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
LINEAR ALGEBRAIC GROUPS
Andrew Browder
INTRODUCTION TO FUNCTION ALGEBRAS
Paul
J. Cohen
SET THEORY AND THE CONTINUUM HYPOTHESIS
Eldon Dyer
COHOMOLOGY THEORIES
Walter Feit
CHARACTERS OF FINITE GROUPS
John Fogarty
INVARIANT THEORY
William Fulton
ALGEBRAIC CURVES
Marvin J. Greenberg
LECTURES ON ALGEBRAIC TOPOLOGY
Marvin J. Greenberg
LECTURES ON FORMS IN MANY VARIABLES
Robin Hartshorne
FOUNDATIONS OF PROJECTIVE GEOMETRY
J. F. P. Hudson
PIECEWISE LINEAR TOPOLOGY
Irving Kaplansky
RINGS OF OPERATORS
K. Kapp and H. Schneider
COMPLETELY O-SIMPLE SEMIGROUPS
Joseph B. Keller Stuart Antman
BIFURCATION THEORY AND NONLINEAR EIGENVALUE PROBLEMS
Serge Lang
ALGEBRAIC FUNCTIONS
Serge Lang
RAPPORT SUR LA COHOMOLOGIE DES GROUPES
Ottmar Loos
SYMMETRIC SPACES I: GENERAL THEORY II: COMPACT SPACES AND CLASSI FICATION
I. G. Macdonald
ALGEBRAIC GEOMETRY: INTRODUCTION TO SCHEMES
George W. Mackey
INDUCED REPRESENTATIONS OF GROUPS AND QUANTUM MECHANICS
Andrew Ogg
MODULAR FORMS AND DIRICHLET SERIES
Richard Palais
FOUNDATIONS OF GLOBAL NON-LINEAR ANALYSIS
William Parry
ENTROPY AND GENERATORS IN ERGODIC THEORY
D. S. Passman
PERMUTATION GROUPS
Walter Rudin
FUNCTION THEORY IN POL YDISCS
jean-Pierre Serre
ABELIAN /-ADIC REPRESENTATIONS AND ELLIPTIC CURVES
jean-Pierre Serre
ALGEBRES DE LIE SEMI-SIMPLE COMPLEXES
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
<
I}, the unit circle.
We shall
as the disk algebra on the
P(~)
per)
to
Is I Is I
t
{s
Let
disk~
as the disk algebra on the circle.
We
easily deduce from the maximum modulus principle that
and
P(~)
per)
are isometrically isomorphic:
this principle tells us that the restriction map of
C(r)
into
C(~)
induces an isometry of
into
P(~)
p(r), and that a sequence of polynomials converging
r
uniformly on on
o
so this isometry is onto.
~,
that
P(~)
r < 1,
<
must necessarily converge
=
For if
A(~).
fr(~)
=
f(r~);
f E
It is easy to see put, for
A(~),
then each
morphic in a neighborhood of
~,
fr
is holo-
hence has a power
series expansion converging uniformly on E P(~);
fr r
~
since
fr
~
f
1, it follows that
If
E P(~).
~,
~
uniformly on f
uniforml~
f
so
as E
C(r),
we know from Fejer's theorem, or Weierstrass's, that in
Per)
f
z
1
is uniformly approximable by polynomials and
z; since
Ref) = C(f).
z
= -1z
on
r,
f
e ReX).
Thus
FUNCTION ALGEBRAS
6
One obvious generalization of the disk algebra is the bicylinder algebra: the two-dimensional torus in closed unit bicylinder.
PC~2)
and
Pcr2)
let
r2 = r x r,
~2, ~2
~
x
the
~,
It is easy to see that
are isometrically isomorphic,
by repeated applications of the maximum principle. Another gQneralization of the disk algebra was first studied by Arens and Singer [2], and has had a great influence in the development of the subject. let
A
Let
be an irrational real number, and
~
be the algebra on
. f unctlons
{n
zl z2 m : n,m
The elements of
A
r x r
generated by the m~ ~
integers, n +
were called "generalized analy-
tic functions" by Arens and Singer; A
is often
referred to as the "big disk algebra". erally, let
G
a}.
More gen-
be a subgroup of the additive group
of real numbers, regarded as a discrete topological group, and let
~
be its compact dual group.
Each
element
g e G defines a continuous function of
modulus
1
phism of
on G
The algebra
G
into A
on
Cin fact, a continuous homomor-
xg •
r), which we denote by G generated by
{x
g
: g > 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
(~)}
<
OC>
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 cnz n , where
f =
With the
00.
-00
pointwise operations and this norm, A
is a commu-
tative Banach algebra with unit, and its generalizations form the subject matter of harmonic analysis. Banach algebras were introduced by Gelfand, in his fundamental paper of 1941 [1].
The main
motivation for the study of Banach algebras was (and is) their applications to harmonic analysis, rather than uniform approximation.
For these
applications, and for more than the most elementary part of the general theory, which is all we describe here, see the books of Loomis, Naimark, or Rickart.
THEOREM 1.2.1. unit.
If
s - f
has an
f
00
series
I
0
s
€
Let A~
s
A €
inverse~
be a Banach ~~ and
If I
<
a~gebra
Is I ,
with
then
given by the norm-convergent
fn n+l f s'
so ~g~ < 1. The assertion of the theorem is that 1 - g has an inverse, given
Proof.
Let
g
12
FUNCTION ALGEBRAS 00
by
Lg 0
n•
n Let
Gn
=
n
g L 0
k•
Then for
m < n, we
L gkll 2. m+nL118 II k 2. ~ 1 _ g ' by . m+l using the additive and multiplicative properties have
IIG n - GmII
of the norm.
=
II
Since
Igil
< 1,
it follows that
{G n }
is a Cauchy sequence, and hence has a limit G e: A. Since (1 - g)G n = Gn(l - g) = 1 - g n+l and g n+l -+ 0 as we have seen, it follows that (1 - g)G = G(l - g) = 1, which was to be proved.
We shall denote the set of invertible elements in an algebra with unit
COROLLARY 1.2.2. unit.
Suppose
A
Let f
Then
1 -
Proof.
We have
p -
f-lg
has an inverse
f-lgh
= 1
g e A,
g e A-1 , and
i c 111 2 I f
<
- g II
Ilf-llilif - gil
f- 1gll 2. IIf-lll ~f - gil < 1, so h.
Multiplying the equation
on the left and right by
respectively, we find that hf-lg = 1, hf- 1
A-1 .
by
be a Banach aZgebra with
A-1 , and
e:
A
ghf- l
is an inverse for
f
and
1; since
g.
Now
f- l
BANACH ALGEBRAS
13
00
h
=
I
00
(1 - f-1 g )n
I
0
[f- 1 (£ _ g)]n, and
0
IIf- 1 _ g -111
II f- 1 - hf- 1 11, so
Ilf- 1 _ g-1 11 < 111 - h 1111 f - 1 II 00
< Ilf-111I1If-lllnllf _ glln 1
II f - 1 11211 f - g II 1 - I f - 11111 f - g II ' as was to be proved.
COROLLARY 1.2.3. unit.
Then
A-I
Let is
is a homeomorphism of
Proof.
A be a Banach algebra with
open~ and the map A
-1
f
+
f- l
on itself.
This is merely a qualitative statement of
the more precise Corollary 1.2.2. Since the completion of a normed algebra is a Banach algebra, the estimate in 1.2.2 shows that the map f + f- 1 is a continuous map (and therefore a homeomorphism) of A~ onto itself for A a nOTmed algebra; but A- 1 need not be open if A is not complete.
DEFINITION. f
€
A.
Let
We define
A be a normed algebra with unit, spec f, the spectrum of
f, by
14
FUNCTION ALGEBRAS
spec f We observe that if with unit, f e A, then spec f
A
is a Banach algebra
spec f
is compact:
for
is bounded by Theorem 1.2.1, and closed by
Corollary 1.2.2. Before going on with the general theory, we pause to give some non-trivial applications to some of the examples described in Section 1. As an application of Theorem 1.2.1, we give here Paul Cohen's [2] proof of a famous result of Wermer, known as Wermer's maximatity theorem.
v
Let
THEOREM 1.2.4.
r.
oirote
Let
A
be the disk atgebra on the
B be a otosed subatgebra of
c(r). and suppose
B
~
A.
B = A or
Then either
B=C(r). Proof.
Suppose there exists
feB, f
f
A.
Let
00
L
cnz n
be the Fourier series of
-00
some
n > 0, c
theorem. assume
-n
r
c_ l
=
1.
exist polynomials
f
Then for
f e A
0, else
Replacing
f.
(c
by
-n
by Fejer's -1 n-l ) z f, we can
Then by Fejer's theorem, there g
and
h
in
z
such that
BANACH ALGEBRAS zf where
Ilkll zF
and
1
=
1
+
<
2·
15
zg
1
+
+
zh
Put
k,
+
f - g - h.
F
zh - zh
k e B.
+
purely imaginary, we have 1
t 2 11zh - Zlill 2
+
P
+
t - tzFl1
j1
+
t > 0
Hence
111
<
111 + t (zh - zh) II + II tk II 1
+
is
<
=
+
t(zh
t.
F e B,
zh - Zli
t(zh - Zli) II
+
for every real
~ Choosing
111
Since
Then
zh) - tkll
t 211 zh - zh I
+
t/2.
small enough, we then have
t - tzF II < 1
+
zF e B-1 , and hence
t, and hence by Theorem 1. 2.1, z -1 e B.
It follows that
B = C(f), and the theorem is proved. We shall give another proof of this theorem in the next chapter.
Wermer's theorem initiated
a spurt of research on the subject of maximality (see for instance Hoffman and Singer [1] and [3]). It has had some striking applications, which this volume is too small to contain.
An earlier result
found by Rudin is an immediate consequence of Theorem 1.2.4: A~
Zet
A
be a function aZgebra on
with the properties: z e A
~
and every function
16 in
A
FUNCTION ALGEBRAS
A attains its maximum moduZus on
f.
Then
is the disk aZgebra.
We next use Theorem 1.2.1 to prove the following result: ~n., ~
X
if
is a compact subset of
ReX)·~s genera t e d b y
then
n +
1
f unc t·~ons.
For the proof, choose a sequence of polynomials {gm}' such that each
{JL: p
such that
R(X).
tive constants
{c}
with the properties:
m
1 < k < m. 00
us that
f - \
-i
ck
gk
= 1,2, ... }
Now choose a sequence of posi-
for every whenever
X and
has no zero on
a polynomial, m
gm
is dense in
gm
€
m, and
The first condition assures
ReX); the second that
00
c by
zl'
Il gk II. m gm
...
,zn
Let and
A f.
be the algebra generated A = ReX) ,
To show that
1 for every k. it suffices to show that € A gk ck Then fl = f € A, and if Let f m = I m gk 1 g. € A for j < m, it follows that f m € A. We 00
J
proceed by induction:
if
fm
gm fmg m = c m + I c k- , and gk m+l
€
A, then
00
but
fmg m 00
cm
>
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* :
f,g e A, and
~(f)Hg)}
Spec A
is the intersection of all such sets with the hyperplane
{q,
e A* :
Spec A
22
FUNCTION ALGEBRAS
is a weak-* closed subset of the closed unit ball A* , and hence
in
Spec A
Gelfand topology.
is compact, with the
A = {f :
Thus,
f e A}
is a
separating algebra of functions on the compact Hausdorff space of
A
into
Spec A.
C(Spec A)
A
f
f
+
is called the GeZfand
A.
representation of
If
The homomorphism
is a function algebra on
X, the
Gelfand representation has a simple interpretation . For each
x e X, let
It is obvious that •
: X
Spec A
+
the points of
=
• x (f)
'x e Spec A.
f e
for all
f(x)
A.
