrates the usefulness of Banach algebras in the study of harmonic analysis and function algebras and gives the reader the basic tools necessary for further work in these areas. The first half of the book is devoted primarily to the general theory of Banach algebras, while in the second half the emphasis is on various more specialized topics related to harmonic analysis and function algebras - among which are: Wiener's Tauberian Theorem, the problem of spectral synthesis, the Bishop, Choquet, and §ilov boundaries, representing measures, Wermer's Maximality Theorem, the Commutative Gel'fand-Naimark Theorem, Plancherel's Theorem, the Pontryagin Duality Theorem, almost periodic functions, and the Bohr compactification. intelligent reading of the book presupposes the usual mathematical equipment possessed by second year mathematics graduate students with regard to topology, algebra, and real and complex analysis, as well as a reasonably good knowledge of basic functional analysis. That portion of functional analysis which is necessary forms a subset of my earlier book in this series "Functional Analysis: An Introduction" [L], and I have retained, for the most part, the same notation in this volume as in the previous one. Results with which the reader is assumed to be familiar are frequently cited without comment. However, in almost all such instances an appropriate reference is given. An

v

Preface

vi

Unfortunately, due to a lack of time and endurance, there are no problem sets per!! in this volume. However, I have left unproved results scattered throughout the exposition and the reader is strongly urged to fill in these lacunae in order to test and strengthen his or her understanding of the subject. The conclusion of a proof is indicated by the symbol 0 at the right hand margin. I would like to thank all the graduate students at Wesleyan who passed through my course while this book was evolving for their comments and suggestions. In particular, I wish to thank Hans Engenes and Michael Paul for their often perspicacious observations and questions, and Polly Moore Hemstead for her valuable editorial assistance. I would also like to thank Helen Diehl who typed the original

manuscript for the book. Finally, thanks are due to the editors and staff of Marcel Dekker for their cheerful and expert cooperation during the production of the book. Middletown, Connecticut May, 1973

C01\iENTS PREFACE CHAPTER 1: 1.0. 1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

CHAPTER 2: 2.0. 2.1. 2.2. 2.3.

CHAPTER 3: 3.0. 3.1. 3.2. 3.3. 3.4.

3.5.

v FUNDAMENTALS OF BANACH ALGEBRAS Introduction Basic Definitions and Some Algebraic Preliminaries Examples of Normed Algebras A Theorem of Gel'fand Regularity and Quasi-regularity The Gel'fand-Mazur Theorem Topological Zero Divisors SPECTRA

1 3 16 23 26 35 40 53

Introduction Definitions and Basic Results The Polynomial Spectral Mapping Theorem and the Spectral Radius Formula A Theorem of §ilov on Spectra TIlE

1

GEL' FAND REPRESENTATION THEORY

Introduction Maximal Regular Ideals and Complex Homomorphisms The Maximal Ideal Space The Gel'fand Representation The Beurling-Gel'fand Theorem Semisimplicity

vii

53 S3 S7 60

64 64 6S

68

74 79

81

Con tents

viii

THE

CHAPTER 4:

GEL'FAND REPRESENTATION OF SOME S PEC I FI C

ALGEBRAS

85

4.0.

Introduction

85

4.1.

C(X)

85

4.2.

n C ( f a, b ] )

4.3.

L (X,S,�)

92

4.4.

A(D)

95

4.5.

Finitely Generated Banach Algebras

98

4.6.

AC(r)

101

4.7.

LI (G)

104

4.8.

M(G)

127

and

C (X) o

92

CD

CHAPTER 5:

SEMISIMPLE COMMUTATIVE BANACH ALGEBRAS

5.0.

Introduction

5.1.

The Gel'fand Representation of S em i s impl e

129

129

Commutative Banach Algebras

130

-

5.2.

A - as a B anach Algebra

5.3.

Homomorphisms and I s omorphi s ms of Commutative

5.4.

Banach Algebras Characterization of S i ngul ar Elements in

135

Sel f - adjoint Semisimp1e

13 9

CHAPTER 6:

133

C ommu t ati v e

Banach Algebras

ANALYTIC FUNCTIONS AND BANACH ALGEBRAS

142

6.0.

Introduction

142

6.1.

Analytic Functions of Banach Algebra Elements

142

6.2.

S om e Consequences of the Preceding Section

151

6.3.

Zeros of Entire Functions

154

6.4.

The Conn ected Component of the Identity in

CHAPTER 7:

A

-1

REGULAR COMMUTAT IVE BANACH ALGEBRAS

7.0.

Introduction

7.1.

The Hull-Kernel Topology and Regular Commutative Banach Algebras

155

159 159

160

Contents 7.2. 7.3. CHAPTER 8:

ix Some Examples

167

Normal Commutative Banach Algebras

174

IDEAL THEORY

8.0. 8.l.

Introduction Tauberian Commutative Banach Algebras

8.2. 8.3.

Two Tauberian Theorems The Problem of Spectral Synthesis Local Membership in Ideals Ditkin's Theorem Ll(G) Satisfies Ditkints Condition Some Further Remarks on Ideals

8.4. 8.5. 8.6. 8.7. CHAPTER 9: 9.0. 9.l. 9.2. 9.3. 9.4. 9.S. 9.6. 9.7. 9.8. CHAPTER 10: 10.0. 10.l. 10.2. 10.3. 10.4. 10.5.

BOUNDARIES Introduction Boundaries The Silov Boundary Some Examples of Boundaries Extreme Points, the ~i1ov Boundary, and the Choquet Boundary Some Applications of Boundaries Representing Measures and the Choquet Boundary Characterizations of the Choquet Boundary Representing Measures and the ~i1ov Boundary B*-ALGEBRAS Introduction B*-Algebras The Gel'fand-Naimark Theorem The B*-Algebra A (G) 0 P1ancherel's Theorem The Pontryagin Duality Theorem

180 180 181 189 192 200 204 209 21S 218 218 219 222 227 232 243 249 259 269 272 272 273 277 280 297 308

Contents

x

10.6. 10.7.

REFERENCES

INDEX

A Spectral Decomposition Theorem for Self-adjoint continuous Linear Transformations

316

Almost Periodic Functions

323

333

337

"Those as hunts treasure must go alone, at night, and when they find it they have to leave a little of their blood behind them." Unknown woman from Bimini

CHAPTER I

FUNDAMENTALS OF BANACH ALGEBRAS 1.0. Introduction. One of the central objects of study in functional analysis is the normed linear space, that is, a linear space A over a scalar field, usually the real or complex numbers, together with a function from A to the real numbers, called the norm on A, which is positive definite, homogeneous, and subadditive. In elementary functional analysis the focus of attention is generally restricted to the linear structure of normed linear spaces. However, a goodly number of the specific spaces that occur in functional analysis come equipped, in a more or less natural way, with additional algebraic structure beyond that of a linear space. In particular, many of these spaces are algebras, and the multiplication in these algebras is continuous with respect to the given norm. The goal of ~his and the succeeding chapters is to introduce the reader to the study of such algebras, generally called normed algebras, and, in particular, to examine what additional information the algebraic structure of certain specific spaces can reveal about them. The development we shall present is intended to be neither the most general possible nor the most exhaustive of the theory discussed. Rather it is designed to expose the reader to the study of normed algebras as abstract mathematical entities and to various questions that arise in such a general setting~ and at the same time to investigate in some detail a limited selection of quite specific normed algebras. In order to accomplish such a program within a reasonable amount of space we have been forced to be rather selective in the scope of our development. Thus, for example, we shall only consider algebras over the complex numbers and shall focus most of our 1

1.

2

FundamentalS of Banach Algebra s

ed algebras where the underlying normed linear attenti on on those norm

space is

a

Banach space, that is,

on

Banach algebras.

Moreover,

although no assumption of commutativity is ge nerally made in the

first twO chapters, almost all of the subsequent material is concerned only with commutative Banach algebras.

Similarly, with regard to

specific Banach algebras, we have placed primary emphasis on algebras of continuous functions and algebras that occu r in harmonic analysis,

and given secondary status to algebras of continuous linear trans formations and of functions of several complex variables.

More

compendious treatments of the general theory can be found, for instance, in

Ri].

[N,

The present chapter is concerned with developing some of the fundamental theory of normed algebras. definitions and

a

We begin with the basic

collection of essentially algebraic results.

Many

of the latter may be already familiar to the reader, whereas others may seem strange.

The utility of some of the concepts and theorems

discussed may not be apparent until subsequent chapters.

This is

true also of some of the other results to be considered in this chapter.

Although some of these results are considered only to

round out the topic under discussion, the majority will find appli cation at later stages of the development.

After this section on fundamentals we shall present a number of concrete examples of normed algebras, theorem due to Gel'fand.

followed by a discussion of a

The next two sections contain

a

treatment

of the important topic of inversion in Banach algebras with identity and its counterpart, quasi-inversion, in Banach algebras without identity.

We sha ll see, for example, that in the former case the

invertible elements form

an

open subgroup of the algebra and that

the operation of inversion is continuous.

These results will then

be applied to prove the Gel'fand-Mazur Theorem, which asserts that every Banach algebra with identity in which each nonzero element is invertible is isometrically isomorphic to the complex numbers.

Our

standing assumption tha t all algebras are over the complex numbers

1.1. Definitions and Preliminaries

3

is crucial for the validity of this result. The chapter concludes with a rather extensive discussion of the notion of topological zero divisors, including a proof of Arens' Theorem, which asserts that an element in a commutative Banach algebra A with identity is a topological zero divisor if and only if it is not invertible in any Banach algebra B with identity in which A can be embedded. We note that throughout this volume the letters C, lR, and Z _ shall stand for the complex numbers, the real numbers, and the integers, respectively. The basic notation and terminology from functional analysis is as in [L]. 1.1. Basic Definitions and Some Algebraic Preliminaries. As the heading of this section indicates, we shall concern ourselves here with introducing some of the basic definitions needed in the succeeding development and in proving several essentially algebraic results. Some of the definitions and results are probably already familiar to the reader, and we include them primarily in the interests of completeness. Definition 1.1.1. A linear space A over C is said to be an algebra if it is equipped with a binary operation, referred to as multiplication and denoted by juxtaposition, from A X A to A such that (i) (ii) (iii)

x(yz) = (xy)z, x(y+ z) = xy + xz; (y+ z)x = yx+ zx, a(xy) = (ax)y = xray).

It is said that and

(x,y,z E A; a E C).

A is a commutative algebra if A is an algebra

(x,y E A), (iv) xy = yx whereas A is an algebra with identity if A is an algebra and there exists some element e E A such that (v)

ex = xe = x

(x E

A).

1. Fundamentals of Banach Algebras

4

It is evident that, if A is an algebra with identity, then the identity element e is unique. To avoid triviality we shall always assume that our algebras !!!. ~ just !!!!. ~ element. Definition 1.1.2. A nonned linear space (A,II·11l said to be a nonned algebra if A is an algebra and

\IxYIl

~

IIxllllYIl

over C is

(x, yEA) •

A nonned algebra A is said to be a Banach algebra if the normed linear space (A,II·II) is a Banach space. A number of examples of normed algebras will be discussed in the next section. Definition 1.1.3. Let A be an algebra. Then I C A is said to be a left (right) ideal in A if I is a linear subspace of A such that xl C I (Ix C I), x E A; I C A is said to be a two-sided ideal if I is both a left and a right ideal in A. An ideal I C A is proper if I ~ A, and a proper left (right, two-sided) ideal I is said to be maximal if, whenever J C A is a left (right, two-sided) ideal in A such that I C J, either I J or J = A. Furthermore, I C A is said to be a subalgebra if I is a linear subspace such that x,y E I implies xy E I.

=

Clearly every ideal in an algebra A is a subalgebra, and in a commutative algebra all ideals are two-sided. In the latter case we shall generally speak only of ideals, and drop the adjective "twosided." Note also that, by definition, a maximal ideal is always proper. The following result concerning normed algebras is easily proved. The details are left to the reader. Theorem 1.1.1. (i)

Suppose A is a normed algebra.

The completion of A is a Banach algebra.


5

(ii) If I C A is a subalgebra, then I is a normed algebra, whereas if A is a Banach algebra and I is a closed subalgebra, then I is a Banach algebra. (iii) If I C A is a closed two-sided ideal, then the quotient space A/I is a normed algebra with the usual quotient norm llIx + I III

=

inf IIx + yll yEI

(x E A)

and with the multiplication (x + I) (y + I) = xy

+

I

(x,y E A).

(iv) If A is a Banach algebra and I C A is a closed two-sided ideal, then A/I is a Banach algebra. If an algebra A has an identity, then it obviously is meaningful to discuss the algebraic notion of inversion. Moreover, as we shall see, it is always possible to consider an algebra without identity as a subalgebra of an algebra with identity, thereby allowing a discussion of inversion even in the case of algebras without identity. However, although this embedding process is very useful, it is also desirable to be able to discuss some sort of concept of inversion in algebras without identity without recourse to this process of embedding. This fact will lead us to the notion of quasi-inversion. We shall discuss inversion and quasi-inversion for normed linear algebras in greater detail in Section 1.4. We now wish to concentrate mainly on the algebraic aspects of these ideas. Definition 1.1.4. Let A be an algebra with identity e. An element x E A is said to have a left (right) inverse if there exists some yEA such that yx = e (xy = e), whereas x is said to have an inverse if there exists some yEA such that xy = yx = e. If x E A has an inverse, then x is said to be regular, or invertible, and x is said to be singular if it is not regular. It is easily verified that, if x E A has both a left inverse y and a right inverse z, then y = z is an inverse

I. Fundamentals of Banach Algebras

6

and that inverses are unique. As usual, we shall denote the inverse of x by x-I. Obviously, if the algebra A is commutative, then left and right inverses are inverses. There is a simple connection between the existence of left and right inverses and membership in ideals, as evidenced by the next theorem. Theorem 1.1.2. Let A be an algebra with identity and suppose x E A. Then the following are equivalent: (i) (ii)

x has a left (right) inverse. x does not belong to any proper left (right) ideal in A.

Proof. Suppose x has a left inverse y. If I is any left ideal such that x E I. then e = yx E I. and so A = Ae C AI C I shows that I = A. Thus x belongs to no proper left ideal. Conversely, if x has no left inverse, then it is easily seen that I = (yx lyE AJ is a left ideal in A such that x E I. Furthermore, I is proper since e ~ I. The proof of the remainder of the theorem is similar.

o

Corollary 1.1.1. Let A be an algebra with identity. If x E A is regular, then x does not belong to any proper left, right, or two-sided ideal in A. In general for results involving left, right, and two-sided concepts we shall prove only one case, leaving the others for the reader. In particular, we note that the identity e belongs to no proper ideal. This observation, combined with Zorn's Lemma [OSl' p. 6]. yields the following result, the details of which are left to the reader:


7

Theorem 1.1.3. Let A be an algebra with identity. If I is a proper left (right, two-sided) ideal, then there exists a maximal left (right, two-sided) ideal MeA such that I C M.

C

A

Corollary 1.1.2. Let A be an algebra with identity and suppose x E A. Then the following are equivalent: (i) (ii)

x has a left (right) inverse. x does not belong to any maximal left (right) ideal in A.

Moreover, if x is regular, then x does not belong to any maximal two-sided ideal in A. In particular, for a commutative algebra A with identity the preceding results say that x E A is regular if and only if x does not belong to any proper ideal in A if and only if not belong to any maximal ideal in A.

x does

Theorem 1.1.3, though algebraically quite elementary, is a crucial tool in studying normed algebras with identity. The verbatim counterpart of the theorem is, however, not valid for algebras without identity. The appropriate analog can nevertheless be obtained if we restrict our attention to so-called regular or modular ideals. Definition 1.1.5. Let A be an algebra. A left (right, twosided) ideal I in A is said to be regular if there exists some u E A such that xu - x E I (ux - x E 1, xu - x E I and ux - x E I), x in A. The element u is called an identity modulo I. Clearly, if A has an identity e, then every ideal is regular, and e is an identity modulo any ideal. The reason for this terminology is apparent from the third portion of the next proposition. Proposition 1.1.1.

Let

A be an algebra.

(i) If I C A is a proper regular left (right, two-sided) ideal and u is an identity modulo I, then u ~ I.


8

A is a regular left (right, two-sided) ideal and A is a left (right, two-sided) ideal such that J ~ I, then (ii)

J C

I

If

C

is regular. Moreover, if u is an identity modulo J.

J

is an identity modulo

I.

then u

(iii) If I C A is a proper regular two-sided ideal, then is an algebra with identity.

If

A/I

I

is a proper regular left ideal and u is an identity modulo I, then the assumption that u E I entails that xu E I, x E A, whence x = xu - (xu - x) E I, x E A, contradicting the properness of 1. Thus u~I. Proof.

For part (ii) of the proposition we note that, if u is an identity modulo the regular left ideal I, then xu - x E I C J, x

E A, and so

J

is regular and

u

is an identity modulo J.

A/I is an algebra, and we claim is an identity for A/I, where u E A is an identity

Finally, it is evident that

that u + I modulo I. The latter assertion is immediate on noting that (u

+

I) (x + I) = ux + I = x + I

= xu =

since ux - x E I

and

xu - x E I, x E A.

+ I

(x+I)(u+I)

(x

E

A)

o

We can now state and prove the indicated analog of Theorem 1.1.3. Theorem 1.1.4. Let A be an algebra. If I C A is a proper regular left (right, two-sided) ideal, then there exists a maximal regular left (right, two-sided) ideal MeA such that I C M. Proof. Suppose I is a regular left ideal and let u be an identity modulo I. Denote by J the collection of all proper left ideals J C A such that J ~ I. Clearly J ~~, each J E ] is


9

u is an identity modulo J. Moreover, u ~ J, J E J. The last two observations are consequences of Proposition 1.1.1 (i) and (ii). We introduce a partial ordering in J by setting J l > J 2 if and only if J I ~ J 2 , J 1 ,J 2 E J. If (J 1 is a linearly ordered subset of J, then it is easily verified a that J = U J is a proper regular left ideal in A that contains a a I. The properness follows from the fact that u l J. a proper regular left ideal, and

Consequently we may apply Zorn's Lemma [DS I , p. 6] to deduce the existence of a maximal element M E J. Evidently M is a maximal regular left ideal such that M~ I. D

Next we wish to examine in detail how to embed an algebra without identity in an algebra with identity. Suppose that A is an algebra without identity and denote by A[e] the set of all pairs (x,a), x E A, a E C, that is, as a point set A[e] = A x £. The point set becomes an algebra if one defines the linear space operations and multiplication as follows: (i) (ii) (iii)

(x,a) + (y,b) = (x+y,a+b) b(x,a) = (bx,ba) (x,a) (y,b) = (xy + ay + bx,ab)

(x,y E A; a,b E C).

The routine verification will be left to the reader. Furthermore, the element e = (0,1) E A[e] is an identity for A[e]. Indeed, (x,a)(O,I)

= (xO+ lx,a)

= (x,a) = (0,1)

(x,a)

(x E A; a E C).

Moreover, it is easily shown that the mapping L: A - A[e], defined by ~(x) = (x,O), x E A, is an algebra isomorphism of A onto the maximal two-sided ideal L(A) = ((x,O) I x E AJ C A[e]. This discussion, plus some additional. argument, leads to the following theorem. The details are omitted.


10

Theorem 1.1.5. Let A be an algebra. Then A[e] is an algebra with identity and the mapping ~: A - A(e] is an algebra isomorphism of A onto the maximal two-sided ideal ~(A) = [(x,O) l x E A} C A[e].

If A is a normed algebra. then A[e1

is a normed algebra under

the norm

IICx.a)ll

= llxll

+

lal

(x E Aj a E C),

and ~: A - A[e] is an isometric algebra isomorphism onto the closed maximal two-sided ideal ~(A) ~ A[e]. Moreover, in this case, the quotient algebra Are]/~(A) is isometrically isomorphic to c,

and A[e]

is a Banach algebra provided A is a Banach algebra.

Evidently A is a commutative algebra if and only if A[e] a commutative algebra.

is

As a rule we shall speak of A itself as a maximal two-sided ideal in ACe]. Similarly we shall often write elements (x,a) E A[e] as (x,a) = x + ae. This makes sense on identifying x with ~(x) = (x,D) and recalling that e = (0,1). Hopefully the simplification in notation gained by these conventions will outweigh the loss of precision.

In the succeeding chapters we shall always use the symbol Are] to denote the algebra with identity obtained from an algebra without identity by the previously developed construction. We shall occasionally speak of this process as that of adjoining ~ identity to A. It should be noted that even when A has an identity we can still construct A[e] and Theorem 1.1.S is valid. The identity for A is, however not the identity for Are]. This observation will at times be useful, but the majority of our applications of A[e] will be to algebras A without identity. I

The general utility of ~he algebra

Are1

lies in the fact that algebras with identity are often easier to deal with than algebras without identity, and one can often deduce properties of A


11

A[e]. .~ a first example of this let us examine the relationship between re~lar ideals in A and ideals in Are].

by examining a related property in

Theorem 1.1.6.

Let

A be an algebra. let

1 denote the e family of all proper ieft (right. two-sided) ideals in Are] that are not contained in A, and let I denote the family of all proper regular left (right, two-sided) ideals in A. only if I = I n A for a unique I E 7 • e e e

Then lEI

if and

Proof. Suppose I C A[e] is a proper left ideal such that e I fI. A and let I = InA. Clearly I is a left ideal in A. and, e e moreover, I is proper, because if I = A, then I ~ A, and so e Ie = A[e], since Ie ~ A~ contradicting the properness of Ie. Also. since I r:f. A, there exist some x E A and some a E C, a ~ 0, e such that x + ae E I . Furthermore, x # 0, as if x = 0, then e again contradicting the properness of t by Theorem 1.1.2. eEl ~ e e Let u = -x/a. Then u - e = -x/a - e = (-l/a) (x + ae) E I . We e claim that u is an identity modulo I. Indeed, if yeA, then in A[e] we see that is a left ideal. But A yu - y = yu - ye = y(u - e) E Ie' as I e is a two-sided ideal in Are], so that yu - y = y(u - e) E A. Consequently yu - y E I n A = I, and we see that u is an identity e modulo the left ideal I~ that is, I is a proper regular left ideal. Conversely. suppose I is a proper regular left ideal in A and let u E A be an identity modulo I. Set Ie = {y , y E Are]. yu Ell. We claim that I is a proper left ideal in Are] such that e

I ~ A and I = InA. Clearly I is a linear subspace of Are1. e e e Further, we note that I is also a left ideal in A[e] because if x E I and y = z + ae E A[e], then yx = (z + ae)x = zx + ax E I, as J is a left ideal in A. It then follows at once that I is e a left ideal in A(e]. Moreover. Ie is proper, since if I = A[e], then eu = u E I, contrary to the conclusion of e


12

Proposition l.l.l(i). Next we note that u - eEl e 2 (u - e)u = u - u £ I, as u is an identity modulo I

e

because I. Thus

f/. A.

Finally, if x E I, then xu - x E I, as u is an identity modulo I, and so xu E I. Thus I C InA. Conversely, if e x E I n A, then xu E I and xu - x E I, from which we conclude e that x E I. Therefore I = InA. Similar arguments, of course, e establish the result for right and two-sided ideals. To complete the proof of the theorem we must show that, if I C A is a proper regular two-sided ideal, then there exists a unique proper two-sided ideal I (A[e), I ~ A, such that I = InA. So e e e suppose I and J are two such ideals in A[e]. Then, as above, e e we can deduce the existence of identities modulo I, call them u and v, such that u - eEl, v - e £ J. Since u and v are e e identities modulo I, it follows that vu - v £ I and vu - u E I, whence v - u E I. Now suppose y = z + ae E I. Then since I e and A are both e two-sided ideals in A[e], we see that uy = uz + au E I e n A = I. Thus z + au = (z - uz) + (uz + au) E I + I as

z - uz

= I,

E I. Consequently y

=z

+ ae

=z

+ au + aCv - u) + aCe - v) £I+I+J e

CJ,

e

as J n A = I. Hence r C J. A similar argument mutatis mutandis e e e demonstrates that J C I, and so I = J . 0 e e e e Corollary 1.1.3. Let A be an algebra. Then MeA is a maximal regular two-sided ideal if and only if there exists a unique maximal two-sided ideal M C A[e] such that M ~ A and M= M

e

n A.

e

e


13

This corollary will be useful when we discuss the complex homomorphisms of commutative Banach algebras in Chapter 3. If A has an identity, then the adjective "regular" in the statement of Theorem 1.1.6 and Corollary 1.1.3 is obviously redundant. Next let us see how the algebra A[e]

can be used to introduce

a concept of inversion in algebras without identity. Suppose A is an algebra without identity and let x E A. One could clearly ask whether or not x has a left (right) inverse in A[e], but it turns out for our purposes that it is more useful to ask this question for e - x, rather than for x. Thus we see that e - x has a left inverse in A[e] if and only if there exist some yEA and a E ~ (ae - y)(e - x)

= ae

- ax - y + yx

= e,

that is, if and only if (a - 1 ) e

Since

ax + y - yx E A,

= ax

+ y -

}'x.

it follows at once that

a = 1,

as

A is

without identity. lienee we see that e - x has a left inverse in Are] if and only if there exists some yEA such that x + y - yx = o. This element y ~ A will be the substitute we seek for the left inverse of x in an algebra A without identity. Note also that the equation defining this element algebras with identity as well.

y

is meaningful in

With these observations in mind we make the following definition: Definition 1.1.6. Let A be an algebra. An element x E A is said to have a left (right) guasi-inverse if there exists some yEA such that y 0 x = y + x - yx = 0 (x 0 y = x + y - xy = 0) , and x is said to have a 9,uasi-inverse if there exists some yEA such that y 0 x = x 0 y = O. If x E A has a quasi-inverse, then x is said to be quasi-regular, or quasi-invertible, and x is said to be quasi-singular if it is not quasi-regular.


14

Evidently Y is a left (right) quas1-1nverse for x if and only if x is a right (left) quasi-inverse for y. Some other elementary results concerning quasi-inverses are collected in the next proposition. Proposition 1.1.2.Ci). If A is an algebra with identity e, then x E A has a left (right) quasi-inverse if and only if e - x has a left (right) inverse, and x is quasi-regular if and only if e - x is regular. Moreover, yEA is a left (right) quasi-inverse for x if and only if e - y is a left (right) inverse for e - x, and y is a quasi-inverse for x if and only if e - y is an inverse for e - x. (ii) If A is an algebra and x E A has a left quasi-inverse y and a right quasi-inverse z, then y = z is a quasi-inverse for x. In particular, quasi-inverses are unique. Proof. The proof of part (i) of the proposition is a routine computation and will be omitted. For part (ii), considering the context of our computations to be A[e] in the case that A is without identity, we see from part (i) that e - y and e - z are left and right inverses for e - x, respectively, and so, by the remarks following Definition 1.1.4, we conclude that e - y = e - z is a two-sided inverse for e - x, whence y = z is a two-sided quasi-inverse for x by part (i).

o

In general, if x E A is quasi-regular, we shall denote its quasi-inverse by x_I' There are analogs for quasi-inversion to Theorem 1.1.2 and Corollary 1.1.2. Theorem 1.1.7. Let A be an algebra and following are equivalent: (i) (ii)

x EA.

x has a left (right) quasi-inverse. (-z + zx I z E A) = A({-z + xz I z E AJ

= A).

Then the


15

Proof. Suppose x has a left quas1-1nverse y. Then x = -y + yx E {-z + zx I zEAl. Clearly (-z + zx I z E AJ left ideal in A, and so for any w E A we have w = (w - wx) + wx E {-z + zx

that is,

{-z + zx

I

is a

zEAl,

zEAl = A.

Conversely, if (-z yEA for which x = -y left quasi-inverse.

+ +

zx I z E AJ = A, then there exists some yx, that is, y 0 x = o. Thus x has a

This result, combined with Proposition 1.1.2(i), yields the next corollary.

o immediately

Corollary 1.1.4. Let A be a commutative algebra with identity and suppose x E A. Then (i) x is regular if and only if {zx I z E A} = A. (ii) x is singular if and only if (zx I zEAl is a proper ideal in A. The analog of Corollary 1.1.2 is the next theorem. is left to the reader.

The proof

Theorem 1.1.8. Let A be an algebra and suppose x E A. the following are equivalent:

Then

(i) x has a left (right) quasi-inverse. (ii) If MeA is a maximal regular left (right) ideal, then there exists some yEA such that y 0 x E M (x 0 y EM). Of course in commutative algebras left and right quasi-inverses are quasi-inverses and the previous results assume a somewhat simpler form. Finally we note that some authors define, for example, the left quaSi-inverse of x as any element y such that x + y + yx = O. This is the case, for instance, in [HIP, p. 680; N, p. 158]. The

1.Fundamentals of Banach Algebras

16

definition of quasi-inverses given here, however, seems to be the

more common one (see, for example, [HR l , p. 471; Lo, p. 64; Ri, p. 16; Wa, p. 19]). The relationship between the two definitions is that y is a left (right) quasi-inverse for x in the sense used here if and only if -y is a left (right) quasi-inverse for -x in the sense used in [HIP, N]. 1.2. Examples of Normed Algebras. In this section we wish to list a number of standard examples of normed algebras occurring in functional analysis. The examples can be divided, loosely speaking, into three classes: function algebras, convolution algebras, and algebras of linear transformations. We shall not prove any of the assertions made about the following examples, but instead leave the, generally routine, verifications to the reader. Example 1.2.1. The complex numbers C ~ith the usual algebraic operations and with absolute value as the norm are a commutative Banach algebra with identity. Example 1.2.2. Let X be a locally compact Hausdorff topological space. By C(X), C (X), and C (X) we denote, respectively, o c the algebras of all continuous complex-valued functions on X that are bounded, vanish at infinity, or have compact support. The algebra operations are the usual ones of pointwise addition, multiplication, and scalar multiplication. With the usual supremum norm

lIfH

= CD

sup t (X

If (t) I

(f E C(X»,

the algebras C(X) and C (X) are commutative Banach algebras, o whereas C (X) is a commutative normed algebra. If X is noncompact, c then only C(X) is an algebra with an identity, whereas if X is compact, then C(X) = Co(X) = Cc(X) is an algebra with identity. The algebra Cc(X) is a Banach algebra only in the case that X is compact.

17

1.2. Examples of Normed Algebras

0 denote the closed unit disk in C, that is, 0 = (, I , E C, 1'1 < 1), and let A(O} denote the family of all f E CeO) such that f is analytic on int(D) = {, , , E c, "I < 13. With the usual pointwise operations and the supremum norm ~ample

1.2.3.

Let

IIfli CD

it is easily verified that with identity.

