Introduction to function algebras (Mathematics lecture note series)

MATHEMATICS LECTURE NOTE SERIES E. Artin and

J. Tate

CLASS FIELD THEORY

Michael Atiyah

K-THEORY

Hyman Bass

ALGEBRAIC K-THEORY

Melvyn S. Berger Marion S. Berger

PERSPECTIVES IN NONLINEARITY

Armand Borel

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Andrew Browder

INTRODUCTION TO FUNCTION ALGEBRAS

Paul

J. Cohen

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Eldon Dyer

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Walter Feit

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John Fogarty

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William Fulton

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Marvin J. Greenberg

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Serge Lang

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Serge Lang

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Ottmar Loos

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I. G. Macdonald

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George W. Mackey

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Andrew Ogg

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Walter Rudin

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jean-Pierre Serre

ABELIAN /-ADIC REPRESENTATIONS AND ELLIPTIC CURVES

jean-Pierre Serre

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jean-Pierre Serre

LIE ALGEBRAS AND LIE GROUPS

Shlomo Sternberg

CELESTIAL MECHANICS

A Note from the Publisher This volume was printed directly from a typescript 'prepared by the author, who takes full responsibility for its content and appearance. The Publisher has not performed his usual functions of reviewing, editing, typesetting, and proofreading the material prior to publication. The Publisher fully endorses this informal and quick method of publishing lecture notes at a moderate price, and he wishes to thank the author for preparing the material for publication.

INTRODUCTION TO FUNCTION ALGEBRAS

ANDREW BROWDER Brown University

W. A. BENJAMIN, INC. New York

1969

Amsterdam

INTRODUCTION TO FUNCTION ALGEBRAS

Copyright © 1969 by W. A. Benjamin, Inc. All rights reserved

Library of Congress Catalog Card number 68-59248 Manufactured in the United States of America 12345MR32109

The manuscript was put into production on February 13, 1969. this volume was published on March 15,1969

w. A. BENJAMIN, INC. New York, New York 10016

INTRODUCTION

The subject of function algebras has been receiving an increasing amount of attention in Several excellent survey articles

recent years. have appeared,

perhaps others,

by Wermer,

Hoffman,

Royden,

and

but there seems to be a place for

a volume which gives a detailed account of some of the more important results and methods,

with­

out attempting the depth of coverage of a treatise such as Gamelin's book

[2],

soon to appear.

The theory of function algebras draws from two sources:

functional analysis,

and the theory

of analytic functions of several complex variables. The reader may turn to the first chapter of the book of Gunning and Rossi [1],

[1],

or to Hormander

for an introduction to function algebras

from the point of view of several complex vari­ ables.

This volume is meant to expound on some

of the applications of functional analysis,

vii

with

viii

FUNCTION ALGEBRAS

only a few indications given as to the relevance of several complex variables. We envision the reader as a graduate student, or mathematician with another specialty, who knows a reasonable amount of integration theory and functional analysis in the abstract, without necessarily having seen many applications. Among our purposes is to show him the theorems of Riesz, Banach and their successors being applied, and to show him some of the Banach space framework in which classical function theory rests. The beautiful idea of Frigyes Riesz, to study approximation problems by passing to the dual space, still has a large amount of energy left, and the pages that follow will show some of the ways in which this idea has flourished a half century after its inception. The Poisson formula, the Schwarz-Pick lemma, Jensen's inequality, Hadamard's three circles theorem, these and other classical results have abstract formulations, an acquaintance with which can only enrich one's mathematical culture. This book is based largely on lectures given at Brown University in the spring of 1966, to a class consisting mainly of second-year graduate students. The background necessary for reading the book is then the background of these students, who had taken the usual year courses in real and complex analysis, plus a semester course in functional analysis. The usual introductory courses in analysis are by now sufficiently standardized so that little need be said about them. Today,

INTRODUCTION

ix

every second year graduate student can be expected to know the basic results of Banach space theory: the Hahn-Banach theorem, open mapping and closed graph theorems, uniform boundedness principle. We shall expect, in addition, a familiarity with the separation theorem (the best form of the Hahn-Banach theorem), the weak-* topology in the dual space of a Banach space, the Banach-Alaoglu theorem (if B is a Banach space, the closed unit ball in B* is weak-* compact) and its converse, the Banach-Krein-Smulian theorem (a subspace of B* is weak-* closed if its intersection with the closed unit ball is weak-* compact). At one or two places, we use the theorem that an operator between Banach spaces has closed range if and only if its adjoint does; the range of the adjoint is closed if and only if it is weak-* closed. We will also need the Krein-Milman theorem. A convenient source for all this is the book of Dunford and Schwartz [1]. Because of the limited size of this book, and my desire to give arguments in full detail whenever possible, many important topics are only mentioned in passing, with a reference, and many interesting results, and their authors, do not get mentioned at all. The choice of what to include or leave out has been highly subjective. In addition, the history of the subject that the reader may glean from this book is distorted. While I have made an effort to attribute theorems correctly, I am aware that important contributions

x

FUNCTION ALGEBRAS

have been slighted. Many results appear without attribution. In some cases, these are results which were found independently by a large enough number of people, or received enough oral circulation prior to publication to merit being placed in the category of mathematical folklore. In other cases, I have simply not been able to determine who exactly proved what, and when. I wish to thank Alfred Hallstrom, Eva Kallin, Kenneth Preskenis, and John Wermer for reading the manuscript and making valuable comments. My thanks go also to Mrs Roberta Weller and Mrs. Evelyn Kapuscinski, who carried out the arduous job of preparing the camera copy.