The map
is one-to-one, since. A
separates
X, and continuous by a remark we
made immediately following the definition of function algebra.
Thus
•
is a homeomorphism of A
onto a closed subset of for each
f e A.
Spec A, and
f
Thus we may regard
X
= f
as a closed
subset of
Spec A, and
f
The Gelfand representation, which is
to
X.
f
o.
X
as the restriction of
always norm-decreasing by Lemma 1.2.9, is seen in A
this case to be an isometry:
each
A
f e A
takes
its maximum modulus on
.eX), this maximum being
Ilf II.
Spec A
Thus we may view
as the larges t
compact space to which the functions in
A
may
BANACH ALGEBRAS
23
be extended so as to form a function algebra.
The
example to keep in mind is the disk algebra on the circle, which can be extended to form the disk algebra on the disk. Let unit.
A
be a commutative Banach algebra with
We say that
is a linear subspace f e A
and
gel.
is an ideal in
I of An
A
and
ideal I
I ~ A, or equivalently, if
if
the unit ideal.
A
fg e I
if
I
whenever
is called proper
1 , I; A
is called
A maximal ideal is a proper ideal
not properly contained in any proper ideal.
If
LEMMA 1.2.10. unit~
I
and
A
is a aommutative algebra with
a proper ideal in
,
for some maximal ideal Proof.
Let
J e
f
J
oJ}
c
M
1.
Order
.t
by inclusion.
oJ,
If then
is clearly an ideal, and proper since
for each
Zorn's lemma.
I
M.
is a linearly ordered subcollection of
V{J
1
then
be the collection of all proper
ideals containing
J
A~
J
e~.
The lemma now follows from
24
FUNCTION
LEMMA 1.2.11.
unit~
the cl,osure
I
ticul,ar~
Proof.
A be a commutative Banach
Let
al,gebra with
ALGEBRAS
is a proper ideal,.
I
of
A.
a proper ideal, in
I
Then In par-
every maximal, ideal, is cl,osed. That
I
is an ideal is trivial.
is proper, then no
f e I
I
is invertible; it follows
III -
from Theorem 1.2.1 that f e I.
and hence for all
If
f~
> 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
a maximal, ideal, in
A.
If
in
M = ker
A" then
ker
M is a maximal, ideal,
for some
(Here, ker
{f e A :
If
ker
= 0 and g e A, then
is an ideal; since
= 0,
is
proper; it is maximal since it has codimension (if
g ~ ker
f = (f - cg) + cg, where and
ker
f e A,
c =
g
generate the unit ideal). M is a maximal ideal in
is a Banach algebra with unit. in
1
AIM, then
7T-
1 (I)
If
A, then I
is an ideal in
M, hence either
M or
proper ideal in
AIM.
A.
Thus
{O}
AIM
is an ideal A
containing
is the only
It follows that if f e (AIM) -1 , and by the
f e AIM, f i' 0, then
Gelfand-Mazur theorem (Corollary 1.2.6) it follows that
AIM
is one-dimensional.
Thus, for each
f e A, there is a uniquely determined such that
e Spec A
f
-
and
It is clear that
= M, and the proof is
BANACH ALGEBRAS
27
concluded.
Let
COROLLARY 1.2.13. unit~
algebra with
J
g l' ...
~
j
f1'
... ,fn e A.
Then
such that
¢ e Spec A
n), or else there exist
If.g. = 1. In particJ J if and only if ¢(f) ~ 0 for every
,gn e A
ular~ f e A- 1
~
be a commutative Banach
and let
either there exists ¢(f.) = 0 (1
A
such that
¢ e Spec A. Proof.
Let
1= qf.g. : g. e A, 1 < j J J J ideal generated by f l , ... ,f n . If I
2. n}, the is a
proper ideal, it is contained in a maximal ideal by Lemma 1.2.10, and so ¢ e Spec A
1 e I.
COROLLARY 1.2.14. spec f = f (Spec A).
= ¢
s e spec f
(s - f)
ker ¢
=
s
for some
If
I
is not
The Corollary is proved.
For every Hence
f e
A~
Ilf I
if and only if
if and only if there exists
o
c
by Theorem 1.2.12.
proper, then
Proof.
I
s - f ~ A- 1 ,
¢ e Spec A
¢(f) = s - f(¢).
with
The other
assertion is now a consequence of Theorem 1.2.8.
28
FUNCTION ALGEBRAS In view of Theorem 1.2.12, we shall refer to
Spec A, when
A
is a commutative Banach algebra
with unit, as the maximal ideal space of
A.
The spectral radius formula gives an answer to the question:
when is the Gelfand transform an
or in other words, when is
isometry~
function algebra?
IIfl12 = IIf211
The answer:
for every
f e A.
A
a disguised
if and only if
Similarly, it gives
a description of the kernel of the Gelfand transform:
f
=
a
only if
if and only i f f
limllfnlr l / n
=
0, i.e., i f and
is a generalized nilpotent.
tative Banach algebra with unit
A
A commu-
is called
semi-simple if and only if the Gelfand transform
is an isomorphism, i.e., if and only if the intersection of all the maximal ideal of called radical) is
A
(the so-
{a}, i.e., if and only if
possesses no non-zero generalized nilpotents.
A
BANACH ALGEBRAS 1-3
29
THE MAXIMAL IDEAL SPACES OF SOME EXAMPLES
Let
THEOREM 1.3.1.
space. ~
be a compact Hausdorff
X
Spec C (X) = X, i. e., for every
Then
there exists
e Spec C(X)
Hf) = f (x)
for every
x e X
such that
f e C (X) .
Proof.
It suffices to show that all the functions
in
~
ker
ker ~
(recall that
ker LX
c
tion functional at c e
a:, ~
set of
ker
zero, then so
Let
= LX' ~
p
.
x) , so ~
and since
that
x e X, for then
have a common zero
If
x
LX
(1) = LX (1) {f l' fl ,
= If-f.
J J
p-l e CeX), so
L
0 f
is the evaluac~
for some
1, it follows
... ,fn} be ... , fn have
a finite subno common
is strictly positive on
X,
~(p) = I~efj)~efj) = O.
This contradiction shows that every finite subset of
ker
~
has a common zero.
It follows from the
finite intersection principle that all the functions in
ker
~
have a common zero, concluding
the proof. Given two compact Hausdorff spaces X and Y, when are C(X) and C(Y) isomorphic algebras? Clearly. if X and Yare homeomorphic. the map T : C(X) -+- C(Y) given by Tf = f * n.
30
FUNCTION ALGEBRAS where n is a homeomorphism of Y onto X, is an isomorphism onto. Conversely, Theorem 1.3.1 shows that every isomorphism of C(X) onto C(Y) is of this form. For if T an isomorphism of C(X) onto C(Y), then for each y e Y the map f + (Tf)(y) is a multiplicative linear functional on C(X), hence there exists n(y) e X such that Tf(y) = f(n(y)) for all f e C (X). It is easy to check that n is a homeomorphism of Y onto X.
THEOREM 1.3.2 Zet
X be a topoZogiaaZ
Let
spaae~
and
A be the Banaah aZgebra (with pointwise oper-
at ions and the sup norm) of aZZ bounded aontinuous
X.
funations on
Then the GeZfand representation
A onto
is an isometry of
Proof.
iifnll = iifii n
Since
positive integer
C(Spec A). for every
f e A, every
n, it follows from the spectral
radius formula that the Gelfand map is an isometry. If wi th
f e A 1m s
is real-valued, then for every
s e
-1
s
does
f
is real-
r
0, f - seA
not vanish on valued. Thus
"'-
A
Hence
; hence
f -
Spec A; it follows that
(1)
(f)-
for every
~
f e A.
is closed under complex conjugation, and
it follows from the Stone-Weierstrass theorem that
A=
C(Spec A).
31
BANACH ALGEBRAS Recall that a topological space
X
is called
aompletely regular if for every closed subset
X and any
of
function f
= 0
f
on
F.
on
x e X\F
F
there exists a continuous
X with
~
0
~
f
1, f(x)
= 1,
and
A subspace of a compact Hausdorff
space is completely regular, by Urysohn's Lemma.
THEOREM 1.3.3. spaae.
X be a aompletely regular
Then there exists a aompaat Hausdorff
X~ aalled the Stone-Cech compactification
spaae X~
of
Let
with the following properties: i)
there exists an imbedding
a dense subspaae of
T
of
X onto
)(..
,y
ii) X
~
i~
spaae~
X
is a maximal "aompaatifiaation" of
and
Proof.
X onto a dense
Y.. then there exists a aontinuous map
...,
of
is a aompaat Hausdorff
a aontinuous map of
T
subspaae of ~
Y
the sense that if
X onto We take
Y.. suah that ,.,
,
T
=
X = Spec A, where
~
0
A
T.
is the alge-
bra of all bounded continuous functions on define
T
by
T(x)(f)
=
From the hypothesis that it follows easily that
f(x) X
T
for all
X, and
f e A.
is completely regular,
is a homeomorphism of
32
FUNCTION ALGEBRAS
X
onto
,(X).
We now observe that
dense; for suppose
g e C(X)
By Theorem 1.3.2, g
f
A
then f
vanishes on
for some 0
ishes on
,(X)
is dense in X
must vanish identically, so
..,
X.
Now suppose
But
,(X)
is a continuous
onto a dense subspace of the compact
liausdorff space
Y.
Then
f.,
e A ,
f e C(Y), so the map
f
-+
whenever
A
(f., )
plicative linear functional on
(x)
C(Y)
is a multifor every
...,
xe
X.
x e X
Applying Theorem 3.1.1, we find for each
A
f(~ (~))
particular, taking
,
f(, (x))
Since
such that
,.,
(f . , ) (x) =
=
~(x) e Y
there exists ,
,
,(X).
x e X, i.e., ,., g e C(X) which van-
0; thus every
f
f e A.
is
for every
A
= 0, so
map of
=
f(,(x))
f(x)
,(X)
=
~. To
f(~.
for every
x = ,(x)
,(x))
for
for every
It is easy to see that
~(X) ~ ,
,
,
eX), ,eX)
compact, it follows that
f e C (Y) .
x e X, we have f e C(Y), so ~
is continuous.
is dense, and ~
maps
In
X
onto
~eX) Y.
proof is finished.
THEOREM 1.3.4.
Let
~
be a positive measure.
Then there exists a totally disconnected compact
is The
BANACH ALGEBRAS
33
Hausdorff space
X and an order-preserving iso-
metric isomorphism of Proof.
We take
L
00
(~)
X = Spec L
00
onto (~),
C (X) •
and show that the
Gelfand representation has the properties listed. Since
f
every positive integer
e: L
00
(~)
and
n, it follows from the
spectral radius formula that the Gelfand representation is an isometry. 00
f e: L
(~)
real-valued if some (and hence every)
representative of If
00
f e: L
(~)
f
is real-valued, then
real-valued. 00
f e: L
(~),
s e: L
f" - s
f 0, whence
(£)
It follows that
00
f
1m s f 0, and thus
X when
zeroes on
(~).
is real almost everywhere
s e: t, 1m s
whenever
every
We call an element
(~)
has no "f
is
" = (f)
for
so the algebra of Gelfand trans-
forms is closed under complex conjugation, and hence equals then
f
=
g2
C (X) •
If
f
e: L
00
(~)
for some real-valued
and g e: L
f.:: 0, 00
(~),
so
"2 f = g .:: 0:
the Gelfand representation is order-
preserving.