= sup If (C) , 0

,E

A(O)

(f E

A(D)),

is a commutative Banach algebra

Example 1.2.4. More generally, let K CC be any infinite compact set. We define P(K) and RCK}, respectively, as all those f E C(K) that can be approximated uniformly on K by polynomials or by rational functions with no poles in K. In analogy to A(D), A(K) denotes all those f E C(K) that are analytic on the interior of K, that is, on int(K). With pointwise operations and the norm of C(K), it is easily seen that peK), R(K), and A(K) are all commutative Banach algebras with identity. Moreover, it is apparent that P(K) C R(K) C A(K) C C(K). For K = 0 it is a classical fact that P(O) = A(O), whereas for K = r = (, I , E C, I" = 1) the StoneWeierstrass Theorem implies that R(f) = C(f). In general, the question of when various pairs of algebras in the chain P(K) C R(K) C A(K) C C(K) are equal, as well as the analogous problem with C replaced by ~, is a very difficult one. Although we shall not investigate this question in any great detail, we shall make some additional comments on it in later chapters. For more detailed discussions the reader is referred to [B, Ga, Lb, S, Wm l ,

Wm 2]· Example 1.2.5. Let a,b E~ a < b, and for each nonnegative integer n let Cne[a,b]) denote the family of all n-times continuously differentiable complex-valued functions defined on the closed interval [a,b], with the usual convention about one-sided derivatives at the end points of the interval. With pointwise operations


18 and the norm n

n (f ~ C ([a,b]»,

sup [1: a 0 was arbitrary, we conclude that and hence lim IIxnU 1/n exists and n

lim sUPnllxnlll/n < a,

1.4. Regularity and Quasi-regularity

limllxnll l/n n The fact that

= infllxnll l/n . n

lim /lxnll l/n < IIxll n

33

-

is immediate on observing that

lixnll < IIxlin.

o

The sequence

( Ilxnlll/n)

and its limit will recur repeatedly in

the succeeding chapters and play an important role in the study of Banach algebras. We have already seen some uses of it in Theorem 1.4.2 and Corllary 1.4.1. At this juncture it also seems appropriate to introduce a new definition. Definition 1.4.2. Let A be a normed algebra. Then x E A is said to be nilpoten~ if there exists some nonnegative integer n such that xn = 0, and x E A is said to be topologically nilpotent if lim IIx n ll l/n = o. n

Evidently every nilpotent element is topologically nilpotent, but the converse need not be so. Topological nilpotents shall also make recurrent appearances in the sequel (e.g., in Section 1.6). Here we wish to prove only one result about such elements.

e.

Proposition 1.4.1. Let A be a Banach algebra with identity If x E A is topologically milpotent. then x is singular. Proof.

Suppose x

is regular.

Then

1 = lIell = lI(x- l x)nlll/n

=i1(x- l )n(x)nIl 1/n

< Iix-lllllxnlli/n xx- l = x-Ix. lim IIxnll 1/n = O.

where we have used the fact that leads to a contradiction, as Therefore

x

is singular.

en

= 1,2,3, ... ),

But this clearly

n

o


34

OUr last concern in this section will be to utilize the previous discussion of regularity and quasi-regularity to establish a fundamental result about ideals in Banach algebras. It is easily seen that. if 1 is a left (right, two-sided) ideal in a Banach algebra A. then the norm closure of I, denoted by e1(I). is again an ideal of the same sort and that, if I is regular, then so is cl(I). It is, however, not as obvious, although it is true, that the closure of a proper regular ideal is again a proper ideal. Theorem. 1.4.5. Let A be a Banach algebra. If I C A is a proper regular left (right, two-sided) ideal, then cl(I) is a proper regular left (right, two-sided) ideal. Proof. We need only prove that cl(I) is proper. Since I is regular, there exists some u E A such that xu - x E I, x E A. From Proposition l.l.l(i) we see that u ~ I. We claim, moreover, that u ~ eICI). To show this it clearly suffices to prove that lIu - xII ~ l, x E I. So suppose the contrary--that is, suppose x E I is such that lIu - xII < 1. Then. by Corollary 1.4.2.(i). u - x is quasi-regular, and so there exists some yEA, namely, y

= (u

- x)_l'

such that y

0

eu - x)

=y

+ u -

=y

- yu - (x - yx) + u

x - yeu - x)

= O.

Hence u =yu - y + ex - yx}. But yu - y E I. as u is an identity modulo the regular left ideal I, and x - yx E I, as I is a left ideal and x E I. Consequently, u E I, which contradicts the fact that u ~ I. Therefore elCI)

is proper.

An immediate and important corollary is the following resul t:

D

1.5. The Gel'fand-Mazur Theorem

3S

Corollary 1.4.4. Let A be a Banach algebra. If MeA is a maximal regular left (right, two-sided) ideal, then M is a closed maximal regular left (right, two-sided) ideal. Thus maximal regular ideals are always closed. It shoUld be noted that the closure of a proper ideal in a Banach algebra need not be proper; that is, the assumption of ~egu larity cannot in general be dropped. For example, if G is a compact Abelian topological group, then L (G), I < p p) for p -1 some p > O. Thus we really need only prove that Ap C A for each p > 0; that is, A-I = A-I n A = A for each p > o. p

p

p

We note first that each Ap is connected. Indeed, suppose x,y (Ap. If Y = -ax for some a > 0, then it is easily seen that the circular arc from y = -ax to ax, defined by fl(t) = aeintx, -1 ~ t ~ 0, followed by the straight-line arc f 2 (t) = (1 - t)ax + tx, 0 < t < 1, determines a cantinuous arc lying completely in Ap whose end points are y = -ax and x. In the case that y ~ -ax for any a > 0, easy arguments show that

1.5. The Gel'fand-Mazur Theorem

39

f (t) = (1 - t) IIxll + tllyll [(1 _ t)x 11(1 - t)x + ty

+

ty]

(0 < t < 1)

defines a continuous arc lying in Ap whose end points are x and y. Thus Ap is connected. Furthermore, A-I is an open subset of p Ap in the relative topology since A-I is open by Corollary 1.4.3 and A-I ~~, as pe E A-I. Moreover, A-I is closed. Indeed, -1

p

P

P

suppose (xk} c: Ap and x E A are such that li~lIxk - xII = O. Since each xk E A-I nAp' we see that IIx;lll < 1/lIxkll < IIp, k = 1,2,3, ... , and so for each nand k we have

IIx~1 _ x~111 = IIx~l(xn - xk)x~lll < IIx;lllllxn - xkllllx~ll1 II xn - xkll

max[l/s,l/(l - s)] we can consider gk as an element of C([O,l]) since 0 < s < 1. Moreover, we claim that limkllfgkllCl) = 0 on the following grounds: if e > 0 and 6 > 0 are so chosen that If(t)1 < e whenever It - sl < 6, 0 < t < 1, then for k > max[l/s,I/(1 - s),l/b] we see that for

It - 51 > ~,

lfgk(t)1 < e for

It - sl ~ ~.

lfgk(t)1

=0

=0

Thus f is a topological zero divisor. If s similar argument using the half-tent functions for

0< t

~

I

k'

or

1,

then a


42

when

s

=0

and

=0

gk(t)

for

gk(t) = k(t - 1) when s

=1

0 < t < 1 -

for

+ 1

t

k' 1 -

1 r< t

~

1,

yields the same conclusion.

Conversely, suppose f E C([O,l]) and f(s) ~ 0, 0 ~ s < 1. Since f is continuous, for each s, 0 ~ s ~ 1, there exists some 6s > 0 such that If(t)I > If(s)I/2, It - sl < 6 s ' 0 ~ t ~ 1. Obviously the open intervals (s - 6 ,s + 6 ), 0 < s < I, cover [0,1], s s -and hence there exists a finite set of points sl-s2, ..• ,sn for which [O~l] cuP. 1(5. - 6 ,5. + 6 ). Thus, if J=

Sj

J

6

=

J

Sj

min

If(sj)l

j=1,2, .•• ,n

2

we see easily that If(t)1 > 6 > 0 for all t, 0 ~ t ~ 1. But then the function l/f is clearly defined and continuous on [0,1), that is, f- l = l/f E C([O,l]). Consequently, if

(gk) C C([O,l])

is any sequence such that

limkllfgkllCD = 0, then one would have also limkllgklt., = 0 since lIgkll CD = IIf-IfgkllCD < IIf-IIlCDllfgkIlCD' Hence f cannot be a topological zero divisor. Furthermore, we note that, if f E C([O,l]) is regular, then f(s) ~ 0, 0 < s < 1, because if f is regular, then f- l E C([O,l]) -1and so (f f)(s) = 1, 0 ~ s < 1, whence f(s) ~ 0, 0 < s < 1. This observation allows us to conclude that f E C{[O,l]) is a topological divisor if and only if f is singular. Moreover, the previous arguments, with the aid of Urysohn's Lemma [W2, p. 55], can be carried over mutatis mutandis to C(X), where X is any compact Hausdorff topological space. We summarize this discussion in the next theorem.

1.6. Topological Zero Divisors

43

Theorem 1.6.1. Let X be a compact Hausdorff topological space and suppose f € C(X). Then the following are equivalent: (i)

f

is a topological zero divisor.

(ii)

f

is singular.

(iii)

There exists some

sEX such that

f(s)

= O.

It is well to note that such a simple characterization of topological zero divisor is not valid in general. For a second specific example we consider the commutative Banach algebra ll(f), where convolution is the algebra multiplication. r is, as it will be throughout the book, the compact Abelian group under multiplication of complex numbers of absolute value one; that is, r = {, I , E~, 1'1 = 1) = {e it I -u < t < u). The convolution of two elements

f,g E Ll(f)

f * g(e is )

naturally has the following form:

= 2~ J~n f(ei(s - t))g(e it ) dt.

We claim that every element of LI(f)


gk ( e it) = e ikt , k = 1 " 2 3 ,.... Cl ear 1y {gk ) C LI (f) IIgkil l = 1, k = 1,2,3, .... Moreover, for any f E Ll(f),

Suppose

and

IIgk

* fill = 2~ J~n 12~ J~n eik(s - t)f(e it ) dtl ds 1

= 2n J~n

=

I•f(k) I

iks . .k le 2u S~n f(e1t)e- 1 t dtl ds

(k

= 1, 2 , 3, • . • ) •

However, from the Riemann-Lebesgue Lemma [E 2 , p. 36], we know that lim k _ +ClJ f (k) I = 0, whence we conclude that f is a topological zero divisor in LI{f).


44

If one wishes to avoid the use of the Riemann-Lebesgue Lemma in this example, one can instead consider the commutative Banach algebra L2 (f) in place of LICf). The same argument then shows that every element of L2 Cf) is a topological zero divisor. The assertion • that limk _ If(k)1 = 0, f E. L.,Cf), is now a consequence of the +00 . . . . kt complete orthonormality of the set eel I k € £] in L2 Cf). (See, for example, fL, p. 408].) As a final, and more abstract, example of topological zero divisors we have the following proposition: Proposition 1.6.1. Let A be a Banach algebra with identity e. If x ~ A is topologically nilpotent, then x is a two-sided topological zero divisor. Proof. Let (akl be any sequence of distinct nonzero complex numbers such that limklakl = 0 and consider the sequence (x/~J contained in A. Since x is topologically nilpotent, it follews at once that limll (~ )nlli/n = Ii~ x = 0 n ak n ak l/n

Ck

= I. 2 , 3 , •.. ) ,

and so, by Theorem 1.4.2. we deduce that each x/a k is quasi-regular. Thus, by Proposition I.I.2(i), e - x/ak = Cake - x)/a k is regular, -1 whence ake - x is regular, k = 1,2.3, .... Let Yk = Cake - x) , k = 1,2,3, ••.• We note next that xYk = akYk

(ake - x)y k

= akYk - e = akYk - YkCake - x)

= Yk x

(k

= 1,2,3, ... ).

Furthermore, since x is topologically nilpotent, it is singular.


45

by Proposition 1.4.1, and so it is easily seen that xYk = Ykx is also singular~ k = 1.2~3~ .... But this implies that !Iaky k ll > I, k = 1,2,3,.... Indeed, if lIakykll < 1, then, by Corollary 1.4.2, akYk is quasi-regular, whence xYk = Ykx = a kYk - e is regular, contrary to the previous observation. Hence "akyk" > 1, for all positive integers k, from which it follows at once that limkllYkll =

GI,

since

limklakl = O.

Consequently the estimates

ll~Yk - ell lIyk ll (k =

1,2,3, ..• )

reveal that

that is,

x is a two-sided topological zero divisor.

o

Now let us turn to some general results about topological zero divisors. We first state a simple proposition whose proof we leave to the reader. Proposition 1.6.2.

Let

A be a Banach algebra.

(i) The set of left (right, two-sided) topological zero divisors in A is closed. (ii) If A has an identity and x E A is a left (right, twosided) topological zero divisor, then x is singular. The converse of Proposition 1.6.2(ii) is not generally valid. For example, consider the commutative Banach algebra with identity A(D) introduced in Example 1.2.3. Evidently the element fez) = z, z E 0, is singular. However, f is not a topological zero divisor as can be seen from the following: suppose (gk] C A(D) is such

I. Fundamentals of Banach Algebras

46

that li,\lIgkfllCD = o. Since If(z)l = 1 when lzl = 1, we see at once that limk(suPlzl =llgk(z)l) = O. whence by the Maximum Modulus Theorem of complex function theory [A, p. 134] we conclude that limkllgkllCD

= o.

The next theorem will provide us with some necessary and sufficient conditions for an element to be either a left or right topological zero divisor. We need one definition and some notation before we can state the result. Definition 1.6.2. Let A be a normed algebra and x E A. left (right) modulus of integrity of x is defined by

~ (x)

= inf IIxyll y ~ 0 llYlr

The

ell (x) = inf ~). y~ 0

llYlr

Given a Banach algebra A, for each x E A we shall denote by T and TX the elements of L(A) defined by T (y) = xy and x x ~(y) = yx, yEA. Theorem 1.6.2. Let A be a Banach algebra and the following are equivalent: (i) (ii)

x

~

A.

Then

x is not a left (right) topological zero divisor. ~(x)

> 0

(~(x)

> 0).

(iii) There exists some constant cllyxll > Kllyll) , yEA.

K > 0 such that

lIxyll > KlIYll

(iv) x is not a left (right) zero divisor and {xy lyE A) ({yx lyE Al) is a closed right (left) ideal in A. (v) Tx (~) has a continuous inverse when considered as a continuous linear transformation from A to T (A) (Tx(A». x Proof. It is apparent from a standard result concerning continuous linear transformations [L, p. 65] that parts (iii) and (v) of the theorem are equivalent, and the equivalence of parts (ii) and


47

(iii) follows at once from the definition of the moduli of integrity. If x were a left topological zero divisor, then there would exist some sequence (Yk} C A such that IIYkll = 1, k = 1,2,3, ... , and limkllxYkll = o. Clearly, in this case, we would have ~(x) = 0, and so part (ii) implies part (i). The converse assertion is equally easy and is left to the reader. It is obvious that, if part (iii) holds, then x is not a left (right) zero divisor. Moreover, for instance, the set (xy lyE A) is clearly a right ideal in A. Suppose (XYk) is a sequence that converges to z (A. Then, since IIx (y,. y. ) II > KIIYk - Y·II, ~ J J k,j = 1,2,3, .•. , we deduce that (Yk) is a Cauchy sequence in A, and hence it converges to some w E A. It is then apparent that xw = z E (xy lyE Al, and so (xy lyE A) is a closed right ideal. Thus part (iii) implies part (iv). Finally, suppose =

Tx(A)

x is not a left zero divisor and

is a closed right ideal.

(xy lyE A)

Then we note first that Tx

is

injective, since Tx(Yl) = Tx (Y2) implies x(Yl - Y2) = 0 implies Yl = Y 2 ' as x is not a left zero divisor. Thus Tx is an lnJective continuous linear transformation of A onto the Banach space T (A). Hence, by a consequence of the Open Mapping Theorem x [L, p. 187], we conclude that T has a continuous inverse on its x range. Consequently, part (iv) implies part (v). Therefore parts (i) through (v) are equivalent. Note, in particular, that the theorem asserts that a left (right) topological zero divisor if and only if

o x E A is ~(x)

= 0

(l1(x) = 0).

The second portion of Proposition 1.6.2 asserts that topological zero divisors in Banach algebras with identity are always singular. The converse of this result, as we noted, need not be valid. However, if A is a commutative Banach algebra with identity and


48

x is singular in every superalgebra of A, then it is a topological zero divisor. This is the important part of the next theorem. We first need to define "superalgebra" precisely. Definition 1.6.3. Let A be a Banach algebra with identity. A Banach algebra B with identity is said to be a superalgebra of A if there exists an isometric algebra isomorphism of A into B. Theorem 1.6.3 (Arens). Let A be a commutative Banach algebra with identity e and let x £ A. Then the following are equivalent: (i) (ii)

x is a topological zero divisor in A. x is singular in every superalgebra

B of A.

Proof. If x is a topological zero divisor in A, then x is clearly a two-sided topological zero divisor in every superalgebra B of A, whence, by Proposition 1.6.2(ii), x is singular in B. Thus part (i) implies part (ii). Conversely, suppose x is not a topological zero divisor in A. Then we shall construct a superalgebra B of A in which x is regular. First, we note, by Theorem 1.6.2, that since x is not a topological zero divisor, we have ~(x) > O. Let p > l/~{x) and consider the commutative algebra Bl consisting of all formal power series in t, yet) = Ik=oYkt k , Yk E A, k = 0,1,2, .•• , such

that lIy(t)1I = ~=OIlYkllpk is finite. FO~ :xample, if y (A is such that IIYII < lip, then yet) = ~=oY t belongs to Bl , where, of course, yO = e. The algebra operations in Bl are the usual formal operations of addition, multiplication, and scalar multiplication applied to power series. Moreover, it is not difficult to ~rifY that 1:·11 defined above is a norm on 81 under which BI is a commutative normed algebra. By Theorem l.l.l(i), the completion of B1 , denoted B~, is a commutative Banach algebra. Let I be the closed ideal in B~ generated by the element e - xt; that is, I is the closure in B~ of the ideal


49

{(e - xt)w I w E B~l. Then 8 is defined to be the quotient algebra B = B~/I with the usual quotient norm lIIw + IIlI = infvEIllw + vU. By Theorem l.l.l{iii), B is a commutative Banach algebra. We claim that B is a superalgebra of A. Indeed, it is evident that the mapping CD

q:(z) = 1: l/~(x), and the penultimate inequality utilizes the fact that IIxyli ~ ~(x)IIYII, y ~ A. It is then apparent from the previous inequality and the fact that {(e - xt)y(t) I yet) E Bil is dense in I that IIlcp(z) III =

III

CD

k

+ I III > IIzlI (z E A). k=O The inequality in the opposite direction is trivial, so we conclude

1: ~(z)kt

so


that 11Iep(z) III = IIzll, z E A; that is, cp is an isometric algebra isomorphism of A into B. Furthermore, an elementary argument reveals that epee) = e + I is an identity for B. Thus B is a commutative superalgebra of A. Finally, we claim that x is regular in B; that is, cp(x) = x + I is regular in B. Indeed, since e - xt E I, we have (x + I)(et + I) = xt + I = e + I, that is, (x + 1)-1 = et + I. Therefore part (ii) sf the theorem implies part proof is complete.

ei)~

and the

o

Returning again to Proposition 1.6.2(ii), we can rephrase the result there to say that in a Banach algebra with identity e, if e - x is a left (right, two-sided) topological zero divisor, then x is quasi-singular. A partial converse of this observation is contained in the next proposition. Proposition 1.6.3. Let A be a Banach algebra with identity e. If x E A is the limit of a sequence of quasi-regular elements in A, then either x is quasi-regular or e - x is a two-sided topological zero divisor. Proof. Suppose {xkJ C A_I is such that limkllxk - xII = o. If x is quasi-singular, we must show that e - x is a two-sided topological zero divisor. We claim first that limkllexk)_lll = CD. Indeed, suppose the sequence {II Cxk)_lIlJ is bounded. Now xk 0 eXk)_l = 0 implies that (xk)-l = -x k + xk (x k)_I' whence

Ck

=

1,2,3, ... ).


51

Since limkUx k - xU = 0 and (lie - (xk)_IIlJ is bounded. it follows that limkllYkll = O. In particular, there exists some ko such that, for k::: ko. IIYkll < 1, and so, by C.,rollary 1.4.2, Yk is quasiregular for k > k. Thus, by Proposition 1.1.2{i), e - Yk is 0 regular, k > ko. However,

Consequently, since e - Yk and e - {xk)_l are regular for k > k , -1 - 0 and A is a group, we deduce that e - x is regular, that is, x is quasi-regular, contrary to the hypothesis that x is quasisingular. Hence {1I{xk)_IIlJ is unbounded. The same argument, mutatis mutandis, shows that no subsequence of (II {Xk)_ll!l can be bounded. and so we conclude that limkUCxkJ_III = CD. Thus we see that for each k liCe - x)(xk)_lll U{xkJ_llI

=

=

=

= 1,2,3, ...