TABLE

Chapter One:

OF

CONTENTS

Banach Algebras

1-1

Function algebras

1

1-2

Banach algebras

8

1-3

The maximal ideal spaces of some examples

29

1-4

The functional calculus

46

1-5

Analytic structure

56

1-6

Point derivations

63

Chapter Two: 2-1

Measures

Representing and annihilating measures

79

2-2

The Choquet boundary

87

2-3

Peak points

96

2-4

Peak sets and interpolation

102

2-5

Representing measures and the Jensen inequality

114

xi

xii

TABLE OF CONTENTS 2-6

Representing measures and Schwarz's lemma

127

2-7

Antisymmetric algebras

136

2-8

The essential set

144

Chapter Three:

Rational Approximation

3-1

Preliminaries

149

3-2

Annihilating measures for R(X)

158

3-3

Representing measures for R(X)

170

3-4

Harmonic functions

179

3-5

The algebras A(X) and AX

195

Chapter Four:

Dirichlet Algebras

4-1

Dirichlet algebras

207

4-2

Annihilating measures

215

4-3

Applications

228

4-4

Analytic structure in the maximal ideal space

Appendix:

243

Cole's Counterexample to the Peak Point Conjecture

Bibliography

255 263

CHAPTER BANACH

1-1

ONE

ALGEBRAS

FUNCTION ALGEBRAS

Let X be a compact topological space. We denote by C(X) the set of all continuous functions from X to the complex field

¢.

For s e

¢,

we shall

denote by the same letter s the constant function which takes only the value s on X. If we define addition and multiplication to be the pointwise operations, C(X) is a commutative associative algebra over

¢.

For f e C(X), we define ~ f ~

sup{lf(x)1

x e X}; if K

sup{ I f(x)

x e K}

I

c

=

X, we put ~f~K to be

(thus Ilfll

= Ilfllx). It is an

elementary theorem of analysis that if {f } is a n

Cauchy sequence with respect to this norm, i.e., if ~f n - f m "

+

0 as n, m

+

00, then {f n } converges, 1

2

FUNCTION ALGEBRAS

i.e., then there exists f e C(X) such that ~fn -

fl

O.

+

Thus

this sup norm. all

is a Banach space with

We note that

separates the points of

'}

x,y e X, x

that

r

f(x)

DEFINITION: X

I f II· I g I for

~ is a set of functions on a space

we say that every

II fg II .::.

11111 = 1.

f,g e C(X), and that If

on

C(X)

r y,

there exists

X

X,

if for

f e rsuch

fey).

We say that

A

is a function algebra

if: i)

X

ii)

A

is a compact topological space; c

C (X) , and

A

separates the points of

X·, iii) i v)

1 e A;

A

is a closed subalgebra of

C (X) .

We observe that ii) implies that

X

is

Hausdorff, and that i) and ii) together imply that the given topology of

X

is equal to the weak

topology (the weakest topology on every function in and

A

X

is continuous).

A

for which For if ~

~ denote the topologies, the identity map of

(X,~)-+ (X,~

is continuous, and

(X,Yj

is Haus-

BANACH ALGEBRAS

3 (X,~)

dorff by ii); since

is compact, the map is

a homeomorphism. If

X

is a compact Hausdorff space, C(X)

is a function algebra on

X.

Here are some other

examples: Let by

P(X)

t.

X be a compact subset of

We denote

the set of all continuous functions on

X by poly-

which can be uniformly approximated on nomials in

z.

X

Here, and throughout this book, Z

denotes the identity function of

C

+

t, or its

restrictions. More generally, if of

is a compact subset

tn, the n-dimensional complex Euclidean space,

we denote by tions on on

X

P(X)

the set of all continuous func-

X which can be uniformly approximated

X by polynomials in

zl'

...

Here

,zn.

denotes the j-th coordinate function on Zj(SI'

••.

,sn)

=

Sj. ~

of C(X) which separates the points of

X.

X

smallest closed subalgebra tains

7

A

of

C(X)

a subset The which con-

and the constant functions is called the

function algebra generated by

?

J

tn:

be a compact space, and

Let

z.

is a set of generators for

1 ; A.

we alse say that If

-:r

is a

FUNCTION ALGEBRAS

4

7- =

finite set,

{f l , ... ,f n }, let

y = {(f I (x), ... , fn ex)) : x e X}. compact subset of

en

Then

homeomorphic to

is isometrically isomorphic to

pey).

Y

is a

X, and

A

Thus the

study of finitely generated function algebras can be regarded as a branch of the theory of functions of several complex variables.

This approach leads

to some of the deepest and most interesting results in the subject.

However, in this book we will not

adopt this viewpoint, but limit ourselves to what can be learned by the methods of functional analysis. If A(X)

X

is a compact subset of

to be the set of all

f e C(X)

holomorphic in the interior of all those functions on

tn, we define which are

X, and

ReX)

to be

X which are uniformly

approximable by rational functions with no poles on p

X, i.e., by functions of the form and

q

are polynomials in the coordinate func-

tions, and

q

P(X)

c

c

R(X)

p/q, where

has no zeroes on A(X).

X.

Evidently,

Each of the inclusions may be

proper; we will study the situation in more detail in Chapter 3 for

n = 1; for

n > 1, the questions

become much more difficult, and the subject remains

BANACH ALGEBRAS

5

in a primitive state. The example that will serve as a touchstone throughout this volume is the following.

=

~

r

E

{s

E ~

refer to

I}, the closed unit disk, and


O}

-

is also known as the algebra of generalized analytic functions, or the big disk algebra.

If

G

is

BANACH ALGEBRAS

7 ~

the group of integers, then

= r,

and

A

exactly the disk algebra (on the circle).

G =

n,m

{n + rna

tional, then

G

integers}

=r

x

r,

where

a

is If

is irra-

and we recover the big

disk algebra first described, which might more properly be called the little big disk algebra. Further generalization is possible:

G

might be

any ordered group, not necessarily a subgroup of the reals.

We will not pursue the matter here.

Given a function algebra natural question to ask is:

A

does

on A

X, the most

=

C(X)?

The

fundamental theorem here is due to M. Stone, and is known as the Stone-Weierstrass theorem: A

=

C(X)

if and onZy if the reaZ functions in

separate the points of and only if

I

€

A

X

(or equivalently, if

whenever

f e A).

This theorem

by no means ends the discussion, however.

For

instance, we shall find in Chapter 3 that if is a compact subset of

A

X

([, with empty interior and

connected complement, then

R(X)

=

C(X); but there

is usually no direct way to exhibit any non-constant real function in

R(X).