If
f = XE' the characteristic func-
tion of a ~-measurable set E, then f2 = f, so "2 " , if we put " = {x e: X f = f· E f(x) = I} , it follows that
f = XE'
Since
f"
" is continuous, E
-1
34
FUNCTION ALGEBRAS
must be both open and closed. combinations of such
Since finite linear
. L "" (ll), lt
are dense in
follows that linear combinations of the characteristic functions of open-and-closed subsets of are uniformly
dense in
C(X), which is easily
seen to be equivalent to connected. If
X
X being totally dis-
All is proven. Y
is any set, the algebra
B(Y)
of all
bounded functions on
Y
phic to
X may be regarded as either
C(X), where
is isometrically isomor-
the Stone-Cech compactification of
Y with the
discrete topology, or the maximal ideal space of L"" (ll), where Let
II
is counting measure on
Y.
U be the open unit disk in the complex
plane, and let
H""(U)
be the algebra of all
bounded holomorphic functions on ideal space of
H""(U)
U.
The maximal
is an extremely complicated
object, extensively studied (see especially Hoffman [3]).
The map
s
easily seen to imbed Spec H""(U).
Spec H (U).
(evaluation at
s)
is
U as an open subset of
It is a deep theorem of Carleson [1]
that the image of 00
+
U under this map is dense in
BANACH ALGEBRAS Let
be Lebesgue measure on the circle
0 00
r, and let H 00
35
be the weak-* closure in
(0)
r.
of the disk algebra on
L (0)
00
00
H (U) and
H (0)
The algebras
are identified by means of a
classical theorem of Fatou, which states: 00
f
H (U); then for almost all
€
s
€
le t
r,
f * (s) = lim f(rs) exists; f * € H00 (0); for every r+l g € HOO(o) , there exists f € Hoo(U) (given by the Cauohy integral formula) suoh that everywhere.
f
*
=
g almost
Via the Gelfand representation,
00
H (0) may be regarded as a closed subalgebra of C(X), where Loo(o).
X
is the maximal ideal space of
We shall discuss
4, and find that X
Hoo(o)
Hoo(o)
separates the points of
(so that it is a function algebra on It is easily seen that if
algebra, Spec A disk
~.
f
For if
is the disk
€ Spec A, and we put
~
s€~
since
is a polynomial in
g = f - f(s) z - 5 f(s) + (Hz) -
A
X).
can be identified with the closed
s=~(z),wehave
If
again in Chapter
Hence s)~(g)
f,and hence for all
1~(z)I2.llzll=l.
z, then so is
~(f) = ¢[f(s) = f(s) f € A.
+
(z - s)g] =
for all polynomials On the other hand,
FUNCTION ALGEBRAS
36
for each
s e
longs to
Spec A.
the map
~,
f
z
Thus
morphically onto
~.
f(s)
+
maps
clearly
be~
Spec A homeo-
This remark can be generali-
zed.
DEFINITION.
Let
A
be a commutative Banach alge-
bra with unit, and spectrum of
...
. ..
,f
,f
is defined by
n
, f n) =
spec(f l ,
e Spec A}.
Thus, spec(f l , ... ,fn) of
~n; if
The joint
e A.
n
is a compact subset
n = 1, this notion agrees with the
other definition by Corollary 1.2.14.
DEFINITION. and by
S S
If
A
a subset of
is a Banach algebra with unit, A, by the aZgebra generated
we mean the smallest closed subalgebra
(with unit) of
A
such that
a set of generators for
If
Ul'
...
,fn} A
A, it is clear that points of
Spec A.
{f 1 '
B
~
S.
We call
B S
B. is a set of generators for
...
A
,fn}
Thus the map
separates the
BANACH ALGEBRAS
37
~ ~ (fl(~)' ... ,fn(~)) Spec A onto Let
is a homeomorphism of
spec(f l , ... ,fn ).
A be the algebra of functions with
absolutely convergent Fourier series, introduced on pp. 10-11. cation that circle
We leave to the reader the verifiSpec A may be identified with the
r, the Gelfand representation being the
identity.
Applying Corollary 1.2.13, we then find
the classical theorem of Wiener:
r
are continuous functions on
,f n
if
with absoZuteZy
convergent Fourier series and no common there exist
gl' ... ,gn
zero~
then
with absoZuteZy conver-
gent Fourier series such that
If.g. = 1. J J
This
result is not easy to prove by direct arguments, even for
n
1
(in fact, using the argument in the
proof of Theorem 1.3.1, the general case follows immediately from the case
n = 1).
Corollary 1.2.13 tells us that if
Similarly, f l , ... ,fn
are
functions in the disk algebra with no common zero on
~,
then there exist
algebra such that for
If. g. J J
gl' ••. ,gn 1.
in the disk
This result, obvious
n = 1, is not easy to prove in general without
recourse to the Gelfand theory.
Carleson's theorem,
FUNCTION ALGEBRAS
38
referred to above, has the formulation: . ..
,f
are bounded holomorphic functions on
n
the open disk zero on
U~
if
I I f./ J
and
U~
is bounded away from
then there exist bounded holomorphic
gl' ... ,gn
U with
on
If.g. = 1 J J
This is
easily seen to be equivalent to the formulation: U
00
is dense in
DEFINITION.
Let "K,
We define
Spec H CU).
K be a compact subset of
the polynomial convex hull of
" K
Ip(s) I ~ ~P~K polynomial
It is obvious that of
~ containing
of
C(K)
onto
onto
K, by
for every
p}
" K is a compact subset
K, and that the restriction map
C(K)
induces an isometry of
P(K)
P(K).
THEOREM 1.3.5. algebra with
Let
unit~
A be a commutative Banach generated by
X be a closed subset of for each K =
(n.
,f n .
Let
Spec A with the property:
f e A~ 11£ II = Ilf Ilx.
{CflC
f l , ...
Let
e
X}.
Then
BANACH ALGEBRAS '"
K = spec(f l , Proof.
39 ,f n ).
Let
s e K.
Then for every polynomial
p,
Ip(s) I < /lP/lK = /lp(f l , ... ,fn) /Ix = I/p(f l , ... ,fn)"'/I.:: IIp(fl' ... ,fn)/I· Thus, the map
p(fp ... ,fn)
+
p(s), which is
obviously a multiplicative linear functional on the algebra of all polynomials in
f l , ..• ,fn'
is bounded, and hence extends to a multiplicative linear functional on s
=
(£l(CP) ,
whenever
•••
A.
'£n(CP))
'" '" s e K, i.e., K
Thus for some c
spec(f l , ..• .,f n ). s e spec(f l ,
On the other hand, if then for every polynomial '"
Ip(s)1 = Ip(fl(CP),
so
...
,fn(CP) I =
'"
,f n ) (CP) I .:: lIP(f l ,
lip(f l ,
,fnllx = /lP(zl'
...
...
, f n II
,zn)Ii K,
This concludes the proof.
COROLLARY 1.3.6 then
,fn) ,
p, we have
Ip(f l ,
s e '"K.
e Spec A
cP
If
'" Spec P(X) = X
X is a compact set in
~n,
(with the usual identification
of points with the associated evaluation homomor-
FUNCTION ALGEBRAS
40 phisms).
We call a compact set '" X = X.
iaZZy convex if
X
po~ynom-
tn
in
We see from the last
corollary, and Theorem 1.3.1, that a necessary condition that ia~~y
P(X) = C(X)
X
is a compact subset of
to determine.
'" X
'" X
X,
then
f
(z - s)-l
=
X=
the distance from
s
X.
of
t\X.
to
If (t) I > 8' while Now if
s
€
We can see
X.
If
P(X), in view of the
spec z.
2
we have
f
of
contains nothing else, as follows.
identification of
t
is easy
contains, along with
X, every bounded component that
'" t, X
From the maximum modulus principle,
it follows at once that
f
po~ynom-
convex.
If
s
X be
is that
Let If
~f~ =
0
=
Is - X I ,
It - sl
i,
0/2,
<
and thus
is in the unbounded component
4:\X, we can choose a fini te set
sO' ... ,sn
s = sO' ISnl > Iz~x' and 1 IS k - sk-l l < Zlsk - Xl for k 1, ... ,no
such that
sn
f
'" X = spec z
(Theorem 1.2.1), applying the
argument above repeatedly, we arrive at Thus
'" X
Since
is the union of
s
f
'" X.
X and its bounded com-
BANACH ALGEBRAS
41
plementary components; X
is polynomially convex
if and only if it has connected complement. in order that X
=
P(X)
Thus
C(X), it is necessary that
have connected complement and empty interior.
It is a theorem of Lavrentiev that these conditions are in fact sufficient; we shall obtain this result in Chapter 3. A
When
n > 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 <
with
met >
+
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
<
r
=
I a (s - c)n,
o
Now
n
spec f ~ D
c
{s : I s I < R}, whence
<
R.
It follows that
is majorized by the geometric series
\L(R r)n ,hence converges to an element we denote by
F(f).
0
f
A which
It is easy to see that this
definition agrees with the one given by Cauchy's integral formula.
Power series in a Banach algebra
element may be manipulated with the same ease as scalar power series; the crucial fact being that absolute convergence
(Ilanl II(f - c)nll
implies unconditional convergence.
<
00
)
We illustrate
with the most important special case.
BANACH ALGEBRAS DEFINITION.
49
Let
A
be a Banach algebra with unit. fn f e A, we define exp f = I --nl 00
For each
o .
If
LEMMA 1.4.1.
f
g
and
commute, then
exp(f + g) = exp f'exp g. (f + g)n nl
00
Proof.
I0
exp(f + g)
00
nl
1
I
I ""Ti.(T n=O i1T j+k=n J. . fj
co
(
fjg k
k
h)
00
(I I }') . k=O
j=O
= exp f'exp g,
the binomial theorem being purely algebraic, and the rearranging of sums being justified by absolute convergence.
If
LEMMA 1.4.2. exists
g e A
f e A
such that co
111
and
-
£II
< 1, there
exp g = f.
- f)n. the series is maj orn 1 ized by the convergent series I III - fll n , so g is Proo f.
Put
well defined.
g =
g
a nk
(1
To see that
argue as follows: k
-I
for each
exp g = f, we may k
~
1,
00
I1
a k(l - f)n, where the complex coefficients n
are determined by the Cauchy rule for multi-
FUNCTION ALGEBRAS
51)
plying power series, hence do not depend on
f.
Then k L L k! = 1
00
exp g
00
a
00
L ( L k~k)(l
+
- f) n,
n=l k=l
0
the interchange in order of summation being justified by absolute convergence. that
exp g = f
cient of when
when
(1 - f)n
n = 1.
f e [, and hence the coeffi-
above is
Thus
Now it is well known
when
0
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
<
is connected.
Ilf- 1 11- 1 , then
BANACH ALGEBRAS
~l - f-lgl/ f
-1
<
Sl
1, so by Lemma 1.4.2 we have
g? exp k
for some
g = exp h exp k = exp(h is open.
Finally, if
closure of IIg - fll gf
-1
€
<
k
€
+
k)
f
€
A, and hence exp A.
€
A-1
and
exp A, there exists ~ gf -1
llf-lrl, so
exp A, and so
f
g
- 11
exp A.
€
· connec t e d , open an d c 1 ose d ln
Thus f
€
exp A
is in the
exp A with so
< 1,
Thus
exp A
A-I, and
1
is
exp A,
€
so the theorem is proved. Since ~(exp
f) f 0
exp A
c
-1
A
, it is clear that ~
for every
€
Spec A.
It is a
remarkable fact that this property singles out the multiplicative linear functionals on a commutative Banach algebra with unit.
The theorem is due to
Gleason [3], and the proof uses an elementary result of complex function theory which is sometimes omitted from introductory courses.