IICXk)_1 - x{xkJ_Ill

IICxkJ_ I II IIYk - xII II {xk)_lll

~~~""'"

lIex - xk)[e - (xk)_l] - xII

II (xkl_lll

from which it follows at once that limkl\Ce - xJexk)_lIl/IlCxk)_11l = Similarly limkl\cxkJ_ICe - xlll/llCxkJ_11I = o. Therefore e - x is a two-sided topological zero divisor. The proposition has the following simple corollary. the topological boundary of a set E by bdy(E).

o.

[j

We denote


S2

Corollary 1.6.1.

Let

A be a Banach algebra with identity e.

(i) If x (bdy(A_ I ), zero divisor.

then

e - x is a two-sided topological

(ii) If x (bdy(A- 1), zero divisor.

then

x is a two-sided topological

Proof. Part (i) follows immediately from Proposition 1.6.3, and part (ii) is apparent on noting that x E bdy(A- l ) if and only if e - x E bdy(A_ l )·

0

... The converse of part (ii) may fail; that is, there exist Banach algebras A with identity which have topological zero divisors not in bdY(A- 1). We shall return to the notion of a topological zero divisor at various points in the succeeding chapters.

CHAPTER 2 SPECTRA 2.0. Introduction. This chapter is devoted to introducing the concept of the spectrum of an element of a Banach algebra and to proving various results connected with this concept. The concept will be seen to be precisely the extension to the context of Banach algebras of the notion of the spectrunl of a continuous linear transformation on a Hilbert space, and the reader acquainted with the Hilbert space theory will find many of the following results and proofs familiar. As with many of the topics in the preceding chapter, the contents of the following sections will appear repeatedly in the sequel. We begin with the definition of spectrum and then prove some fundamental theorems, the most important of these being that the spectrum of any element of a Banach algebra is a nonempty compact subset of C. The third section cor.tains proofs of the Polynomial Spectral Mapping Theorem and the Spectral Radius Formula. The former theorem asserts that polynomials map spectra onto spectra, whereas the latter provides us with a formula for computing limnllxnlli/n in terms of the spectrum of x. The final section discusses the relationship between the spectra of an element when computed in different algebras. Once again the reader should observe the role played by the theory of functions of a complex variable in the study of Banach algebras. 2.1. Definitions and Basic Results. We have already noted that, if V is a Hilbert space over ~, then Lev), the space of S3

54

2. Spectra

continuous linear transformations from V to itself, is a Banach algebra with identity. In studying such transformations an important role is played by the notion of the spectrum of an element T E LeV), that is, by the set oCT) of all C E ~ for which T - ,I is singular. Here, of course, I denotes the identity transformation on V. It is evident that this definition of spectrum can be carried over verbatim to the context of any Banach algebra with identity. However, we also wish to define the spectrum of an element in a Banach algebra without identity. The motivation for the definition of the spectrum in this case comes from the fact that in a Banach algebra with identity e, x - Ce, , # 0, is quasi-singular.

is singular if and only if

xl'

Definition 2.1.1. Let A be a Banach algebra and let x E A. If A has an identity e, then the sEectrum of x, denoted by a(x) , is the set of all 'E~ such that x - Ce is singular; if A is without identity, then a(x) is the set of all C E~, , ~ 0, such that xl' is quasi-singUlar, together with , = o. Note that, if A is a Banach algebra without identity, then o E o(x), x E A. If A has an identity, then 0 E o(x) if and only if x is singular. In discussing the spectrum of an element x 1n a Banach algebra A it is often important to emphasize that the spectrum is being computed with respect to a particular algebra. When this is the case, we shall write o(x) = 0A(x) to highlight this point. Such a distinction is important, for example, in the next theorem. Theorem 2.1.1. If x E A,

then

Let A be a Banach algebra without identity.

0A(x)

= 0A[e] (x).

Proof. We note first that 0 E 0A[e] (x), because if x were regular in A[e], then there would exist some yEA and a E ~ such that (x,O)(y,a) = (xy + ax,O) = (0,1), which is clearly impossible.

ss

2.1. Definitions and Basic Results

,E

,=

Now suppose aA(x). If 0, then, from the preceding paragraph, we see that aA[e] (x). If ,~o. then xl' is quasi-singular in A and hence in A[e]. The latter follows on observing that, if xl' were quasi-regular in A[e], then there would exist yEA and a (~ for which

,E

x

= (, + y =

ax ) cxy - C,a

(0,0).

Consequently a = 0 and xl' 0 y = o. Similarly y 0 xl' = 0, and so xl' is quasi-regular in A. a contradiction. Thus xl' is quasi-singular in Are], whence x - 'e is singular in A[e]. Hence aA(x)

C

aA[e] (x).

Conversely, the preceding argument shows that, if ,~O and x - 'e is singular in A[e], then xl' is quasi-singular in A, and 0 E aA[e] (x) from the first paragraph of the proof. Therefore aA(x) = aA[e] (x).

o

Thus we see that in a Banach algebra A without identity we may compute a(x) with respect to A or A[e], whichever is most convenient, and obtain the same result. Note that the theorem fails if A has an identity.

Indeed,

as seen above, 0 E aA[e] (x), x E A, whereas 0 l aA(x) whenever x E A is regular. The most that can be said in this case is that aA(x) c aA[e] (x). Suppose X is a compact Hausdorff topological space and f E C(X). Then, by Theorem 1.6.1. a(f) if and only if there exists some t E X such that f(t) = ,. Thus a{f) is precisely R{f), the range of f. If X is a locally compact noncompact Hausdorff topological space, then it is easily verified that a(f) for fEe o (X)

,E

2. Spectra

56

is just R(f) U (O}. We shall see in Section 3.4 that an analog of these observations is valid for any commutative Banach algebra. If V is a Hilbert space over ~ and T E L(V), then a fundamental theorem from the study of Hilbert spaces asserts that aCT) ~ ~. This result is also valid for arbitrary Banach algebras. Theorem 2.1.2. a(x)

Let

A be a Banach algebra.

is a nonempty compact subset of {C, e E

\l;,

If x E A,

then

Ie' < IIxli}.

Proof. Since, by Theorem 2.1.1, aA(x) = aA(e] (x) when A is without identity, we may assume, without loss of generality, that A has an identity e. If a(x) =~, then x - Ce is regular for each C E~. Then, as in the proof of the Gel'fand-Mazur Theorem (Theorem 1.5.1), let x* be a continuous linear functional on A such that x*(x- l ) = I and consider the function g: ~ - ~ defined by g(C) = x* ( (x - Ce) -1 ], e E~. Precisely the same arguments as used before show that g is a bounded entire function such that lim'C,_eog(C) = O. Consequently, again applying Liouville's Theorem (A, p. 122], we deduce that gee) = 0, C E~, contradicting the fact that g(O) = 1. Thus a(x) ~ ~. Furthermore, if e EIC is such that 'el > IIxll, then whence, by Corollary 1.4.2, x/e is quasi-regular. Hence

IIx/CII 0 such k. k 0 that 'e' fix II 0, n is a positive integer, and x l ,x 2 , .•• ,xn are arbitrary.

in A

The main theorem concerning the Gel'fand topology is the following result: Theorem 3.2.2.

Let A be a commutative Banach algebra.

(i) If A has an identity, then ~(A) with the Gel'fand topology is a compact Hausdorff topological space. (ii) If A is without identity, ~h~n A(A) with the Gel'fand topology is a locally compact Hausdorff topological space, ~nd

72

3. The Gel'fand Representation Theory

A(A[e]) with the Gel'fand topology is the one-point compactification of A(A). Proof. Since the weak* topology is Hausdorff, it is evident that the Gel'fand topology is always Hausdorff. Suppose that A has an identity e and, as usual, assume that lieU = 1. From Theorem 3.1.2(i) we see that UTU = 1, T E A(A). Since the closed unit ball in A* is compact in the weak~ topology, to show that A(A) is compact in the Gel'fand topology it suffices to prove that A(A) is weak* closed. So suppose (T 1 C A(A) is a net and x* E A*, a IIx*1I < 1, are such that (T) converges to x* in the weak* topoa logy on A*; that is, lim T (x) = x*(x), x E A. We must show that aa x* is multiplicative and IIx*1I = 1. To this end let x,y E A, x ~ 0, y ~ 0, and ~ > O. Since converges weak* to x*, there exists some a such that, o a > a , T belongs to the weak* neighborhood o a U(X*;b;x,y,xy) ·{y*ly* EA*, lx*(x)-y*(x) I 0 for ..,-a.lmost all t E X, then f(T) ~ 0, ,. E 4 (L.) • Moreover, ACLCD) provides us with an example of a rather complicated maximal ideal space in that it is zero-dimensional. For the sake of completeness we define this term explicitly. if

f E L (X.S,Il)

A

and

f(t)

CD

-

Definition 4.3.1. Let X be a topological space. Then X is said to be zero·dimensional if the topology has a base that consists of clopen sets -- that is, a base consisting of sets that are both clos eel and open.

Cle~rlY~ sional~

if X is a discrete topological space, it is zero-dimen-

but the converse need not be the case.

The previously indicated results are contained in the next theorem. Theorem 4.3.1.

(X,S~~)

Let

be a positive measure space.

Then

The Gel' fand transformation is an isometric orier preserving isomorphism of L (XJS,~) onto C[6{LCD (X.S.~))]. (i)

~

(ii)

ACLCD (X,S,I'»)

Proof.

is zero-dimensional.

By the same argument mutatis mutandis as used in proving .

Theorem 4.1.3, we see that the Gel'fand transformation is an isometric isoaorphisll of L (X,S, .. ) onto C[A(L )}; and, in particular, if CD CD .. .. f E L.. (X,S,p) is real valued, then so is f E C[A{tJ '1 = L_ (X,S,~) •

..

is greater than or equal to zero ...-aIWlOst everywhere, then one can evidently write f = g2, where lEt (X.S.~) ~ • 2 ~ 0, or E ACL.), and 50 •the is real valued.. Hence feT) = [geT)] Gel'£and transformation is order preserving. But if

f E L (X.S,p)

..

finally. we shall show that 6(L) is %ero-cliJDensional. To this end let E c: X be measurable and le~ Xs E L.(X,S,I') denote

4. The Gel'fand Representation of Specific Algebras

94

the characteristic function of E. Since X~ = XE, it is apparent A2 A A that ~ = Xs' whence we deduce that RexE) C (O,l). Let A A • E = {T I T ( ~(Lm)' XE(T) = I}. Evidently XE = XE; that is, the Gel'fand transform of XE is the characteristic function of the • • • set E C ~(Lm). The continuity of XE shows at once that E must be clopen. Conversely, suppose Ec A(L) is a clopen set. Then, A m since L (X,S,~) = C[A(L )], there exists some f E L (X,S,~) m A co 2 CD such that f = XE' from which it follows that f = f. Thus f is the characteristic function of some measurable set E C X, and A obviously we must have E = E. Since in the preceding paragraph we allow measurable sets E of possibly infinite measure, we see that the characteristic functions of the measurable subsets of X generate a norm-dense subalgebra of LCD (X,S,~). Consequently, since the Gel'fand transformation is an isometry, we deduce that the set {X lEe X, E measurable} c C[A(Lm)] generates a norm-dense subalgebra of C[ACLCD)]. It is then easily seen that the set {E lEe ACLCD), E clopen) = {E , E c X, E measurable: forms a base for a zero-dimensional topology T on ACL) that is CD weaker than the Gel'fand topology on ~CL). Since ACLCD) is a m compact Hausdorff topological space in the Gel'fand topology, we see, as in the proof of Theorem 4.1.1~ that in order to show that T coincides with the Gel'fand topology it suffices to prove that T is Hausdorff.

e

So suppose Tl ,T 2 E A(Lm), Tl ; T2 • Then there exists some " • A h E Lm(X,S,~) = C[ACLCD) ] such that h(T 1) ; hCT 2), as C[ACLCD) ] separates the points of A(L). Without loss of generality we may A CD • A. assume that h(T1); O. It is then easily verified that f = gg, where g = [h - h(T 2)]/[hCT l ) - h(T 2)], belongs to C[ACLCD) ] and • •• • fCT I ) = 1, f(T 2) = 0, f(T) > 0, T E A(Lm). Next let k be a finite linear combination of characteristic functions of clopen subsets of " ACL) such that kCT) > 0, T E ACLm), and for which IIf - kll < 1/3. m This is possible because {XE lEe ACLCD) , E clopen) generates a norm-dense subalgebra of C[ACLm)], as seen in the preceding paragraph.

.

..

".

.

.

4.4. A(D)

9S

• But then it is apparent that (T I T E 6(L ), k(T) > 2/3] is a m • clopen set containing T I , whereas (T I T E 6(Lm), k(T) < 1/3] is a clopen set containing T2, and these clopen sets are disjoint. Thus T is Hausdorff. Therefore T coincides with the Gel'fand topology on 6(L) m [W 2 , p. 84], and so the Gel'fand topology on 6 (Lm) is zero-dimensional.

o

It is, of course, once again clear that

Lm (X,S,~)

is semisimple.

describe the Gel'fand topology on 6(LCD} in somewhat different terms. Suppose T E 6(Lm) and let C denote the maximal connected subset of 6(Lm) that contains T. Since the closure of a connected set is connected, we see that C is closed. Suppose wEe and w ~ T. Since 6(L) is zero-dimensional and Hausdorff, m there exists a clopen set Ec 6(L) such that T € f, w l E. But m then in the relative topology on the connected set C we see that Ene is open, C - (f n C) is open, and C = (E n C) U [C - (E n C)] thereby contradicting the connectedness of C. Consequently C = (T]. One-~n

A topological space with this property, that is, a topological space such that the maximal connected set C that contains a given point T of the space is just C = (T), is said to be totally disconnected. In particular, 6(L) is totally disconnected. m The argument above actually shows that, if a topological space of at least two points is zero-dimensional and To' that is, given any two distinct points there exists an open neighborhood of one that does not contain the other, then it must be totally disconnected. Conversely, it can be shown that a totally disconnected locally compact Hausdorff topological space is zero-dimensional [HR 1 , p. 12]. 4.4. ~. Next we wish to consider the commutative Banach algebra with identity A(D), that is, the algebra of all continuous

96

4. The Ge1'fand Representation of Specific Algebras

complex-valued functions on D = (z I z E~, Izl < 1) that are analytic on the interior of o. We shall see that ~(A(D)) can be identified with D, which on the surface may not be very surprising. This will, however, provide us with an example of a commutative Banach algebra A that contains a subalgebra B whose maximal ideal space properly contains that of A, that is, such that ~(B) ~ ~(A) and ~(8) ~ 6(A). Some further observations about ~(A(D)) will set the scene for a discussion of finitely generated algebras in the next section. As in the preceding sections, it is apparent that, if z ( 0, then Tz(f) = fez), f € A(D), defines a complex homomorphism of = (f I f E A(D), fez) = 0). ACO) and that Mz = T-I(O) z On the other hand, let h E A(D) denote the identity function, that is, h(z) = z, zED. If T E ~(A(D)), we set T(h) = ,. Since IiTIi ~ 1, we see that I"~ < 1. We claim that Tef) = f(C), f E A(D). To see this let f (A(O) and for each r, 0 < r < 1, define fr(z) = f(rz), zED. Then fr is continuous on (z I z (~, Izi < l/r). In particular, f E ACD) and the power series expansion of f r k r about z = 0, say ~=Oak(r)z, converges uniformly to fr on (z I z E~, Izi < 1). Thus we see that T(f ) r

= T[

CD k 1: ak(r)z ] k=O

CD

= 1: ak(r)T(h) k=O

k

CD

=

1: ak(r)c k

k=O

= f r (e)

(0

< r < 1).

Moreover, an elementary argument involving the uniform continuity of

4.4. A(O) f

on 0

97 reveals that

limr_lllfr - flieD = O. T(f) =

=

Thus we have

lim T(f ) r-l r lim f (C) r-l r

= f(C), which is what we set out to prove. Therefore we see that ~(A(O)) and 0 can be identified as point sets, and the usual argument employed twice before concerning the comparability of compact Hausdorff topologies shows that the usual topology and the Gel'fand topology on D coincide. The validity of the next theorem is now apparent. D = {z

Iz

Izl

~ 1).

Then

(i) The mapping z - Mz = {f I f E A(O), fez) a homeomorphism of 0 onto ~(A(D)).

= 0),

Theorem 4.4.1. 0 such

G, 0 ~ ~, 0 open) > p.

But now suppose that 0 eGis open and that 0 has compact closure. The regularity of Haar measure entails that ACO) A( U Ok) = t A(Ok) > np. k=l k=l Since A(O) < m, it follows at once that must be finite. Hence every open subset of G with compact closure is finite. In particular, every compact neighborhood of a point in G is finite, whence, since the topology of G is Hausdorff, we conclude that single points are open sets.

°

109

Therefore G is discrete, and part (i) of the theorem implies part (ii).

o

On the other hand, Ll(G) always contains what is kno~~ as an approximate identity, whether G is discrete or not. We give the definition of an approximate identity in the context of an arbitrary commutative Banach algebra.

net

Definition 4.7.2. Let A be a commutative Banach algebra. (u a ) C A is said to be an approximate identity if (i)

(ii)

sup lIu II < a a

go.

(x E A).

lim a lIu ax - xII = 0

Obviously, if

A

A has an identity

mate identity on setting

e,

then

(u) is an approxia The converse need not

u = e for all a. a be valid, as seen from the next theorem and Theorem 4.7.1.

in

Before we can prove the existence of an approximate identity we need one preliminary result. Ll (G)

Proposition 4.7.3. Let G be a locally compact Abelian topological group and let A be a Haar measure on G. If U is a nonempty open subset of G, then A(U) > o. I

Proof. Since A is translation invariant, we need consider only nonempty open subsets that contain the identity of G. Suppose U eGis such a set and A(U) = O. Then A(t + U) = 0, t ~ G. KeG is compact, then (t + U l t E K) forms an open covering of K, and so there exist t l ,t 2, ... ,t in K such that n n K C vk=l(t k + U). Consequently A(K) < ~=lA(tk + U) = O. Since K is arbitrary, it follows at once from the regularity of A that A = 0, which is a contradiction. If

Therefore

A(U) >

o.

o


110

Theorem 4.7.2. group. Then LI(G)

Let G be a locally compact Abelian topological contains an approximate identity.

Proof. Let (u 1 denote the family of open neighborhoods of a o in G that have compact closure and are such that Ua = -ua . Such neighborhoods exist because G is locally compact. Then define denotes the characteristic ua = XujA.(U~, where as usual Xu function of U . In view of Proposftion 4.7.3, we see that each u ex a is well defined, and, as is easily verified, each u defines a a nonnegative element of Ll (G) n LCD(G) such that lIuaU I = 1. Moreover, (u) becomes a net provided we set a > e if and only if U C Uc • a a ~ 10 see that (ual is an approximate identity in Ll(G) we need only show that limJlua * f - fill = 0 for each fELl (G) • Let f E L1 (G). Then we see, on applying Fubini's Theorem [Ry, pp. 269 and 270], that lI

ua * f - fill

= IGIIG

f(t - s)ua(s) dA,(s) - f(t)1 dA,(t)

= IGIIG

[f(t - s) - f(t)]ua(s) dA,(s)1 dA,(t)

~IG[fG 'f(t - s) - f(t)luex(s) dA,(s)] dArt)

= IG[IG =

IG

11s(f)(t) - f(t)l dArt)] ua(s) dA,(s)

111s(f) - fIl1ua(s) dA,(s).

However, by Proposition 4.7.1(ii), 111s(f) - fill is a uniformly continuous function on G, whence, given e > 0, there exists some Uao such that, if s E Ucro' then 111s(f) - fli l < e. Consequently, if a > a, then o

lIua

* f - fill ~
0, there exists some compact set KeG such that If(y)1 < e, y E G - K.

..

-

.

lIS Because of the identifications indicated in Theorem 4.7.3 we " as the maximal ideal space of L1 (G) shall generally speak of G " and of the Gel'fand transform of f E LlCG) as a function on G. Note also that the norm-decreasing nature of the Gel'fand trans-

" IIfIlCD:5 IIflll, f E Ll(G).

formation means explicitly that

For the classical groups, for example, r and ~ the Gel'fand transform on LICG) is precisely the familiar Fourier transform. " For this We shall see this momentarily when we describe r" and ~ reason we shall use the terms "Gel'fand transform" and "Fourier transform" interchangeably in discussing Ll!G). Also, recalling Example 1.2.9, it is natural to write LICG) = FLl(G). Before we proceed any further with the general investigation " "r. Suppose y E ~ " of LICG), let us describe in detail Rand Since (O,y) = 1, there exists some 6 > 0 such that

J~ (s,y) ds ~ O. Consequently we deduce from

~ (t

+

s,y) ds = (t,y)~ (s,y) ds

(t E IR)

that

Ct,y)

=

s~ (t

=

y' (t)

=

s,y) ds

Jo (s,y)

Jt+6 t Thus we see at once that y

+

6

ds

(s,y) ds

J6o Cs,y)

(t E lR) •

ds

is differentiable on

lim (t h-O

+ h,y) -

h

(t,y)

~

and so

116


= (t,V)

lim (h,V) ~ (O,y)

h-O

= V' CO)Vet)

(t (lR).

Solving this elementary differential equation and using the facts vCO) = I and IVCt) I = I, t (!h, we see that there exists some ~ E lR such that y (t) = (t, V) = ei~t, t E iRe Conversely, it is obvious that for each ~ E 1f\. the function ei~. belongs to R, so that ~ = {e igo I ~ E u~. Since the correspondence ~ - e ig ., g E ~ that

" with the point set ~, and is bijective, we can also identify lit we shall often do this. It is now evident that the Gel'fand transform of

f E

Llor~

is indeed the classical Fourier transform .

..

The description of f

is obtained most easily by utilizing the

preceding results. To do this we note that r is isomorphic to iV2nZ, where 2n£ = (2nk I k ~ Z). Then it is easy to check that .. " V ( f if and only if V E~ and V is periodic with period 2n. Thus we see that = (e ik . IkE if). As in the case of lri, we

r ..

generally identify f k E oZ.

with L

.

by means of the mapplng

k - e

ike

,

..

" If V E Z, To round out this discussion we shall describe L. then there is some f such that (1,V) = ,. Clearly then we must

,E

have (k,V) = ,k, k E L. Conversely, given ,E r, the mapping k - ,k, k E Z, is obviously an element of Thus we see that 1 = (,Co) I , E fl. As usual, we generally identify ~" with f.

Z.

.."..

Allowing for a certain amount of imprecision, we observe that ..,," "" (JR.) = l[( = lR, (f) = It. = f, and (~) = f = lL.. These observations are indicative of a general phenomenon for locally compact Abelian " topological groups G, which, loosely speaking, says that (G) = G. We shall return to this important result, known as the Pontryagin Duality Theore~ in Section 10.5, where we shall discuss it in detail.

..

..

It is obvious that for the algebras previously discussed in

117 this chapter the Gel'fand transformation is injective, that is, the algebras are semisimple. The algebra L (G) is also semisimple, 1

but the proof of this fact is not entirely trivial. Theorem 4.7.4. group.

Then

LI(G)

Let

G be a locally compact Abelian topological

is semisimple.

Proof. In view of Theorem 3.5.1 and Corollary 3.5.1, it suffices to show that, if f E L1 (G) is such that limnllfnll~/n = 0, then f = 0, where by fn we mean the n-fold convolution of f with itself. We claim first that we may assume, without loss of generality, that f E LI(G) n Lm(~). Indeed, suppose we know that f E LI(G) n LmCG) and limnllfDlI~/n = 0 imply f = O. Then suppose g E LI (G) is such that lim IIgnlll1/n = O. Let {u 1 C LI (G) n L CG) be an approximate n a m identity; for example, we may choose {u) as in the proof of a Theorem 4.7.2. From Theorem 3.5.1 we see that, for each a, limnliCua * g)nll~/n = 0, as Rad[L I (G)] is an ideal. However, by the assumption just made and the fact that lIu a * gil 0 ) 0 and each compact set KeG define U(K,a)

= (y

•

lyE G, I(t,y) -

11

< a, t E K).

Then the family of sets (U(K,e)), where e > 0 is arbitrary and KeG is an arbitrary compact set, forms a neighborhood base at • for the Gel'fand topology on G. • y EG o

Proof. It suffices to show that every such set U(K,a) is • open in the Gel'fand topology on G and that every open neighborhood of y in the Gel'fand topology contains some such set. o

• - f, defined by We first observe that the function from G X G • (t,y) - (t,y) = yet), t E G, yEO, is continuous. Here (t,y) • and G X G • is given denotes a point in the product space G X G, • the product topology. Indeed, suppose (t) C G and (~) C G ~ a are nets that converge to t (G and y E OJ respectively. If


124

A

f (Ll(G)

is such that

exists some

eto

fCy)

a >

such that

A

then, since

0,

~

implies

eto

A

f ( C (0),

A

0

f(y~ ~

O.

there

Elementary

computations reveal that for each s E G we have A

= [T-s (f)]

(s,y)f(w)

A

A

(w)

(w

E G),

whence, recalling the development preceding Theorem 4.7.3, we conclude that A

(s,y)

=

[T -s (f)] (y)

(s ( G)

A

fey) and A

(s'YJ

=

[T -s (f)] -(Y~

(s E G; a > a ).

•

0

f(YJ

Clearly then to show that lim (t ,y ) = (t,y) we need only prove • a a.a • • that lima[T_ta(f)] (Ya) = [T_t(f)] (y) since limaf(Ya) = fey). But

+

I [T_t(f)]

•

A

(Ya) - [T_t(f)] (y)1

< II T_t (f) - T_ t (f)1I 1 a

from which we at once deduce the desired conclusion via Proposition • to r is 4.7.I(ii). Thus the mapping (t,y) ~ (t,y) from G X G continuous. Now suppose s > 0 and KeG is compact, and consider U(K,s). If w E U(K,s), then from the continuity of the function (t,y) ~ (t,y) t E G,y E G, and the fact that I(t,w) - 11 < s, t E K, we deduce that, for each t E K, there exists an open neighborhood Vt of t A

125

in G and an open neighborhood

W

u"

,.

of w in G such that, if

t

(s,Y) ~ Vt x Ww,t' then I(s,y) - 11 < i . The sets {V t I t E K) clearly form an open covering of the compact set K, and so there t l ,t 2, ..• ,tn in

exist

K C ~=IVtk'

K for which

Let

W= nk=IWw,tk.

Evidently W is an open neighborhood of w in the Gel'fand topology, and if t ~ K, then I(t~y) - 11 < i, y (W; that is, We U(K,i). Since ~ E U(K,s) is arbitrary, we conclude that U(K,e) is open in the Gel'fand topology. the other hand, suppose that W is an open neighborhood of Yo in the Gel'fand topology. Then~ since the GeI'fand topology is the relative weak* topology on A(L 1 (G)), we may assume without loss of generality that there exist f l ,f2 , ••• ,fn in LI(G) and £ > 0 such that ,.,. ,. ,~= {y I y.E. G, Ifk(y) - fk(yo)1 < e, k = 1,2" •.• ,n). On

Let gk ~ ec(G) and consider

If yEW. o

be chosen so that

llfk - gklll < e/3, k = l,2" ...• n,

then

+

~

II +

,. ,. Igk(y o) - fk(y o ) I

ll

,.

,.

f k - gk l + Igk(Y) - gk(Y o) I

ligk

- fklll

e e e 333

0 such that

IIxll 2 < Kllx 211,

A.

Proof. exists some

If 11·11 and 11.11 0 are equivalent norms, then there Kl > 0 such that IIxli < Klllxllo' x E A. Consequently (x ~

setting part (ii). On

we see that part (i) of the theorem implies

Conversely, suppose

lIxU 2 ~ Kllx 2U, x € A,

for some

K>

o.

A).

5.1. The Ge1'fand Representation

131

Then applying this estimate repeatedly, we see that, for each x E A, IIxll ~ KI/2I1x 2111/2

< Kl/2[Kl/2UX4111/2] 1/2 =

222 Kl/2+l/2 IIx 2 111/2

< •.. < Kl / 2+1/22 +... +1/ 2n ll x2nlll/2n

-

-

Cn = 1,2,3, ••• ).

Thus, by the Beurling-Gel'fand Theorem (Corollary 3.4.1) or the Spectral Radius Formula (Theorem 2.2.2), we deduce that IIxll < KUxll , a x E A. This estimate, combined with the fact that IIxU a < IIxll, x E A, shows that 11·11 and 1I'lIa are equivalent norms. Hence part (ii) implies part (i).

o

The first application of this theorem is the next corollary. Recall that a mapping is said to be a topological isomorphism if it is both an isomorphism and a homeomorphism. Corollary 5.1.1. Let A be a semisimple commutative Banach algebra. Then the following are equivalent:

• of A is a closed subal(i) The Gel'fand representation A gebra of (Co[fl(A)],U·Um). (ii)

There exists a constant

K > 0 such that

IIxU 2 ~ Kllx 2U.

x E A. (iii) The Gel'fand transformation is a topological isomorphism • of A onto A. Proof. If part (i) holds, then the Gel'fand transformation is evidently a continuous bijective linear mapping of A onto the Banach space •A C C [fl(A)]. Thus an application of the Open Mapping o Theorem [L, p. 187] reveals that the inverse of the Gel'fand trans• to A, and so formation is a continuous linear mapping from A • there exists some Kl > 0 for which IIxll ~ K111xllm = K1llxlla. x E A.

S. Semisimple Commutative Banach Algebras

132

Consequently 11·11 and 11.11 0 are equivalent norms on A, whence, by Theorem 5.1.1, we conclude that there exists some K > 0 such that IIxll 2 < Kllx 2l1, x 'A. Thus part (i) implies part (ii). If part (ii) holds, then Theorem 5.1.1 shows that 11·11 and 11.110 are equivalent norms, from which it follows immediately that the Gel'fand transformation is a topological isomorphism, as

•

IIxlia = lIxllm, x E A. (iii).

Hence part (ii) of the corollary implies part

To show that part (iii) implies part (i) is even easier and is left to the reader.

o

Before we can state the next corollary we need a definition. Definition 5.1.1. Let A be a commutative Banach algebra. •• A is said to be self-adjoint if whenever • x E• A then x E A, where the bar denotes complex conjugation. Thus A is self-adjoint if and only if the Gel'fand representa• tion A is closed under complex conjugation. It is left as an exercise for the reader to verify that the Banach algebras C (X), X o being a locally compact Hausdorff topological space, and Ll(G), G being a locally compact Abelian topological group, are self-adjoint, whereas A(D) is not. It should be observed tha~ if A is a self-aajoint commutative Banach algebra, then an appeal to the Stone-'Veierstrass Theorem • [L, p. 332] shows that the Gel'fand representation A is dense in

•

•

(Co [~(A)] , I;· ltD). In particular, Ll (G) is dense in Co (G) , G is a locally compact Abelian topological group.

where

The proof of the next corollary utilizes the general fact about self-adjoint algebras just mentioned. The details are left to the reader.

..

S.2. A as a Banach Algebra

133

Corollary S.1.2. Let A be a self-adjoint semisimple commutative Banach algebra. Then the following are equivalent: (i) x € A.

Cii)

K > 0 such that

There exists a constant

..

A

Ilxll 2 < Kllx 211,

= Co [deAl].

The final result of this section gives a necessary and sufficient condition for the Gel'fand transformation to be an isometry. Theorem S.1.2. Let A be a commutative Banach algebra. the following are equivalent:

(ii)

Then

-

The Gel'fand transformation is an isometry of A onto A.

Proof. If IIxU 2 = IIx 2U, x E A, then clearly for each x E A I 2n II = IIxll 2n , n = 1,2~.3, .... Hence, by the 8eurlingwe would have Ix Gel'fand Theorem (Corollary 3.4.1), we deduce that n

IIxll

= limUx 2

II

1/2n

n

.. = IIxll

(x E A),

G)

from which it is apparent that the Gel'fand transformation is an isometry. The converse is immediate on noting that 2

IIx II

-2

= IIx

II CD

.. 2

= IIxll = IIxll

2

(x E A)

CD

whenever the Gel'fand transformation is an isometry.

c

It is worth remarking that, if the Gel'fand transformation is an isometry, then A must be semisimple and A must be a closed subalgebra of C [A(A)).

-

..

o

A ~~ Banach Algebra. If A is a commutative Banach algebra, then the Gel'fand Representation Theorem (Theorem 3.3.1) 5.2.

s.

134

Semisimple Commutative Banach Algebras

• is a normed subalgebra of C [A(A)]. It need not, shows that A o however. be a Banach algebra with ~he supremum norm. Nevertheless, • can be normed in such a way that it does if A is semisimple, A become a Banach algebra. This technique is often useful in studying various problems. Theorem 5.2.1.

• bra and define IIxli. (iJ

(ii) (iii)

11·11.

••

= IIxll, x E A.

Then

•

is a norm on A.

• (A,II·II.) • A(A)

Let A be a semisimple commutative Banach alge-

is a semisimple commutative Banach algebra .

is homeomorphic to

A(A).

(iv) The Gel'fand transformation is an isometric isomorphism of • A onto (A,II·U.).

• and that The verification that is a norm on A (A,II·II.) is a commutative Banach algebra is routine and is left to the reader. The semisimplicity of A is required to show that II·U. is well defined. Proof.

n·lI.

•

•

•. If T E A(A), define wT on A by w (x) = T(X), •x E A •T • Clearly wT is a complex homomorphism of A; that is, wTEA(A). • Moreover, the mapping from A(A) to A(A) so defined is bijective. • • •• Indeed, if wT = W , then X(T I ) = TI(X) = T 2 (X) = X(T 2), x E A, 1 T2 • which implies that TI = T2 , as A separates the points of A(A) (Theorem 3.3.1). Thus the mapping is injective.

• and one defines T(X) = w(x), • w E A(A) • • • • x E A, then obviously T E A(A) and wT(x) = T(X) = w(x), x E A. Hence the mapping is surjective. On the other hand, if

Furthermore. if (T J C A(A) is a net that converges to T E A(A), • a • then, since A C C [A(A)], we see at once that {w (x)] converges Ta to wT(x) for each x E A. from which it follo~s by the definition • that (w ] converges to w • of the Gel'fand topology on A(A) T T a

.0. .

5.3. Homomorphisms and Isomorphisms

135

.

Consequently the mapping T ~ wT of 6(A) onto 6(A) is continuous. A similar argument proves the continuity of the inverse mapping . Therefore 6(A) and 6(A) are homeomorphic •

..

The semisimplicity of are now apparent.

..

(A,II·II .. )

and part Civ) of the theorem

o

One of the advantages of this construction is that it enables us to replace a semisimple commutative Banach algebra A by another such algebra B, where the algebra B is an algebra of continuous functions under pointwise operations that is isometrically isomorphic to A and has the same maximal ideal space as A. The algebra B, as we have just seen, is B = A with the norm 11·11.. Thus, for example, if A = Ll(G), G being a locally compact Abelian topological group, then LI(G) is isometrically isomorphic to the semisimple commutative Banach algebra L1CG) of functions in C CG) with the .. 0 norm IIfll .. = IIf1l1, f (; LI(G). A special case of this was mentioned in Example 1.2.9 when G = f and G = Z. There we denoted LICf) by FLICf). Similarly in this case we may write LICG) = FL1CG) when G is an arbitrary locally compact Abelian topological group.

..

..

..

..

..

..

5.3. Homomorphisms and Isomorphisms of Commutative Banach Algebras. In this section we wish to prove several results concerning homomorphisms and isomorphisms between commutative Banach algebras. A corollary of the first of these results, which asserts that an algebra homomorphism of any commutative Banach algebra into a semisimple commutative Banach algebra is continuous, will show that the norm on a semisimple commutative Banach algebra is essentially unique. This assertion will be made precise in Corollary 5.3.1. First, however, we prove the indicated theorem. Theorem 5.3.1. Let A and B be commutative Banach algebras and suppose A is semisimple. If T : B ~ A is an algebra homomorphism, then T is continuous.

S. Semisimple Commutative Banach Algebras

136 Proof.

Since

T is a linear mapping

and

A and

Bare

Banach spaces, we may appeal to the Closed Graph Theorem [L, p. 189] to deduce the continuity of T. To this end, suppose {xk ) is a sequence in B and x E Band yEA are such that

and lim/lT(X k) - yllA k

where II'II B and We must show that

= 0,

U'IIA

denote the norms in T(x) = y.

B and

A,

respectively.

Let T E ~(A) and set wT = ToT. Since T is a homomorphism, it is easily seen that either wT = 0 or wT E ~(B). In either case we have limkw,,(x k) = wT(x). lienee we see, on the one hand, that for each T E A(A) =

T(x) " (T)

and, on the other hand,

" lim T(xk ) " (T) = yeT), k

as simplicity of A,

" whence, from the semiT(x) " = y, we conclude that T(x) = y. Thus

Therefore, T is a closed linear mapping and hence, by the Closed Graph Theorem [L, p. 189], a continuous mapping. Corollary 5.3.1.

Let

c

A be a commutative Banach algebra under

the norms 11'11 1 and 11'11 2 . If (A,II·II I ) and 1i'11 2 are equivalent norms on A.

is semisimple, then

11'11 1

Proof. Consider the mapping T : (A,II'1I 2) - (A,li'H I ), defined by T(x] = x, x E A. F.vidently T is an algebra homomorphism and, by Theorem 5.3.1, is continuous. Thus there exists some K > 0 such

5.3. Homomorphisms and Isomorphisms

137

that IIxlll < Kllxll2, x ~ A. However, since (A,II·lI l ) and (A, 11.11 2) are both Banach spaces, this implies, via the Two Norm Theorem [L, p. 190], that

11.11 1 and

11.11 2 are equivalent.

0

Thus the meaning of our previous remark concerning the essential uniqueness of the norm in a semisimple commutative Banach algebra should now be clear. Corollary 5.3.1 is actually valid even if the assumption of commutativity is dropped, but the proof requires more of the general theory of Banach algebras than we have at our disposal. A proof of this result can be found in [Jo]. In Theorem 4.1.4 we proved that two compact Hausderff topological spaces X and Yare homeomorphic if and only if C(X) and C(Y) are algebraically isomorphic. Our next theorem shows that one~half of this equivalence is valid in general for commutative Banach algebras. Theorem 5.3.2.

Let

A and

B be eommutative Banach algebras.

If there exists an algebra isomorphism of is homeomorphic to 6(B). Proof. Suppose As before, for each

B onto

A,

then

6(A)

T: B ~ A is a surjective algebra isomorphism. T E 6(A) we define wT E 6(B) by

WT(X) = (T

0

T)(x) = T[T(x)]

(x E B).

Note that wT ~ 0, because if wT(x) = T(T(x)] = 0, x E B, then T = 0, as T is surjective, contradicting the fact that T E 6(A). Evidently the mapping ~(T) = wT ' T E 6(A), maps 6 (A) into 6(B). r.toreover, ~ is bijective. Indeed, if ~(Tl) = ~(T2)' then Tl[T(x)] = T2 [T(x)], x E B, whence we conclude that Tl = T2 , as T is surjective; and if w E 6(B), then, as above, we see that T (y) = W[T-l(y)], yEA, W defines an element TUI E A(A). Clearly ~(T) = w, and so ~ is W bijective.

138

5. Semisimple Commutative Banach Algebras

Furthermore, ~ and ~ -1 are continuous. For instance, suppose (T ) C A(A) is a net that converges to T E A(A). Then, since • a A C Co[A(A)], we see that for each x E A lim a

~(T~(X)

a

a

a

whence we conclude that ~ is continuous. tinuous. Therefore

•

•

= lim T [T(x)] = lim T(x) (T ) = T(x) o(T) = ~(T)(X)

~:

A(A)

~

A(B)

~

Similarly ~-l

is a homeomorphism.

is con-

o

A few remarks about this theorem are in order. First, if A is semisimple, then, by Theorem 5.3.1, T is continuous, and T is completely determined by the equation T(K) • (T) = •X[~(T)] where T E A(A) and x E B, and ~ is the homeomorphism defined in the proof of the theorem. Thus, if Tl and T2 were surjective algebra isomorphisms from B to A which defined the same homeomorphism ~J then Tl = T2 " This follows at once from the semisimplicity of •• • A and the equations Tl(x) (TJ = X[~(T)] = T2 (x) (T), T E A(A) and x E B. Contrary to the situation for C(X) discussed earlier, the converse of Theorem 5.3.2 need not be valid. In particular, if ~ : A(A) ~ ACB) is a homeomorphism between A(A) and A(B), then • T : B ~ A, defined by T(x) • (T) = X[~(T)], T E A(A) and x E B, need not define a surjective algebra isomorphism. The crux of the difficulty is that, given a homeomorphism ~: A(A) - A(B), it is not at all evident that x• 0 ~ (A• when • x E•B. The question of precisely which homeoMorphisms between A(A) and ACB) induce isomorphisms between B and A is generally a rather intricate one, even in the case that A = B. For some specific algebras the answers are known, for instance, when A = C(X) and B = Cry), X and Y being compact Hausdorff topological spaces. We shall not discuss the problem any further, but content ourselves ~ith describing the situation for LIOR) " Recall that A(LIOR» = ~

5.4. Characterization of Singular Elements Theorem 5.3.3. If T is a mapping from then the following are equivalent: (iJ

T

(ii)

~

Ll (lH) - Ll (lR)

139 LIOR)

to itseff,

is a surjective algebra isomorphism.

R-R is a homeomorphism of R onto itself of the form C9(t) = at + b, t E IR, for some a,b E IR, a ~ 0, and .. T(f) (t) = f[C9(t)], t E Rand fELl (lR).

.

A proof of this result can be found in [Ka, pp. 217-219]. Some further discussion of such isomorphism problems is available in [Ru l , pp. 77-96]. 5.4. Characterization of Singular Elements in Self-adjoint Semisimple Commutative Banach Algebras. In our earlier discussion of topological zero divisors we proved (Proposition 1.6.2) that, if x is a topological zero divisor in a commutative Banach algebra A with identity, then x is singular. For the special algebra C(X), X being a compact Hausdorff topological space, we sa\.; in Theorem 1.6.1 that f E C(X) is a topological zero divisor if and only if f is singular. For an arbitrary commutative Banach algebra A with identity it need not be the case that a singular element is a topological zero divisor, but if A is self-adjoint and semisimple, then it is true. We shall prove this in the theorem of this sectioR. Before we can do this, however, we need some preliminary results about self-adjoint semisimple commutative Banach algebras. Suppose A is such an algebra. Given x € A, we know that there .. .. exists some yEA such that y = x. Since A is semisimple, this element y is clearly unique. ~hese observations lead to the following definition: Definition 5.4.1. Let A be a self-adjoint semisimple commutative Banach algebra. If x ~ A, then we denote by x* the uni... que element of A such that x* = x.

.

5. Semisimp1e Commutative Banach Algebras

140

We shall need the next proposition. Proposition 5.4.1. Let A be a self-adjoint semisimple commutative Banach algebra. Then (i) x,y (A

(x + y)* and

a €

= x*

+

y*, (xy)*

= x*y*,

and

(ax)* = ax*,

~.

(ii) The mapping * : A - A, defined by * : x - x*, x E A, is an antilinear topological isomorphism of A onto itself. Proof. Part (i) and the fact that linear isomorphism are easily verified.

* is a surjective antiWe shall prove explicitly

only that * is a topological isomorphism. To this end we introduce a new norm in A by defining IIxlll = IIx*lI, x (A. It is a routine exercise to check that 11'11 1 is indeed a norm. _For example,

•

if IIxlll = 0, then IIx*1I = 0, and so x* = O. Hence x = 0, which • implies that X = 0, and so x = 0, as A is semisimple. Moreover, 11'11 1 is a complete norm on A. Indeed, suppose {xnJ is a Cauchy sequence in (A, U·I;l)' Then evidently {x*) is a Cauchy sequence in (A,II'II), and so there n

exists some yEA such that x = y*, we see that limllxn - xIII n

lim IIx* - yll n

n

= limllx*n

= O.

But then, if

x"l1

n

= limllx*n

- yll

n

=

Thus

(A,II'II I )

o.

is a Banach space.

Consequently we see that (A, Ii '11 1) and (A,II'II) are both commutative Banach algebras and (A, Ii 'Ii) is semisimple, whence we conclude, by Corollary 5.3.1, that 11'11 1 and 11·11 are equivalent norms. Thus, in particular, there exist Kl > 0 and K2 > 0 such that K2lixli < IIx* II ~ Klllxll, x E A, and so * is a topological isomorphism. c

5.4. Characterization of Singular Elements

141

One further observation is in order before we can state and prove the indicated theorem. If A is a commutative Banach algebra, then x ~ A is a topological zero divisor if and only if there {y} C A such that inf lIy II > 0 and n n n The proof of this assertion is left to the reader.

exists a sequence lim lixy II = O. n n

Theorem 5.4.1. Let A be a self-adjoint semisimple commutative Banach algebra with identity e. If x E A, then the following are equivalent: (i)

x


(ii)

x

is singular.

Proof. The implication from part (i) to part (ii) is contained in Proposition 1.6.2. Conversely, suppose x E A is singular. A

Then _0 ( o(x) = R(x),

by Theorem 3.4.1.

Set

y

= xx*.

Clearly

1 = ~i = '~l2,

and so 0 C R(y) and R(y) C [O,~). Thus we see that -lIn ~ oCy), n = 1,2,3, ... , so that y + (l/n)e is regular, n = 1,2,3,.... Since lim liy + (l/n)e - yli = 0, we conclude that n y (bdy(A- 1 ), and so, by Corollary 1.6.1, y is a topological zero divisor. infn Ilyn II > 0

Hence there exists some sequence and

1imn lIyyn II

= 1imn Iixx*yn II = O.

(y) n

C

A such that

If inf IIx*y II > 0, then clearly x is a topological zero n n divisor. On the other hand, suppose inf lix*y II = O. Then there n

n

exists a subsequence (y ) of (y) such that limkllx*y II = O. nk n nk However, in view of Proposition 5.4.1, there exist Kl > 0 and K2 > 0 such that K2lizll ~ IIz*1I < KlllzlI, z E A. see that lim.!Ixy* II = O. Furthermore, k. nk

Thus, once again, we see that

x

Consequently, we


U

CHAPTER 6 ANALYTIC FUNCTIONS AND BANACH ALGEBRAS 6.0. Introduction. Let A be a commutative Banach algebra with identity and suppose p is a polynomial. Then it is obvious that p(x) makes sense for any x E A and belongs to A. Moreover, the Polynomial Spectral Mapping Theorem (Theorem 2.2.1) entails • • that p(x) (T) = p[X(T)], T E ~(A). Our main concern in this chapter will be to show that such a phenomenon is valid for a much larger class of functions than polynomials. Thus, we shall see that, if x E A and f is analytic in some open set containing a(x), • • then there exists some y = f(x) E A such that f(x) (T) = f[x(T)], T e 6(A). The discussion and proof of this result will take up Section 6.1. The topics in the succeeding sections are essentially applications of the results of Section 6.1. Besides a generalization of the Polynomial Spectral Mapping Theorem, we shall establish a sufficient condition for a commutative Banach algebra with compact maximal ideal space to have an identity; prove a generalization of Wiener's Theorem (Theorem 4.6.2); show that, if f is a nonconstant entire function and x E A is such that f(x) = 0, then there exists a nonconstant polynomial p such that p(x) = 0; and give a description of the connected component of the identity in the set of regular elements in A. 6.1. Analytic Functions of Banach Algebra Elements. Let A be a commutative Banach algebra with identity, let x·e A, and suppose f is a complex-valued function defined and analytic on some open set 0 ~ a(x). Our main goal in this section will be to

142

6.1. Analytic Functions

143

show how to define f(x) as an element of A such that A A f(x) (T) = f[x(T)], T E 6(A). We shall accomplish this through a series of lemmas culminating in the desired result. Afterward we shall briefly discuss, without proof, some generalizations of the development. To begin we first need a number of elementary definitions.

a

~

Definition 6.1.1. A £omplex-valued function t ~ b, is said to be a smooth arc if (i) (ii)

dC/dt exists and is continuous on d~/dt

and dll/dt

Crt) =

~(t) + i~(t),

[a,b].

have no common zero on

[a,b].

Furthermore, C is said to be a regular curve if there exists a = to < tl < ... < t n _l < tn = b such that C restricted to [tk,t k+ l ] is a smooth arc, k = O,I,2, ••. ,n - 1. In the definition of a smooth arc we mean, of course, that the appropriate one-sided derivatives exist at t = a and t = b. Definition 6.1.2. Let C{t), a < t ~ b, be a regular curve. Then C is said to be simple if there exist no points a ~ tl < t z < b such that ,(tl) = '(t 2), and , is said to be a simple closed regular curve if C(a) = C(b) and there exist no points a < tl < t2 < b such that ,(tl) = C(t 2). The disjoint union of a finite number of simple closed regular curves is said to be a regular contour. Thus a simple regular curve , does not cross itself, and a simple closed regular curve meets itself only at its end points. Now suppose A is a commutative Banach algebra with identity and let x (A. Then o(x) is a nonempty compact subset of C, which may, however, consist of several disjoint pieces and whose complement need not be connected. Nevertheless, if 0 C C is an open set such that 0 ~ a(x), then it is not difficult to verify

144

6. Analytic Functions and Banach Algebras

that there exists a regular contour y

that satisfies the following

properties, where n(Y,a) denotes the winding number of the regular contour y with respect to the complex number a [A, pp. 114-117]: (1)

yeO - o(x).

If a E o(x), then n(Y,a) > 0; inside the regular contour y. (2)

(3)

If

a E o(x),

then

n(Y,a)

= 1;

that is,

o(x)

lies

that is, each point

a E o(X) lies inside precisely one of the simple closed regular curves that make up y. (4)

a

E~ -

If a ~ 0, then n(y,a) = 0; that is, no point 0 lies inside of the regular contour y.

Definition 6.1.3.

Let

A be a commutative Banach algebra

with identity, x E A, and suppose 0 ~ o(x) is an open set. If Y is a regular contour that satisfies properties (1) through (4) , then Y is said to be a spectral contour for

a(x)

lying in

O.

application of the Cauchy Integral Formula [A, p. 119] immediately establishes the first lemma. An

Lemma 6. 1. 1.

Let

A be a commutative Banach algebra with

identity and let

x C A.

o~ a(x)

is any spectral contour for

and

y

If f

is analytic in some open set a(x)

lying in

0,

then f(a) -- _1_ '" , f(,) d'" 2ni 'y - a ~

(a E a(x)).

We wish Rext to extend this result so that we can replace a in the above equation by x. To do this we clearly need to define the integral

where e denotes the identity of A,

and show that it belongs to

A.

6.1.Analytic Functions

145

With this in mind suppose A is a commutative Banach algebra with identity e, x E A, O~ a(x) is open, and f is analytic on 0, and let y be a spectral contour for a(x) lying in O. To make the argument a bit simpler we shall assume for the moment that is a simple closed regular curve then C ~ a(x), and so (Ce - x)-l

y

mapping sion in

C = C(t), a o(x). In view of Theorem 3.4.1 we see that, if r > !lxII, then the circle y = (C I Ici = r) is a spectral contour for o(x). Moreover, if C (V, then, by Corollary 1.4.3 and the comments following it or by direct computation, we see that CD

= ,-lee _ X)-l = ,-1 1:: (x)n C n=O ,

(Ce - x)-l

and that the convergence is absolute and uniform in norm on V. Hence we deduce that 1 f(x) = 2ni

.

.I y

f(C)(Ce - x)

[~J fCC) dC]x n n=O 2n1 V Cn+ l

=

f(n) (0) I n= O n.

1:

1:

CD

=

dC

CD

=

n

CD

\

-1

X

n

a x , n=O n 1:

by an application of the Cauchy Integral Formula [A, p. 120].

c

It is apparent from the definition of f(x) that the mapping from f - f(x) is linear. However, more can be said, as seen in the next lemma.

6.1. Analytic Functions

149

Lenuna 6. 1. 5 . Let A be a conunutative Banach algebra with identity e and let x E A. If f and g are analytic on an open set O:Ja(x), then (i) (ii) (iii)

(f

+

= f(x)

g) (x)

= af(x),

(af) (x)

+

a

g (x). E; \l;.

(fg) (x) = f (x) g (x) .

Proof. We need only prove part (iii). Let VI and Y2 be distinct spectral contours for a(x) lying in 0 such that YI lies inside V2 , that is, such that n(v 2 ,C) > 0, , E VI' We note that, if z (Y 2 and C E Yl , then (Ce - x)-l(ze - x)-l(z - C)

= (Ce

- x)-l(ze - x)-l[(ze - x) - (Ce - x)]

= (Ce - x)

-1

- (ze -x)

-1

.

Using this observation we see that I. f(x)g(x) = [-2

J'

TTl

= (2;i)2

SY

I

1

x) -1 dz]

SV l JV2 f(C)g(z)CCe - x)-lCze - x)-1 dz dC l

- 2ni

&

1

= (2ni)2 1 J'Y JY2 1 -- 2TTi

rY2 g(z) (ze -

1 f(C) (Ce - x) -1 dC] [2n'

YI

f

-1

f(C) ( l[CCe - xl - (ze - x) g z z - ,

-1

] dz dC

1 J~ (C)(Ce - x) -1[ 2ni Y2 zg(z) _ C d z ] dC

JY

2 g(z)(ze - x)

-1

J

1 f(C) [2ni Yl z _ CdC] dz.

Now g is analytic on and inside Y2 , so for each by the Cauchy Integral Formula [A, p. 119], gCC)

C E Y1 we have,

=~ J g(z) dz 2TTI Y2 z - C

as Y l lies inside Y2' On the other hand, f(C)/(z - ') is analytic on and inside VI as a function of C for each z E Y2' whence


150

we conclude, from the Cauchy Integral Theorem [A, p. 145], that for each

z E Y2

1 j. f(C) dC = 0 2ni Y1 z - C •

Consequently we have f(x)g(x)

I = 2ni

SY

f(C)g(C)(Ce - x) -1 dC = (fg)(x),

1

and the lemma is proved.

0

The reader should note the crucial, if tacit, role plated by Lemma 6.1.3 in the proof of the preceding result. We can summarize the preceding development in Theorem 6.1.1. The proof of part (iii) is left to the reader. Definition 6.1.4. Let A be a commutative Banach algebra with identity and let x € A. The collection of all complex-valued functions f that are defined and analytic on some open set o ~ a(x) will be denoted by A(x). The open set 0 in the preceding definition may, of course, depend on f. Evidently A(x) is an algebra under pointwise operations of addition, scalar multiplication, and multiplication. Theorem 6.1.1. Let A be a commutative Banach algebra with identity and let x E A. Then the mapping f - f(x), f E A(x), is an algebra homomorphism from A(x) into A such that (i)

If

f E A(x) ,

then

•

•

f(x) (T) = f[X(T)], T E ~(A).

(ii) If f(C) = ~=oanCn, C E~, is an entire function, then f(x) = ~=oanxn, and the convergence is absolute in norm. (iii)

Suppose

(f) c A(x) n

and

f E A(x)

are analytic on an open

set 0 ~ a(x). If the sequence (f) converges uniformly to n each compact subset of 0, then lim IIf (x) - f(x)1I = o. n

n

f

on

6.2. Consequences of the Preceding Section

151

Portions of this theorem can be extended to the case where a(x) is replaced by the joint spectrum O(x l .x 2••••• xn) of finitely many elements of A. We shall not give a proof of this theorem since it involves the techniques of the theory of functions of several complex variables, and a complete discussion would necessitate a rather lengthy digression. We shall, however, state the indicated result. Proofs are available. for example, in [HOI' pp. 101-114; Hr, pp. 68-70; Ri, pp. 156-162; S. pp. 60-72]. Theorem 6.1.2 (~ilov-Arens-Calderon). Let A be a commutative Banach algebra with identity and let x l ,x 2 ' .•.• xn be in A. If f is a function of n complex variables that is defined and analytic on some open set 0 ~ a(x l 'x 2 ' ..•• x). then there exists some • •• n. yEA such that yeT) = f[x 1 (T),X 2 (T) •••.• Xn (T)], T E A(A). Theorem 6.1.1 also gives a partial answer to a more general question. If A is a commutative Banach algebra and f is defined • on some open set 0 c~, then we say that f operates in A if • • • fox E A whenever x E A and R(x) c O. Theorem 6.1.1 then asserts • that functions that are analytic on open sets in ~ operate in A for any commutative Banach algebra A with identity. On the other hand, if A = C(X), X being a compact Hausdorff topological space, then it is easily verified that any continuous function defined on • an open subset of ~ operates in A. The question of determining • for a particular algebra in A and we shall not pursue it here. to [Ka, pp. 235-249; Ri. p. 167;

precisely which functions operate A is in general a difficult one, The interested reader is referred Ru l , pp. 137-155].

6.2. ~ Consequences of the Preceding Section. In this section we wish to establish a number of results that are simple consequences of the development in the preceding section. In particular we shall obtain generalizations of the Polynomial Spectral Mapping Theorem (Theorem 2.2.1) and Wiener's Theorem (Theorem 4.6.2).

152


Theorem 6.2.1 (Spectral Mapping Theorem). Let A be a commutative Banach algebra, let x E A, and suppose f is a complexvalued function that is defined and analytic on some open set O~ a(x). Then (i)

If

A has an identity, then

f[a(x)]

= a(f(x».

(ii) If A is without identity and f(O) = 0, then there exists some f(x) E A such that f[a(x)] = a(f(x». Proof. Part (i) follows immediately from Theorem 6.l.l(i) on • and a(f(x» = R[f(x)] • sin~e A has recalling that a(x) = R(x) an identity. To establish part (ii) we need first to prove the existence of a suitable element f(x) E A. The development of the preceding section is not directly applicable, as A is without identity. However, by Theorem 2.1.1, we know that a(x) = aA(x) = aA[e] (x), so we can apply Theorem 6.1.1 to the algebra A[e] to deduce the existence • of an f(x) E A[e] such that f(x) • (T) = f[X(T)], T E ~(A[e]). However, if TCD E ~(A[e]) denotes,as usual, the unique complex homomorphism on A[e] that vanishes identically on A, we see that • ) ] = f(O) = 0 by the assumptions on f. Hence f(x) • (TCD) = f[X(T CD f(x) E A.

• The definition of f(x) also reveals that f(x) • (T) = f[x(T)], T E ~(A), which, combined with the facts that f(O) = 0, • • a(x) = R(x) U {oj, and a(f(x» = R[f(x) ] U {oj, implies that f[a(x)] = a(f(x». o A similar result is the following corollary: Corollary 6.2.1. Let suppose ~(A) is compact. and f is a complex-valued • on some open set 0 ~ R(x), •yeT) = f[X(T)], • T E A(A).

A be a commutative Banach algebra and If x E A is such that •X(T) ~ 0, T E ~(A), function that is defined and analytic then there exists some yEA such that

6.2. Consequences of the Preceding Section

153

Proof. If A has an identity, then ~(A) is automatically compact, and the corollary follows at once from either Theorem 6.l.l(i) ,. or 6.2.1. So assume that A is without identity. Since x is ,. continuous, we see that R(x) is a compact subset of ~ that does not contain zero. Thus there evidently exist open sets 01 and 02 in q; such that

,. (a)

R(x)

C

(b)

0 €

02.

(c)

01 n 02

01

C

0.

= ~.

Having chosen such sets, define g(C)

= f(C)

g(C) = 0 for

for

C ~ 01'

C E 02.

Clearly g is analytic on the open set

01 U 02

~

a(x),

and

g(O) = O.

Consequently, by Theorem 6.2.l(ii), there exists some y = g(x) ,. ,. in A such that yeT) = g[X(T)), T E ~(A), from which it follows ,.,. ,. at once that yeT) = f[X(T)), T E ~(A), as R(x) cOl· 0 As we know from Theorem 3.2.2, if A is a commutative Banach algebra with identity, then ~(A) is compact. It is natural to ask ~hether or not the converse assertion is valid. Some support for such a conjecture is given by the next result. CorGllary 6.2.2. Let A be a semisimple commutative Banach algebra. If ~(A) is compact and there exists some x E A such ,. that X(T) ~ 0, T E A(A), then A has an identity.

,.

Proof. Since R(x) is a compact set in ~ that does not con" such that 0 ~ o. tain zero, there exists an open set ~ R(x) Then f(,) = I'" , E 0, is analytic on 0, and so, by Corollary

°

154


• • • 6.2.1, there exists some yEA such that yeT) = f[X(T)] = l/X{T). T ~ A(A). It is then immediate from the semisimplicity of A that xy is an identity for A. o

Actually. one can prove that, if A is a semisimple commutative Banach algebra and A(A) is compact, then A has an identity. The proof. however, requires the use of the theory of functions of several complex variables and will not be discussed here. The interested reader is referred to [Ri, pp. 167-169; 5, pp. 75-77]. An appropriate analog of the §ilov-Arens-Calderon Theorem (Theorem 6.1.2) for algebras without identity and the §ilov Idempotent Theorem [Ri, p. 168; 5, p. 73] are the main tools needed to establish the indicated result. The semisimplicity of A is necessary for the validity of the result. Another easy application of Theorem 6.1.1 yields the following generalization of Wiener's Theorem (Theorem 4.6.2): "t

Theorem 6.2.2 (Wiener-Levy Theorem). Let f = (e l I -n < t < n). If h E AC(f) and f is a complex-valued function defined and analytic on some open set 0 ~ R(h), then there exists some g € AC(f) such that g = f 0 h. Proof. The result follows immediately from Theorem 6.I.l(i) on recalling that the Gel'fand transformation on AC(f) is the identity mapping of AC(f) into C{f) (Theorem 4.6.1). Wiener's Theorem is, of course, the special case of this result it where h(e ) ~ 0, -n ~ t < n. and f(C) = lIe. , E ~. 6.3. Zeros of Entire Functions. As another application of the results of Section 6.1 we shall show that, if x is an element of a commutative Banach algebra A with identity and f is a nonconstant entire function such that f{x) = 0, then there exist some nonnegative integer nand bk €~, k = 0,l,2, ... ,n such that

6.4. Connected Component of the Identity tk=Obkxk = 0,

155

where not all of the bk are zero.

Theorem 6.3.1. Let A be a commutative Banach algebra with identity and let x (A. If there exists a nonconstant entire function f such that f(x) = 0, then there exists a nonconstant polynomial p such that p(x) = o. Proof. From Theorem 6.1.1 we know that f(x) E A for any • • entire function f and that f(x) (T) = f[x(T)], T E ~(A). Hence, • since f(x) = 0 by assumption, we see that f[x{T)] = 0, T E A(A); that is, f vanishes identically on a(x). Since a(x) is compact and the zeros of a nonconstant entire function are isolated [A, p. 127], we conclude that a(x) is finite, say

Let mk denote the multiplicity of the zero of f k = 1,2, .•• ,r, and define