Again, in cases where

A f C(X), it is not usual to prove this by exhibiting a function in

A

whose conjugate does not

FUNCTION ALGEBRAS

8

belong to

A, or a pair of points in

distinguished by any real function in More generally, we can ask:

if

X not A. A

~

CeX),

or possibly with some stronger hypothesis, to what extent do the functions in morphic functions?

A

behave like holo-

I.e., to what extent is the

disk algebra, for instance, prototypical?

On the

most superficial level, we observe for instance that there is a shortage of real functions among holomorphic functions, and this persists (StoneWeierstrass) whenever

A

~

CeX).

More interesting

is the appearance of such phenomena as the Jensen inequality, Schwarz's lemma, and the maximum modulus principle, for instance, or the existence of point derivations, in situations of great generality; and it is especially interesting to deduce the presence of genuine analyticity from hypotheses of a general character, as we shall do in §4-4.

1-2

BANACH ALGEBRAS

DEFINITION:

We call

A a normed aZgebra if

has the structures of normed linear space and associative algebra over

~,and

if

A

BANACH ALGEBRAS IIfg II

.::.

IlfJI ~g II

9

for all

f ,g

A.

€

If

A

has a

multiplicative unit, which we may denote by

1

11111 = 1.

without danger, we require also that

A

Banaah aZgebra is a complete normed algebra.

It is clear that a function algebra is a commutative Banach algebra with unit.

Our interest

in the more general category has two sources. Firstly, many theorems concerning function algebras have proofs that go over, word for word, to Banach algebras.

Secondly, Banach algebras which are not

function algebras may arise in the course of studying function algebras. Here are some examples of Banach algebras which are not function algebras. Let

~

be a positive measure (non-negative

extended-real valued countably additive function defined on a a-algebra of subsets of some set). We recall that

L

00

(~)

is the set of equivalence

classes of essentially bounded measurable functions, when

f

and

g

are called equivalent if

almost everywhere

(~),

tially bounded if

II f II =

inf{t

.. I fl < t -

and

f

f

=g

is called essen-

esssupifl

almost everywhere

(~)}




FUNCTION ALGEBRAS

10

With the pointwise operations and the essential 00

sup norm, L

is a commutative Banach algebra with

unit. Let

Y be a set, and let

B(Y)

be the set

of all bounded complex valued functions on

Y.

With the pointwise operations and the sup norm, B(Y)

is a commutative Banach algebra with unit.

This example is contained in the last one: 00

B(Y) = L

(~),

where

~

is counting measure.

These two examples, as we shall see in the sequel, are merely function algebras in disguise but the disguise is only penetrated with the aid of some general Banach algebra theory.

Here are

two other examples, which are not realizeable as function algebras:

let

C(n)[O,l]

be the space

of n-times continuously differentiable functions on the closed unit interval.

With the pointwise

operations and the norm n

IIfll

=

Io K\.

sup{lf(K)(t)1 : 0 ~ t ~ l}, c(n)[O,l]

is a commutative Banach algebra with unit, the prototype of a class extensively studied by Shilov (see Merkil [1]).

Let

the unit circle, and let

A

r

denote as before

be the algebra of all

BANACH ALGEBRAS

11

continuous functions on

r

convergent Fourier series:

which admit absolutely f

€

A

if and only if

00

~f~ = Ilcnl


I m+l I

Ck

€

A,

gm gk II,

17

BANACH ALGEBRAS -1 mm e A

f g

so

by Theorem 1.2.1, whence

gm-

1

e A.

This accomplishes the inductive step and completes the proof. We now return to the general theory of Banach algebras.

THEOREM 1.2.5.

unit, f e A. Proof. -

S

Let

A

Then

spec f

Suppose that

f .: '- A-I

be a Banach aZgebra with is not empty.

spec f

f or· every

is empty, so that

seC.

continuous linear functional on F(s)

ep((s - f)-I)

=

for each

Let

ep

be any

A, and put Since

set.

(s - f)-l - (t - f)-l = (t - s)(s - f)-let - f)-l s, t e t, as can be seen by multiplying

for all

both sides by

(t - f)(s - f), we have

F(s) - F(t) = -ep[(s _ f)-let _ f)-I]. It follows s - t from Corollary 1.2.3 that F is entire, and I

F (s)

= -ep((s - f)

-

2

)

for all

Since

set.

(s - f)-l

as

s

-+

00

by Corollary 1.2.3,

F(s)

-+

But an entire function which vanishes at

0

as 00

be identically zero, by Liouville's theorem.

s

-+

must Thus

00.

FUNCTION ALGEBRAS

18 ¢((s - f)-I) =

°

for all

Since

5 •

¢

was an

arbitrary linear functional, it follows that -1 (s - f) 0, for all s. But this is impossible for any

s, and this contradiction proves the

theorem. Since the completion of a gebra is a Banach algebra, that spec f is not empty in any normed algebra with

A normed algebra with the prop-

COROLLARY 1.2.6.

erty

A-I

Proof.

is one-dimensional.

A\{O}

If

f e A, there exists

Theorem 1.2.5, such that esi~,

normed alit follows for any f unit.

this implies

(5 - f)

s e

~,

f

A-I.

by By hypoth-

f = s.

This result is known as the GelfandMazur theorem.

LEMMA 1.2.7.

Let

A

unit~

Then

5

5

= tn

Proof.

f

e A.

for some tl ,

Let

Then

fn -

that

fn - s

5

be a Banaoh algebra with e spec f

n

if and only if

t e spec f.

.. .

,tn

be the n-th roots of s .

= (f - t l ) ... (f - t n ) , and

i t is clear

fails to be invertible if and only if

BANACH ALGEBRAS f - t.

19

fails to be invertible for some

J

j, which

is the assertion of the Lemma. We combine this lemma with a refinement of the argument used in Theorem 1.2.5 to prove a quantitative version of this theorem, known as the speotral radius formula.

THEOREM 1.2.8. unit~

e A.

f

Let

A

be a Banaoh algebra with

Then

lim Ilf n Ill/n.

sup{ I s I : s e spec f}

n+ oo

Proof.

Let

s e spec f.