LEMMA. there
Let exist
G be an entire function, and suppose constants
ReG(s) ~ Klsl N 7,arge.
Then
K, N
for every
S
such that e:
a:, I s I
sUfficient7,y
G is a po7,ynomia7, of degree
<
N.
52
FUNCTION ALGEBRAS 00
Proof. Then
We can assume G(O) = o. Let G = L a zn. 1 n i8 00 n ReG(re ) = r (Rea cosn8 - Ima sinn8).
L 1
1
J27t,
n
n
i8
k
Then
TI
1. J21T 1T 0
ReG(re i8 )sink8d8 = rklma k , for
0
21T Jo
while
ReG(re
o
ReG (re
~ 1Tr
=
k = 1,2, ... ,
i8
)
IImakl
It follows that
O.
J21T Re G(rei8)(1~sin k8)d8~~, r
0
N and Kr
~
< 1 ± sink8 < 2.
IReakl
r Rea k , and
"8
ReG(re 1 )d8
and + 1m a k = since
=
)cosk8d8
=
0 <1 ± cosk8
Letting
0
for
r
+
~
2,
we find that
00,
k > N, which was to be
proved. We refer to this lemma as the "real part Liouville theorem".
THEOREM 1.4.4. algebra with
Let
unit~
A
be a aommutative Banaah
and let
~
be a linear funa-
tional on A with the property that for all f e A, Proof.
~(l)
f) f 0
= 1. Then ¢ e Spec A.
We first observe that
linear functional,
~(exp
in fact that
~
is a continuous ~~~
1.
For if
BANACH ALGEBRAS f e A
and
S3
Ilfll < 1, for every
we have by Lemma 1.4.2 that hence IIfll
<
~
I~(f)
Thus
= 1.
1 -
is e exp A, and I < 1
whenever
In view of the identity
g)2 - (f - g)2], in order to show that
+
is multiplicative it suffices to show that
[~(f)]2 each
for all
sell:, put F(s)
so
O.
1, i.e., II~I
fg = t[(f ~
~(f)
s -
sell:, lsi..:. 1,
=
F(s)
00
n n
o
n!
f e A. ~(exp
Let
sf).
f e A; for Thus
HI~)
is an entire function without zeroes, and
F
IF (s) I
.:: I II~ II n I sin
where
G
s e ft.
= exp I s III fl·
is entire and
Since
ReG(s) <
F = exp G,
Thus
I f II s I
F(O) = 1, we can assume
for all G(0) = O.
By the "real part Liouville theorem" above, it follows that
G = az for some constant a, so n F = e az = 2 ~ zn Equating coefficients, we thus n! 0 have ~ (fn) = a n [Hf)] n for all n, and in 00
particular for
n
=
2, which proves the theorem.
An obvious corollary is that a linear functional
~
on
A, with
~(l)
= I, is a multiplica-
tive linear functional if and only if for all
f e A-I.
~(f)
f 0
FUNCTION ALGEBRAS
54 I
f we assume to start with that
.cj>
is con-
tinuous, we can apply Gleason's argument with an otherwise weaker hypothesis.
THEOREM 1. 4.5. ~
Let
:r
be a set of generators for
a commutative Banach aZgebra with unit.
cj> e A* , cj>(l) for every
=
1
and suppose that
cj>(exp f) f 0
in the linear span of~.
f
Let
Then
cj> e Spec A. Proof.
Let
g
of ~,so
g
Define
by
F
be any element in the linear span n
Is. f., where
1
As before, F
J J
F(~) =
cj>(exp
s. e It, f. e J
~g) =
J
7.
Io cj>(gk)
~k.
k!
is an entire function with no
zeroes, F(O) = 1 and
IF(~) I < 11cj>llexp(llflll ~I), so
applying the real part Liouville theorem we have F = exp az for every
for some k.
a, and thus
cj>(g k )
=
[4>(g)] k
Now (1)
and ( 2)
BANACH ALGEBRAS
55
where the sums are taken over all
= (j l' ... ,j n)
j
c~ J
with
j 1 + ••••• + j n _. k,
denotes the multino~ial coefficient k!
~.~I------~.~I'
J n·
J 1·
jl
fj
fl
and we use the abbreviations
.....
f
jn n
sj = sl
.....
jn sn
j
[
jl
Since the polynomials (1) and (2) in are equal for all values of
sl' ... ,sn their
coefficients are equal, i.e.,
It follows immediately that all polynomials
p
and
q
for
in the generatQrs, and
hence (again taking account of the continuity of
is a multiplicative linear functional.
The theorem is proved. If
X
is a compact subset of
a continuous linear functional on fUnction
C(X), the entire
variables defined by n is known as the 4>(S1' ••• , s n ) =
of
(n, and
n
1
LapZaoe transform of
]
]
Thus Gleason's theorem
56
FUNCTION ALGEBRAS
tells us:
if
restriction of) functional on transform of
e C(X) *
and
cp
is a multiplicative linear
cP
P(X)
if and only if the Laplace
never vanishes.
cp
(1) = 1, then (the
It seems that
there should be applications of this result, but we know of none.
1-5
ANALYTIC STRUCTURE
In the handful of examples that we have looked at, when
A was a function algebra on
and
Spec A was strictly larger than
son
was that the functions in
on
X, the rea-
A were analytic
Spec A\X . (or, as in the big disk algebra, on
substantial parts of
Spec A\X).
was conjectured that whenever than
X
At one time, it
Spec A
is larger
X, there must be "analytic structure" some-
where in
Spec A.
As to what "analytic structure"
should mean, it would seem that a minimum ment might be: disk in
there is an imbedding
~
Spec A, such that the functions
analytic, for every
f e A.
requireof a f.~
are
Besides the known
examples, support for this conject.ure came from the work of Wermer in two different contexts.
One
BANACH ALGEBRAS
57
of Wermer's theorems was this: analytic closed cu~ve in
P(J) = C(J), o~ else
finite
bo~de~ed
finite
numbe~
Riemann
J
be an
Then eithe~
(n.
=
polynomially convex (i.e., J case
let
J
J
is
Spec P(J), in which
=
su~face,
Spec P(J) with
of points identified.
is a
pe~haps
a
(This theorem
has been improved, to weaken the requirement of analyticity to continuous differentiability:
the
major step due to Bishop, and important contributions due to Royden and Stolzenberg.
See Stolzen-
berg [4] for the best current version, and for the references.) The other theorem of Wermer's to which we referred is this: functions in
if
ReAl
the
A, is dense in
CR(X),
¢ e Spec A, one of the following i)
ii) ~
II ¢ - 1); II the~e
=
2
~eal
f 0 ~ eve ~ y
pa~ts
of the
thmfo~
each
alte~natives
holds:
1jJ e S p e c A, 1); f ¢;
exists a continuous one-one map
of the open unit disk onto
P = {1jJ e Spec A : II¢ - 1jJ1I < 2}, bicontinuous i f
P
is given the
holomo~phic fo~
met~ic
each
theorem in Chapter IV.
topology, such that
f e A.
f.~
is
We shall prove this
FUNCTION ALGEBRAS
58
However, the hope that analytic structure is always present when
Spec A
r
X was shattered
by an example devised by Stolzenberg [1]; he found A
a set
X
in
such that
X
X
properly containing
X,
contains no "analytic disks", in the
sense described above. If one cannot expect to have analytic disks in
Spec A, one can still ask if there are more
primi ti ve aspects of analyticity left.
In Chapter
II, we discuss the presence in function algebras of such phenomena as anti-symmetry (real valued holomorphic functions are constant on connected sets), Schwarz's lemma, and the Jensen inequality. In the next section, we discuss a more direct aspect of holomorphic functions, viz., their differentiability.
In this section, we present a
theorem of Gleason which deduces the presence of analytic structure from algebraic properties. We shall need the notion of analytic variety.
DEFINITION. subset in
G
V
Let of
G
G
be an open subset
f
",n. ~
is called an analytic variety
if for each point
open neighborhood
0
U of
s e G
there exists an
s, and a finite set
A
BANACH ALGEBRAS {f l , ... ,f r } that
V n V
=
59
of functions holomorphic in {~
e V :
=
f.(~)
J
0 for 1
2
V, such
j
< r}.
The finiteness of the defining family of functions is not essential. subset
V
variety in s e G\V ishing on
Thus, to show that a
of the open set
G
is an analytic
G, it suffices to show that for every
there exists V, with
f
f(s)
holomorphic in
=
1.
G, van-
(This is a simple
consequence of the fact that the ring of convergent power series in
n
variables is Noetherian; for
the proof, we refer the reader to the book of Gunning and Rossi [1].) It is not necessary to know any of the deep theory of analytic varieties to understand Gleason's theorem, but to help appreciate its import in our context, we state the following fact: an analytic variety in
G.
let
V
Then for each non-
isolated point
s e V, there exists a one-one
continuous map
~
V, with in
~
~(O)
=
whenever
of the open unit disk
s, such that f
be
fo~
~
into
is holomorphic
is holomorphic in
G.
Again,
we refer to Gunning and Rossi for the proof.
FUNCTION ALGEBRAS
60 THEOREM 1.5.1 (Gleason [2]).
A
Let ~
tative Banaah algebra with unit,
be a aommu-
e Spec A .
M = ker ¢
Suppose that the maximal ideal
is
finitely generated in the algebraia sense, i.e.,
f l' ... ,fn e M suah that
there exis t
M = {\' f . g .
g. e A, 1 < j < n}. J -
J J
L
Then there exists
> 0, and an analytia variety
£
V in the polydisk 1
~ j
~
n}, with the following property:
be the neighborhood of
Spec A
in
~
let
U
defined by
A
U = {ljI e Spec A : If. (ljI) I < £ , 1 < j J
let
F : U+lI(O,£)
...
A
F(ljI)
(f 1 (ljI),
g
there exists A
T : An
Evidently, T hypothesis
T
F
is a homeo-
V, and for eaah
holomorphia in for all
g e
lI(O ,E)
A,
suah that
ljI e U.
Consider the Banach space
with the norm Define
Then
,fn(ljI)) .
'V
g(ljI) = g(F(ljI)) Proof.
be defined by A
U onto
morphism of
n}, and
~
An = Ax>
:"'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 =
. .. +
such that
,n)
I g j II ~ KIi& II
and
Lf.g· . J J
Similarly, for each j we find gjk € A with n g. =
k = 1, ... ,no
Thus
n
g =
+
L
IIg jk ll ~ K211g II
where
for
j,k=l, ... ,no
By
induction, we find for every positive integer a polynomial
Pr
in
n
r,
variables, with complex
coefficients, of degree 2.. r - 1, such that ,f )
g
n
+
with
for
every mUlti-index that
E:nK
n
I Iji (f.) J
I
<
E:
Let
< l.
for
Choose
j 1 ..... j r'
j
= 1,
U = {Iji
€
,n} ,
E: > 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 <
I/J e IT,
, fn)
Pr(fl'
U.
IT, the
separates the points of
...
fl ,
Hence the map
,fn
F
separate the points
defined in the statement
of the theorem is a homeomorphism of image, and hence of
U onto its image
observe that for each
Spec A,
IT
onto its V.
We
r,
n
Pr + 1 - Pr = where
L
C. .
jl, ... ,jr=l ... ,zn
JI"'J r
zJ. • •••• z. I
Jr
are the coordinates of
(n
(or indeterminates), and so
Icjl.,.jr l
(functions)
..::
Kr~g~,
Pr(zl""
Thus, the polynomial ,zn)
converge uniformly
on the closed polydisk
X(O.E), and hence their
limit is a function
holomorphic in
g
~(O,E).