at

~,

r

p(C) =

n

(C - a )mk. k=l k

Then, as is easily verified, the function g defined by gCC) = fCC)/pCC), C (C, is an entire function that never vanishes • on o(x); that is, g[X(T)] ~ 0, T ~ 6(A). However, by Theorem 6.1.1 and Corollary 3.4.2(ii), we see that g(x) (A is regular and f{x) = p{x)g{x) = O. Therefore p(x) = 0. 0 6.4. The Connected Component of the Identity in A-I. If A is a commutative Banach algebra with identity e, then we have seen previously that A-1 , the set of regular elements in A, is an open subgroup of A. We now wish to show, at least in the case that A is semisimple, that the connected component of e in A-I is precisely the set of those elements of A of the form ~=oxn/n!, x E A. We recall that the connected component of point in a topological space is the largest connected set that contains the given point.


156

In what follows, exp will denote the entire function m n exp(C) = ~n=O' In!, C (Cj that is, exp is the complex exponential function. We need two lemmas before the indicated theorem. The proof of the first lemma is left to the reader. Lemma 6.4.1. Let A be a semisimple commutative Banach algebra with identity. If x,y € A, then (i) (1• 1· )

(iii)

exp (x) € A. w» n l n .. I exp ( x) -- Ln=OX

exp(x

+

y) = exp(x)exp(y).

Lemma 6.4.2. Let A be a semisimple commutative Banach algebra with identity e. If x E A and lie - xII < 1, then there exists some y (A such that exp(y) = x. Proof. Since lie - xII < 1, we see from Theorem 1.4.1 that • is a compact x is regular, whence, by Corollary 3.4.2{ii), R(x) set that does not contain zero. Let 0 be an open subset of C • such that 0 ~ R(x) and 0 ~ O. Then fCC) = 10gC, C € 0, is analytic on 0, and so, by Theorem 6.1.1, y = f(x) E A and yeT) = log X(T), T (6(A). But exp{y) E A and

..

.

• exp{y) • (T) = exp[Y(T)] = exp[log • X(T)] • = X(T) from which we conclude that exp(y)

= x,

as

(T

E 6(A»,

A is semisimple.

o

We can now state and prove the desired theorem. Theorem 6.4.1. Let A be a semisimple commutative Banach algebra with identity e and let exp(A) = (exp(x) I x € Al. Then exp(A) is the connected component of e in A-1 . Proof. From Lemma 6.4.1 we see that exp(x)exp(-x) = exp{O) = e, -1 x ~~ A, and so e ~~ exp(A) cA. Moreover, if x E A, t h en"1t lS easily verified that wet) = exp{tx), 0 < t < 1, defines a continuous

6.4. Connected Component of the Identity

157

curve in exp(A) from e to exp(x), from which it follows at once that exp(A) is connected. Thus exp(A) is contained in the connected component of e in A-1 . To show that exp(A) is this connected component it suffices to prove that exp(A) is both open and closed in A-I.

IIY -

To this end suppose y = exp(x) zll < lIlly-III. Then lie - y-l zl1

and let

= lIy-l y

z (A be such that

_ y-l zl1

~ lIy- 111l1y -

zll

< 1,

and so, by Lemma 6.4.2, there exists some w E A such that exp(w) = y-I z. Consequently

z whence

z

~

= y[exp(w)] = exp(x)exp(w) = exp(x

exp(A),

and

exp(A)

+

w),

is open.

On the other hand, suppose y E A-I and y E cl[exp(A)]. Then there exists some z = exp(x) such that lIy - zll < lIlly-III. Arguing as before, we conclude that y-I Z (exp(A), and so y-l E exp(A). Thus y E exp(A) and exp(A) is closed. Therefore exp(A)

is the connected component of

e

in A-1 0

Actually the theorem and the lemmas are valid in any commutative Banach algeb~a with identity. We have proved the results only for semisimple algebras because the arguments in Lemmas 6.4.1(iii) and 6.4.2 are less technical in this case. The lemmas can be established in the general case via a power series argument. See, for example, [8, pp. 49 and 50]. We shall not pursue the discussion of A-I and exp(A) any further, but content ourselves with mentioning one of the most

158


important theorems on the subject. a subgroup of A-I.

Note that

exp(A)

is evidently

Theorem 6.4.2 (Arens-Royden). Let A be a commutative Banach algebra with identity. Then the quotient group A-1/exp(A) is isomorphic to HI[a(A),Z], the first ~ech cohomology group of a(A) with integer coefficients. Using this result one can prove, for instance, that A-1 is either connected or has an infinite number of connected components. Discussions of the Arens-Royden Theorem and some of its consequences can be found in [Ga, pp. 88-91; 5, pp. 98-104; Wm 2 , pp. 88-96]. More generally, other treatments of the role of analytic functions in the study of Banach algebras are available in [B, Ga, Ho l , Hr, Lb, Ri, 5, wm l , Wm 2].

CHAPTER 7 REGULAR COMMUTATIVE BANACH ALGEBRAS 7.0. Introduction. If X is a compact Hausdorff topological space, then it is well known that X is a completely regular topological space and a normal tOP91ogicai space. The first assertion is equivalent to saying that for each closed set E C X and each point t E X, t t E, there exists some f E C(X) such that < f(s) < 1, s ( X, fet) = 1, and fes) = 0, sEE. The second assertion, combined with Urysohn's Lemma, reveals that, if El C X and E2 C X are disjoint closed sets, then there exists some f E C(X) such that 0 ~ f(s) ~ 1, s ( X, f(s) = 1, s (E l , and f(s) = 0, s E E2 . Recalling that the Gel'fand representation of • = C(X), we see the commutative Banach algebra A = C(X) is just A • that these observations say precisely that A has the indicated properties. In particular, if A = C(X), X being a compact Haus• separates disjoint closed subsets dorff topological space, then A of A(A). Our goal in this chapter is to determine sufficient conditions on an arbitrary commutative Banach algebra A such that the • possess separation properties analogous to those elements of A described here.

°

In order to accomplish this we shall introduce a new topology on ~(A), the hull-kernel topology, and show that, when this topology coincides with the Gel'fand topology on A(A) , then, whenever E C A(A) is closed and T E A(A), T ~ E, there exists some x E A • • such that X(T) = 1 and xew) = 0, wEE. Furthermore, in this case, we shall also see that, if K C A(A) is compact, E C A(A) is closed, and

• X(T) = 1,

T E K,

KnE and

=~,

•X(T)

then there exists some

= 0,

T

~

159

E.

x

E A such that

7. Regular Commutative Banach Algebras

160

A commutative Banach algebra A that satisfies either of the foregoing separation properties will be termed regular or normal, respectively.

The hull-kernel topology and regular commutative

Banach algebras play a central role in the study of the ideal structure of commutative Banach algebras, as we will see in the next chapter. 7.1. The Hull-Kernel Topology and Regular Commutative Banach Algebras. The main concerns of this section will be to define and investigate the elementary properties of the hull-kernel topology on A(A) and to determine when this topology coincides with the Gel'fand topology. We begin with a number of preliminary definitions.

EC

Definition 7.1.1. Let A be a commutative Banach algebra. If A(A) , then the kernel of E, denoted by keEl, is defined as k(l:)

= n

M

=

MEE if E ~~, while k(~) = A. If I C A is an ideal, then the hull of I, denoted by h(I), is defined as h(l)

= [M I M £ A(A), = (T I

For each x E A, as

M~ IJ

T E A(A), T-I(O) ~ IJ.

the

~

set of x,

denoted by Z(x),

Z(x)

= {T

T E. A(A), X(T)

= {M I

ME A(A),

..

x

whereas if I C A is an ideal, then the by Z(I), is defined as Z(I) =

fl Z(x) = {T

= 0)

EM), ~

set of

..

I,

T (A(A), X(T) = 0, x ( I)

xEI

= {M

is defined

M E a(A), I eM).

denoted

7.1. Hull-Kernel Topology

161

It is evident from the definitions and the continuity of x• that Z(x) and Z(I) are closed subsets of ~(A), and Z(I) = h(I). Moreover, we have the following proposition: Proposition 7.1.1. (i)

If I

(ii)

heAl

(iv)

If E C

keEl

= (x I

Let A be a commutative Banach algebra.

A is an ideal, then h(l)

C

=~

and h«(O)

~(A),

= h[cl{I)].

= ~(A).

then

..

= {x I

x € A, X(T) = 0, TEE) then keEl

x € A, Z(x) ~

(v)

If E C

~(A),

(vi)

k[~(A)]

= Rad(A).

(vii)

If I

C

A is an ideal, then cl(I)

(viii) h = hkh.

If I

C

A is an ideal, then h{I) = h(k[h(I)]);

is a closed ideal in A.

C

k[h{I»).

(ix)

If E C

~(A),

then keEl = k(h[k(E)]);

(x)

If E C

~(A),

then h[k(E)]

(xi) If El C ~(A), E2 and h[k(E l )] C h[k(E 2)]. (xii)

If El

C ~(A)

C ~(A),

and E2 c

El.

~

k

= khk.

E.

and El

~(A),

that is,

that is,

C

E2,

then k(E l )::> k (E 2)

then h[k(E l ) n k(E 2)]

=

h [k (E l U E2 )].

Proof. All of the proofs are rather elementary. only one proof and leave the rest to the reader. Suppose h (k [h (I) ] ) •

We shall give

I C ~(A) is an ideal. We shall show that h(l) = If M E h(k[h(I»)), then M~ k[h(I)] ~ I by part (vii),


162

whence M E h(I). Conversely, suppose T ~ h(k[h(I»)) = Z(k[h(I»)). " Then there exists some x E k[h(I)] such that X(T) ~ 0, while A from part (iv) we see that x(w) = 0» W E hel). Hence T ~ h(I)" and so h(l) = h(k[h(I)]), which proves part (viii).

o

As the reader should guess from the preceding proof, in discussing hulls and kernels it is often of considerable advantage to keep in mind the descriptions of these objects both in terms of maximal regular ideals and in terms of complex homomorphisms. The comment preceding Proposition 7.1.1 shows that h(l) is a closed subset of 6(A) for each ideal leA. The idea behind the hull-kernel topology is to use such closed sets, that is, hulls of ideals, as the closed sets in a topology. With this in mind we make the following definition: Definition 7.1.2. Let A be a commutative Banach algebra. If E c 6(A) , then the hull-kernel closure of E, denoted by E, is defined to be E = h[k(E)]. The reader should recall that the closure of E in the Gel'fand topology is denoted, as usual, by cl(E). In order to show that the hull-kernel closures of sets in 6(A) are actually the closed sets for some topology on 6(A) we must show that the correspondence E - E, E C 6(A) , is a closure operation [DS l , pp. 10 and 11]; that is, we must show that the operation of forming the hull-kernel closure satisfies the following conditions:

= 6(A).

(a)

6(A)

(b)

E

(c)

E = E.

Cd)

(E l U E2 )-

C

E.

= El

U E2 •

7.1. Hull-Kernel Topology

163

Evidently conditions (a) and (b) hold by Proposition 7.1.I(vi) and (x), respectively, and

E = h[k(h[k(E)])] = h[k(E)] = E, by Proposition 7.1.1(ix). So suppose EI we see that

C

A(A)

and

E2

C

A(A).

From Proposition 7.1.I(xi)

(k = 1,2),

whence El U E2 C (E I U E2)-. Conversely, suppose M belongs to and let u be an identity modulo M. Clearly ~(A) - (E I U E2) k(E k ) ~ M, k = 1,2, and u ~ M, 9y Proposition l.l.l(i). Thus, since M is the null space of a nonzero continuous linear functional and hence is of codimension one [L, p. 68], we see that there exist v,w ~ M, x e keel)' and y e k(E 2) such that u = x + v = y + w. 2 2 However, u - xy = u - (u - v,(u - w) = (v + w)u - VW, which 2 shows that u - xy € M, as M is an ideal, while xy E keEl) nk(E 2). Now, if M E (E I U E2) = h[k(E l U E2)], then, from Proposition 7.1.1(xii) we see that M € h[k(E 1) n k(E 2)]. Thus, in this case, we conclude that xy € M, whence u2 E M, as u 2 - xy € M. However, since u is an identity modulo M, we have u2 - u € M, and so u E M, which is a contradiction. Therefore M ~ (E I U E2) and (E I U E2)- = El U E2 . Thus we see that the correspondence E - E, E C ~(A), is a closure operation and so it can be used to define a topology on

~(A).

Definition 7.1.3. Let A be a commutative Banach algebra. Then the topology on ~(A) determined by the closure operation E - E = h[k(E)], E C ~(A), is called the hull-kernel topology. The relation of the hull-kernel topology to the Gel'fand topo109)' on ~(A) is contained in the next theorem. We recall that a topology is said to be Tl if every singleton set is closed.


164

Theorem 7.1.1. Let A be a commutative Banach algebra. Then the hull-kernel topology on A(A) is a Tl topology which is weaker than the Gel'fand topology. Moreover, if A has an identity, then A(A) in the hull-kernel topology is compact. Proof. Since h[k([M))] = (M), M (A(A), and h[k(E)], E C A(A), is always closed in the Gel'fand topology, we see at once that the hull-kernel topology is Tl and weaker than the Gel'fand topology. A bit more work is required to show that 6(A) is compact in the hull-kernel topology when A has an identity. If A has an identity e, then in order to show that deAl is compact in the hull-kernel topology it suffices to prove that, if (E) C A(A) is a family of hull-kernel closed sets with the a finite intersection property, then n E #~. Evidently, if (E) aa a is such a family of sets, then E # ~ and k(E) is a proper a a closed ideal in A. Let I denote the closed ideal in A generated by U k(E); that is, I is the norm closure of the linear subspace a a in A generated by Uak(Ea'. It is easily seen that I is a closed ideal, and, moreover, I is proper. Indeed, if I = A, then there would exist a l ,a2 , ••. ,a and xk (k(E ), k = 1,2, ... ,n, such ~

n

~

that lie -1:k=lxkl: < 1. Hence, by Theorem 1.4.1, ~=lxk is regular. Let y be the inverse of Ik=IX k . Clearly e = Ik=IXkY and xky E k(E Ok ), k = 1,2, ... ,n. Thus we see at once that the norm closed ideal J generated by U~=lk(EOk) is all of A, as e € J. We claim that ~=lh[k(Eax)] =~. If this were not so, there would exist some M E A(A) such that M~ k(E ), k = 1,2, ... ,n, and ~ hence M~ J = A. This, ho~ever, contradicts the properness of M. Thus (~=lh[k(ECZk)] = D, whence r~=IECZk = D since EG\ = EOk = h[k(E )], k = 1,2, ... ,n. The latter assertion is contrary to the ~ assumption that (E) has the finite intersection property, and so a we conclude that I is proper.

M~

Hence, by Theorem 1.1.3, there exists some M E A(A) such that I ~ k(Ea' for each a. Consequently from Proposition 7.1.1(iii)

7.1. Hull-Kernel Topology we see that M= heM) c Therefore a (A)

nah[k(Ea)]

165

=

naa E,

whence

naa E

~~.

is compact in the hull-kernel topology.

o

The next most obvious question to raise is: ~~en do the hullkernel and Gel'fand topologies on ~CA) coincide? From Theorem 7.1.1 and an argument used several times previously we see that, if A has an identity, then the topologies will coincide precisely when the hull-kernel topology is Hausdorff. This observation is actually true in general, as is shown by the next result. Theorem 7.1.2. Let A be a commutative Banach algebra. the following are equivalent:

Then

(i) The hull-kernel topology and the Gel'fand topology on a(A) coincide. (ii)

The hull-kernel topology on a(A)

is Hausdorff.

(iii) If E c a(A) is closed in the Gel'faBd topology and A T € a(A), T ~ E, then there exists some x E A such that X(T) = I A and x(w) = 0, wEE. Proof. Obviously part (i) implies part (ii), and from the remark preceding the theorem we see that part (ii) implies part (i) if A has an identity. If A is without identity, then consider the algebra A[e]. By the comment following Definition 3.2.2 and Theorem 3.2.2(ii) we have a(A[e]) = a(A) U (T), where TCD E a(A[e]) co is the unique complex homomorphism of A[e] such that Tco (x) = 0, X (A. Denoting hulls and kernels computed with respect to aCA[e]) by hand k, respectively, it is easily verified that e

e

h [k (E)]

e

e

= h(k[E n 6(A)])

U (T co)

(E

c:

a (A [ e ]) ) .

From this it is apparent that the hull-kernel topology on a(A) coincides with the relative hull-kernel topology inherited from A(A[e]), and the same is true of the Gel'fand topology by Theorem

166


3.2.2(ii). Since the hull-kernel topology on A(A) is assumed to be Hausdorff, it follows that the hull-kernel topology on A(A[e]) is Hausdorff, whence, as before, the hull-kernel and Gel'fand topologies on A(A[e)) coincide. Hence these topologies on A(A) also coincide, and part (ii) implies part (i) in the case that A is without identity. Now suppose the hull-kernel and Gel'fand topologies on A(A) coincide and let E CA(A) be closed and T E A(A) - E. Since E = E = h[k(E)], we see that T l h[k(E)], and so there exists • some x E keEl such that X(T) ~ O. However, since x € keEl, • = 0, wEE, and part (i) implies part (iii). x(w) Conversely, suppose part (iii) holds. Thus, if E C A(A) is closed in the Gel'fand topology and T E A(A) - E, then there exists • • = 0, wEE. Hence some x E A such that X(T) = 1 and x(w) x E keEl and T ~ h[k(E)] = E, from which we deduce that, if T ~ E, then T ~ E; that is, if E is closed in the Gel'fand topology, then E is closed in the hull-kernel topology. Consequently the Gel'fand topology is weaker than the hull-kernel topology, which, combined with Theorem 7.1.1, shows that the two topologies coincide. Therefore part (iii) implies part (i), and the proof is complete. O Recalling the discussion in the introduction to this chapter • of a commutative Banach we see that the Gel'fand representation A algebra A separates points and closed sets precisely when the hull-kernel topology is Hausdorff. We wish to single out such algebras with a special name. Definition 7.1.4. Let A be a commutative Banach algebra. Then A is said to be regular if the hull-kernel topology on A(A) is Hausdorff. In view of the topological remarks in the introduction the reader may wonder why we do not call such Banach algebras completely

7.2. Some Examples

167

regular instead of regular. Actually this is done by some authors (see, for instance, [Ri, pp. 83-96]), but the term "regular" seems to be more widely used, and for this reason we prefer it. Before we show that the Gel'fand representation of a regular commutative Banach algebra also separates compact sets from closed sets, we shall consider the question of regularity for several specific algebras. 7.2. Some Examples. As mentioned in the introduction, every compact Hausdorff topological space X is normal, whence we see that the commutative Banach algebra C(X) is regular as the Gel'fand transformation on C(X) is just the identity mapping (Theorem 4.1.1 and Corollary 4.1.1). Similarly, if X ~s a locally compact Hausdorff topological space, then C (X) is a regular algebra. To see o this, in view of Theorem 7.1.2, we need only show that, if E C X = ~(Co(X)) is closed, then it is closed in the hull-kernel topology; that is, E = h[k(E]). Denoting the o~e-point compactification of X by X+, we know that C (X)[e] = C(X+), and so, by the regularity + 0 of C(X), we deduce that E

= cl(E)

=

cl (E) n X e

=

he[ke(E)] n X

=

(h[k(E)] U {Tm )) n X

= h [k (E)] ,

where cle(E) denotes the closure of E in X+. Thus Co (X) is regular. Using Theorem 7.1.2(iii), it is also easily seen that, if a,b ~~ a < b, then Cn([a,b]), n = 1,2,3, .•• , is regular. On the other hand, the commutative Banach algebra

A(D) is not regular. Indeed, consider the set E = {lIn I n = 2,3, ..• ). Evidently E C D and cl(E) = E U (oj. But recalling that the Gel'fand transformation on A(D) is again the identity mapping,


168

we see that keEl

= (f I =

f E A(D), f(l/n)

= 0,

n

= 2,3, •.• }

(0),

since the zeros in an open set of a nonconstant analytic function are isolated [A, p. 127]. Hence E = h[k(E)] = herO)) = A(A(D)) = D ~ cl(E). Thus the Gel'fand and hull-kernel topologies do not coincide for A(D), and so A(D) is not regular. As our final example we wish to show that LI(G), G being a locally compact Abelian topological group, is regular. However, this result is not as simply proved as the previous assertions. In order to do it we shall have to appeal to another fundamental theorem of harmonic analysis, Plancherel's Theorem, whose proof will not be given until Section 10.4. Before stating this result we wish to remind the reader of some of the discussion in Section 4.7. We saw there that the maximal ideal space A(L I (G)) is identifiable with the dual group G of G, that is, with the group of continuous homomorphisms of G into r = {, I , (Q;, 1,1 = I). Moreover, with the Gel'fand topology, G is even a locally compact Abelian topological group, so, in particular, we can speak of Haar measure ~ on G. Furthermore, the Gel'fand transform, which in this instance we refer to as the Fourier

..

..

..

transform, is defined by

..fey)

= SG

f(t) (t,y) dArt)

(y

.

E G; f E L1 (G)).

.

.

Plancherel's TheoTem assures us of the existence of a dense linear subspace Vo of L2 (G) such that Vo C LI(G) n L2 (G) , Vo C L2 (G), and on which the Fourier transform is an L2 isometry. More precisely we have the following theorem:

7.2. Some Examples

169

Theorem 7.2.1 (Plancherel's Theorem). Let G be a locally compact Abelian topological group and let A be a given Haar measure on G. Then there exists a Haar measure ~ on G and a linear subspace V of L2 (G) such that 0

.

(i)

Vo

(ii)

V

0

C

LI(G)

n L2 (G).

is norm dense in

L2 (G) .

is norm dense in

• L2 (G).

,. (iii)

V

0

,.

(iv)

IIfll2 = II f 1i 2 , f E V0 .

..

..

.

(v) The mapping f - f, f E Yo' from V0 to V0 can be uniquely extended to a linear isometry of L2 (G) onto L2 (G) . The extensiop of the Fourier transformation on Vo to all of L2 (G) will be called the Plancherel transformation, and we shall ,. denote the Plancherel transform of f once again by f. Due to the way the Plancherel transform is defined, it shares a number of the formal properties enjoyed by the Fourier transform. In particular, it is not difficult to show that, if f E L2 (G), then &,. ,.,. ,. fey) = f(-y) and [(.,w)f] (y) = fey - w), y,w E G, where, of course, the identities are interpreted to hold only almost everywhere with respect to 11. The details are left to the reader. As we shall subsequently see, the proof of Plancherel's Theorem is a nontrivial matter. However, in the case of compact Abelian topological groups the result is an easy consequence of our discussion in Section 4.7 and standard results of Hilbert space theory. Indeed, if G is compact Abelian and Haar measure A on G is normalized so that A(G) = 1, then from Corollary 4.7.4 we see that ,. G is a complete orthonormal set in the Hilbert space L2 (G). By ,. Corollary 4.7.5 the dual group G is discrete, and we may assume that Haar measure ~ on G is normalized so that ~((y}) = 1, Y (G. Then, since L2 (G) C LICG), we see that the Fourier transformation is defined on all of L2 (G), and, by a standard theorem

.

..


170

of Hilbert space theory [L, p. 405], we have IIfll2 = [ t.lf(Y)12]1/2 yEG =

[Sa

If(y)1 2 dTJ(y)]1/2

• = IIfll2 Moreover, the Riesz-Fischer Theorem [L, p. 404] shows that the mapping f - •f, f E L 2 (G), is surjective, and so Plancherel's Theorem is proved in the case of compact Abelian topological groups. Here, of course,

Vo

= L2 (G).

OUr concern now, however, is not to prove Plancherel's Theorem, but to use it. The first consequence of the theorem we wish to mention is the next corollary. Corollary 7.2.1. (Parseval's Formula). Let G be a locally compact Abelian topological group and let A be a given Haar measure • is so chosen that Plancherel's on G. If Haar measure ~ on G Theorem is valid, then

SG Proof.

f(t)g(t) dA(t) =

Sa

•f(y)g(y) • dTj(y)

Apply Plancherel's Theorem to the identity

o The next consequence we need -- one that is of considerable interest in its own right -- says that the Gel'fand representation of Ll(G) is precisely L2 (G) * L~CG); that is,. f E Ll(G) if and only if there exist g,h E L 2 (G) such that f = g * h. Note that the equality here holds pointwise since both f• and g * h belong to Co(G).

_.

..

-

Theorem 7.2.2. Let G be a locally compact Abelian topological group and let A be a given Haar measure on G. If Haar measure ~ • is so chosen that Plancherel's Theorem is valid, then on G • • • LICG) = L2 (6) * L2 (6).

7.2. Some Examples

171

Proof. If f E Ll(G), then we can write f = glh l , where gl,h l E L2 (G). For instance, define hI and gl almost everywhere as hl(t)

= If(t)1 1/ 2 ,

= e-iargf(t)lf(t)ll/2.

gl(t)

Let g and h

A

in L2 (G) denote the Plancherel transforms of gl and hI' respectively. Then, by Parseval's Formula and the observations following A Plancherel's Theorem, we have, for each w E G,

IG = IG

A

few) =

=

Ia

f(t) (t,w) 4A(t) gl{t)h1(t) (t,w) dArt) g(y)h(w - y) d~(y)

* hew).

= g

A

Conversely, suppose g,h € L2 (G) such that the Plancherel transform of is h. Evidently f = gl h1 E Ll(G), A ceding argument reveals that f = g * Therefore

LI(G)

..

A

=

and let gl,h 1 (L 2 (G) be gl is g and that of hI and a repetition of the preh.

A

L2 (G) * L2 (G).

The regularity of L1 (G) theorem.

o

will be a corollary of the next

Theorem 7.2.3. Let G be a locally compact Abelian topological group, let A be a given Haar measure on G, and suppose Haar A measure ~ on G is so chosen that Plancherel's Theorem is valid. A A If KeG is compact and U eGis measurable and such that o < ~(U) < GO, then there exists some f € L1 (G) such that (i) (iiJ (iii)

A

= 1,

fey)

Y E K.

A

fey) = 0, ylK+U A

o ~ fey)

A

~

1, Y E G.

- u.


172

..

Proof. Clearly Xu/ij(U) and XK_U belong to L2 (G) , where, as usual, XE denotes the characteristic function of E. Hence, by Theorem 7.2.2, there exists some f (LI(G) such that

.f = [Xu/Tj(U)]

* XK- U. Thus we see that • fey)

1 = ~(U)

=

Hence, if y E K,

faxK-u(y - w)Xu(w) d~(w)

~tU) Ju xK-U(y

then y - w

~

.. I fey) = Tl{U)

..

(y ~ G).

- w) dl1(w)

K - U, w E U,

J"U dTj(w)

and so

= I,

..

..

..

whereas, if Y ~ K + U - U, then y - w ~ K - U, w E U, whence fey) = O. Finally, it is obvious that 0 ~ fey) ~ 1, y E G.

.

..

o

Corollary 7.2.2. Let G be a locally compact Abelian topological group. If KeG is compact and WeGis open and such that W:J K, then there exists some f E LlCG) such that (i) (ii) (iii)

..fey)

..fey) 0

=

..

~ f

1, Y E K•

= 0, (y)

y

~

w.

..

5 1, Y £ G.

Proof. In view of Theorem 7.2.3 and the fact that the Haar measure of a nonempty open set is positive (Proposition 4.7.3), it suffices to find an open neighborhood U of y in G such that o .. K + U - U C W. As usual, y denotes the identity of G. To do o this we obviously may assume, without loss of generality, that Yo is in K. Moreover, we claim that it suffices to show only that there exists some open neighborhood V of y in G for which o

..

.

K + V C W.

Indeed, suppose such a neighborhood V exists. Then, by the continuity of the group operations, for each y E V there exists • such that U C V and an open neighborhood Uy of y0 in G y

7.2. Some Examples

173

Y + U c V. Set U = U (VU. Clearly U is an open neighborhood Y. Y Y of y in G and U C V. Furthermore, o

U C V = U (y yE.V whence we conclude that as we wished to show.

+

U) c V+ U U y y(V y

U - U C V.

Thus

= V + U,

K + U - U C K + V C W,

Hence it remains only to prove the existence of an appropriate neighborhood V. Again from the continuity of the group operations one deduces for each y E. K the existence of an open neighborhood • such that y + V + V C W. Evidently the family Vy of y0 in G y y {y + Vy lyE K} forms an open covering of K, and so, since K is compact, there exist v1.v 2•...• yn in K such that K c k l(Y k + V ). Let V = nP IVy' Then V is an open neighbor= Yk. °K= k hood of Yo in G, and if y E K, there exists some k, 1 ~ k < n, such that Y (Y k + VYk ' Thus

ur

Y + V C Yk

+

V Yk

from which we conclude that

+

V C Yk

K+ VC

+

w.

V Yk

+

V C W, Yk

o

Corollary 7.2.3. Let G be a locally compact Abelian topological group. Then Ll(G) is a regular commutative Banach algebra.

• • - E. Then K = {w} Proof. Suppose E eGis closed and w E. G is compact, and there exists an open neighborhood W of w such that Wn E = " as E is closed. By Corollary 7.2.2 there exists some f E LI(G) such that •few) = 1 and •fey) = 0, Y E E, whence, by Theorem 7.1.2, we conclude that L1(G) is regular. 0 For particular groups G it is possible to prove the regularity of LI(G) by direct arguments, thereby avoiding the use of Plancherel's Theorem. For example, suppose G = ~ If E c~ is closed l E. then let 0 6 > 0 be such that (t - 6,t + 6) n E = ,. and t0 0 Define h to be the continuous tent function


174

h{s)

1 = 2n

(o + s)

for

-6 < s < 0,

h{s)

1 = -2n

(o - s)

for

o
0,

and set feu) = e f E LlOR),

Then

it u 0

~

J-~

Hence

that

Ll (IR) 7.3.

ius

ds

f{t) = 6 ; 0 o is regular.

•

and

f{t)

= 0,

tEE,

Normal Commutative Banach Algebras.

(u

.

and direct computation reveals that

A

t E IR.

h{s)e

f{t)

E lR).

= h{t

- to)'

which shows

In this section we

shall show that the Gel'fand representation of a regular commutative Banach algebra A separates not only points and closed sets in ~(A) but also compact sets from closed sets. Before we can prove this result, however, we need to establish some facts about the Gel'fand representation theory of ideals and quotient algebras. Once again these results are of considerable interest by themselves. Theorem 7.3.1.

I

let

C

Let

A be a commutative Banach algebra and

A be a closed ideal. Then

(i)

a(l)

is homeomorphic to

a(A) - h{I).

o

(ii) If x E I and x denotes the Gel'fand transform of x o as an element of the commutative Banach algebra I, then x is the

· 0 to 6(A) - h(I); that is,· x = xI 6 {A)-h{I).

restriction of x o

o

(iii)

If.1

•

denotes the Gel'fand representation of

I,

then

I = Il~{A)_h(I). (iv)

6(A/I) If

(v)

A/I,

ex

+

then o

I)

•

is homeomorphic to o

h(I).

(x + I) denotes the Gel'fand transform of (x + I) o is the restriction of x- to hel);

= xih(I)'

x + I

in

that is,

7.3. Normal Algebras (vi) If o then (A/I) (vii)

A/I

(A/I)

•

o

175

denotes the Gel'fand representation of A/I,

= Alh(I)' is semisimple if and only if

I

= k[h(I)].

Proof. Suppose T E ~(A). Then it is evident that T restricted to I is either a complex homomorphism on I or is identically • zero. But T(X) = 0, x E I, if and only if X(T) = 0, x E I, that is, T E h(I). Thus every T E A(A) - h(l) defines a complex homomorphism on I via restriction to I. On the other hand, suppose T (A(I). Then there exists som~ y E I such that T(Y) = 1. Define w: A - C by w(x) = T(xy), x E A. Clearly w is linear, and if x,z E A, then w(xz)

= T(XZY) = T(XZY)T(Y) = T(XYZY) = T(xy)TCZY) = w(x)w(z),