By Lemma 1.2.7,

sn e spec f n , and by Theorem 1.2.1 it follows that Isnl < Iif n ll lsi 2. ~fnlil/n Let

R

=

for every positive integer for all

lsi 2. inflfnli l/n .

n, so

sup{ I s I : s e spec f}.

uous linear functional

~

on

n, i.e.,

For each continA, put

F(s) = ~((s - f)-I); as we observed in the proof of Theorem 1.2.5, F hence in

is ho10morphic in

{s ell:: I s I

>

R}.

(\spec f,

Since (Theorem 1.2.1)

20

FUNCTION ALGEBRAS

for

I s I > Ilf II, it follows from the elements of

complex function theory that the series converges n

for

In particular, {H f n)}

I s I > R.

~

sequence for any

* A , 151

e

s

> R.

is a bounded

By the Banach-

Steinhaus theorem (uniform boundedness principle), fn it follows that is a bounded sequence in A, n s

for

lsi> R.

constant

Thus, if

K such that

lsi> R, there exists a ~fnll < Klsl n , for all

lim sup Ilf n Il lln .::. I s I.

and hence

n,

Thus

lim supllfnli l/n .::. R'::' infllfnli lln , and the theorem is proved.

DEFINITION.

Let

We say that

~

on

A

if

~

be a Banach algebra with unit.

is a multiplicative linear functional is a non-zero linear functional on

~

A, such that i.e., if

A

=

~(fg)

for every

~(f)~(g)

is a homomorphism of

A

onto

f,g e A;

t.

We

denote the set of all multiplicative linear functionals on

A

by

Spec A.

It is obvious that if

~(l) = 1, and for particular,

~(f)

~

e Spec A, then

f e A-I, ~(f-l) = (~(f))-l; in

1 0

for all

f

-1

eA.

BANACH ALGEBRAS LEMMA 1.2.9.

21

Let

~

e Spec A.

Then

tinuous "linear functional .. and

Proof.

Let

f e A.

II ~ II

~

(f) :f s.

for any

¢(s - f) :f 0,

s, I s I > IlfL by Theorem 1.2.1, so i. e. ,

1.

=

-1 s - f e A

Then

is a aon-

~

I ~ (f) I ~ Ilf II, for any

Thus

f e A, which was to be proved.

DEFINITION. For each

Let

A

be a Banach algebra with unit.

f e A, we define the Gelfand transform A

f ~

of

f, f

Spec A

e Spec A.

-+

t, by

f(~)

=

~(f)

for

We define the Gelfand topology on A

Spec A

to be the weak

A

topology, i.e., the

weakest topology on

Spec A

functions

are continuous.

f(f e A)

for which all the

Lemma 1.2.9 shows that of the closed unit ball of the map of

A

f

-+

into

f

Spec A

* A.

is a subset

We observe that

consists of the canonical injection

A** ,followed by restriction to

The Gelfand topology on

Spec A

weak-* topology.

{~e

Since

is weak-* closed for each

Spec A.

is the relative

A* : 1

Hence

1

for all

f I,

f e I,

and the

lemma is proved. We recall that if

I

is a linear subspace

A, the quotient space

of a linear space

the set of cosets of

by

and

g.-k, then

'IT.

SeC; thus If

A

'IT 1

f + g -h + k, and

A/I

sf -sh

onto f-h for any

has the structure of linear space. I

A, it is trivial that also

is a unit for

is a homomorphism. I

A

It is trivial to verify that if

is a commutative algebra.

then

and

if and only if

is a commutative algebra, and

ideal in A/I

f -g

Denote the canonical map of

A/I

is

I, i.e., the set of equivalence

classes under the relation: f - gel.

A/I

If

A/I. A

If

is an

fg ""hk, so A

has a unit,

It is clear that

'IT

is a normed linear space,

a closed subspace, then

A/I

is a normed

BANACH ALGEBRAS

25

linear space with the quotient norm Ilnf II = in£{ Ilg II : ng = nf}.

If

A

is a normed al-

gebra, it is trivial to verify that

II (nf) (ng) II if

A

< IInf

is, and

II ling II, I

so

A/I

is a normed algebra

is a closed ideal in

A.

If

A

is a Banach space, and I a closed subspace, it is well known that

A/I

is a Banach space.

If

is a commutative Banach algebra with unit, and a proper closed ideal, we have then that

A/I

a commutative Banach algebra with unit, for lin 111

1 by Theorem 1. 2.1.

=

Let X be a compact Hausdorff space, K a closed subset of X, I = {f e: C (X) : f = 0 on K}. Clearly I is a closed ideal in C(X). The reader may verify that C(X)/I is naturally isomorphic to C(K). If A is a function algebra, and I a closed ideal in A, the Banach algebra A/I need not be realizeable as a function algebra. For example, if A is the disk algebra, and I = {f e: A : f(O) = fICO) = O}, one easily verifies that A/I is a twodimensional algebra, with basis {l,nz}, and (nz)2 = 0 (of course, in a function algebra, f2 = 0 only if f

=

0).

A I is

26

FUNCTION ALGEBRAS

THEOREM 1.2.12.

A

Let

be a oommutative Banaoh

al,gebra with unit.

If

1, the problem of finding

even of determining if

X

X, or

is polynomially convex,

is much more difficult, and has only been solved for A

special cases.

Again, X

must contain every bounded

component of the complement of n > 1, much more is true:

X

(in fact, when

a theorem of Hartogs

states that every function holomorphic in a neighborhood of

X

extends to a function holomorphic

in every bounded complementary component). more is true:

if

R

is a finite bordered Riemann

surface holomorphically imbedded in aR

c

R

X, then

ciple. such an

c

X

tn, and

by the maximum modulus prin-

One might conjecture that. if R

whenever

ially convex.

Even

X

::>

aR, then

X X

contains is polynom-

But Stolzenberg [1] has given an

ingenious counter-example.

It is known that if

FUNCTION ALGEBRAS

42

the compact set

X

~

c

has connected complement,

then so does any homeomorphic image of

X

in

~.