BANACH ALGEBRAS
63
Evidently, g(1ji)
=
g(F(1ji))
remains to show that
for each
V
and
h =
0
for every
,fn
fl - sl'
- sn
(by definition of nition of
Put
U).
h
t e V.
Now if
Spec A\U
such that
J
J
U
(by defi-
J
1: (£.J
- s.) g. J
J
It is clear that
h
= l.
The proof is finished.
POINT DERIVATIONS
DEFINITION.
Let
A be a commutative Banach alge-
bra with unit, and let 1ji
1
=
Hence (Corollary 1.2.13) there
fulfills the requirements.
1-6
h(s)
then
V), nor in
}:(z. - s .)g ..
1
with
there
have no common zero in
,gn e A
gl'
~(O,£)\V,
~(O,£)
~(O,£)\V,
,sn) e
exist
s e
holomorphic in
h(t)
It
is an analytic variety,
i.e., to show that for any exists
1ji e U.
~
e Spec A.
is a point derivation on
linear functional on 1ji(fg) whenever
= 1ji(f)~(g)
f,g e A
A
at
We say that ~
if
1ji
is a
A, and if +
~(f)1ji(g)
(Leibniz' rule).
A bounded
pqint derivation is a point derivation which is
64
FUNCTION ALGEBRAS
bounded (continuous) as a linear functional. ~
If
~,it
is a point derivation at
clear that
~(l)
~(f)
=
= ~(g)
=
O.
o
~(fg)
0, and that
~
Conversely, if
is
whenever
is a linear
functional with these properties, it is easily ~
seen that ~[(f
is a point derivation at ~(f))(g
-
-
o
~(g))]
~:
for
for any f,g
A,
€
and multiplying out and using linearity, we obtain Leibniz' rule.
We can formulate this remark in
terms of ideals.
Let
M
~, i.e., M = ker ~, and let
associated with
denote the linear span of M2
is an ideal in
t
and
M2
+
{fg: f,g
~
+ M2;
at
~
M}.
Then
A point deriA
which
it follows that there exists
a non-zero point derivation at
r M,
A.
is a linear functional on
annihilates
M2
€
M2
A, though not necessarily closed,
is a subalgebra of ~
vation at
be the maximal ideal
~
if and only if
and there exists a bounded point derivation if and only if
If
A
=
derivation on
M2
is not dense in
M.
C(X), there exists no non-zero point A.
For if
x
is any point of
X,
BANACH ALGEBRAS and
65
=
f E A, f(x)
g(x)
=
hex)
0, we can write
0, g,h E A:
=
f
gh, where
= /fIT,
for instance, g
h = gsgnf.
If
A
is the disk algebra, there is an ob-
vious point derivation open disk, given by can write so if
=
D
Ds at each point
Dsf
any derivation at
=
Dz
= 1, and
=
=
D(z - s)
whenever
f
D
If
of the
f E A, we
for some
s
,
(Dz)f (s)
g E A;
=
(Dz)Dsf.
is a multiple of
a derivation at
0, since
s
s, Df = D[(z - s)g]
=
(Dz)'g(s)
Thus every derivation at lsi
f (s).
f = f(s) + (z - s)g
D(z - s)g(s)
If
,
s, then
{z - sEA, so
a polynomial in
z.
Ds'
Df
=
0
This does not
immediately exclude the possibility of a non-zero (unbounded) point derivation at
s.
By using
deeper results of function theory (the Nevanlinna "inner-outer" factorization) one can show that for any
f E A, f(s) = 0, there exist
g(s)
=
h(s)
=
0, such that
f
=
no non-zero point derivation at
g,h E A,
gh; thus, there is s.
We shall see
this as a result of Banach algebra methods later in this section. Point derivations may occur even in the absence of any analytic structure.
It was only
66
FUNCTION ALGEBRAS
very recently shown, by Brian Cole, that there exist function algebras, other than
C(X), which
admit no non-zero point derivation.
One of Cole's
examples is given in the appendix. The next theorem has been known by various people for many years, but apparently never published.
THEOREM 1.6.1.
unit~
aZgebra with
A
Let
be a commutative Banach
and
a, Zinear functionaZ on
$
A (not necessariZy continuous). ker $
A.
is a subaZgebra of
i)
Then
$(1) , 0, then
if
Suppose that
$
is a scaZar mul-
tiple of a multipZicative linear functionaZ;
ii)
$(1) = 0, then either there exist
if
6~ ljJ
e Spec A
of
6 -
$
ljJ~
$
such that
or there exists
is a point derivation at
Proof.
If
<j>
(1)
Then, for every cP
Since
(fg)
=
r 0,
i,s a scalar multiple
6 e Spec A
such that
6.
we may assume
cP
(1)
1.
f,g e A, we have ([f -
$(£ - <j>(£))
cP
o
(f)][g =
cP
(g)])
+
<j>
<j>(g - $(g)), and
(f)
BANACH ALGEBRAS
67
is an algebra, it follows that
~(fg)
and i) is proved. Now suppose
If there exist
8,1/1 e Spec A, 8 f 1/1, such that every
f e ker
ker
is a scalar multiple of
and on
~ (e 2) . = O.
A
f O.
1, and put e
=
(eO)
~
by
8 (f)
8 - 1/1.
have for every
We are left with 8~1/1
with
e Spec A
8(f) f 1/I(f).
Choose eO e A such that 1 2 eO - 2
Since
ker
f -
~(f)e
8
e ker ~
is an algebra, we
f,g e A:
=
~(f)e]
[g - Hg)e])
= (f)Heg) +
~
Define the linear functional
f e A, and
for every
o
=
for
ker(8 - 1/1), so
f e ker
there exists
We may assume ~
c
the case: (8 f 1/1)
8(f) = 1/I(f)
~(f)
so
J
= ~(f)8(g)
= (ker
(*) that
8(1) = 1.
J
+
is an ideal in
(*)
We see at once from A, proper since
We note that for every
f e A,
FUNCTION ALGEBRAS
68
e (f) - 4> (f) e e
f -
4> (1) =
as we see from Now J
(**)
J,
e (e) =
0, 4> (e) =
e (1) =
is contained in a maximal ideal of
(Lemma 1.2.10) , and thus there exists such that we have
J
ker
c
IjJ
for every
e Spec A
f e ker 4>, and
it follows that
IjJ
is
We wish to show that
IjJ
= e.
from our assumption on uniquely determined.
A
From (**)
(Theorem 1. 2 .12) .
= e (f)
IjJ (f)
IjJ
1.
4>
Using (**), we see this is equivalent to showing that e
ljJ(e) ljJ(e)
+
O.
=
f
ker
Now if
ljJ(e)
since
ker
1jJ;
imal ideal containing e
ljJ(e)
+
and
+
ljJ(e))g
e - ljJ(e) , we have Now
IjJ (e 2 )
e2
IjJ (e 2)
(*)
But then, multiplying by
(e 2 - ljJ(e 2))g
4>(e)
shows that
=
e -
IjJ
1, 4>(1)
=
ljJ(e) 4>
e, concluding the proof.
+
e - ljJ(e) e J.
e 2 e ker 4>, so
since
e J, and hence
diction shows that and
such that
g e A
e (e 2)
is impossible:
is the unique max-
IjJ
J, it follows that
1 e J.
+
0, then
generate the unit ideal, i.e.,
J
that there exists (e
r
=
O.
(e) e J. O.
Thus
But this
This contra-
e
= IjJ
e Spec A,
is a point derivation at
69
BANACH ALGEBRAS
A similar theorem, concerning subalgebras of finite codimension in
A, was found by Gamelin
[1] .
A simple-minded approach to the problem of constructing point derivations might run as follows. Suppose
$
Spec A
€
is not isolated in the metric
. ..
topology, so there exist 11$ - $n l
with ~n
€
A* ,and
point of
-+-
~~n~ =
{~n}'
since
=
$(g) ,.,
't'n
= 0,
-+-,., 't'
,
Let
1.
be any cluster
~
in the weak-* topology of ~(l)
is clear that $(f)
Put
O.
= 0,
then hence
~(fg)
~
=
$.
O.
Thus
It
If
I~n(fg) I ~ I$n(f) Ilg~
bounded point derivation at this argument is that·
I~~ < 1.
and
A* •
-+-
0
is a
The trouble with
might be the zero deri-
vation, which we can obtain with less trouble.
In
fact, there need not exist bounded point derivations (other than zero) even at non-isolated (metrically) points of
Spec A.
However, we do have a theorem
for the existence of (possibly unbounded) non-zero point derivations. author [2].
This theorem is due to the
,
FUNCTION ALGEBRAS
70
THEOREM 1.6.2.
LetA
unit~
algebra with
be a commutative Banach
e Spec A. Suppose there
~
let
~.
exists no non-zero point derivation at ~
is an isolated point of
Spec
A~
Then
in the metric
topology.
Proof.
As above, let
linear span of is then that U and let
M
=
ker ~
and
{fg: f,g eM}. M2
=
M.
M2
the
The hypothesis
Let
{fg : Ilfll < 1,lgll.2. l,f,g eM},
=
K be the convex hull of
K
{ L
t.f.
I
finite, t. > 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 :
€
<
71
c}.
M, 11£/1 -<
€ €
M, IIf II -< IS}
-<
sup{le(f) 1
f
€
K}.
Spec A.
ljJ €
to
ljJ
Let
M.
denote the
e
Then
lIel < IIcp -
K, then there exist f. , g. J
t.
e (f)
->
0, It. J
ljJlI· ~ j
=
-< n)
1,
It.e(f.g.) = It.e(f.)e(g.) J
le(f)1
<
IS
J J
J
J
J
\' t . II e 1 2, II f. II /I g . II < II e 112 • /. J
IS lie II ~ lie 11 2 , so ljJll >
J
M, (1
€
J
< 1 and 11£.11 -< 1, IIg.11 J J such that f = LtjJjg j • Then
IIcp -
1.}
f
with
so
M* ,
€
sup{le(f)1
restriction of €
: f
e
=
Now suppose
f
I
sup{lSle(f)
IS lie II
If
Hence for any
'J
lie I 2: IS
for
ljJ
€
J-
Thus'
unless, e ;:: 0, therefore
Spec A,
f cpo
ljJ
The theorem
is proved. The sophisticated reader will observe that the proof consisted of applying the open mapping theorem to the natural map of the tensor product M ® Minto
M.
The converse of Theorem 1.6.2 does not hold. In fact, Sidney [2] has constructed an example where
~cp
-
ljJ~
=
2
for all
ljJ
€
Spec A,ljJ f cp, yet
FUNCTION ALGEBRAS
72
admits a bounded point derivation.
~
The reader
is referred to this paper (and Sidney [1]) for more about point derivations. We do
have~
however, a condition which ensures
the non-existence of point derivations, due to Paul Cohen.
Let
DEFINITION.
A be a commutative Banach alge-
bra with unit, M that
M
a maximal ideal of
A.
We say
has an approximate identity if there
exists a constant
K such that for every
every
E
f l , ... ,fn
IleII
< K,
j
1, ... ,no
=
such that
bounded net
every
f
M, there exists lef. - f·1I < J
J
£
e
0,
>
£
M,
E
for
(In other words, there exists a
{e a }
in
M
such that
ea f
+
f
for
M.)
E
The notion of approximate identity is a familiar one in harmonic analysis.
Let us see
what it amounts to for function algebras.
LEMMA 1.6.3. and
x E X. i)
Let
A be a function aZgebra on
The the foZZowing are equivaZent:
The maximal ideaZ associated with
x,
X,
BANACH ALGEBRAS M
=
73
= a}, has an approximate identity.