whence w E ~(A). Obviously w(x) = T(X), x E I, and w ~ h(I). Moreover, it can be shown that w is unique: Suppose w' E A(A) is such that w'(x) = T(X), x E I. Then for any x E A we have w'(X)

= W'(X)T(y) = w'(x)w'(y) = w'(xy) = T(xy) = w(x),

and so w' = w. Thus we see that the restriction of T E A(A) - h(I) to I defines a bijective mapping between A(I) and A(A) - h(I). Furthermore, from the definition of this mapping it is apparent that

o

•

0

x = xIA(A) -h(I)' x E I, where x denotes the Gel'fand transform of x E I. Finally, it is now clear that the weakest topology on o ACI) that makes all of the functions x continuous is just the relative Gel'fand topology on A(A) - h(I), from which we conclude that A(I) and A(A) - h(l) are homeomorphic. This proves parts (i) through (iii). Next let ~: A - A/I denote the canonical homomorphism; that is, ~(x) = x + I, x E A. If T E ~(A/I), then it is evident that W =T 0 ~ is a complex homomorphism on A, where 0 denotes the usual composition of mappings. Moreover, if x E I, then


176

w(x)

=T 0

q>(X)

= T(X

+

I)

= T(I) = 0,

whence we see that w E h(I). Conversely, if w E h(I), then we define T on A/I by setting T(X + I) = w(X) , x E A. This is well defined since, if x,y E A and x - y E I, then T[(X

+

I) - (y

+

I)] = T(X - Y + I)

= w(x

= 0,

- y)

as w E h(I). Thus T E 6(A/I), and w = T Q q>. In this way we obtain a bijective mapping from 6(A/I) to h(I). Furthermore, it is easily verified that, if

(x

+

I)

o

o

denotes ..

the Gel'fand transform of x + I E A/I, then (x + I) = xlh(I) and that 6(A/I) and h(I) are homeomorphic. This proves parts (iv) through (vi). o

Finally, A/I is semisimple if and only if (x + I) (T) = 0, Q T (6(A/I), implies x E I. But it is readily seen that (x + I) if and only if x E k[h(I)]. Since k[h(I)] ~ I, by Proposition 7.1.1(vii), we conclude that A/I is semisimple if and only if I=k[h(I)]. o Before we establish the theorem indicated at the beginning of this section we shall make a definition and prove some preliminary results. Definition 7.3.1. Let A be a commutative Banach algebra. We say that A is normal if, whenever K C 6(A) is compact, E C 6(A) is closed and K n E =~, there exists some x E A such that

..

(i)

X(T)

= 1,

T

(ii)

X(T)

•

= 0,

TEE.

E K.

Corollary 7.2.2 says precisely that I.I(G), G being a locally compact Abelian topological group, is normal. We shall see that the

=

°

7.3. Normal Algebras

177

same is true of every regular commutative Banach algebra. Lemma 7.3.1. Let A be a regular commutative Banach algebra. If T E ~(A), then there exist some x E A and some open set U C b(A) containing T such that •x(w) = 1, w E U.

• Proof. Clearly there exists some yEA such that yeT) ~ o. Let U be an open neighborhood of T with compact closure such that • yew) ~ 0, w € cl(U). Such a neighborhood exists, as ~(A) is locally compact. Since A is regular, we have cl(U) = h(k[cl(U))). Set I = k[cl(U)). Then I is a closed ideal, and from Theorem 7.3.1

=

o

•

we see that A(A/I) h(l) = cl(U) and (y + I) = yICl(U). In particular, (y + I) never vanishes on A(A/I), and ~(A/I) is compact. Moreover, A/I is semisimple, as k[h(I}) = k[cl(U)) = I. Consequently we may apply Corollary 6.2.2 to deduce that A/I has an identity. • Thus !here exists some x E A such that (x + I) o = xlh(I) that is, x(w) = 1, w E cl(U), which completes the proof.

= I, o

Corollary 7.3.1. Let A be a regular commutative Banach algebra. If K C ~(A) is compact, then there exists some x E A such • that X(T) = 1, T E K. Proof. From Lemma 7.3.1 we see that for each T E K there exist some xT E A and some open neighborhood UT of T such that •xT(w) = I, w E UTe Obviously the family {U I T E K) is an open T covering of K, and so there exist TI ,T 2, .•. ,Tn such that n K C Uk=IU Tk . A straightforward calculation then reveals that x = x 0 x 0 ••• 0 x in A is such that •X(T) = 1, T E K, Tl T2 Tn where x 0 y = x + y - xy. o It is perhaps worthwhile pointing out that this corollary, combined with Corollary 6.2.2, gives us another sufficient condition for a Banach algebra to have an identity.

178 ...-


Corollary 7.3.2. Let A be a semisimple regular commutative Banach algebra. If A(A) is compact~ then A has an identity. This corollary, combined with Theorem 4.7.1, yields a partial converse to Corollary 4.7.5. Corollary 7.3.3. Let G be a locally compact Abelian topolo" is compact, then G is discrete. gical group. If G Proof. Since LI(G) is a semisimple regular commutative " we conclude from Corollary 7.3.2 Banach algebra and A(Ll(G)) = G, that LI(G) has an identity. But, by Theorem 4.7.1, this occurs if and only if G is discrete. o The normality of regular commutative Banach algebras is an immediate consequence of the next theorem. Theorem 7.3.2.

Let

A be a regular commutative Banach algebra.

If K C A(A) is compact, E C A(A) is closed, is any closed ideal in A for which h(l) = E, some x E I such that (i)

(ii)

" X(T)

= 1,

K n E =~,

and I then there exists

T E K•

• X(T) = 0, TEE.

Proof. Let I be a closed ideal such that E = h(I); for example, I could be keEl. By Theorem 7.3.1, A(I) is homeomorphic to A(A) - h(l) = A(A) - E. Since this latter set is open in A(A), it follows at once from the regularity of A that the hullkernel topology on A(I) is Hausdorff, and so, by Theorem 7.1.2, 1 is a regular commutative Banach algebra. Clearly K C A(A) - E = A(I) is compact, and therefore from Corollary 7.3.1 we deduce the existence of some x € I whose Gel'fand o transform x is identically one on K. But, by Theorem 7.3.1, o " " = 1, T € K. However, x = xlA(A) -h(l)' from which we see that X(T)

7.3. Normal Algebras since x E I, complete.

179

•

we have X(T)

Corollary 7.3.4. then A is normal.

= 0,

T

E h(l) = E, and the proof is

o If A is a regular commutative Banach algebra,

-

In the case that 6 (A) is compact it is immediate that the Gel'fand representation A of a regular commutative Banach algebra A even separates disjoint closed sets. This observation should help explain the origin of the term "normal" as applied to Banach algebras.

CHAPTER 8 IDEAL THEORY 8.0. Introduction. One of the more interesting portions of the theory of commutative Banach algebras is the study of the structure of closed ideals. As we shall see, any attempt to completely describe the closed ideals in an arbitrary commutative Banach algebra is hopeless, although the task may be relatively easy for certain specific algebras. Nevertheless a number of questions concerning closed ideals can be investigated fruitfully. The first such question we shall take up is to determine sufficient conditions under which a proper closed ideal I in a commutative Banach algebra A is contained in some maximal regular ideal. That is, when is it the case that h(l); ~ for each proper closed ideal I in A? Of course, as we saw previously, if A has an identity, then this is certainly the case, and similarly every proper regular closed ideal is always contained in a maximal regular ideal. However, in algebras without identity it is not entirely clear when the desired phenomenon occurs. We shall prove in the first section that, if A is a semisimple regular commutative Banach algebra such that the elements of A whose Gel'fand transforms have compact support are norm dense in A, then h(l) ~ ~ for each proper closed ideal I in A. We shall see that one important example of such a Banach algebra is LI(G), where G is a locally compact Abelian topological group. This result has a number of interesting consequences. For example, we shall show that the linear subspace of LI(G) spanned by the translates of some f E LI(G) is dense in LI(G) if and only if 180

8.1. Tauberian Commutative Banach Algebras

181

A

f never vanishes. Moreover, the indicated result will be used to establish some Tauberian theorems. We shall prove two such theorems, one due to Wiener and one to Pitt. The second question about ideals that we shall consider at some length is the determination of conditions under which the hull of a proper closed ideal I uniquely determines I. Since we know that in a regular commutative Banach algebra h(I) is closed and h(k[h(I)]) = h(l), we see that I and k[h(I)] are two closed ideals with the same hull, and we shall see that the question of when h(l) determines I is equivalent to determining when I = k[h(I)]. In other words, the problem will be to determine when a proper closed ideal I is the intersection of the maximal regular ideals containing I. We shall refer to this as the problem of spectral synthesis. Although it is, in general, a very difficult problem, some general results are available. For instance we shall see that, if A is a semisimple regular commutative Banach algebra that satisfies certain additional requirements known as Ditkin's condition, then I = k[h(I)] for each proper closed ideal I in A such that the topological boundary of h(l) contains no nonempty perfect set. The proof of this result, which appears in Section 8.5, is rather intricate and involves a number of results of independent interest. Although the problem of spectral synthesis is, in general, difficult, its solution for certain algebras is quite elementary. We shall, for instance, solve the problem completely for C(X), X being a compact Hausdorff topological space, and for Ll(G), G being a compact Abelian topological group. The problem for Ll(G) when G is noncompact is, however, exceedingly complex. In the final section of this chapter we shall briefly mention some other questions and results concerning closed ideals. 8.1. Tauberian Commutative Banach Algebras. We have already observed that, if A is a commutative Banach algebra, then every proper regular ideal in A is contained is some maximal regular

182

8. Ideal Theory

ideal. However, it need not generally be the case, when A is without identity, that every proper closed ideal is contained in a r-aximal regular ideal. The main result of this section will be to show that this does occur whenever A is a semisimple regular commutative Banach algebra such that {x I x E A, x• E C (A(A))) is norm dense c in A. In particular, we shall see that, if G is a locally compact Abelian topological group, then every proper closed ideal in Ll(G) is contained in some maximal regular ideal. Some further applications of these results will be discussed in the next section. To begin we make some definitions and some elementary observations. Definition 8.1.1. Let A be a commutative Banach algebra and suppose E C A(A) is closed. Then I (E) will denote the set of • 0 all x E A for which x vanishes identically on some open set Ox C A(A) such that Ox ~ E; and Jo(E) will denote the set of • € Cc(A(A)). x E IoCE) such that x Given a closed set E C A(A), it is evident that J o (E) C I 0 (E), that Jo(E) and I o (E) are ideals in A, and that cl[Jo(E)) = cl[Io(E)). Moreover, it follows from the definition of k[h(I)) that, if I is any closed ideal such that h(l) = E, then k[h(I)) ~ Cl[Io(E)). If A is semisimple and regular, then cl[Io(E)] is actually the smallest closed ideal in A whose hull is E. This is the content of the second part of the next theorem. Theorem 8.1.1. Let A be a regular commutative Banach algebra and let E C 6(A) be closed. Then (i)

h(cl[I o (E)])

= h(cl[Jo(E)]) = E.

(ii) If A is semisimple, then cl[Io(E)] is a closed ideal such that h(l) = E.

C

I

whenever

I

C

Proof. It is apparent that E C h(cl[Jo(E)]). On the other hand, if T ~ E, then let U be an open neighborhood of T with

A


183

compact closure such that cl(U) n E = ,. Since A is regular, there exists some x E A such that •X(T) ~ 0 and •x(w) = 0, w E ~(A) - U. Thus the support of •x lies in cl(U) and so is • compact, and x vanishes identically on the open set

ox

= ~(A)

- cl(U)

which contains E. Hence x E J eE) C cl[1 (E)], and •X(T) o 0 Consequently T ~ h(cl[Jo(E)]), and so E = h(cl[Io(E)]).

~

o.

To prove part (ii) of the theorem, suppose I C A is any closed ideal such that hel) = E. If x E Jo(E), then •X E C (~(A)), and • c there exists an open set 0 ~ E on which x vanishes identically. x • If K denotes the compact support of x, we see at once that K n E = ,. Thus, by Theorem 7.3.2, there exists some y E I such that •yeT) = I, T E K, and •yeT) = 0, TEE. However, it is easily • • verified that X(T) =• X(T)y(T). T E ~(A), whence, since A is semisimple, we conclude that x = xy E I.

I

Therefore Jo(E) is closed.

C

I,

and so cl[Io(E)]

= cl[Jo(E)]

C

I,

as

o

The result indicated in the introduction is a simple corollary of this theorem. Definition 8.1.2. Let A be a commutative Banach algebra. Then A is said to be Tauberian if (x l x E A, ~ E C (~(A))l is c norm dense in A. Corollary 8.1.1. Let A be a Tauberian semisimple regular commutative Banach algebra. If I C A is a proper closed ideal, then there exists some maximal regular ideal MeA such that M ~ I.

Proof. Suppose I is a closed ideal that is contained in no maximal regular ideal; that is, E = h(I) = ,. Then clearly Jo(E) = (x I x E A. x• E Cc(~(A»], and so, by Theorem 8.1.1, we have A = cl[Jo{E)] C I, as A is Tauberian. Hence I is not proper. 0

8. Ideal Theory

184

Since all ideals in an algebra with identity are regular, Corollary 8.1.1 has nontrivial content only in the case of algebras without identity. An important collection of such algebras that are Tauberian are the algebras Ll(G). Theorem 8.1.2. group. Then Ll(G) Banach algebra.

Let G be a locally compact Abelian topological is a Tauberian semisimple regular commutative

Proof. We need only prove that Ll(G) is Tauberian, as the other assertions were established in Theorem 4.7.4 and Corollary 7.2.3. Appealing to Plancherel's Theorem (Theorem 7.2.1), we see at once that the set of f E L2 (G) whose Plancherel transform •f • is equal almost everywhere to some element of C (G) is a norm-dense c linear subspace of L2 (G). We denote this subspace by L~(G). Moreover, if g,h E L2 (G), then g * h E C (G), whence we deduce that • • • c • gh E Ll(G) n L2 (G) and (gh) = g * h. Note that (gh) here is actually the Fourier transform of gh. Furthermore, each f ( Ll(G) can be written as f = f l f 2• where fk E L2 (G) , k = 1,2, from which it follows easily that (gh I g,h € L~(G)] is norm dense in L1(G), since L~(G) is norm dense in L2 (G).

c··

However, (f I f E LI(G), whence we conclude that LI(G)

·

f € Cc(G)) ~

(gh is Tauberian.

I

g,h € L~(G)J, 0

Corollary 8.1.2. Let G be a locally compact Abelian topological group. If I C Ll(G) is a proper closed ideal, then there exists some maximal regular ideal Me LI(G) such that M ~ I.

as

Of course, if G is discrete. then the corollary is trivial, Ll(G) has an identity.

Recalling the definition of the translation operators Ts defined in Definition 4.7.1, we can apply Corollary 8.1.2 to obtain the following interesting theorem:


185

Theorem 8.1.3. Let G be a locally compact Abelian topological group, let f E LI(G), and suppose I denotes the closed linear subspace of LI(G) spanned by {Ts(f) I s E G). Then the following are equivalent:

(ii)

•

~

fey)

~

0,

•

E G.

Proof. We claim first that I is a closed ideal in To see this it suffices to show that g * f £ I for each since g * Ts(f) = Ts(g * f), s E G, and I is invariant the translation operators T , s E G. To this end we note s ~ ( L (G) is such that CD

IG

h(t)~(-t) dArt)

=0

LI(G). g £ L1 (G) under that, if

(h E I),

then f * ~(s)

= IG

f(s - t),(t) dA(t)

= IG

f(s

=

IG

+ t)~(-t)

dArt)

T_s(f)(t)~(-t) dArt)

=0 that is,

f *

~

= O.

iG g *

(s

Hence for any g € LI(G)

f(t)~(-t) dArt)

= (g *

E G),

we have

f) * ~(O)

= g * (f *

~)(O)

= o. Thus, since the dual space [OSI' pp. 289 and 290; El , of the Hahn-Banach Theorem each g E LI(G). Hence I

of pp. [L, is

LI(G)

can be identified with Lm(G) 215-220, 239 and 240], a consequence p. 90] shows that g * f £ I for a closed ideal in L1 (G).

8. Ideal Theory

186

Now if I ~ Ll(G), then I is a proper closed ideal in Ll(G), whence by Corollary 8.1.2 there exists some maximal regular ideal Me L1(G) such that M~ I. Thus from Theorem 4.7.3 we see that A • there exists some y E G such that M = {g I g E Ll(G), g(y) = 0). A In particular, we would have fey) = 0, contradictiong the assumption that •f never vanishes. Thus part Cii) of the theorem implies part (i).

• Conversely, suppose fCY) = 0 for some y E G. An elementary A • A computation reveals that T (f) (w) = (-s,w)f(w), s E G and w E G, s. whence we deduce that T (f) (y) = 0, s E G. It is then obvious A s that hey) = 0, h E I, and so I is contained in the maximal regular A ideal M = {g I g E L1CG), g(y) = oj. Hence I is proper, and so part Ci) implies part (ii). o A

The fact established in the preceding proof that the closed linear subspace I spanned by {Ts(f) I s (G) is a closed ideal is a special case of a generally valid result in Ll(G): a closed linear subspace I of LleG) is an ideal if and only if it is translation invariant; that is, if and only if Ts(g) E I, s E G, whenever gEl. We state this result as the next theorem leaving the proof to the reader. Theorem 8.1.4. Let G be a locally compact Abelian topological group and let I C Ll(G). Then the following are equivalent: (i)

I

is a closed ideal.

Cii)

I

is a closed translation-invariant linear subspace.

This is also an appropriate point to mention some approximation results for LI(G) that are easy corollaries of the fact that Ll(G) is Tauberian. Corollary 8.1.3. Let G be a locally compact Abelian topological group. If f E L1(G) and & > 0, then there exists some A A V E LI(G) such that v E Cc(G) and IIf - f * vIII < &.


187

Proof. From Theorem 4.7.2 we know that LlCG) contains an approximate identity, and so there exists some u E LICG) such that IIf - f * ull l < 1/2. Then, s!nce LICG) is Tauberian, there exists some v E LI CG) such that v ( CC C~) and lIu - vIII < 1/2l1f Il 1 • Note that we may assume that f ~ 0 since the result is trivially valid in this case. Hence IIf - f * vIII ~ IIf - f * ull l

+

IIf * u - f * vIII

~ IIf - f * ull l

+

IIflllliu - vIII

£

£

2

2

0, then there exists a finite linear combination of continuous characters of G, say Ey EG ayC· ,y), ay E~, where only finitely many ay are nonzero, such that IIf - f * [~EG ayc·,y)]U I < c. Proof. 8y Corollary 8.1.3 there exists some v E LlCG) such • • • that v £. Cc (G) and IIf - f * vIII < £. Since G is discrete by Corollary 4.7.5, the support of must be finite, say Yl'Y2' ••• 'y • •

v•

n

It is then evident that v = Ik=lvCYk)X(Yk)' where X(Yk) denotes the characteristic function of (Yk), whence, from the semisimplicity of Ll(G) (Theorem 4.7.4) and Corollary 4.7.4, we conclude that

•

v = ~=lvCYk)(·'Yk)·

0

A finite linear combination of continuous characters of a locally compact Abelian topological group G is usually called a trigonometric polynomial. The origin of the terminology is clear on considering the group G = f. Utilizing these corollaries it is not difficult to prove the following result, the details being left to the reader:

8. Ideal Theory

188

Corollary 8.1.5. Let G be a locally compact Abelian topological group. Then LI(G) contains an approximate identity (u a ) such that (~) c C (~). Moreover, if G is compact, then LI(G) Ot c contains an approximate identity consisting of trigonometric polynomials. Theorem 8.1.2 and the idea of the proof of Corollary 8.1.3 also provide us with the following useful result: Corollary 8.1.6. Let G be a locally compact Abelian topolo• gical group. If KeG is compact and e > 0, then there exists some f E LI(G) such that (i)

(ii)

•f E C (G). • c

..

= 1,

fey)

Y E K•

•

Proof. If G is compact, then from Corollary 7.3.3 we see that G is discrete, and so Ll(G) has an identity e. Clearly

..~(y)

.

and lIeU I < 1 + e. On the other han~, suppose G is noncompact. Then there exists an open set WC G such that • - W, then E is closed W~ K, and cl(W) is compact. If E = G and K n E whence, by Theorem 7.3.2 or Corollary 7.3.4, we • deduce the existence of some h £ Ll(G) such that hey) = 1, Y E K,

= I,

.

Y E G,

="

and hey) = 0, y E E. In particular, be such that 6 < min(cllhlll,e).

..

.

h E Cc(G).

6 >

Now let

°

From the proof of Theorem 4.7.2 we see that there exists some u C LI(G) such that lIulil = 1 and IIh - h * ull l < 6/3, and, since Ll(G) is Tauberian (Theorem 8.1.2), there exists some g E Ll(G)

•

•

such that g E C (G) and !lu - gli l < 6/3I1hIl1. c .. • Evidently f E LI(G), f E Ce(G), and

- .

fey) = hey) = 1

+

+

Set

f

=h

+

g - h * g.

. ..

g(y) - h(y)g(y)

• g(y) - •gCy) = 1

(y E K).

8.2. Two Tauberian Theorems

189

Moreover,

< IIglll

+

IIh - h * gill

< IIglll

+

IIh - h * ull l

< lI u ll l

+

< I

6

6

+ -

311 h l;1 6

6

3 311hlll e- + e- + e < 1 +3 33

=1

+

+

+

3

+-

+

IIh * u - h * gill IIhlllliu - gill

6

+3

e.

c

8.2. Two Tauberian Theorems. The theorem of Tauber, from which the name "Tauberian theorem" originates, is the following: Suppose (ak ) is a sequence of complex Rumbers such that the power series

~=O ak,k converges in (,

1,

E

Q;,

1'1

< I).

If

limk_CDkak = 0

and lim Ek=O akrk = a, r-I O 0, since g is slowly oscillating, there exist a compact set KI C G and a compact neighborhood U of 0 € G such that Ig(u) - g(v)1 < e/2, u - v E U and u E G - KI . Consequently, if fo

= Xu/A.(U),

then

Ig(t) - g * fo (t) I

fo (L1(G) -

rlur JU get - s) dA(s) I

Iu

19(t) - get - s) I dA.(s)

= 19(t) ~ 1(~)

and

e

(t E G - KI ).

3, and suppose E C ~=IUk. Then F = E n (X - Un) is closed and F C ~:~Uk. Again appealing to the normality of X, we deduce the existence of an open set U such that n-I Feu

U Uk" k=l Then, by the induction hypothesis, there exist C

cl(U)

C

gk E A, k = 1,2, ... ,n-I,

~-I such that ~=Igk(t) = 1, t E cl(U), and gk(t) = 0, t ( X - Uk' k = 1,2, •.• ,n - 1. Moreover, E C U U U, and so, applying the n lemma with two open sets, we see that there exist f and h in n A such that f(t) + h (t) = I, t (E, and f(t) = 0, t E X - U, n and hn(t) = 0, t ( X - Un. Hence on setting hk = fg k , k = 1,2, •.. ,n-l it is easily verified that tk=lhk(t) = 1, tEE, and ~(t) = 0, t E X - Uk' k = 1,2, •.. ,n.

This completes the proof of the lemma.

o

8.4. Local Membership in Ideals

203

The reader should observe that the lemma is actually valid without any assumptions of continuity on the functions in A. However, our applications of the lemma will only be in the case where A consists of continuous functions. Theorem 8.4.1. Let A be a semisimple regular commutative Banach algebra and suppose I C A is an ideal. If x E A belongs locally to I at every point of A(A) and at infinity when A(A) is noncompact, then x E I. Proof. We claim that, without loss of generality, we may assume that A has an identity. If this were not so, then I would still be an ideal in A[e] and x would belong locally to I at all points of A(A[e]) = A{A) U {TCDJ. Indeed, it is obvious that x belongs locally to I at each point of A(A), and x belongs locally to I at T since x belongs locally to I at infinity • CD and Y(T) = 0, yEA. Thus if the theorem is valid for algebras CD with identity, we conclude that x E I. Consequently we may assume that A has an identity. Then A(A) is a compact Hausdorff, and so normal, topological space, and A is a normal Banach algebra by Corollary 7.3.3. Thus • C C(A(A)). we may apply Lemma 8.4.1 to A Now x belongs locally to I at each point in A(A). Hence, given T E A(A), there exist an open neighborhood U of T and •• T some YT E I such that x{w) = YT(w), w E UTe Clearly the sets (U T I T E A{A)) form an open covering of A(A), and so there exists a finite number of the UT, call them Ul ,U 2 , .•. ,Un , such that A(A) = Uk=IU k • For the sake of notational simplicity we denote the YT corresponding to Uk by Yk , k = 1,2, ... ,n. Applying Lemma 8.4.1 to the closed set E = A(A), we deduce the existence of xk E A, k = 1,2, •.. ,n, such that ~=I;k{T) = 1, T E A(A), and ~k(T) = 0, T (~(A) - Uk' k = 1,2, •.. ,n. Evidently ~=lxkYk E I. Moreover, we claim that

8. Ideal Theory

204

" (1')

X

Indeed. let

(1' (

~(A)).

~(A)

and let Uk ,Uk •.•. ,U k denote those " ..1 2 m . Uk such that l' E Uk. Now Yk.(w) = x(w). W (Uk.' J = 1,2, ... ,m, J J so in particular these equations hold for W = 1'. On the other hand, l' (

x" k (1') = 0 if k ~ kj , j = 1,2, ... ,m, Hence we see that

= x" (1')

" since xk(w) = 0,

W

E

~(A)

- Uk.

n "

1: x k . (1')

j=l

J

" = x(1'). Therefore, since A is semisimple, we conclude that x = ~=IXkYk belongs to 1.

o

8.5. Ditkin's Theorem. We are now almost in a position to prove the theorem alluded to at the beginning of the preceding section. To describe the relation between Ditkin's Theorem and spectral synthesis we first need to make another definition. Definition 8.5.1. Let A be a commutative Banach algebra. Then A is said to satisfy Ditkin's condition at l' in ~(A) if, whenever x E A and l' E Z(x), there exist a sequence (x k ) C A and open neighborhoods Uk of T such that (i) (ii)

li~lIxxk -

xII

=

o.

" xk(w) = 0, wE Uk' k = 1,2,3, ....

If d(A) is noncompact, then A is said to satisfy Ditkin's condition at infinity if, whenever x E A, there exists a sequence {xkl C A such that

8.5.

Ditkin's Theorem

A

(b)

xk E

205

Cc(A(A)), k

= 1,2,3, •.•. A

We remind the reader that Z(x) = (T I T E a(A), X(T) = 0) (Definition 7.1.1). It is apparent that, if A satisfies Ditkin's condition at infinity, then A is Tauberian. An immediate corollary of Ditkin's Theorem will show that, if A is asemisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of A(A) and at infinity when A(A) is noncompact, then a closed set E C A(A) will be a set of spectral synthesis for A provided that bdy(E) contains no nonempty perfect set. Similarly it follows that, if I is a closed ideal in such a Banach algebra, then I is the intersection of the maximal regular ideals containing I, that is, I = k[h(I)], provided bdy[h(I)] contains no nonempty perfect set. For the sake of completeness we recall that, if subset of a topological space X, then bdy(E) = E n E n [X - int(E)], and a set E is perfect if E is every point of E is a limit point of E, that is, lated points.

E is a closed cl(X - E) = closed and E has no iso-

One further lemma is necessary before we can prove Ditkin's Theorem. Lemma 8.5.1. Let A be a semisimple regular commutative Banach algebra, let I be a closed ideal in A, and suppose x E A. Then x belongs locally to I at each T in a(A) that satisfies either of the following conditions: (i) (ii)

T

E int[Z(x)].

T E

Proof.

a(A) - h(I).

Suppose

T E

int[Z(x)].

Then there exists an open

8. Ideal Theory

206

neighborhood U of T such that U c Z(x), and so •x(w) w E U. Thus x belongs locally to I at T.

= •O(w) = O.

Now suppose T ( A(A) - h(I). Then, since A(A) is locally compact, there exists an open neighborhood U of T such that cl(U) is compact and cl(U) C A(A) - h(I). Since cl(U) is compact, h(l) is closed, and cl(U) n h(l) we see from Theorem 7.3.2 that there exists some y E I such that •yew) = 1, w E cl(U). Hence • •• • xy E I and (xy) (w) = x(w)y(w) = x(w), w E U. Therefore x belongs locally to I at T.

="

o

Theorem 8.5.1 (Ditkints Theorem). Let A be a semisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of A(A) and at infinity when A(A) is noncompact, let I be a closed ideal in A, and suppose x E A is such that h(l) C Z(x). Then (i) If E is the set of T E A(A) such that x does not belong locally to I at T, then E is a perfect subset of h(l) n bdy[Z(x)] = bdy[h(I)] n bdy[Z(x)]. (ii) If bdy[h(I)] n bdy[Z{x)] set, then x ( I.

contains no nonempty perfect

Proof. From Lemma 8.5.1 and the fact that Z(x) is closed we see at once that E C h(l) n (~(A) - int[Z(x)]). However, since h(l) C Z(x), we have bdy[h{I)] nbdy[Z(x)] = (h{I)

ncl[~(A)-h(I)])

n (Z{x)

ncl[~(A)-Z{x)])

=h(l) n (Z{x) ncl[~{A)-Z(x)]) = h{l) nbdy[Z(x)] • h (I)

n [Z (x) n (6 (A) -

int [Z (x)])]

= h{l) n (6(A) - int [Z{x)]).

8.5.

Ditkin's Theorem

207

Consequently E C h(l) n bdy[Z(x») = bdy[h(I)] n bdy[Z(x»). Moreover, from the definition of local membership in an ideal we see at once that E is closed. Thus to show that E is perfect we need only prove that E has no isolated points. So suppose that T € E is isolated. Then there exists some open neighborhood U of T such that cl(U) is compact and (cl(U) - (T) n E = ,. Since E C bdy[Z(x)] c Z(x), we see that T E Z(x). Thus, by Ditkin's condition, there exist a sequence (x k ) • in A and open neighborhoods Uk of T such that xk(w) = 0, wE Uk' k = 1,2,3, •.. , and limkllxxk - xII = O. Let W be an open neighborhood of T such that cl(W) C U. This is possible because A(A) is locally compact. Then cl(W) is compact, A(A) - U is closed, and cl(W) n [A(A) - U] = " whence, since A is normal, • we deduce the existence of some y £ A such that yew) = I, w E cl(W), • and yew) = 0, w ~ A(A) - U. We claim that the elements yxx k , k = 1,2,3, ••• , belong locally to I at each point of A(A) and at infinity.

•

Indeed, since xk(w) = 0, w E Uk' and T E Uk' we see that yxx k belongs locally to I at T; and if w £ cl(U) - (T), then by the definition of E we see that x belongs locally to I at w, whence yxxk belongs locally to I at w, k = 1,2,3, .... Thus • each yxx k belongs locally to I on cl(U). Finally, since y vanishes identically on A(A) - U and cl(U) is compact, we see at once that yxx k ' k = 1,2,3, •.. , belongs locally to I at each w E A(A) - cl(U) C A (A) - U and at infinity. Hence yxxk belongs locally to I at each point of A(A) and at infinity, k = 1,2,3, .•.• Therefore (yxx k ) C I by Theorem 8.4.1. Consequently yx £ I, as I is closed and limkllyxxk - yxil = O. • ••• • But if w £ W, then (yx) (w) = y(w)x(w) = x(w), as y(~) = I, w E cleW). Since T E W, this says that x belongs locally to I at T, contradicting the fact that TEE. Therefore

E

is perfect and part (i) of the theorem is proved.

8. Ideal Theory

208