Thus polynomial convexity is a purely topological property for plane sets. in higher dimensions:

This is no longer true

consider the circles

Xl

{s e 1[:2

Isll

1, s2

X2

{s e q:2

Isll

1, sl s 2

see that X2

Xl

= O} ,

and

H.

It is easy to

{s e «;2 : IS11 < 1, s2

is polynomially convex

algebra, while

(P(X l )

= O} ,

while

is the disk

P(X 2) = C(X 2)).

It can be shown quite easily that compact convex sets in

q:n

are po1ynomial1y convex, and

it is not hard to show that the union of two disjoint convex compact sets is also po1ynomial1y convex.

Need the union of three disjoint compact

convex sets be polynomia1ly convex?

This question,

apparently 'first raised by the German school of several complex variables in the early 30's, was only answered in 1965, by Kallin [2].

She showed

that the answer is yes, for disjoint closed balls, but no, for polycylinders. To study the maximal ideal space of we introduce the

following~

ReX),

BANACH ALGEBRAS DEFINITION.

43

Let

X be a compact set in

rationaZ convex huZZ of

s e [n p

X

is the set

with the property: pes) = 0

such that

(n.

XR

The

of all

there exists no polynomial

while

p

has no zeroes on

X. It is trivial that the rational convex hull of

X

is a compact set containing

X.

We call

X

rationaZZy convex if it coincides with its rational

convex hull. in

(

of

It is obvious that every compact set

is rationally convex.

From the definition A

XR we see that if

s e XR , f(s)

is well-

defined for every rational function poles on functions f

X; further, If(s) I ~ If Ix f.

p/q, where

q f 0

on

For if p

and

q

X, we see that

~

e Spec R(X), let

Then

~(p)

= pes)

s =

for all such

are polynomials and p - f(s)q

is a poly-

s, but at no point of

This shows that each point of Spec R(X).

with no

If(s) I > ~f~x, and

nomial which vanishes at

element of

f

XR

X.

define5 an

On the other hand, if (~(zl')

,

...

,~(zn))·

for any polynomial

polynomial with no zeroes on

X. q

~(p/q) = ~(p)/~(q) = *(5); thus

~

-1

p; if

q

e R(X), so

is determined

a

44 by

FUNCTION ALGEBRAS s,

~(p)

pes) = 0, then

Finally, s e XR' for if p ¢ R(X)-l, so

0, so

=

p

must vanish on

A

X,

Spec ReX) = XR'

Thus

A

Suppose

X

A

=

XR'

Then any polynomial

q

X

X

A

which has no zeroes on el·th er, so only if

q-l apeX), '-

A

A

X

XR'

has no zeroes on peX) = ReX)

Thus

if and

Let us examine the maximal ideal space of the big disk algebra ~

A, introduced on Page 6,

e Spec A, and put


+

ISlns2ml ~ 1, whenever

~

i = 1,2,

for

e z.) 1

1, i = 1,2, and since for all

Then Is. I 1 n

s. = 1

we have l~ezlnz2m) 1 = ° it follows that nloglsli

+

n + met ~ 0, so

IS21 = Isll et ,

Since

~

10gls2 1

Let

n,m

mlogls21 ~

etloglsll, i.e"

=

is clearly determined by

(sl,s2)' we have a one-one continuous map of into

{(sl'sZ) e 1[2 : 1511 ~ 1, IS21

Let now

(sl,s2)

be any point of

=

Isllet}

Spec A = K.

K, other than

s2 = eits l et for some real t, some et . determination of sl,l,e., there exists 1;;, ReI;; < 0, r s with s 1 = el;; ' 2 = e it e et "', Let f be any (0,0),

Then

"trigonometric polynomial" in sum of the form

'\

Lcnrnzl n Zz m'

A, i.e., a finite Put

°

BANACH ALGEBRAS F( s)

45

imt = ILcnme (n+ma)s e ,

Then

F

for

sea:, Res.::. O.

{Res < O}, continuous

is holomorphic in

and bounded in principle,

{Res < O}.

By the maximum modulus

(or more precisely, by a

Phragm~n-

Lindelof theorem) it follows that IF(~)I

F(in)

r

x

< sup{IF(il1)1

=

r.

:

real}.

11

IC e1nne1m(an+t) L nm '

Thus

But for

a value of

n

f

real,

on

IF (~) I .::. II £II; but the map

f

+

F (~)

is evidently a multiplicative linear functional on the algebra of such trigonometric polynomials, and hence the extension to linear functional on Spec A

A A.

Hence the image of

~ +

under the map

K\{(O,O)}, and hence

is a multiplicative

=

K.

(~(zl),~(z2))

We observe that

decomposes into the following parts: original space on which

A

contains

r

x

r,

Spec A the

was defined; (0,0),

the "origin" of the big disk; a union (uncountable) of continuous images of the left half-plane, on each of which the functions in

A

(or more precisely,

their Gelfand transforms) are holomorphic. observe that each half plane is dense in

We Spec A.

46

FUNCTION ALGEBRAS

1-4

THE FUNCTIONAL CALCULUS

If

A

is a function algebra on

X, and

f e A, it is immediate from the definitions that Fof e A if either F e P(f(X)) or F e R(spec f). One of the notable features of the Gelfand theory is a version of this for general Banach algebrai. Let

A

and suppose U

be a Banach algebra with unit, f e A, F

of spec f.

spec f

is holomorphic in some neighborhood Choose a contour

(this means that

y

y

spec f is

1).

F(f)

f

2;i

y

U

enclosing

is a finite union of

rectifiable closed curves lying in winding number of

in

U, and the

with respect to any point of

We define F(z;;)(r - f)-ldZ;;, where the vector-

y

valued integral is defined, as to be expected, as the limit of Riemann-Stieltjes sums n

L F(Z;;.)(Z;;. - f)

1

J

J

-1

(Z;;. - Z;;. 1). J

J-

The same argument

that shows that the Riemann-Stieltjes integral of a continuous scalar function exists shows that F(f)

is well-defined (the vector-valued integrand

is continuous by Corollary 1.2.2).