{f e A : f(x)
ii)
for any neighborhood
f e A
exists
If(y)1 < Proof.
E
of
U
M has an approximate identity.
Suppose
K
such that for any
f1' .•. ,fn e M, there exists J
borhood of
J
If
E
If
f
y
I (e£.) (y)- f. (y) I <
E"
whenever
y
f = 1 - e.
f
e (x) =
f e y
f
for
with
A
U.
Put
e e M.
since
0
E
Put
Thus i) implies i i) . f1'
Ifj(Y)1 < ElK}.
neighborhood of E
Ie (y) - 11 <
, hence
U, and
U = {y :
611f J·11 <
j,
J
If ii) holds, and put
Choose
U, then for some
I£. (y) I > 1, and
J
U.
c
1 .::. j .::. n, we have J
is a neigh-
f1' ... ,fn e M such
x, there exist
accordingly.
0,
E >
e e M, U
{y : If j (y) I < 1, 1 < j .::. n} -
that
there
> 0 ,
E
y e X\U.
for aZZ
Ilell
e
x, any
Ilf II < K" f (x) = 1, and
with
Then there exists any
K, such that
There exists a constant
for
x.
Choose
j = 1,
...
...
,fn e M, E>
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
< <5
1 < j ~ n.
Thus
I ef.J (y) - f.J (y) I lef. - £.~ < E for J J
f
y
U,
Thus
< E.
=
M has an approximate identity.
The proof is completed.
COROLLARY 1.6.4. x~
x e
X~
A
If
is a function aZgebra on
and there exists
f e
A which assumes
x~
then the maximaZ
its maximum moduZus onZy at
=
ideaZ {f e A: f(x)
Proof. y
r x,
If
O} has an approximate identity.
f e A, f(x)
then for
n
=
1
I fey) I
and
<
sufficiently large, f n
fies condition ii), with
1
for satis-
K = 1.
THEOREM 1.6.S (Cohen [1]). Let A be a commutative Banach aZgebra with
unit~
M a maximaZ ideaZ of
A~
and suppose M has an approximate identity. Then for every f e M, there exist g, heM such that
f
=
gh. For any
<5
>
0, h can be chosen in the
cZosed ideaZ generated by f, and Ih - f 1<
Proof.
Let
K be as in the definition of approxi-
mate identity, and choose c, 0 assume
K > l.
that whenever
o.
Then
0 < c <
e e M and
<
1
4'
1 4K· We can and it follows
c
<
Ilell ~ K, we have
BANACH ALGEBRAS 1 - c
+
75
-1
ce e A
; indeed,
(1 - c + ce)
-1
=
00
1
(
1 _
the series converging since Furthermore, if
lieF - FII
c )k k c - 1 e J
ce 1/4 1 Ilc - 111 <3]4 = 3' is small for some F,
-1 IEF - F II, where E = (1 - c + ce) , 1 c k 1 = 1) , we have 1 - c L (c
then so is
00
For since
0
liEF
-
FII
=
11
- c 1
lIekp - pil
k-l
=
II I
o
k-l <
I
o
6
c (c - l)k(e k F - P)
8 ( r :cc ) k ~e k P
00
< 1 - c
but
00
1
- F~,
. e J + 1 p - ejFI
. lIeJ~~eP - FII k-l
<
I
lIeP - pil
L o
. lell J
k < lieF - P 11K K_ 1 ;
hence 00
1 1 1 liEF - pI < leF - F~l - c L K - 1 [4(1 0 <
c)]
2 K - 1 ieF - P ~ ,
This estimate is the key to constructing h, which we do as follows,
g
and
k
FUNCTION ALGEBRAS
76
We shall define inductively a sequence ~e
en e M, gn
c
n
L
n
~
<
-
K, in such a way that if
c)
(1 -
{en}'
k-l
1
e k + (1 - c)n, we have
for
n, gn e A-1 , ~gn -If - gn+l -1 fl < £ and Zn <5 - gl-lfll < 2· Put h n = gn -1 f. Then h e M n
every
Ilf
(in fact, h n is in the ideal generated by f) and {h n } is a Cauchy sequence, so h n + heM. 00
Also, gn
C
+
L
(1 - c)
k-l
1 ~ek~ ~
converges since
e k = geM
K and
Hence, gh = lim gnhn = lim gngn
(the series
0 < 1 - c < 1). -1
f
= f, and the
theorem is proved as soon as this construction is made. Now
gl
ce l + 1 - c, so
=
-1 gl e A
and
<5 IIf - gl -1 fl 1 < 2' provided only that lelf - £I < <5 -(K - 1) . Suppose that e l , ... ,en have been 4 chosen so that gn e A -1 , etc. We assert that -1 and IIgn+l- lf - gn -1 f II < <5Z- n provided gn+l € A only that ~ en+1f - f~ and ~en+lek - ek~ (1
<
k
<
n) are sufficiently small. For if E
(1 - c + ce n +l ) g and
=
n
-1
E-lc
,we have n
L
1
(1 - c)k-1 Ee k + (1 - c)n
BANACH ALGEBRAS
77
n
c) k-l Ee k + (1 - c)n. 1 n (1 - c) k-l IIG n - gnll < II Ee k - ekl c
Let
Gn
c
I
(1
Then
t
max lEek - e k II < l-
<
Hence
Gn e A-1 , and
IIG n
-1
2
K -
1 maxllen+lek - e k I·
gn -111
is small,
Ile n + l e k - ekll is small for E-IG k = 1, we have ,no Since gn+1 n' then gn+l e A-1 , gn+l -1 = Gn -IE ' so
provided only that
...
Ilg +l-l f - g -1£1 = IIG -IEF - g -lfll n n n n -1 -1 -1 -1 < IIG n Ef - gn Efll + IIg n Ef - gn fll <
Thus if
IIG n - l - gn-1111Efll + Ilg n - 1 11l Ef - fl.
Ile n +1 f -
fl
and
Ile n +1 e k - ekll(l < k < n)
are sufficiently small, we will have IIg n +1 - l f - gn-1fll
as small as we please.
The
inductive construction can thus be accomplished, and the theorem is proved. We note that the commutativity of A was never used, and that it sufficed that M have a "left approximate identity". See Cohen's paper for the original ap-
FUNCTION ALGEBRAS
78
plications to harmonic analysis, and other factorization results.
COROLLARY 1.6.6. Let A be a commutative Banach aZgebra with unit~ ~ e Spec A. If ker ~ has an approximate
identity~
then there exists no non-
zero point derivation at
~.
This corollary was first pointed out by Curtis and Figa-Talamanca [1]. Combining it with Corollary 1.6.4, we have
COROLLARY 1.6.7. Let A be a function aZgebra on
X, Zet x e X. If there exists f e A such that f assumes its maximum moduZus onZy at
x~
exists no non-zero point derivation at
then there
x.
CHAPTER
TWO
MEASURES
2-1
REPRESENTING AND ANNIHILATING MEASURES Let
measure
X be a compact Hausdorff space. on
By a
X, we shall understand a complex,
regular Borel measure, unless otherwise qualified. A probability measure is a positive measure of total mass
1.
If
~
is a measure, I~I
denotes
the associated positive total variation measure, and we wri te
II~
II
for
I~ I (X).
the unit point mass at
x.
For
x eX, Ox
For each measure
X we can determine a smallest closed set on which
~
denotes
"lives", its support.
on
~
supp
Thus, supp
~ ~
is the complement of the union of all open sets such that
I~I
if and only if
(U) = 0, or equivalently, x e supp I~I
(U)
>
0 79
for every
U ~
80
FUNCTION ALGEBRAS
neighborhood
U
of
x.
Our interest in measures derives from the Riesz representation theorem: linea~
uous
dete~mined measu~e
uniquely
~(f)
ffd~ fo~ eve~y
Let
A
C(X)~
funational on
~
if
~
is a con tin-
the~e
X such that
on
f e: C(X), and
be a linear subspace of
exists a
II~I
C(X).
=
II~II.
Then the
Hahn-Banach theorem and the Riesz representation theorem yield the following two remarks: 1)
~ II~
II~
=
measu~e
II·
of all ~
~
Such a
fo~
for all
~.
f e: C(X)
i
e: A , where
g e: A}.
A
Ai
=
fgd~ =
{~
Ai
The elements of
~
If ciated
A for
i
~
for every
for every
will be called
A, and we sometimes
i
is a probability measure, the asso-
~(l)
=
HI)
~~~ ~
as obviously, i f and
0
0
e: A •
~
linear functional
property:
C (X)
~
is precisely the set
Jfd~ =
such that
annihilating measures for write
f e: A, and
will be called a ~ep~esenting
The closure of
2)
there exists a measure
~(f) = ffd~
such that
II
e: A* ,
~
For each
=
h II
=
1.
is a
on
C(X)
has the
Conversely, but not linea~
functional on
= 1, the associated
measu~e
MEASURES is a
81
probabi~ity
~(u)
suffices to show that u e C(X),O < u < 1. a
and
ut
=
b
u
1~(ut)12 1
+
1
<
b 2t 2 . b
hence
-
I~(l)
If if
real. ibt.
+
O.
=
Ilu t 12
Now
111 - ull
+
<
11
a
ib,
+
t, put b 2 t 2 , so b 2 (t
1) 2
+
<
<
1 - a2 _ b2,
1, so al .s. 1, so
is a linear subapce of and
=
~(u)
t, 2tb 2
C (X) ,
a > O. 1 e A,
~ ~ II = ~ (1) = 1, it follows that
any representing measure for measure.
for every
< 1 +
b 2 t 2 , i. e. , a 2
~(u) I < 1, i. e. ,
A
0
Suppose
Thus for all real
e A*
~
~
For each real
Then +
To see this, it
measure.
~
In particular, for each
associated evaluation functional
is a probability x e X, the Lx' defined by Lx e A* ,
f(x), has the property:
Lx (1) = IILx II = 1, so any representing measure for Lx
is a probability measure.
A
repr~senting
measure for
Lx
will also be called a representing
measure for
x.
If
see that if
~
for
~
A
is a function algebra, we
e Spec A, any representing measure
is a probability measure. Occasionally we will have use for measures
which represent extensions other than Hahn-Banach extensions.
If
~
e A* , we say that the measure
82
FUNCTION ALGEBRAS represents
~
or is a complex representing
~J
~,if ffd~
measure for
Thus a measure
II
=
1I'l> II. If
g
for every Borel set whenever
a closed space If
+
Jgfd~.
E.
If
f e A
(f~)(E) = fEfd~
Thus, A
and
is an algebra, then ~ e AL.
Conversely,
A with this property must be an A
is a function algebra, and
represents some and
X, and
the measure corresponding to the.
f~
linear functional
f e A
is a
~
if and only if
~
is a measure on
~
algebra.
f e A.
We apologize for this language.
we denote by
f~ e AL
for all
which represents
~
representing measure for II~
~(f)
=
e Spec A, then
~
JfdjJ
o.
f~
~
A whenever
L
These trivial remarks have
extensive applications. Let us look at our standard example, the disk
r.
algebra on the circle Lebesgue measure on
r.
Since
k > 0, it follows that measure for write \I
so
=
A.
f = f(s)
z o. z - s J£d\l
=
If +
Then £(s)
+
Let
zo
be the normalized
a
JzkdO = 0
is an annihilating
lsi < 1, and
f e A, we can
(z - s)g, where Jd\l
=
J~
for
g e A.
(sz)n do
fgZdO = £(s).
=
Thus
~
Put
snJZn dO \I
is a
1,
MEASURES
83
complex measure representing Now
integral formula). 1 1 -
€
v
s
€
A,
sz
I
f(s) = f I
i. e. ,
I -
A
and
f(O) = 0, then
Jfzkda = 0 f
A~
fo, f
for
€
fa
If
A; such measures
~
, in fact.