To prove part Cii) we suppose that bdy[hCI)] n bdy[Z(x)] contains no nonempty perfect sets. Then from part (i) we see that x belongs locally to I at each point of a(A) , as E =~. If aCA) is compact, then x E I by Theorem 8.4.1. On the other hand, if a(A) is noncompact, then, since A satisfies Ditkin's condition • at infinity, there exists a sequence (xk ) C A such that (xk) is contained in Cc (6(A» and limklixxk - xII = O. For each k let Ek be the set of T E a(A) such that xXk does not belong locally to I at T. We claim that Ek = ~, k = 1,2,3, ..•. Indeed, since x belongs locally to I at each point of a(A), we see that, if T (A(A), then there exist an open neighborhood U • • of T and some y E I such that x(w) = yew), w (U. Clearly • • then we must also have (xx k) (w) = (YX k ) (w), w E U, and YXk E I, k = 1,2,3, ... ; that is, xX k belongs locally to I at T, k = 1,2,3, •••• Hence Ek C E = ~, k = 1,2,3, •.••

-

Thus xX k belongs locally to I at each point of 6(A), k = 1,2,3, •... Moreover, since (x k) C CcCaCA», it is apparent that xXk also belongs locally to I at infinity, k = 1,2,3, ...• Consequently by Theorem 8.4.1 we conclude that (XX k ) C I, whence x E I, as I is closed. It should be noted that the hypothesis that A satisfies Ditkin's condition at infinity was only utilized in proving part (ii) of the theorem. Thus Theorem 8.S.1(i) remains valid under the weaker assumption that A satisfies Ditkin's condition only at each point of a(A). Some easy consequences of Ditkin's Theorem are the following corollaries: Corollary 8.5.1. Let A be a semisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of a(A) and at infinity when a(A) is noncompact. If E C a(A) is a

8.6. LI(G) Satisfies Ditkin's Condition

209

closed set such that bdy(E) contains no nonempty perfect set, then E is a set of spectral synthesis for A. Proof. Let I be any closed ideal such that h(I) = E. If x E k[hel)] = keEl, then obviously E = hel) c Zex), and bdy(E) = bdy[h(I)] ~ bdy[h(I)] n bdy[Z(x)]. Since bdy(E) contains no nonempty perfect set, it follows that bdy[h(I)] n bdy[Z(x)] has the same property, whence we conclude that x E I by Ditkin's Theorem. Hence I = k[h(I)] = k(E), and E is a set of spectral synthesis' D Corollary 8.5.2. Let A be a semisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of a(A) and at infinity when a (A) is noncompact. If I C A is a closed ideal such that bdy[h(I)] contains no non empty perfect set, then I is the intersection of the maximal regular ideals containing I; that is, I = k[h(I)]. It is perhaps worthwhile mentioning explicitly that, if A is a semisimple regular commutative Banach algebra, then a (A) is compact if and only if A has an identity. This is the content of Corollary 7.3.2. Thus in the theorems of this and the preceding section a(A) is noncompact precisely when A is without identity. In the next section we shall show that LI(G), G being a locally compact Abelian topological group, satisfies Ditkin's condition, and so we may apply Ditkin's Theorem to this algebra. 8.6. L1(G) Satisfies Ditkin's Condition. Before we prove that LI(G), G being a locally compact Abelian topological group, • satisfies Ditkin's condition at each point of a(LI(G)) = G and • is noncompact, we shall establish two lemmas . at infinity when G • and As usual, ij will denote Haar measure on the dual gruop G, • have been so we shall assume that the Haar measures on G and G chosen that Plancherel's Theorem (Theorem 7.2.1) is valid.

210

8. Ideal Theory

Lemma 8.6.1. Let G be a locally compact Abelian topological • be an open neighborhood of the identity y in group, let WC G • 0 G, let 6 > 0, and let E C G be compact. Then there exist a compact set KeG• with nonempty interior and an open symmetric • neighborhood a of Yo in G such that (i) (ii) (iii) (iv) (v)

cl(U)

is compact.

y € int{K). o

~(K

- U)
I

-

C(X) X a compact Hausdorff topological space

I

Yes

Yes

Yes

Yes

2

No

No

No

Yes

3

Yes

No

No

Yes

4

No

Yes

No

Yes

More generally the answer to question I is affirmative either when A has an identity or A is a Tauberian semisimple regular commutative Banach algebra. However, if A is a radical algebra, then the answer to question 1 is negative. For further material on ideal structure in addition to that cited in this chapter the reader is referred to [N,Ri).

CHAPTER 9 BOUNDARIES 9.0. Introduction. Recall that A(D) is the commutative Banach algebra with identity of all those complex-valued functions that are defined and continuous on the closed unit disk D and analytic on int(D). The Maximum Modulus Theorem of analytic function theory asserts that, if f E A(D) , then IIf llCl) = sup, €. rl f (C) (. where r = (e Ie€. D, lei = 1), and there exists some int(D)

,€

such that IIfllCD = If(C)1 only if f is a constant function. A second central result of analytic function theory, the Cauchy Integral Formula, tells us that, if f € A(D), then f(C)

= ~S 2nl r

fez) dz

z -

C

(C E int (D) ) .

In this chapter we shall examine the question of obtaining generalizations or analogs of these classical results in the context of commutative Banach algebras. Actually, it is advantageous to shift the investigation away from commutative Banach algebras or, more precisely, away from the Gel'fand representations of such algebras to a more general class of algebras, which we shall call separating function algebras. In the following sections we shall develop a portion of what can be said about the Maximum Modulus Theorem and the Cauchy Integral Formula for such algebras. The development falls roughly into two parts. In Sections 9.1 through 9.4 we shall concern ourselves primarily with analogs of the Maximum Modulus Theorem. This leads to the introduction of the concept of a boundary for a separating function algebra, which will be seen to be a suitable replacement for r in the study of A(D). We shall investigate three specific boundaries: the ~ilov, 218

9.1. Boundaries

219

Bishop, and Choquet boundaries. In Sections 9.6 through 9.8 our primary, but not our only, concern will be counterparts of the Cauchy Integral Formula. This development leads to the notion of representing measure. From the classical situation, as exemplified in A(D), it should come as no surprise that the notions of boundaries and representing measures are intimately connected. This connection will be perhaps most apparent in Sections 9.6 and 9.7. Between the two portions of the chapter just indicated is a section devoted to several applications of the material on boundaries developed up to that point. However, the range of applications of the material introduced in the chapter is substantial, particularly in the study of function algebras, and we have made no attempt to provide a general introduction to these applications. The reader who is interested in such applications should consult the appropriate references cited in this chapter. 9.1. Boundaries. Consider the commutative Banach algebra with identity A(O) consisting of all those functions defined on the closed unit disk 0 in C that are continuous on 0 and analytic on int(O). Then the classical theory of analytic functions contains a number of results concerning the maximum modulus of elements of A(O). To be precise, the following results are valid. Recall that

r = (C ICE D, Ici = 1). If f E A(D), then there exists some C E r such that IfCC) I = II f Il CD • In particular, we see that IIfU. = suPC E rl fCC) I and that If I assumes its maximum on f. 1.

2. If E c r is closed and if for each f E A(D) there exists some C E E such that If(C)1 = IIfll, then E = r. In particular, CD r is the smallest closed subset of D on which If I assumes its maximum for each f E A(D). 3. If f E A(D) and there exists some If(C)1 = UfU, then f is a constant. CD

C E int(D)

such that

9. Boundaries

220

4. If f (. A(D) is nonconstant and e E int(D) , then for each p > 0 such that (z I Iz - el < p] C int(D) there exists some w, Iw - el < p, such that If(w)1 > If(e)l· In other ,~ords, if f ( A(D) is nonconstant, then I f I cannot have a local maximum at any point in int(D).

The first result or some variation of it is usually referred to as the Maximum Modulus Theorem or the Maximum Principle (see, for example, [A, pp. 133-135; Ru 2 , p. 213]). The fourth result is called the Local Maximum Modulus Theorem. Our main concern in this and the following four sections will be to examine some extensions of these results for A(D) to general commutative Banach algebras. In particular we shall see that there always exist valid analogs of result~ 1 and 2, but result 3 may very well fail to hold. A valid analog of result 4 also exists, but a proof would involve considel'nble machinery which we have not developed. The curious reader is referred to [Ga, pp. 91-93; S, pp. 89-98; Wm 2 ' pp. 52-55] for these generalizations of the Local Maximum Modulus Theorem. There is some advantage in the sequel in working with algebras of continuous functions rather than an arbitrary commutative Banach algebra. Moreover, the Gel'fand Representation Theorem (Theorem 3.3.1) asserts that every commutative Banach algebra A is continuA ously homomorphic to a subalgebra A of C (A(A) that separates o the points of A(A). This theorem will then allow us to translate results about such subalgebras of C (X), X being a locally compact o Hausdorff topological space, into results about commuative Banach algebras. With this in mind we make the following definition: Definition 9.1.1. Let X be a locally compact Hausdorff topological space and suppose A is a subalgebra of C (X). Then A o is said to be a separating function algebra on X if (i) (ii)

A separates the points of X. ZeAl = {t

I

t

E X, f(t) = 0,

f

E A] = ,.

9.1. Boundaries

221

Clearly, if X is compact and Ac C(X) is a subalgebra that separates the points of X and contains the constant functions, then A is a separating function algebra. Furthermore, if A is • a commutative Banach algebra, then A = Ace (a(A)) is a separating o function algebra. A useful observation about such algebras is contained in the next proposition. The proof is left to the reader. Proposition 9.1.1. Let X be a locally compact Hausdorff topological space. If A is a separating function algebra on X, then the topology on X is the weakest topology on X such that all the functions in A are continuous. We conclude this section with several additional definitions. Definition 9.1.2. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. A set E c X is said to be a boundary for A if for each f E A there exists some

tEE

such that

If(t)1

= IIfllCD = sUPsExlf(s)l.

Thus a boundary for a separating function algebra A is a subset of X on which If I assumes its maximum for each f E A. For obvious reasons boundaries are often called maximum modulus sets. If A = A(D), then the Maximum Modulus Theorem for analytic functions says precisely that r is a boundary for A(D) and that r is even the smallest closed subset of D that is a boundary for A(D). We shall see shortly that every separating function algebra has a smallest closed boundary. On the other hand, a smallest boundary need not exist. Definition 9.1.3. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. A point t E X is said to be a peak point for A if there exists some f E A such that If(t)1 = IIfll CD = 1 and If(s)1 < 1, s ~ t. The set of all peak points for A is called the Bishop boundary for A and is denoted by pA.

9. Boundaries

222

It is apparent that, if t € X is a peak point for A, then t belongs to every boundary of A, and so pA is the minimal boundary for A. However, there exist separating function algebras A that have no peak points -- that is, such that pA =~. We shall see an example in Section 9.3. Peak points are also called strong boundary points and unique maximum points (see, for instance, [HOI' p. 61; Ri, p. 141]). The term "strong boundary point", however, is usually applied to points that we shall call weak peak points. The reader is warned that there seems to be no fixed terminology for these concepts and is advised to check the definitions in the individual references cited. 9.2. The §ilov Boundary. The main theorem of this section is that every separating function algebra possesses a unique minimal closed boundary, that is, a closed boundary that is contained in every closed boundary. We shall also show that this is true, in a suitable sens~of any commutative Banach algebra and that, if such an algebra A is either regular or self-adjoint, then the minimal closed boundary is precisely A(A). This unique minimal closed boundary will be called the §ilov boundary. I

Theorem 9.2.1 (§ilov). Let X be a locally compact Hausdorff topological space. If A is a separating function algebra on X, then A has a unique nonempty minimal closed boundary. Proof. Let E denote the family of all closed boundaries for A. Clearly E ~~, as X € E. We introduce a partial ordering in E by setting EI > E2 , EI ,E 2 E E, if and only if El C E2 . If (E) is a linearly ordered subset of E, then obviously E = n E a 0 a a is a closed subset of X. We claim that E is also a boundary for

A.

o

Indeed, suppose f E A and f is not identically zero. Define Ef C X as Ef = {t l t E X, If(t) 1 = IIflt'). We claim that Ef is a nonempty compact subset of X. Since A c C (X), there exists o

9.2. The §ilov Boundary

223

some compact set K c: X such that If(t) 1 < IIfll CD/2. t E X - K. Thus f restricted to K is continuous, and so Ifl assumes its maximum modulus on K. Hence Ef is a nonempty closed subset of K, and so Ef is a nonempty compact subset of X. Moreover, since each Ea is a closed boundary for A, we see that Ef n Ea is a nonempty compact subset of K for each a. Since the family {E) is lineartl ly ordered, we see that the family {E f n EaJ has the finite-intersection property, whence we conclude that na(Ef n Ea1 = Ef n Eo is nonempty. In particular, this also shows that E itself is nono empty and that, if t E Ef n Eo c: E, then If(t)l = IIfll. Evident0 CD ly, if f E A is identically zero. then there exists some tEE o such that If(t)l =Ufll. Hence we see that E0 = naa E is a closed CD boundary for A. Thus every linearly ordered subset of E has an upper bound, and so, appealing to Zorn's Lemma [D5 l • p. 6], we deduce the existence of a maximal element in E. call it E. Clearly E is a minimal closed boundary for A, and to show that it is unique it suffices to prove that, if F is any closed boundary for A, then E c: F. To see t~is suppose there exists tEE - F. Now, by Proposition 19.1.1, the topology on X is the weak topology genera~ed by the elements of Aj that is, a neighborhood base for the topology of X at the point t consists of sets of the form

where s > 0 and f l ,f 2, •.. ,fn in A are arbitrary. In particular, since F is closed, there exist some s, 0 < s < 1, and f l ,f2, •.• ,fn in A such that t E U(tjeifl'f2 ••.• ,fn) = U and U n F = ,. Moreover, we may assume, without loss of generality. that IfkCs) - fk(t)1 < 1, k = 1,2, •.•• n; sEX, on the following grounds: If sUPk=1,2, •.• ,n[suPsEXlfk(s) - fk(t)il = M> 1, then set gk = fk/M, k = 1,2, .••• n. Clearly gk £ A, Igk(S) - gk(t)1 < 1, k = 1,2, ••. ,n; 5 ~ X. and U(t;c/M;gl,g2, .•• ,gn) c: U(t;c;f l ,f2 ••.• ,fn).

9. Boundaries

224

Moreover, since E is a minimal closed boundary, there evidently exists some f E A such that IIfli CD = 1 and If(s)1 < 1, sEE - U. Note that, if E - U is empty, then the second assertion is trivially satisfied. Let g E A be a sufficiently high power of f such that Ig(s) I < I, sEE - U. Clearly one still has Then we see that, for each k = 1,2, •.. ,n,

= 1.

IIgli CD

(s E U)

<e

and

n, ~ o

9. Boundaries

228 f(s) Therefore t

=

no

~

f k (s)/2

k

O.

f(5) ~(s)

(f £ C(X))

9. Boundaries

238

Now we recall that the extreme points of the closed unit ball in M(X), that is, of the set (v I v E M(X), IIvll < 1), are precisely those measures of the form aCt' t (X, where a (~, lal = 1, and 6t E MeX) is the measure with unit mass concentrated at t; that is, if E C X is a Borel set, then 6t (E) = 1 if tEE and 6t (E) = 0 if t ~ E. A proof of this fact is available, for instance, in [L, p. 338]. Since the closed unit ball in M(X) is weak* compact and convex, by the Banach-Alaoglu Theorem [L, p. 254], we deduce, from the Krein-Miltman Theorem [L, p. 322], that it is the weak* closed convex hull of its extreme points. Combining these observations with the fact that ~ > 0 and II~II = 1, one easily verifies that ~ is a weak* limit of convex linear combinations of the extreme points (6 t I t E xl. However, since the restrictions to A of the functionals determined by ~ and 6 t , t E X, are just x* and Tt , t E X, respectively, it follows immediately that x* is a weak* limit of convex linear combinations of the Tt' t E X. That is, x* E -COrTeX)], and so -CO[T(X)] = B11 • Next we shall show that ext(B II ) C T(X). We observe that, since Bi is weak* closed and norm bounded, it must be weak* compact by the Banach-Alaoglu Theorem, whence we conclude from the Krein-Mil'man Theorem that ext(B 1l ) ~~. If x~ E ext (B II ), then, by the argument used in proving part (ii), we know that there exists a measure ~o E M(X), lI~oll = 1, ~o ~ 0, x~(f)

such that

= Ix

f(s) ~o(s)

If E C X is any Borel set such that

0
- 0, there exists some t E X such that ~o = 6t . Thus x~(f) =

whence we have x~

Ix

= Tt .

f(s) d~o(s) = f(t) = Tt(f)

(f E A)'

Hence ext(B~) C T(X).

Finally, suppose F A* - C is a linear functional that is weak* continuous; that is, F is a continuous linear transformation from (A*,rw*) to ~. We claim that

Note that the supremum is finite, as F is continuous on the weak* compact set B~. Clearly the left-hand side of the equation is always greater than or equal to the right-hand side. Suppose that the inequality is strict, let p > 0 be such that sup lIF(x*)1 > x* E 8 1

p

>

sup I IF(x*)I, x* E ext (B l )

and let x* E Bli be such that IF(x*)1 > p. By the Krein-Mil'man o _ 1 0 1 Theorem, we have that co[ext(B I )] = Bl , and so there exist ~ E ext(B~) and a k E~ a k > 0, k = 1,2,3, ... ,n, such that ~=lak = 1

and n IF(x~)' IF(x*) - F( E a x*)1 < 2 o k=l k k

Hence, on the one hand,

p

9. Boundaries

240

> and~

on the other

p~

hand~

n n IF( E akx,) I < ~ akIF(x,ll

k=l

k=I

< p. This contradiction shows that

for each weak* continuous linear functional

F on

A*.

it is apparent that~ if F is a weak* continuous linear functional on A*~ then we also have Furthermore~

=

sup IIF(x*)1 x* €B 1

sup 1 IF(x*)I, x* ( c 1 [ext (B l )]

1 where cl[ext(B~)] denotes the weak* closure of ext(B I ). Moreover, since weak* continuous linear functiona1s separate the points of A* [L, p. 240], it is easily verified that cl[ext(B~)] is the smallest closed subset E C T(X) C Bl such that 1

sup llF(x*)1 x* (B 1

=

sup IF(x*)1 x* (E

for each weak* continuous linear functional

F.

Now, however, each weak* continuous linear functional F on A* is of the form F(x*) = x*(f), x* (A*~ for some f E A [t, p. 238]. Thus, if f E A and F(x*) = x*(f), x* (A*, then

9.4. Extreme Points

241

IIfli

CD

:: sup If (t) I €X

t

= ::

sup ITt(f)1 Tt ~T(X) sup Tt

IF(Tt)1

€ T (X)

::

sup 1 IF(Tt)l Tt E cl[ext(B l )]

::

sup 1 IF(Tt)l Tt E ext (B 1)

=

Tt

sup 1 If(t)l· ~ ext (8 1 )

Consequently we see that, if f E A, IIfllCD :: ::

then

sup -1 1 If(t) 1 t E cl (T [ext (B 1) ] ) _lsuP 1 If(t)l, t ( T [ext (B l )]

and that cl(T -1 [ext(B 11)]) is the smallest closed subset of X for which the identity is valid. Therefore we see that

oA that is,

I = Cl(T -1 [ext(Sl)])::

-cl(T - 1 [ext(co[T(X)])])~

oA is the closure in X of (t I t E X, Tt € ext(co[T(X)])).

This completes the proof of the theorem. The last portion of the proof reveals that

is a boundary for

A that always exists.

make the following definition:

With this in mind we

o

242

9. Boundaries

Definition 9.4.2. Let X be a compact Hausdorff topological space and let A be a separating function algebra on X that contains the constants. If for each t E X we set Tt(f) = f(t). f E A. and B~ = (x* I x* E A*. IIx*1I = x*(l) = 1), then xA = (t I t E X. Tt E ext(B~)} is called the Choquet boundary for The next corollary follows immediately from Theorem 9.4.1. Corollary 9.4.1. Let X be a compact Hausdorff topological space. If A is a separating function algebra on X that contains the constants. then (i)

(ii) (iii)

xA

~

,.

XA is a boundary for

A.

cl(XA) = aA.

From our discussion of boundaries for specific algebras in the preceding section we see at once that XC(X) = X, X being a compact Hausdorff topological space, xA(D) = f. and xP(D 2) = pP(D 2) = ap(D 2). The Choquet boundary can, however, be a rather complicated set. If X is metrizable, then xA is always a G6-set, but if X is not metri:able, then ~ may not even be a Borel set (see, for instance, [5, pp. S4 and 55, 138 and 139]). In Sections 9.6 and 9.7 we shall discuss a number of equivalent descriptions of the Choquet boundary in the case that A is norm closed. In view of Definition 9.2.2 and Theorem 9.4.1. it is evidently meaningful to consider the Choquet boundary XA = XA• whenever A is a commutative Banach algebra with identity. In particular, the Choquet boundary for such an algebra always exists. Before we contiftue an investigation of the Choquet boundary we wish to look at several consequences of the existence of the §ilov and Choquet boundaries.

A.

9.S. Applications of Boundaries

243

9.S. Some Applications of Boundaries. In this section we wish to prove some results about separating function algebras and commutative Banach algebras whose proofs utilize the notion of a boundary. Theorem 9.S.l. Let X be a space and let A be a separating tains the constants. If T : A ~ A such that T(l) = 1, then T

compact Hausdorff topological function algebra on X that conA is a linear isometry of A onto is an algebra isometry of A onto

A. Proof. Let T*: A* ~ A* denote the adjoint of T, that is, T*(x*)(f) = x*[T(f)], x* E A* and f E A. It is easily seen that T* : A.* -- A.* is also a surjective linear isometry. Moreover, since T(l) = 1 and T*[x*(l)] = x*[T(l)] = x*(l), x* € A*, we see that T* maps B~ = (x* I x* E A*, \lx*1I = x*(l) = lJ onto itself. From this observation it is readily deduced that T* maps ext (B 1l ) onto ext(B 11). Now, if t E XA, then Tt E ext(B~), and so there exists some s E XA such that T*(T) = Ts since ext(B 1l ) C T(X). Consequently t for each f,g E A we have T(fg)(t)

= Tt[T(fg)]

= Ts (fg) = (fg) (5) = f(s)g(s)

= Ts(f)TS(g) = T*(Tt ) (f)T*(Tt ) (g) =

T(f) (t)T(g) (t).

9. Boundaries

244

Since

t E XA

is arbitrary, we conclude that for each

f,g E A we

have T(fg)(t) = T(f)(t)T(g)(t), t E xA. whence T(fg)(t) = T(f)(t)T{g){t), t E X, as XA is a boundary for A. Hence T is an algebra isometry.

o

Next we wish to prove a result about topological zero divisors in separating function algebras and in commutative Banach algebras. Clearly a separating function algebra is a normed algebra, and so the notion of topological zero divisors is meaningful in this context. The results on topological zero divisors will be corollaries of the following theorem: Theorem 9.5.2.

Let

X be a locally compact Hausdorff topolo-

A be a separating function algebra on

gical space and let f E A, then

inf If (s) I s E: oA

X.

If

IIfgll m = inf IIglim gEA g~ 0

Proof.

If

g

E A and

t E oA be such that

19(t)1

g

is not identically zero, we let

= IIglimo

Then

inf If(s)1 < If(t)l -

s E: oA

Since this holds for each

g E A, g

inf If(s)l < a s E: oA

~ 0,

=

we conclude that Ii f

inf

gli m

I' II

g E A Ig

.

Q)

g~O

If

a

= 0,

the proof is complete, so we may assume that

a > O.

Let

E = (s I sEX, If(s)1 > aJ. Evidently E is a compact subset of X, as 0 < a < IIfll. If E were a boundary for A, then E would be a closed boundary for A, and so E ~ oA. From this it would -

Q)

9.5. Applications of Boundaries

245

follow at once that a ~ inf ,f(s)I ~ inf If(s),, s€E s E3A

which combined with the preceding estimate would show that inf If (s) I

s EoA

=

inf

UfgllCD

g€A

II II . g

CD

g~O

Consequently it remains only to show that E is indeed a boundary for A. However, if E is not a boundary, then there exists some g E A, g ~ 0" such that suPsEElg{s)l < Ilg11 CD " as E is compact. Moreover" it is easily seen for each positive integer k that

sup s( E

ISk~S)1 = sup[lg(s)l]k, IIg II m sEE ~

from ""hich it follo\\'s at once that limk[suPsEE(lgk(s)l/llgkUCD)] = O. Furthermore" given k" there exists some tk E 3A such that If(tk)gk(t k)I = IIfgkU CD " whence we deduce that IIfgkUCD a 0 for which IIxll 2 < Kllx 211, x E A. If x E A, then the following are equivalent: (i) (ii)

x is a topological zero divisor. There exists some T E oA

such that

•X(T) = O.

Our final application will be to obtain some results concerning the extension of maximal ideals. In particular, we shall show that every maximal ideal in the §ilov boundary of a closed subalgebra A of a commutative Banach algebra B with identity can be extended to a maximal ideal in B. More precisely, we have the following theorem: Theorem 9.5.3. Let B be a commutative Banach algebra with identity e and let A be a closed subalgebra of B that contains e. If M E oA c ~(A). then there exists some N E ~(B) such that N ~ M.

248

9. Boundaries

Let T denote the complex homomorphism of A such that M = T -1 (0) and suppose there exists no N E A(B) such that N ~ M = T-1(O). Then the ideal I in B generated by M must be all of Bj that is, the set Proof.

m

I = ( E XkYk 1 Xk € M, Yk E B, k = 1,2, .•. ,m),

k=l where m is an arbitrary positive integer, must be all of B. Otherwise, by Theorem 1.1.3, there would exist some N E A(B) for which N ~ I ~ M, contrary to assumption. In particular, there exist xk EM and Yk € S, k = 1,2, •.. ,n, such that e = tk=lx kyk , where, without loss of generality, we may assume that

lI~k IICD = sup

w£A(A) Furthermore, we observe that

I~k (w) 1 < 1

(k

= 1, 2, ... , n) •

sup 1~(UI)1 = limllxnll 1/n = sup l~(w)l wEArS) n w€A(A) by the Beurling-Gel'fand Theorem (Corollary 3.4.1). Now let

(x ( A),

a be so chosen that a>

=

•

sup [sup lYk(w)11 k=l,2, ... ,n w(A(B)

and consider the open neighborhood U of T in A(A), U = (u, = (w

defined by

• • Ixk(w) - Xk(T)1 < 1/2na, k = 1,2, ... ,n) • I Ixk(w)l

< l/2na, k

= l,2, ... ,n).

Since M = T-l(O) E oA, we see from Corollary 9.2.2 that there exists some yEA such that IIri! = 1 and I;(u.) I < 1/2na, w £ A(A) - U. Clearly y = ye = y(tk=lxkYk)' from which it follows that CI)

9.6. Representing Measures •

249

n •

•

ly(w) ~ xk(w)Yk(w)l = w~A(B) k=l sup

=

• lIyll •

= 1.

However, we also see that

•

n •

•

n...

sup ly(w) ~ xk(W)yk(w)l < sup [~ly(w)~k(w)lIYk(w)l] wEA(B) k=l w~A(B) k=l
O. Moreover, we see that, for each f E B(f),

9. Boundaries

256

However, our development at the beginning of this section entails that ~ = Vo since the representing measures in M(r) for the points of 0 are unique. Thus we have f(O)

= T(f) = Ir

't it f(e 1 ) dvO(e ) (f € A).

In particular, if f E A,

then

o = hk(O)f(O) = T(hk)T(f)

= T(hkf) = ~Jn 2n -n

..

= f(-k) that is, However, that the f on r

eiktf(eit) dt (k

= I ~ 2 , 3, ..• ) ;

the negative Fourier coefficients of each f £ A are zero. a classical theorem of Fourier series [E 2 , p. 87] asserts Cesaro means of the Fourier series of a continuous function converge uniformly to f; that is,

o = limllan (f) n

= lim[ n

- fll CD

sup -n 0,

then

~

We must show that, if ~

= 6t . II~ II

However, if

= JX 0, Y E G, then f(O) > o. Proof. Let f = fr. From Lemma 10.3.S we know that f(O) is real. Suppose f(O) < O. Then, since f is continuous, there exists a symmetric open neighborhood U of 0 that has co,pact closure for which If(t) - f(O)1 < If(0)1/2, t £ U. Recall that U symmetric means that U = -U. Let g E Ll(G) n L2 (G) be such that (a)

g * g* E Cc(G).

(b)

The support of g * g*

(c)

IG g * g*(s)

dA(s)

is contained in U.

= 1.

Such an element g can be obtained in the following manner: Let W be an open symmetric neighborhood of the identity in G such that W+ WC U and set gl = Xw' the characteristic function of W. Then g = gl/[iG gl * gies) dl(s)]1/2 has the indicated properties. We claim that f * g * g* £ CAo (G). Indeed, from Proposition 4.7.2(i) we see that f * g * g* E C(G). Furthermore, let (fn) C L1 (G) n C(G) be such that limn 1I1fn - 111 = 0 and limn IIfn - fll CD = 0, and set hn = fn * g * g*, n = 1,2,3, ..• , and S = TTg * g*. Obviously (hnl C LI(G) n C(G), while the estimates - S

II

=

liTf

n

* g * g*

- TTg

< 1I1f - TlIlITg * g*1I n

and

* g* II (n =

1,2,3, ••• )

10.3. The B*-Algebra A (G)

291

o

show that f * g * g* E CA (G) and corresponds to S = TT * o g * g in Ao(G). Moreover, from Lemma lO.3.4(ii) and (v), we see that o

(f * g * g*) (y)

= (TTg

.

•

* g*) (y)

= T(y)(g

-

* g*) (y)

0·2

= f(y)lg(y)1

>0

(y E G).

Thus, in particular, from Lemma 10.3.5 we see that is real. Thus, on the one hand, the estimate If * g * g*(O) - f(O)1

= IIG

(f * g * g*)(O)

f(s)g * g*(-s) d1(s) - f(O)1

< Iu If(s) - f(O)lg * g*(-s) d1(s) < If(O)l 2

reveals that real.

f * g * g*(O)

~

f(0)/2 < 0,

q.

as

f * g * g*(O)

is

• • •

But, on the other hand, try) = T(y) > 0, y E G, and TEe (G) entail that (T)I/2 is a well-defined real-valued function in °Co(G). Since by Theorem 10.3.1 the Gel'fand transformation on Ao(G) is surjective, we deduce that there exists some PEA (G) such that • = (T) • 1/2 . Moreover, since A (G) is semisimple, 0 (T) • 1/2 P is real o valued, and the Gel'fand transformation is an *-isomorphism, we see that P is self-adjoint and p2 = T. Now let (gn) C Ll(G) be such that lim liT - pil = O. Since P is self-adjoint, we may assume, n gn without loss of generality, that g* = g , n = 1,2,3, ... , as was n n done in the proof of Lemma 10.3.5. Then for each n we have

292

10. B*-Algebras

lim liT - T *11 = O. In partin gn * gn cular. we must also have lim IIT£ - T *11 = O. Finally then, n n gn * gn since g * g* (; Cc (G), g E L2 (G), and limn IIfn - fll m = 0, we see that whence we conclude at once that

f * g * g* (0)

= iG

g * g*(-s)f(s) dA(s)

= lim iG

g * g*(-s)fn(s) dA(s)