For any

~

passing through the approximating sums, we find

e A* ,

BANACH ALGEBRAS

47

~(F(f)) = ~J F(~)~[(~ 1T 1. Y

seen, ~[(~ - f)-I] ~

outside

as we have

is a holomorphic function of

is independent of the choice of

e A*

~

Since

f)-l]d~;

spec f; it follows from Cauchy's theorem

~(F(f))

that

-

was arbitrary, it follows that

does not depend on the choice of

~

y. If

y

•

F(f)

e Spec A,

~[(~ - f)-I] = [~ _~(f)]-l, and from the Cauchy integral formula we find that

~ =

Thus

F(spec f). whenever borhood of

F(~(f)).

F(f), and in particular, spec F(f)

It can be shown that F

=

~(F(f))

and

G

FG(f)

=

F(f)G(f) ,

are holomorphic in a neigh-

spec f.

Let us consider the following special case. Let

X

be a compact plane set, with bounded com-

plementary components for each X

n, and let

Choose

e G n n be the uniform closure on

GI,G Z ' . . . . A

a

of the algebra of all rational functions whose

only (finite) poles are among the

{a }. n

The

arguments used in the last section in determining Spec P(X)

show

that

above now shows that

spec z = X. A

contains every function

holomorphic in a neighborhood of if

X

The discussion

X.

In particular,

has connected complement, every function

FUNCTION ALGEBRAS

48

holomorphic in a neighborhood can be approximated uniformly on

X by polynomials.

This is a classi-

cal result known as Runge's theorem, and the proof we have just given is indeed (stripped of the fancy language) the classical proof. In F

spec f

is contained in a disk

is holomorphic in

D, and

D, we can also define

by a Taylor series. For if D

=

F(f)

c I < R},

{se¢: Is 00

then for where implies

seD

we have

F(s)

lim supla I l/n < l/R. n

-

spec (f - c)

limll(f - c)nlll/n Ian(f - c)n


1,

-1

exp g = f, and the lemma is

proved. When

A

is a Banach algebra with unit, we

define

exp A = {exp f

we have

exp f·exp(-f)

so

From Lemma 1.4.1,

exp 0 = 1

for any

f e A,

-1 exp A c A .

THEOREM 1.4.3.

Let

algebra with unit.

A

exp tg (0

+

connecting Now if

exp A 1

in

A

is precisely the -1

•

We observe first that for any

Proof. t

be a aommutative Banaah

Then

conneated aomponent of

map

f e A}.

f

1 =

to

g e A, the

< < is a path in exp A - t - 1)

exp g, so

exp h, and

exp A

I f - gil




:"'XA,

l(gl' ... ,g)lI=maxqg·lI}. J

n

+

M by

T (g l ' ...

,g n )

= L\' f.1 g 1..

is a continuous linear map, and by is onto.

theorem, there exists

Hence, by the open mapping K > 0

such that for every

BANACH ALGEBRAS

61

g e. M there exist

...

g1'

,gn e. A

such that

Lf. g. and Ilg.11 < ~ K~g~ for j = 1, ... ,n. J J J It follows that for every g € A, there exist

g

g. e A (j = 1, J g = 0

Spec

A

so

U

so

is an

FUNCTION ALGEBRAS

62 ~e

¢.

open neighborhood of

note that

closure of

U, is contained in

I ..:: j < n}.

Then for

A

Thus on

IT.

J

-

E,

A

converges uniformly to

g

"-

Since

A

it follows that of

{I/J : 1I/J(f·)1
0, It.

(where

nK = {nf : f e K}).

J J

jeI

U,

J

J

1,

=

f. e U}. 1.

Then

2 M =

00

U nK

Since

1

M2 = M, and

M is a Banach space, it follows from

the Baire category theorem that for some

n, nK

is somewhere dense, i.e., the closure of

nK

non-empty interior.

Since

K

is convex and

symmetric, it follows that the closure of contains a neighborhood of the origin in there exists

0

>

0

such that

has

K

nK M.

is dense in

Thus,

BANACH ALGEBRAS {f

I f II

M :

€




Then 6 > 0

U

is a

so that

,n, and by ii), choose

Ilfll < K, f(x) = 1, If (y) I < 6 e = 1 - f.

o,

Then

for all

e e M, lie I < K

y e U, I efj (y) - f j (y) I = I f (y) f . (y) I < J

+

I",

FUNCTION ALGEBRAS

74

IIf III f j (y)

I

while for

< E,

I £.11 J

I fey) fj (y) I

< 0, as well as

J-zkdll~ = 0

JzkdlJ' so

lJ

~

f or al 1

C(r)

by the

Fejer (or Weierstrass) theorem, i.e., lJ = O.

If

A* , 1111 = (1) = 1, it follows that

admits



€

a unique representing measure on



r, for if

A

FUNCTION ALGEBRAS

84

and

represent

l.l

measure, hence 1

~,

A

- l.l

is a real annihilating

In particular, if

O.

1s 1

< 1,

2

-

lsi (J is the only representing measure for sil 11 s on r, and for I s I 1, Os is the only representing measure for the disk algebra

s

on

r.

A on the disk

If we look at

~,we

1$1 < 1

many other positive measures on

senting

s, for instance

circle of center

=

s.

For put

IIgIi

=

t e

~,

s, or the normalized Lebesgue

1

g = I(l = 1,

and

Then n, and g n

t f s.

integer

s.

+

1 g (t) 1 < 1

gn e A

s, 1

l.l =

for all

converges boundedly to the

a representing measure for l.l

g e A,

for each positive

{s}.

that

But again, if

Then

5z) .

characteristic function of

and since

repre-

is the only representing measure for

1, Os

g(s)

~

0s,or the mean around a

measure on a disk of center 1s 1

find for

Hence, if =

fgndl.l

+

l.l

is

l.l({s}),

is a probability measure, it follows

os.

As an application of representing measures, we give another proof, due to Hoffman and Singer, of Wermer's maximality theorem (Theorem 1.2.4). Recall the statement:

if

B

is a function algebra

MEASURES

85

t, and B

on the circle then either

B = A or

as follows.