For if
f e C(r)
k > I, then the Cesaro means
are polynomials iri
theorem.
'
(Poisson integral formula).
are weak-* dense in and
da
11 - szl 2'
is a (positive) representing
11 - szl2° s
Jf
=
I s I 2 da .
11-sz1 2
Is 12
-
measure for
of
a, and
1 - sz A, so we have for every f
or
€
1
=
f(s) ;:: Jf _ I_ _ dv 2 I - Isl 1- sz
f
(the Cauchy
z, so
f
€
A
by Fejer's
A deeper fact is that the set of measures A~; this
A, f(O) = 0, is norm dense in
is a theorem of F. and M. Riesz, which will be proved in a more general setting in Chapter 4. If
lJ
€
A ~·1S real, t h en
k > 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* , 11<1>11 = <1>(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 ~
<1>,
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)
<
< ~(U)
~(U)
>
~ = ~
~({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 of
a x
<
x e X.
S, suoh that for every neighthere exists
Ref.::. 0, Ref(x) > -a, and y
f
U.
Then
Suppose there exist
x e Ch(A).
f
e
A with
Ref(y) < -S
for
MEASURES Proof.
From
fl"nd ).l
89 -a
).l(U) > 8
for
<
= fRefd).l
Ref(x)
-8).l(X\U), we
<
Sa t "lng or f any represen
x, any neighborhood
U of
measure
x, and the
theorem follows as before.
THEOREM 2.2.3. =
8
= inf { Re$ (f)
f e A, Ref> 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
e C(X) *
for that
with
f e P.
Re
for
feN, Re 0
The latter inequality easily implies
is a positive linear functional, so we
90
FUNCTION ALGEBRAS
may assume also that ~(f
-
Re~(f))
e N, so
Im6(f) = Re6(-if) it follows that Since
~u
e N,
= 1.
~(1)
Re~(f)
~(f)
= Re~(f).
COROLLARY 2.2.4.
Let
for any
= ~(f)
= O.
~(u) ~.
Take
Since 6,
f e A.
to be the measure
~
The proof is finished.
~
e K.
~
Then ~
unique representing measure
u e
f e A,
for any linear functional
which represents
every
If
admits a
if and onZy if for
CR(X)~
IUd~.
sup{ReHf) : f e A, Ref < u}
COROLLARY 2.2.5.
Let
and onZy if for every
x e X. u e
x e Ch(A)
Then
if
CR(X)~
sup{Ref(x) : f e A, Ref < u}
=
u(x).
Combining this last Corollary with Theorem 2.2.2, we obtain the following characterization of Ch(A): THEOREM 2.2.6.
The foZZowing statements are equi-
vaZent: i)
borhood
For every
U of
X3
a~B30
<
a <
there exists
B~
f
and every neighe
A3
with
MEASURES
91
Ref ~ 0, Ref(x) > -a, and
f
y
for aZZ
U. ii)
a~S~O
There exist
A~
iii) Proof.
f
y
U.
x e Ch(A). That i) implies ii) is trivial, and ii)
U a neighborhood of lemma, there exists u < -S
exists
there exists
x
implies iii) is Theorem 2.2.2.
and
such that
Ref < 0, Ref(x) > -a, Ref(y) < -S
with
for aU
S~
< a <
U of
for every neighborhood
f e
Ref(y) < -S
x,
°
Suppose
< a < S. ~
u e CR(X) , u
on. X\U.
f e A with
x e Ch(A) ,
By Urysohn's 0, u(x) = 0,
By Corollary 2.2.5, there Ref
<
u
and
Ref(x) > -a.
Thus iii) implies i), and all is proved.
COROLLARY 2.2.7. Ch(A)
is a
If
Go
X is
metrizeabLe~
then
(countabLe intersection of open
sets). Proof.
Let
topology of
X.
be a metric on
n e:
A with
for all
y
Ref with
X
inducing the
For each positive integer 'n, let
be the set of all
G
f
p
-<
0,
x
e:
X for which there exists
Ref(x)
p(x,y)
> -1 ,
> 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
<
supp
~,
supp
~
V
fd~
and
x
A
= ox'
=
~(U)-lJu fd~. U of
for any
measure for
~
~,
x, then
V, so
f e A, and arbitrar-
V of
x.
It follows that
separates points, it follows that ~ =
{x}, and
Since
But
was an arbitrary point of
every neighborhood ~
Since
~(U))1jJ.
for any smaller neighborhood
Since
=
(1 -
for some neighborhood
~(V)-lJ
= Lx'
+
K, it follows that
~(f)
i.e., that
X\U
~(U)8
ily small neighborhoods ~
-If.
(1 - ~ (U))
e K, and
8,1jJ
fd~,
Iu
is an extreme point of
~
8
~tU)
ox'
U of
~(U)
If
=
1
for
x, we have at once that
was an arbitrary representing
it follows that
~
= Lx' with
x e Ch(A), and the proof is concluded.
The Krein-Milman theorem now assures us that Ch(A) is not empty. DEFINITION. boundary for
A subset A
Y of
if for every
X is called a f e A
there exists
FUNCTION ALGEBRAS
94
y e Y such that If(y)1 = I flo THEOREM 2.2.9.
The Choquet boundapy is a boundapy.
Let f e A, and let x e X be any point
Proof.
where If(x) I =
~
fl. Put L =
{~
~(f)
e K :
= f(x)}.
Clearly, L is a closed convex subset of K, L is not empty since LX e L, so by the Krein-Milman theorem there exists an extreme point
~
= ta
+
then
~
a,
e K, we have
~
is extreme in K: for if
If II = 1~(f)1
-<
tla(f)1
+
so equality holds, so a(£) = a,
~
e L, and since
By Theorem 2.2.8,
~
~
of L. But
(1 - t)I~(f)1 ~(f)
=
~(f)~
is extreme in L,
t)~,
,(1 -
If~,
-<
i.e.,
e =
~
=
~.
Ly for some y e Ch(A) , and
~
the proof is finished. COROLLARY 2.2.10.
The closupe of the Choquet
boundapy is a closed boundapy, contained in evepy closed boundapy.
Proof.
If Y is a closed boundary for A, it is
clear that the restriction map is an isometry of A onto Alx = {fiK : f e A}. Hence, each !\
I "
e K
admits a ~presenting measure whose support is
MEASURES
95
contained in Y; it follows immediately that Y J Ch(A). The unique minimal closed boundary is called the ShiZov boundary; its existence (for algebras of functions) was first proved by Shilov. Other proofs have been given by Arens and Singer [1], and lIormander [1], among others. Hormander's proof does not use the axiom of choice. The main theorem about the Choquet boundary is Choquet's Theorem: if X is
measure
e K, there
~
for each ~
boundary~
~xists
metrizeabZe~
then
a representing
which is concentrated on the Choquet in the sense that
~(X\Ch(A))
=
O. A
beautiful short proof of this theorem was found (independently) by Bonsall [1] and Herve [1]. If X is not metrizeable, the situation is more complicated; since Ch(A) need not be a Borel set, the statement of the theorem must be modified. The generalization of Choquet's theorem to the non-metrizeable case is due to Bishop and deLeeuw [1]. The reader is referred to the book of Phelps [1] for a more complete discussion and further references.
96
FUNCTION ALGEBRAS PEAK POINTS
2-3
Throughout this section, A will be a closed subspace of C(X), X compact Hausdorff, separating the points of X and containing the constant functions. DEFINITION.
A subset K of X is said to be a peak
set if there exists f e A such that K
ix
e X
f(x) = l}
I f (x) I
{x e X
= ~ f" };
any such f is said to peak on K. We call K a peak set in the weak sense if K is the intersection of
some collection of peak sets. A point x e X is called a peak point if {x} is a peak set, or a peak point in the weak sense if {x} is a peak set
in the weak sense. Peak points in the weak sense are also referred to as strong boundary points. We note that a peak set is necessarily a 00
compact Go
(K =
n {x eX: If(x)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
<
t,
and Ifu(y)1
<
~ for all y • U.
We define inductively a sequence {Un} of neighborhoods of x, and a sequence {fn} in A, as follows: let Ul = V, and fl = fV' Having defined Ul"",U n _l and f l , ... ,f n - l , set
1 < j
I},
< n -
00
and K =
and put fn = fUn' Now let f {y eX: fey) = I}. 00
If y • V = Ul = U Un' we have Ifn(y)1
<
1
for all n, so If(y) I
<
j.
Un\U n + l ' we have Ifj(Y) I
t
If for some n, y e <
1 +
j
2- n for 1 < j < n,
and Ifj(Y)1 < ~ for all j > n, so
I f (y) I
<
n-l L
j=l
1 -
00
(1 + 1 2 - n) 2 - j
3
.£ 3
+ 4 2 - n + 1. L 2 - j ~ ~ n+l
4 -n < 1.
00
Finally, if y e ~ Un' then Ifjey) I all n
>
Thus Hfl
j, so If j (y) I <
< 1
<
1 +
~ 2- n for
for all j, so If (y) I
1, f(x) = I, and If(y)1
<
1 for all
< 1.
MEASURES
99
00
y ~ n U 1
1 In particular, K C V, and 2 (1
n
t
f),peaks
on K. The theorem is proved.
LEMMA 2.3.3. sense~
If x e X is a peak point in the hleak
then x beZ6ngs to the Choquet boundary.
Proof.
Let
~
be a representing measure for x, and
U any neighborhood ot x. There exists f e A which peaks on K, x eKe U. Then 0
=I
(1 -
f)d~ =
f Re(l
-
f)d~,
~(X\U)
=
O. It now follows from the regularity of
~
~(X\{x})
that
but Re(l - f) > 0 outside U, so
=
0, i.e., that
~
=
ox. This proves
the lemma. We now return our attention to function algebras. Gathering together the results of this section and the last, we have:
THEOREM 2.3.4.
Let A be a function aZgebra on
X, x e X. Then the foZZohling ape equivaZent: i) x is a peak point in the hleak sense;
ii) x e Ch(A); iiI) If jJ({x})
~
is a representing measure for x,
> 0;
iii) There exist a,
~~
with 0
<
a
< ~ <
1,
FUNCTION ALGEBRAS
100 fo~
suoh that
f
€
A, \\fll
any neighborhood U of x
f(x)
< 1,
> S~ and
If(y)1
the~e
<
exists
a fo~ all
y ~ U.
Proof.
i) implies ii) is Lemma 2.3.3; ii) and
iiI) are equivalent by the remark made immediately following the definition of Choquet boundary; iii) implies ii) is Theorem 2.2.1. If iii) holds -with a
~, S
=
i,
=
then i) holds by Theorem 2.3.2.
We now use the hypothesis that A is an algebra to show that ii) implies iii) with any a, S (in 1 3 particular, 4' 4)' If x € Ch(A) , by Theorem 2.2.6, for any 0
<
a
S
<
there exists g
€
<
1, any neighborhood U of x,
A, with Re g < 0, Re g(x) > log S,
and Re g(y) < log a for all y
~
function algebra, exp g
A. All is proven.
THEOREM 2.3.5. and suppose X is peak points
Proof.
fo~
=
f
€
U. Since A is a
Let A be a funotion algebra on X, met~izeable.
A is a
Then the set of all
bounda~y
fo~
A, and a G6 set.