= lim fG

Tf

n = lim f n * g * g*(O) n = lim Tf (g) * g*(O) n n n

(g)(s)g(s) dA(s) n

= lim

{Tf (g),g)

= lim

{Tg

= lim n

g * g* * g * g*(O) n n

= lim

(g

n n

n * g*(g),g} n n

n

= lim fG

* g) * (g

n

(g

n

= lim

IG

= lim

(lign

n

n

* g)*(O)

n * g)(s)(gn * g)*(-s) dA(s)

(gn * g)(s)(gn * g)(s) dA(s)

= lim rliTg n

n

n

*

g1l2)2

(g)1I 212

= rll p (g)1I 212 > O.

This, however, contradicts the previous observation that f

Therefore

f(O) >

o.

* g * g*(O) < o. [j

10.3. The B*-Algebra A (G)

293

o

We shall have need of one more result about CA (G). But in o order to prove it we need to introduce a new topology on G and establish a preliminary result. Lemma 10.3.7. Let G be a locally compact Abelian topological • group. For each ~ > 0 and each compact set KeG define N(K,~)

= {t

Then the family of sets

•

1 t E G, 1Ct,y) - 11 0 is arbitrary and

KeG is an arbitrary compact set, forms a neighborhood base at the identity of G for a topology that is weaker than the given topology on G. We shall see subsequently that the topology obtained as in the lemma actually coincides with the given topology on G. The proof of the lemma is left to the reader. The argument is essentially the same mutatis mutandis as the one given in the first half of the proof of Theorem 4.7.5, the crucial point being that the mapping • • from G X G ~ r, defined by (t,y) ~ (t,y), t E G and y EGis continuous. Lemma 10.3.8. Let G be a locally compact Abelian topological group. If KeG• is compact, 6 > 0, and a > 0, then there exists some g E C (G) such that c (i)

get) > 0, t E G.

(ii)

• • > 0, y E G. g(y)

(iii)

•

Ig(y) - al < 6, y E K.

Proof. From Lemma 10.3.7 we see that N(K,6/a) is an open neighborhood of the identity in G. Let U c N(K,6/a) be an open neighborhood of the identity in G that has compact closure and let W be a symmetric open neighborhood of the identity such that W + We U. Let g = c(Xw * Xw), where c is so chosen that

10. B*-Algebras

294

IG

g(s) dA(s) = a.

Since Xw * Xw is nonnegative, we see that c > O. Evidently g E Cc(G) , the support of g is contained in W+ W, get) ~ 0, t E G, and

•

•

g(v) = c(Xw * Xw) (V) =

c(Xw *

XW) • (V)

• 2 = cIXw(v)1

•

(V E G).

>0

Finally, if y E K,

•

19(y) - al

then

= IIG

g(s)(s,V) dA(S) -

~Iu g(s)l(s,v) ~

sup

s E N{K,6/a)

< 6.

11

I(s,v) -

IG

g{s) d1{s)1

dA(S)

11 IG

g{s) dA(S)

o

couple of additional remarks about this lemma are in order. Clearly the function g is of the form g = f * f*, where • = I-fey) 12 > 0, V E G. • Moreover, f e Ll(G) n L2 {G) n L.(G) and g(V) it is not difficult to see that we can even demand that f ( C (G). c Indeed, in the proof of the lemma we need only replace Xw by Xwl * XWI ' where WI is a symmetric open neighborhood of the identity such that WI + WI C W. We shall have need of these observations in the next lemma and in the next section. A

Lemma 10.3.9. Let G be a locally compact Abelian topological group. If ho E Cc (G) is real valued, then for each e > 0 there exist h,k E CA (G) such that

-

o

(i)

~,~ E C (G). c

10.3. The B·-Algebra Ao (G) (ii) (iii) (iv)

o

295

0

hand k are real valued.

o '"'"----~

0

..

key) < h (y) < hey), y € G. -

0

-

0 0, t e G, g (y) > 0, y E G, and Ig (y) - 21 < 1/2, o 0.. 0 y E K. In particular, since g is real valued, we have .. 0 go(y) > 3/2 > 1, y E K. Applying Lemma 10.3.8 again, this time with a = I and 6 > 0 arbitrary, we obtain a function gEe (G) • • • c such that get) > 0, t ~ G, g(y) ~ 0, y € G, and 19(y) - 11 < 6, Y E K. In particular, 1 - 6 < g(y) < 1 + 6, y E K. We shall specify 6 more precisely in a moment.

..

..

First, however, since ho E Cc (G) and the Gel'fand transformation on A (G) is a surjective isomorphism by Theorem 10.3.1, we o .. deduce the existence of a unique TEA (G) such that T = h. Let o

0

Tl = TTg+ 6g and T2 = TTg-~go ~ . We claim that Tl and T2 determine elemen~s of CA (G) for any 6. To see this it clearly suffices o to show that TT and TT determine elements of CA (G). But g go 0 suppose {f) c:: Ll(G) is such that lim IITf - Til = o. As observed n n n after Lemma 10.3.8, we may assume, without loss of generality, that g = f • f*, f E Ll(G) n L2 (G) n Lm(G). Consequently (fn • g) belongs to Ll(G) n C(G) and, using HHlder's inequality, we find from the estimates IIfn * g - f m • gilm = lI(fn - f m) • f * f*1I m < lI(fn - fm) • f1l211f*1I2 < IITf - Tf n

II

m

Uf1l211f*1I2

(n,m

= 1,2,3, ... )

and

IITf * g n

11 II < g -

IITf - TIIIITg II n

(n =

1,2, 3, ... )

10. B*-Algebras

296

that lim IITf - 17 II = 0 and that (f * g) cenverges uniformn n * g g n Iy to some function in C(G). Hence TTg determines an element of CAo(G). A similar argument shows that the same is true of TT . go Thus TT 6 and TT 6 determine elements of CA (G), g+ go g- go 9 o. • which we denote by hand k, respectively. Clearly h = T(T +6 ) •• 9 • • 0 g go = h (g + 6g) and k = h (g - 6g), whence we see that hand 000 00. k are real-valued functions in C (G). Moreover, if y € K, then • • c since g (y) > I and g(y) + 6 > 1, we have o

o

hey)

••

= ho (y)[g(y)

+ 6g 0 (y)]

o

0

whereas if y ~ K, then h(y) = 0 = h (y). Thus h(y) > h (y), • 0 .0 0 y (G. Similarly k(y) < h (y), y E G. Hence parts (i) through (iii) 0 of the lemma are established for any 6 > O. However, to obtain part (iv) we must put some additional restrictions on 6. Suppose go inequality,

= f 0* 0 f*, 0 f

E Ll(G)

n Lm (G). Then, by HHlder's

Ifn * go(O)l = Ifn * f 0 * f~(O)1 = ITf (fo> * f*(O)1 0

n = lIG Tf (fo)(s)£o(s) dA(s)1 n (n = 1,2,3, ••• ). < IITf (fo)1I2l\ f oll2 n Since (I\Tf III is a bounded sequence because lim UTf - Til = 0, n n n we conclude that {f * g (0») is a bounded sequence. Set n 0 x = supn Ifn * g0 (0)1 and suppose 0 < 6 < e/6x. Now from the proof that TT 6 and TT 6 determine elements of CAoCG) it is g+ go g- go immediately apparent that

10.4. Plancherel's Theorem

297

limllfn * (g n

+

6g 0 ) - hI! CD

=0

and limllfn * (g - 6g 0 ) - kll n

Q)

Thus there exists some positive integer n

=

o

o. such that, if n>n, -

0

then IIfn * (g

+

6g 0 ) - hll

Q)

< £3

and

JJ f n Therefore, for

Ih(O)

n

>n , -

0

* (g - 6g0 ) - k II CD < £3· we have

- k{O)1 < Ih(O) - f n * {g + 6g o )(0)1 + Ifn * (g + 6g 0 ) CO) - f n * (g - 6g 0 ) (0) I +

Ifn

* (g - 6g0 ) (0) - k(O)1

2£ 0, V E G, and such that F[TV(f~] = F(f), f E Cc(G) and V E G. With this in mind, let g E C (G) be real valued. c Then from Lemma 10.3.9 we see that sup{k(O) , k E CA (G), ~(V) < g(y), V E o -

= inf{h(O)

Gl

, h E CAo(G), ~(V) > g(V), V E Gl.

We denote this common vAlue by F(g). In this way we obtain a map• -~ where CR(G) • ping F: CR(G) denotes the space of real-valued c • c continuous functions on G with compact support. It is easily verified from the definition of F that F is linear and F(g) > 0 • Moreover, if f E- CA (G) whenever g E CR· (G) and g(V) > 0, V E G. oR. c 0 and f E Cc(G) , then, since in this instance we can take h = k = f in Lemma 10.3.9, we see that F(!) = f(O). We can extend F to • in the obvious manner: if gEe (G), • all of Cc (G) then c g = Re(g) + iIm(g), and we set F(g) = F[Re(g)] + iF[Im(g)]. Hence F is a positive linear functional on C (G) such that FC!) = f(O) c whenever f E CA (G) and tEe (G). Furthermore, we claim that o c F is not identically zero. Indeed, by Theorem 10.3.1, there exists some T E AO(G) , T ~ 0, • • such that TEe (G). Let gEe (G) C L2 (G) be such that c c IIT(g)1I2 > 0 and set f = T(g) * T(g)*, where, as usual, T(g)*(t) =

10.4. Plancherel's Theorem

299

From Proposition 4.7.2(ii) we see that fEe (G). Actuo ally, f E CA (G). To see this, let {f) c L1 (G) be such that o n lim IITf - Til = O. Moreover, lim IIT£* - T*II = 0 by Lemma 10.3.1. n n n n Now (fn * f*n * g * g*) c L1 (G) n erG) and for n = 1,2,3, ...

T(g)(-t).

IIfn * f*n * g * g* - T(g) * T(g)*11 m =

lI(fn * g) * (fn * g)* - T(g) * T(g)*11 m

=

IITf (g) * Tf (g)* - T(g) * T(g)*lI co n

n

< IITf (g) * Tf (g)* - Tf (g) * T(g)*lI co n

+

n

n

IITf (g) * T(g)* - T{g) * T{g)*lI co n

:s. IITf (g)* - T{g)*1I 2I1 Tf (g)1I 2

n n + IIT(g)*1I 2i1 Tf (g) - T(g)1I 2 n

= IITf

n

(g) - T(g)1I 2 [IITf (g)1I 2

+

IIT(g)1\2]'

n

Hence we conclude that limllf * f* * g * g* - T(g) * T(g)*11 n n n m Furthermore, for

= O.

n = 1,2,3, ...

IITfn * f*n * g * g*

whence we have limllT f * f* * g * g* - TT*Tg * g*1I = 0 n n n by the continuity of multiplication in a Banach algebra. Consequently f = T(g) * T(g)* ~ CA (G), and f corresponds to TT*T * E A (G). q • ..0. • g * g 0 However, f = T(T*) g(g*) E Cc (G), and so F(t)

= f(O) = T(g)

* T(g)*(O)

10. B*-Algebras

300

= fG T(g) (s)T(g)(s) dA(s)

= [IIT(g)1I 2]2 > O. Hence F is not identically zero. Finally, we claim that F is translation invariant. Indeed, by the same sort of argument as we used several times previously, it is not difficult to show that, if k (CA (G), then (·,w)k ( CA (G), • 0 0 0 0 w E G, and [(·,w)k] = T (k). The details are left to the reader. -w • Thus, since k(O) = [(.,-w)k](O), w £ G, we deduce that, given • • g (C R(G), we have for each w € G c F[Tw(g)]

G)

= sup{k(O)

IkE CA (G), ~(y) < T (g)(y), Y E o - w

= sup{k (0)

g(y), y E G} I k (CAo (G), T-w (k)(y) < -

o

•

o

•

= sup{k (0) I k (CAo (G), [(.,-w)k] (y) O. Then there exists some f E Cc{G) such that IIf - gll2 < 1/2. Moreover, we ~laim that we may assume without loss of generality that f = fl * f2 * f3' fk E Cc(G), k = 1,2,3. Indeed, from the proof of Theorem 4.7.2, if (u 1 is a family of symmetric open a neighborhoods of the identity in G that have compact closures and are such that n(tUa = {oj, then we know that {u] /A(U(t)1 a = {~. 'U a is an approximate identity for LICG). It is easily seen that, if f E Cc (G), then lima''If - f * u(tCII II = 0 and that (ua * ua1 c: Cc (G) is also an approximate identity. After combining these remarks it becomes apparent that we can assume f has the indicated form. We leave the details to the reader.

• and, by Theorem 10.3.1, the Gel'fand Now since •fl E Co(G) transformation on A (G) is a surjective isometry, we see that o •• there exists some TEAo (G) such that T E Cc (G) and

liT - fIll

=

liT - Tf II < 211f : f 1

CII

2

II .

3 2

Thus

c

O. Since Cc(G) is dense in L2 (G) and the Gel'fand transfor_ _ we see that _there exists some mation on Ao (G) is surjective, TEAo (G) such that _ TEec (G) and lIg - TII2 < e/2. Let the compact support of T be K and apply Lemma 10.3.8 with a = 1 and 6 = e/[2l1TII the existence of some fEe c (G: _ CD1)(K)1/2] to _deduce 1/2 for which I fey) - Il < el [211TIl 1)(K) ], y E K. The Haar measure CD 1) on G can, at this point, be any such measure. As indicated in

10. B*-Algebras

304

the remarks following Lemma 10.3.8, we may assume that f = f o * f*, 0 where f o (C c (G). Repeating the same argument as before, mutatis 0 __ mutandis, we see that T(f) = TCfo * f*) E V and TCf) = Tf. Con0 sequently

liT - TfU2

=

[fa

IT(y)1 2 cll - fcy)12) dll(y)]1/2

=

[f K

IT(y)12(ll - f(y)12) dll(y)]1/2

r

~ II ll..ll(K)I/2[ 11.11 2 T

e

c 1/21 1}(K) co

= 2' whence

e

e

0, a straightforward argument using the uniform continuity of h on aCT) reveals that there exist points tk £ o(T) ,

10. B·-Algebras

322 k = 0,1,2, ... ,n, for which

such that

-IITII

~ to < tl < ... < tn ~

IITII

and

n

lh(T) -

t tk[gt (T) - gt

(T)]l
0 and • such that t l .t 2•... ,tn are in G, then there exists some y E G Ih(t k ) - (tk,y)l < I, k = 1,2, ... ,n. Proof. By Corollary 10.7.2 we can consider h as an element • Since each tk (G defines the almost periodic function of bG. • • (t k ,·) E AP(G), k = 1,2, ... ,n, we deduce, on recalling that bG = • • ~(AP(G» and the definition of the Gel'fand topology on A(AP(G»). that (g

I

• Ih(t ) k

g E bG,

g(tk)1 < s, k

= 1.2 ••.. ,n)

10. B*-Algebras

332

•

defines an open neighborhood of h in bG. Consequently, on iden• for • with tCG) •C• tifying G bG, we see that there is some y E G • which Ih{t k ) - (tktY)I < 't k = 1,2, .•. ,n, because G is dense • in bG.

o

This corollary is actually an abstract version of the classical Kronecker approximation theorem, but we shall not pursue this observation since it requires more detailed information about the Bohr compactification of specific groups than we have at our disposal. For this and other general discussions of almost periodic functions and the Bohr compactification we refer the reader to [Bo; BR I , pp. 245-261, 430-438; HR 2 , pp. 310-312; Lo, pp. 165-173; Ru 1 , pp. 30-32; We, pp. 130-139].

REFERENCES

[A]

L.V. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.

[Ba]

G. Bachman, Elements of Abstract Harmonic Analysis, Academic Press, New York, 1964:-

[BaNr]

G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, 1966.

[Bo]

H. Bohr, Almost Periodic Functions, Chelsea, New York, 1951.

[B]

A. Browder, Introduction to Function Algebras, Benjamin, New York, 1969. N. Dunford and J. T. Schwartz, Linear Operators, Part General Theory, Interscience, New York, 1958.

!:

N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Interscience, New York, 1963. (OkRa]

C.F. Dunkl and O. E. Ramirez, Topics in Harmonic Analysis, Appleton-Century-Crofts, New York, 1971. R.E. Edwards, Functional Analysis:Theory and Applications, Holt, Rinehart, and Winston, New York, 1965.

R.E. Edwards, Fourier Series: A Modern Introduction, Vol. Holt, Rinehart, and Winston, New York, 1967. [Ga]

I,

T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969.

[GRk~]

I.M. Gel'fand, O.A. Raikov, and G.E. ~ilov, "Commutative Normed Rings", American Mathematical Society Translations (2)~,

[Go]

115-220(1957).

R.L. Goodstein, Complex Functions, McGraw-Hill, New York, 1965.

[He]

G. Helmberg.

~

Introduction

~

Spectral Theory in Hilbert

Space, North Holland, Amsterdam, 1969.

333

334

[H]

References

E. Hewitt, "A survey of abstract harmonic analysis", Surveys in Applied Mathematics: Some Aspects of Analysis and Probability. Vol. !. Wiley. New York. 1958. pp. 105-168. E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol. Springer-Verlag, Berlin-GHttingen-Heidelberg, 1963.

!,

E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol. II, Springer-Verlag, New York-Heidelberg-Berlin, 1970. [Hy]

H. Heyer, Dualitlt lokalkompakter Gruppen, Lecture Notes in Mathematics, No. 150, Springer-Verlag, Berlin-HeidelbergNew York, 1970.

[HIP]

E. Hille and R.S. Phillips, Functional Analysis and Semigroups, American Mathematical Society Colloquium Publication XXXI, American Mathematical Society, Providence, R.I., 1957. K. Hoffman, Fundamentals of Banach Algebras, Instituto de Matematica da Universidade-do Parana, Curitiba, Brazil, 1962. K. Hoffman, Banach sraces of Analytic Functions, PrenticeHall, Englewood Cli~s, N.J., 1962.

[Hr]

L. HHrmander, An Introduction to Complex Analysis in Several Variables, D. Van Nostrand, Princeton, N.J., 1966.

[J]

B.E. Johnson, "The uniqueness of the (complete) norm topology", Bulletin of the American Mathematical Society,73,537-539(1967).

[Ka]

Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.-----

[L]

R. Larsen, Functional Analysis: An Introduction, Dekker, New York, 1973.

[Lb]

G.M. Leibowitz, Lectures on Complex FURction Algebras, Scott, Foresman, Glenview, Ill., 1970.

[LeRd]

N. Levinson and R.M. Redheffer, Complex Variables, HoldenDay, San Francisco, 1970.

[Lo]

L. Loomis, An Introduction to Abstract Harmonic Analysis, D. Van Nostrand, Princeton,-W.J., 1953.

[Nb]

L. Nachbin, The Haar Integral, D. Van Nostrand, Princeton, N.J. J 1965.

[N]

M.A. Naimark, Normed Rings, Noordhoff, Groningen, The Netherlands, 1964.

References

335

[Ph]

R.R. Phelps, Lectures ~ Choquet's Theorem, D. Van Nostrand, Princeton, N.J., 1966.

[Pi]

H.R. Pitt, Tauberian Theorems, Oxford University Press, London, 1958.

[Po]

L. Pontryagin, Topological Groups, Princeton University Press, Princeton, N.J., 1958.

[Ri]

C.E. Rickart, General Theory of Banach Algebras, D. Van Nostrand, Princeton, N.J., 1960.

[RzNg]

F. Riesz and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.

[Ry]

H. Royden, Real Analysis, 2nd ed., Macmillan, New York, 1966.

[Ru l ]

W. Rudin, Fourier Analysis 1962.

~

Groups, Interscience. New York,

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. [5]

E.L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson, N.Y., 1971.

[Wa]

J.K. Wang, Lectures ~ Banach Algebras, Lecture Notes, Department of Mathematics, Yale University, New Haven, Conn., 1965.

[We]

A. Weil, L'integration dans les groupes topologiques et ~ applications, Actualites scientifiques et Industrielles 8690-1145, Hermann, Paris, 1951. J. Wermer, "Banach Algebras and Analytic Functions", Advances in Mathematics, Vol. 1, fasc. 1, Academic Press, New York, 1961. J. Wermer, Banach Algebras and Several Complex Variables, Markham, Chicago, 1971.

[Wr l ]

N. Wiener, The Fourier Integral and Certain of Its Applications, Dover, New York, 1958. N. Wiener, Generalized Harmonic Analysis and Tauberian Theo~, M.I.T. Press, Cambridge, Mass., 1964. A. Wilansky, Functional Analysis, Blaisdell. New York, 1964. A. Wilansky, TOpology for Analysis, Ginn, Waltham, Mass., 1970. J.H. Williamson,"Remarks on the Plancherel and Pontryagin theorems", Topology, 1,73-80(1967).

INDEX of absolutely convergent Fourier series, 22 radical, 82 separating function, 220 uniform, 255

A

..

A, 74

Ar,

316

A[e], 9 A(K),

17

AP(G), 18

almost periodic compactification, 329 almost periodic function. 18,323 apPlications of boundaries, 243-249

A(x), 150

A ~ (0), assumed after E!&! 4

approximate identity, 109 in LI(G), 109, 188

A_I' 29

Arens' Theorem, 48

A-I, 29

Arens-Royden Theorem, 158

connected component of the identity in, 156,158 B

AC(r), 17

Gel'fand representation of, 102 maximal ideal space of, 102 bG, 329 ~ilov boundary for, 228 bdy(E), 51 A(D), 17 8(a(T)), 317 Bishop boundary for, 228 Gel'fand representation of, 97 is not regular, 167-168 B*-algebra, 273 maximal ideal space of, 97 examples of, 273-275,282,323 ~ilov boundary for, 228 Banach algebra, 4 commutative semisimple, 81 adjoining an identity, 10 conditions to have an identity, algebra, 3 153-154,178 Banach, 4 finitely generated, 98 normal commutative. 176 commutative, 3 division, 35 regular commutative, 166 normed, 4 self-adjoint commutative, 132 337

Index

338

Tauberian commutative, 183 with involution, 273 belong locally to I, point, 201 at infinity, 201

at a

C(V) , 23

XA, 242 XE ' 93-94

C(X), 16 Beurling-Gel'fand Theorem, 81 Bishop boundary when X is compact metric, 227-228 Bishop boundary, example of nonChoquet boundary for, 242 existence of, 230-232 closed ideals in, 195 existence for a uniform algeexample of pC(X) = ~, 230-232 bra, 268 Gel'fand representation of, for A(D), 228 87-88 for C(X), X compact metric, maximal ideal space of, 86,90 227-228 is regular, 167 for a commutative Banach algesets of spectral synthesis for, bra, 226 196 for a separating function alge§ilov boundary for, 227 bra, 221 C (X), 16 o Gel'fand representation of, 88 Bishop-deLeeuw Theorem, 236-237 is regular, 167 Bohr compactification, 329 maximal ideal space of, 88 uniqueness of, 328 ~ilov boundary for, 227 boundary, applications of, 243-249 Bishop, 221 Choquet, 242 existence of, 242 for a commutative Banach algebra, 226 for a separating function algebra, 221 ~ilov, 225 existence of, 222

Cn ( [ a , b] ), 17

Gel'fand representation of, 92 maximal ideal space of, 92 ~ilov boundary for, 228 C*-algebra, 275 character, 331 continuous, 113

C, 3

characterization of singular elements, in C(X), 43 in commutative Banach algebras, 81 in self-adjoint semisimple commutative Banach algebras, 141

Cc(X), 16

Choquet-Bishop-deLeeuw Theorem, 271

CAo(G), 286

Choquet boundary, characterization of, 257,263 existence of, 242 for A(D), 242 for C(X), 242

C

cl(I), cl(E), 34,162 co(E), 233 co(E), 233

Index

339

for separating function algebras, 242

Oitkin's Theorem, 206 division algebra, 35

closed convex hull, 233 dual group, 127 closed ideals in C(X), 195 in L1 (G), G compact, 197

E

closure operation, 162 commutative algebra, 3

E, 162

Commutative Gel'fand-Naimark Theorem, 277

1}, 168 1}(x), 46

compactification, 323 almost periodic, 329 Bohr, 329 Stone-~ech, 90

ext(E), 233

complex homomorphism, 67

extreme point, 233

exp, 156

connected component, 15~1 of the identity in A , 156,158 continuity of inversion, 31 of quasi-inversion, 29 continuous character, 113 contour, regular, 143 spectral, 144 convex hull, 233 closed, 233

F

1, 105 o

f, 287

•f, 21,114,169,307-308 •f(k), 21 f

convolution, 19-20

* g, 19

IIfllAc'

22

IIflln, 18 D

IIfllp'

19

0, 17

IIfIlCD, 16,18

()A, 225

finitely generated Banach algebra, 98

A(A), 69 assumed non empty after E!&! 78 finitely generated commutative Banach algebra, maximal ideal discrete topological group, 106 space of, 99 Gel'fand representation of, 99 Ditkin's condition, 204

340

Index

Fourier coefficient, 21 transformation, 21,115

homomorphism, complex, 67 *-, 277

Fourier-Stieltjes transform, 128

homomorphism space, 69

function, almost periodic, 18,323 hull, 160 slowly oscillating, 190 hull-kernel closure, 162 functions which operate, 151 topology, 163 G

..

I

G, 113

I , 11 e

r, 17,21 • 116 f,

10 (E), 182

int(K), 17

y , 121

o Gel'fand-Mazur Theorem, 35

ideal, 4 left, 4 maximal, 4 modular, 7 primary, 216 proper, 4 regular, 7 right, 4 two-sided, 4

Gel'fand representation, 76 of AC(f), 102 of A(D), 97 of C(X), 87-88 of Co(X), 88 of Cn([a,b]), 92 of a finitely generated commu- identity modulo I, 7 tative Banach algebra, 99 of L1(G), 114 inverse, 5 of L (X,S,~), 93 left, 5 right, 5 Ge1'fand Representation Theorem, 74 Inversion Theorem, 298,301 CD

Gel'fand topology, 71

invertible, 5

Ge1'fand transform, ;6

involution, 273

Gel'fand transformation, 76

isomorphism, topological, 131 J

H

hel), 160

Jo(E), 182

Haar measure, 18

joint spectrum, 98

Index

341

quasi-inverse, 13 topological zero divisor, 40 zero divisor, 40

K

k (E), 160

kernel, 160

linear functional, multiplicative, 67 positive, 259

Kronecker approximation theorem, 332

Local Maximum Modulus Theorem, 220

L

L(V) , 22

M , 12 e •~, 128

A (G), 282 o

II~II, 20

A, 18

l~L 20

Ll(G), closed ideals when G is compact, 197 closed translation-invariant linear subspace of, 186 condition to have an identity, 107 contains an approximate identity, 110,188 failure of spectral synthesis when G is noncompact, 199 Gel'fand representation of, 114 is regular, 173 is semisimple, 117 is Tauberian, 184 maximal ideal space of, 114,126 satisfies Ditkin's condition, 214 sets of spectral synthesis, 215 sets of spectral synthesis when G is compact, 198 ~ilov boundary for, 228

~

Lp (G), 19

L (X,S ,toLL 18

~ Gel'fand representation of, 93

maximal ideal space of, 93 left ideal, 4 inverse, 5 modulus of integrity, 46

*

\I,

20

M(G), 20 is semisimple, 315 maximal ideal space of, 128 Malliavin's Theorem, 199 maximal ideal, 4 maximal ideal space, 69 of AC(r), 102 of A(D), 97 of C(X), 86,90 of Co(X), 88 of cn([a,b), 92 of a finitely generated commutative Banach algebra, 99 of LI(G), 114,126 of L (X,S,~), 93 of MTG), 128 maximum modulus set, 221 Maximum Modulus Theorem, 218-219 Mergelyan's Theorem, 101 modular ideal, 7

Index

342 modulus of integrity, 46 left, 46 right, 46

Polynomial Spectral Mapping Theorem, 57

Pontryagin Duality Theorem, 312 multiplicative linear functional, 67 positive linear functional, 259 primary ideal, 216 N

proper ideal, 4 N(K,e), 293,309-310

Q

nilpotent, 33 topological, 33 Noncommutative Gel'fand-Naimark Theorem, 280

quasi-inverse, 13 left, 13 right, 13

normal commutative Banach algebra,quasi-invertible, 13 176 quasi-regular, 13 normed algebra, 4 quasi-singular, 13 p

R P(K), 17 Parseval's Formula, 170,305

• 115-116

~

peak point, for a commutative Banach algebra, 226 R(K), 17 for a separating function algebra, 221 R(f), 55 weak, for a separating function algebra, 262 Rad(A), 82 peak set, for a separating function algebra, 262

pA, 221

radical, 82 perfect set, 205 radical algebra, 82 Plancherel's Theorem, 169,301, 307-308

regular commutative Banach algebra, 166 P1anchere1 transform, 169,305 contour, 143 curve, 143 Plancherel transformation, 169,305 simple, 143 simple closed, 143

Index

343

element, 5 ideal, 7 representing measure, 253 and the Choquet boundary, 257 and the §ilov boundary, 269-270 Riemann-Lebesgue Lemma, 114 right ideal, 4 inverse, 5 modulus of integrity, 46 quasi-inverse, 13 topological zero divisor, 40 zero divisor, 40

for C(X), 227 for Co(X), 227 for CO((a,b]), 228 for L (G), 228 for a !ommutative Banach algebra, 226 for a separating function algebra, 225 §ilov-Arens-Calderon Theorem, 151 simple closed regular curve, 143 regular curve, 143

a(x), 54

singular element, 5 in C(X), 43 in a commutative Banach algebra, 81 in a self-adjoint semisimple commutative Banach algebra, 141

*-homomorphism, 277

slowly oscillating function, 190

s

self-adjoint, 132,275 smooth are, 143 commutative Banach algebra, 132 element in a B·-algebra, 275 Spectral Mapping Theorem, 152 semisimple, 81

spectral radius, 58

separating function algebra, 220 Spectral Radius Formula, 58 set of spectral synthesis, 194

spectral radius norm, 130

for C(X), 196 for Ll (G), G compact, 198 spectral synthesis, failure in for L (G), 215 Ll(G) when G is noncompact, for a ~emisimple regular commu199 tative Banach algebra that in C(X), 196 satisfies Ditkin's condiin L1(G), G compact, 198 tion,208-209 in Ll (G), 2l~ in a commutatIve B·-algebra, set, of spectral synthesis, 194 278 in a semisimple regular commuperfect, 205 tative Banach algebra that satisfies Ditkin's condi§ilov boundary, 225 and representing measures, tion, 209 problem of, 194 269-270 characterization of, 225 set of, 194 existence of, 222 spectrum, 54,69 for AC (n, 228 joint, 98 for A(D), 228

Index

344 Stone-~ech compactification, 90

strong boundary point, 222,262 structure space, 69 suba1gebra, 4 superalgebra, 48 support of

~,

235

symmetric set, 275 T

IITII,

Ditkin, 206 R.E. Edwards, 38 Gel'fand, 23,74 Ge1'fand and Mazur, 35 Gel'fand and Naimark, 277,280 Ma11iavin, 199 Mazur, 40 Mergelyan, 101 Pitt, 191 Plancherel, 169,301,307-308 Pontryagin, 312 Riemann and Lebesgue, 114 ~ilov, 61,222 ~ilov, Arens, and Calderon, 151 Tauber, 189 Wermer, 255 Wiener, 104,190 Wiener and L~vy, 154

22

T , 46 x TX, 46

topological isomorphism, 131 . zero divisor, 40-41 in C(X), 43 in a separating function algebra, 246-247

Tf' 280 topologically nilpotent, 33 T (f), 18,104

s T(G) , 324

T,

66

totally disconnected, 95 transform, Fourier, 21,115 Ge1'fand, 76 P1ancherel, 169,305

T , 69-70

e

T , 70 CD

Tt , 86

(t,y), 114

transformation, Fourier, 21,115 Gel'fand, 76 order preserving, 93 P1anchere1, 169,305 translation invariant linear subspace, 186

Tauberian commutative Banach algebra, 183

translation operator, 18,104

Tauber's Theorem, 189

trigonometric polynomial, 187

theorem of Arens, 48 Arens and Royden, 158 Beurling and Gel'fand, 81 Bishop and deLeeuw, 236-237 Choquet, Bishop, and deLeeuw, 271

two-sided ideal, 4 topological zero divisor, 40 zero divisor, 40

Index

345

•X,

U

74

o

x, 174

U(K,c), 123

U(T;C;x l ,x 2, •.• ,xn), 71

x*, 139,273

uniform algebra, 255

IIxlla,

58

• lIxll.,

134

unique maximum point, 222

x

y, 13 ~(x), 46

v

0

z

vanish on an open set, 235

w

Z, 3

•

Z, 116

weak peak point, 262 Wermer's Maximality Theorem,

Z(x), 160 255

Wiener's Tauberian Theorem, 190 Wiener's Theorem, 104 Wiener-L:vy Theorem, 154

x x

-1

,6

Z (I), 160

zero-dimensional, 93 zero divisor, 40-41 left J 40 left topological, 40 right, 40 right topological, 40 topological, 40-41 two-sided, 40 two-sided topological, 40 zero set of an element, 160 of an ideal, 160

about the book . . .

This textbook i~ designed to provide an introduction to the theory of commutative Banach algebras. The book not only dea ls with abstract theory, but illustrates the usefuln ess of Banach algeb ras in the s tud y of harmonic analysis and function algehras as we I!. The first half of the book is primarily devoted to the general theo ry of Banach algeb ras. Many spec ific examples are included in this part to illustrate the princi pies discussed. The second half o f -the book cons iders more speciali zed '.opics, among which are: Wie ner's Tauberian Theorem; the problem of spectral synthes is ; th e Bishop. 'Choquet, and Silov boundaries; and Wermer's Maximality Theorem. The second part also examines th e Commutative Gel'fand-Naimark Theo rem , Planchere l's Theorem, th e Pontryagin Duality Theorem, almost periodic fun cli o ns. and th e Bohr com pacti fi Ci_lti on.

Bana('h Algebras i ~ a stimul at ing text ro r g raduate stude nts with a backg ro und in abs trac t algebra. ge neral w[',,:ogy, real and com plex analysis , and fun ctional analys is.

about the author .. . RONALD L ARSEN has been Associate Professor of Mathemati cs at Wcsicyan Universit y in Middletown, Connecticut. since 1970. He previously tau ght at the Un iversity of California al Sanla C ru z (1965- 70).a nJ ~t Yale Univers ity (1963- 65). He received hi s B.S .

(1957) and M.S. ( 1959) from Michigan State University and his Ph. D. (1964) from Stanford University .

Professor Larsen 's rcsC!arl:h interes ts center on harmo ni c analysis and Banach algebras. He is the author of thre,e books. The Multiplier Problem, Allllllroductioll to the Theory of Multipliers, and Functional Allal)'sis: All Iwroc/uction (Marcel D=k ker, In c.), and numerous articles. The author wa ~ the recipient of a Ful bright-Ha ys Act Advanced Research Grant to the University of Oslo ( 1968-69) . He is a mem ber of the Ameri ca n M.Hhematical Society. the Mathe mat ical Assoc iation of Am eri ca , an d the Norsk Malematisk Forening.

Prin(ed ill rltl' Ullired Sf{/Il:'.~ of AIII('rica

ISB N: 0 - 8247- 6078- 6

MARCEL DEKKER , INC., NEW YORK

Banach algebras: An introduction (Pure and Applied Mathematics: A Series of Monographs and Textbooks 24)

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