Suppose

Then

A, the disk algebra,

~

B = C(r).

r0

(z)

The proof runs

for all e Spec B.

z e B, B = C(r)

z-l e B, and since

Weierstrass (or Fejer) theorem.

by the

Suppose on the

other hand that there exists

e Spec B with

(z) = O.

for all f e A.

(f) = f(O)

Then

is a representing measure for

~

that

f(O) =

ffd~

have seen, that measure on for

B

c

=

0,

the normalized Lebesgue

Hence for all

r.

it follows

f e A, hence, as we

for all ~

,

f e B, we have,

n > 1, 0 = (zn)Hf) = (zn f ) = ffZndO. -

B = A.

A, so

If

Thus

The proof is finished.

The argument by which we derived the Poisson formula from the Cauchy formula can be adapted to a more general context.

THEOREM 2.1.1.

x, .

e Spec A,

Let ~

A be a function algebra on

a complex measure which represents

Then there exists a positive representing

measure

0

for

with respect to

, with ~.

0

absolutely continuou8

86

FUNCTION ALGEBRAS

Proof.

Choose a positive measure

~ = Fp

F e LZ(p), such that p

I~I, and F = dl~I).

sure of

A

in

LZ(p).

Let

p, and

(for example, HZ

denote the clo-

By the projection theorem

in Hilbert space, we may write ~ = f + g, where f e HZ and g .L H2 . Then for every h e A, we have

Jhd~

Jhe!'

g)dp = JhTd P . Since f e H2 , there exist fn e A, f n + f in L2 norm; then for any h e A with (h) = 0, we have o = (hfn) = (h) =

=

+

Jhf n1'd P for all n, and hence IhlflZdp = -1 2 o=c If I dp, where c = Jlfl2dp. Then probabili ty measure, and for all JhdO = J [ (h) = (h)

+

+

o. 0

Put is a

h e A, we have

h - (h)]do

J [h - (h)]do

(h) ,

and the theorem is proved. The proof above was found by D. Sarason, and independently, by Konig and perhaps others. The result seems to have been first stated, and a more involved proof given, by Hoffman and Ros si [2]. In the sequel, we shall often deal with the space of all reaZ-vaZued continuous functions on X; we denote it by CReX).

MEASURES 2-2

87

THE CHOQUET BOUNDARY Throughout this section, we consider a

linear subspace such that

A

A

of

C(X), X

separates the points of

contains the constants.

K

{cp e A*

=

compact Hausdorff,

It is clear that

X

and

We set

= IIcp II

HI)

= I}.

K is a convex subset of the

* containing each A,

closed unit ball of (x e X), and that

K

Lx

is weak-* closed, hence

weak-* compact.

The Choquet boundary of

DEFINITION.

x e X

Ch(A) , is the set of all

A, denoted

such that

Lx

admits a unique representing measure; i.e., such that

Ox

is the only representing measure for

If

x

Lx'

measure v

V

f

Ch(A) , there exists a representing

for

Lx

with

V({x})

=

O.

r

is a representing measure for

v({x})

=c

< 1.

Put

V

=

Lx'v ox' then -1 (1 - c) (v - cox), It

is trivial to verify that

V

which represents

v({x})

Lx' and

For if

is a positive measure

= o.

FUNCTION ALGEBRAS

88

THEOREM 2.2.1. a,S,

oonstants

x e X.

Let

Suppose there exist

0 < a < S < 1, suoh that

with

U of

for every neighborhood

x

there exists

f e A

with

Ilfll.::. 1, f(x) > S,and

aZZ

f

Then

y

Proof. and

U. Let

~

If(y) I < a

for

be a representing measure for

Lx'

x e Ch(A).

U a neighborhood of

S

f(x)




~ = ~

~({x})

>

~

Thus so

=

Then for some

fU fd~ + fX\U

ffd~

=

x.

+ a~(X\U)

a

=

fd~

+ (1 - a)~(U).

for any neighborhood

~

f e A,

U

of

x,

The theorem follows from the

remark above. The same sort of result holds if we consider real parts.

THEOREM 2.2.2. a,S,

with

borhood

Let

0


u}, y, a

For any

-

representing measure

Proof.

Replacing

that /y = O.

Let

for some real and let that

Also, N t

u N

t, some

and

fUd).l = y.

with

$

for

there exists a

u - y, we may assume

by

{f e C(X) : Ref < tu g e A

P

P

are disjoint, for if

tu

Re$(g) > -ta > 0 if

t < O.

Reg

It is clear

are convex cones, and

Re$(g) > ItlS ~ 0

+

Re$(g) < A},

with

P

real, g e A, then

and

).l

< 8,

2. y

P = {f e C(X) : Ref > O} •

N and

Let

f e A, Ref < u} and

sup { Re$(f)

a

a < 8.

80

$ e K, U e CR(X).

Let

is open. +

if

Reg > 0, t > 0,

Hence, by the

separation theorem, there exists a non-zero lin.

and

Ref(y)

< -2

It is clear that

FUNCTION ALGEBRAS

92

00

each

Gn

is open, and

If

X

Ch(A) = n Gn

by Theorem

1

2.2.6.

Ch(A)

is not assumed metrizeable then,

need not even be a Borel set.

See Bishop

and deLeeuw [1] for examples.

THEOREM 2.2.8.

Let

~

e K.

K i f and only i f

point of

~

Then ~

= L

is an extreme for some

x

x e Ch(A). Proof. x = Let L

Let

t~

x e Ch(A) , and suppose

+ (1

t)1/I, where

be

r~presenting

~,v

tively.

Then

measure for ox

=

t~

t~

+

measures for

(1 - t)v

Lx' and since

(1 - t)v.

+

a Borel set, x

f

Since

let hood ~

(U)

Let

< 1,

v

are positive

= veE) = 0 whenever

~(E)

~

= v = ox' and hence

is an extreme point of

~

be a representing measure for

~

x e supp U

and

~

is extreme.

Now suppose that

K.

of

~,

x.

define

respec-

~,1/1

is a representing

E, and so

Thus

0 < t < 1.

x e Ch(A) ,

measures, it follows that E

e K and

~,1/1

so

~

(U)

and

1/1

, and

for every neighbor-

> 0

If for some neighborhood

e

~

by

U

of

x,

93

MEASURES

=

8(f)

1jJ (f)

then

=

=

if

1jJ

~,

=

~(U)

~(V)

< 1

~(f)

=

~

1


1 -

~}),

and

1

peak sets in the weak sense are compact. LEMMA 2.3.1.

If the closed subset K of X is a

peak set in the weak sense and a peak set.

In

particular~

Go~

then K is a

the intersection of a

MEASURES

97 .?'

countable family of peak sets is a peak set. 00

Proof.

Suppose K

n Gn , each Gn open, and 1

= n

K

Ka , where Ka is a peak set for each a in

aeI

the index set I. By the finite intersection principle, for each n there exists an e I with 00

KN u.

n

C G • Let f

n

e A peak on K • then f n an '

=r

2- n f

1

e A, since A is uniformly closed, and f evidently 00

peaks on n K = K. 1 an In particular, if X is metrizeable, there is no distinction between peak sets (or peak points) in the weak sense and peak sets (or peak points). THEOREM 2. 3. 2 (Bishop [2]) .

Suppose x e X, and

suppose that for every neighborhood U of x there exists f e A such that

Ilfl~l,

3 and f(x) > 4'

1 If(y) I < 4 for all Y ~ U. Then x is a peak point

in the weak sense.

Proof.

We must show that for every neighborhood

V of x there exists a peak set K with x e K C V. Now our hypothesis may be restated: for every neighborhood U of x there exists fU e A, with

n

FUNCTION ALGEBRAS

98

fU(x) = 1, IfUI




=




G\~(w;

0 small enough so that

~(w;

E). Then G£ is a smoothly

£)

c

152

FUNCTION ALGEBRAS

bounded domain with boundary y - y £ , where y £ denotes the circle {s: Is - wi = d

with positive

orientation. By Green's theorem, we have

I

y

fdz

idz

I

z - w

y

e:

d

z - w

=

II

f

dZ

[z

II

f-z dm. z - w.

- w )

dz

A

dz

G£ 2i

Ge: fdz Now

and

Iy £ II

z -

I2TI o

i

few

+

e:e i6 )d6

W

-

f-z dm £+0 z - W

G£ we have the lemma.

f-z

JI G

Z -

-+

£+0

2TI if (w) ,

dm. Collecting terms, w

Two special cases of Lemma 3.1.2 are of interest: if fz = 0 in G, i.e., f is holomorphic, Lemma 3.1.2 is the Cauchy integral formula. The other special case is:

COROLLARY 3.1.3.

few) = ; f¢ Proof.

w

~z,

Let f

e: C(l) c . Then for all w e:

¢,

dm.

Take y to be a large circle in Lemma 3.1.2.

We next derive some simple facts about com-

RATIONAL APPROXIMATION

153

pactly supported measures in the plane. CAs usual, measure means complex Borel measure.)

Let X be a compact set in ¢,

DEFINITION.

measure on X. For all w e ¢, we put jlCw)

LEMMA 3.1. 4.

~

With X,

as

above~

~ a =

J

dl~

I

Iw - zl

il is summab Z-e

(with respect to m) over any bounded set; in parti-

jl
0 and ~

En

< 00.)

00

Let X = DO\U D , so X is a compact set 1

n

with empty interior. Let Yn be the boundary of Dn'

RATIONAL APPROXIMATION

IYn

Since

~ I

that

1

Idzl

163

2TIr n , and L rn

=

describe by: if f e C(X), 1

fdz. Evidently,

Yn

00, it follows

fdz converges absolutely for any f e

Yn

C(X). Thus there is a measure

~ I




If(t) I fo~

fo~ all

t e U n X. Then s is a peak point

R(X).

Proof.

As we saw in Chapter 2, if s is not a peak

point, there is a representing measure for s other than os' and hence there exists ~({s})

f

o.

~

~

R(X) with

We show this is impossible with our

hypothesis. Choose h e C~l)(U) such that h a neighborhood of s. Put v v

~

on supp v

J fndv

c

U

n

= 1.

Then f n

X, so we have

in

h~ - !TI h-am. Then z = ~({s}).

R(X), as we have seen, and v({s})

may assume that f(s)

lim

=

= 1

+

~({s})

We

X{s} boundedly

= v({s}) =

= O. The theorem is proved.

This theorem admits a generalization to arbitrary function algebras, one of the genuinely deep results of the subject. It was found by Rossi, and asserts: let A be a funotion algeb~a on X = Spec A. Suppose K is a olosed subset of X, U a neighbo~­ hood of K, with K a peak set fo~ Alu. Then K is a peak set fo~ A.

RATIONAL APPROXIMATION

169

For a proof of Rossi's loeal maximum modulus principle, we refer the reader to the book of Gunning and Rossi [1], where it is proved in the first chapter. See also Stolzenberg [2] and Hormander [1]. Our next lemma is due to Bishop [1]. The proof we give was suggested by Hoffman.

LEMMA 3.2.12.

Let II

open sets with X

n c

U.. Then there exist ll·

u

~

j

~

n, with supp

a~d II

2:

1

O.

U. , supp

c

II .

J

n for eaah j,

J.

J

J

1 R(X) , 1

R(X), and Zet Ul"",U n be

J.

J

c

J

U.

J

ll .•

J

Choose h. e C (1) (U . ) such that 2: h.J .= I in c J J a neighborhood of X (i.e., a partition of unity

Proof.

subordinate to the cover {U.}). Choose ll. J

so that

O.

J

=

h.O and supp ll.

J

J

Then 0 = (2: hj)O = 2:(hjO) II

=

=

J

c

J.

R(X),

U. (Lemma 3.2.8).

2: OJ

J

.

=

(2: llj)A, so

L ll. by Lemma 3.1.7. The lemma is proven. J

THEOREM 3.2.13.

Suppose f e C(X), and suppose

eaah point of X has a neighborhood V in X suah that

fiVe R(V) .. Then f e ReX).

FUNCTION ALGEBRAS

170

Proof.

From the compactness of X, we can find n

open U., 1 J

U.

J

II

J

< j