By Theorem 2.3.4 (and the remark following
Lemma 2.3.1), the set of peak points coincides with the Choquet boundary, which is a boundary by Theorem 2.2.9, and a G6 by Corollary 2.2.7. Since any boundary must certainly contain
101
MEASURES
every peak point, under the hypothesis of Theorem 2.3.5 we see that the set of peak points is the unique minimaZ bounda~y.
bounda~y
fo~
A, aontained in
eve~y
If X is not metrizeable, there need not
exist such a smallest boundary. For instance, let X
IT I , where each I , as well as the index set aeI a a
I, is the interval [0,1]. Let K = {x eX: x a = 0 for all but countably many a's}, L = {x eX: xa 1 for all but countably many a's}. Then K and L are disjoint, but each is a boundary for C(X) , since one sees from the Stone-Weierstrass theorem that each f e C(X) depends on only countably many coordinates. We close this section with a few remarks. We can now rephrase a result of Chapter 1 (Lemma 1.6.3): i f A is a funation
aZgeb~a ~n
X, x e X,
the ma$imaZ ideaZ assoaiated with x has an
app~O$-
imate identity i f and onZy i f x is a peak point in the weak sense.
If A = C(X), it is clear that each point of X is a peak point in the weak sense for A. For a long time, it was conjectured that if A is a function algebra on X, with Spec A
X, and if each
point of X is a peak point for A (or more generally,
FUNCTION ALGEBRAS
102
if Ch(A)
= X), then A = C(X). This IIpeak point
conjecture l l was recently smashed (along with other conjectures) by Brian Cole [1]. We describe one of Cole's examples in the Appendix. It is sometimes useful to observe that if A is a function algebra on X, a stronger version of Lemma 2.3.3 holds: if x is a peak point in the weak
sense~
and
senting x, then
~
any (compZex) measure repre-
~({x})
=
1. For if U any neighbor-
hood of x, there exists f e A which peaks on K, x f
€
n
K
c
U. Then 1 = fn(x) =
J fnd~
converges boundedly to XK' so
for all n, and
~(K)
result follows from the regularity of 2-4
= 1, and the ~.
PEAK SETS AND INTERPOLATION Throughout this section, A will be a function
algebra on X. We begin with a theorem of Bishop. THEOREM 2.4.1.
Suppose K is a peak set for A, and
g e A does not vanish on K. Then there exists f e A
such that flK X
= glK, and If(x) I
< If~ for every
e X\K.
Proof.
~ithout
loss of generality, we may assume
ggH K = 1. Let h be a function in A which peaks on
103
MEASURES
K, so hex)
1 for x e K, Ih(x) I
=
<
1 for x e X\K.
For each positive integer n, let Un
=
{x eX:
I g (x) I
<
1 + Z-n} .
Thus each Un is an open neighborhood of K, and Un +1 C Un for every n. Let M
= ~
gLand choose for k
each n a positive integer k n such that Ih n(x) I < -n -1 Z M for all x e X\U n . Put f
=
g
co
L
- n k Z h n . It
1
is clear that f e A and that flK
=
glK. It remains
to verify that If(x) I < 1 whenever x e X\K. If x e X\U 1 , then I h kn (x) I < M-l-n Z for every n, and so If(x)1 ~ Ig(x)1 E Z-n Ihkn(x) I ~ MM- 1 EZ- Zn =
%<
1. If for some n, x e Un\U n+1 , then Ig(x) I < 1 + Z-n, and Ihkj(x) I < M- 1 Z-j for all j > n, so If(x)1
< =
co n (1 + Z-n) E z-j + M E M- 1 2- Zj n+1 1 1 -n (1 + Z-n) (1 - Z-n) + '3 4
1 - 4- n + !3 4- n
< 1.
co
Finally, if x e n Un' and x e X\K, then Ig(x) I
<
1
and Ihex)1
<
1, so Ifex)1 < 1. The proof is
finished. If K is assumed to be only a peak set in the
1
FUNCTION ALGEBRAS
104
weak sense, we have a similar result. THEOREM 2.4.2. sense~ and let
Let K be a peak set in the
g e A, glK
r o.
If L is any Go set
= glK,
containing K, there exists f e A ~ith flK such that
Proof.
1f
(x)
Let L'
~eak
I f I for every x e X\L.
1
<
=
{x eX: Ig(x)1 < IlglI K}. Then L'
is a Go containing K. We can find a pe.k set K' wi th K C K' C L n L', and the theorem follows by applying Theorem 2.41 to K'. COROLLARY 2.4.3.
Let K be a peak set in the
~eak
sense. Then AIK is closed.
Proof.
Suppose g is in the closure of AIK. Then
there exist gn e A such that IIg n - gliK < 2- n . By Theorem 2.4.2, we can find f n e A with
~f
n
I <
00
and flK = lim(gn - gl) IK = g - gllK, so g = (f
+
gl) IK e AIK, completing the proof.
COROLLARY 2.4.4.
If K is a peak set for A, and
L C K is a peak set for AIK, then L is a peak set for A. The same holds
~ith
"peak set" replaced
throughout by "peak set in the
Proof.
~eak
sense".
Obvious from Theorem 2.4.1 (or 2.4.2).
MEASURES
105
COROLLARY 2.4.5.
If K is a peak set (in the weak
sense) containing more than one
point~
there exists
a proper subset L of K which is a peak set (in the weak sense).
Proof.
Since A separates points, there exists
g e A which is not constant on K. We can assume Ig~K
1 and g(x)
{x e K
=
1 for some x e K. Then L
=
g(x) = I} is a peak set for AIK (~ (1 + g)
peaks on L, relative to K), and the Corollary now follows from Corollary 2.4.4. COROLLARY 2.4.6.
Every peak set contains a peak
point in the weak sense.
Proof.
Ordering the peak sets in the weak sense
by inclusion, it follows at once from Zorn's lemma that every peak set contains a minimal peak set in the weak sense. By Corollary 2.4.5, a minimal peak set in the weak sense must be a singleton. This last result yields another proof of the fact that the set of peak points in the weak sense is a boundary, and thus in the case that X satisfies the first axiom of countability, that the set of peak points is a boundary. For if f e A, K {x :
I f ex) I
=
=
II fll} contains a peak set: namely,
FUNCTION ALGEBRAS
106
choose any x e K, then (2Ifl)-1(1 peaks on {y e K : fey)
=
sgn £(x)f)
+
f(x)}. Then K contains,
by.Coro1lary 2.4.6, a peak point in the weak sense, which was the assertion. This is Bishop's original proof, given in [2]. We now turn our attention to the relationship between peak sets in the weak sense and annihilating measures. THEOREM 2.4.7.
If
~
XK~
sense. Then
Proof.
Let K be a peak set in the weak
~
e A 1
wheneve~ ~
e A , and
E >
borhood U of K such that
1
eA.
0, we can find a neigh-
I~I
(UIK) <
There exists
E.
a peak set L with K C LeU. If f peaks on L, then fn
+
XL pointwise and bounded1y, so by Lebesgue's
dominated convergence theorem we have for every g e
A,
J gXLd~ But If gXK dp E
= lim
J gfnd~
= O.
J gXLdPI ~ I~I (L\K)~gl
was arbitrary,
J gXKdp
= 0 for all g
Elg~.
< €
Since
A, which
was to be proved. This result can be used to give another proof of Corollary 2.4.3. Let T be the restriction
MEASURES
107
mapping, Tf
flK, of A into the closure of AIK.
=
We want to show that T has closed range; by a standard result of functional analysis, this will be the case if (and only if) the adjoint map T* has closed range. Let
~
Banach extension of
to C(K). Then
II ~ II and
=
I
~ T*~I =
~
I
~
inf { ~ ~
=
e (AIK)*, let ~ be a Hahn-
~ + 'J
inf{
e A.i , supp
+ 'J ~ : 'J
C K},
'J
J..
'J
e A }, as we see by
considering a Hahn-Banach extension of
to C(X).
T*~
(This is the hopefully familiar argument that A* is isometrically isomorphic to C(X)*/A~). But if ~
'J e A ,
I~
+ 'J .1..
I
=
I~
+ XK'J + (1 - XK) 'J
I
~
I J.1
+ XK'J
I,
and XK'J e A by Theorem 2.4.7. Thus T* is an isometry, hence has closed range, and the proof is completed. For the proof of the converse to Theorem 2.4.7, we shall need a lemma from functional analysis. LEMMA 2.4.8.
Let C be a Banaah
spaae~
and let S
and T be weak-* alosed subspaaes of C*. Then S + T is weak-* alosed i f (and only if) there exists a aonstant k
suah that for eaah A e S + T there
exist J.1 e: S, 'J e T with A
=
J.1 + 'J and 1IJ.111 <
kliAII.
FUNCTION ALGEBRAS
108
Proof.
Let E
{veT: IvO
= {ll
~
1
+
e S:
~lll <
k}
and F
k}. Since Sand Tare weak-*
closed, E and Fare weak-* compact, by the BanachAlaoglu theorem. Since the map (ll, v)
+
II
+ v
is
continuous, and E x F is compact, it follows that {ll
+
v : II e E, v e F} is weak-* compact; but by
hypothesis, this set contains the closed unit ball of S
+
T. Thus the closed unit ball of S
+
T is
weak-* compact, and the theorem of Banach-KreinSmulian now assures us that S
+
T is weak-* closed.
The "only if" part of the lemma follows from the open mapping theorem, since weak-* closed subspaces are norm closed. The next theorem is due to Glicksberg [1]. THEOREM 2.4.9.
Let K be a oZosed subset of X with .1
the property that XK ll e A
whenever
II
1 eA. Then
K is a peak set in the weak sense. Proof. Let Y be the space obtained from X by identifying K to a point Yo' and let TI be the canonical map of X onto Y; thus TI maps X\K homeomorphically onto Y\{yO}' and TICK) fOTI e A}, B
YO' Let B' = {f e C(Y) :
{foTI: feB'}. Thus feB if and
only if f e A and flK is constant. Let M(K) be the
MEASURES
109
set of all measures
~
~(K) =
O. Then Ai
M(K) is weak-* dense in Bi.
Now if
~
XK~ +
1
e A
+
on X such that supp
v e M(K), then
v, and (1 -
XK)~
hypothesis. Since II~
+
1
~
~
C K and
v
(1 - XK)~ +
XK~ +
v e M(K) by
+
e A
and
vII
11(1 -
XK)~~ + IXK~ +
vi,
1 Lemma 2.4.8 applies, with C = C(X), S = A , T = 1
M(K), and k = 1. Thus A and hence.B
1
1
=A
+
+
M(K) is weak-* closed,
M(K).
We next observe that B' is a function algebra on Y. The only point to check is that B' separates the points of Y, i.e., that 0TIX - 0TIy TIX = TIy. If
is a measure on X, let
~
induced measure on Y: thus for all f e C(Y). Then
f fd(TI~)
~
B' only if
TI~
denote the
=
J
fOTI
d~
B if and only if TI~ i B'. 1 e B (since °TIX ° TIy i B' , then Ox If TIX 1 TIOX) , so ° x - y = ~ + v, where ~ e A , v e M(K). Then (0 x - 0y)(K) = 0, so either {x, y} C X\K or
°
~
i
°
{x, y} C K. If {x, y} C X\K, then Ox - ° (1 -
XK)~
1
Y
e A , and since A separates points, x = y.
If {x, y} C K, then TIX = TIy = YO' Thus B' separates points. Next we show that yo is a peak point in the weak sense for B'. It suffices (Theorem 2.3.4) to show that there exists no representing measure a
FUNCTION ALGEBRAS
110 for YO with o({yO}) measure; then
=
0 = TIP,
o.
Suppose
0
were such a
where p is defined by peE)
o (TI (E\K)) for Borel sets E in X. Since I
it follows that Ox - p e: B Ox
-
p =
p (K) =
lJ
for some
lJ + \!
(K) =
that 0x(K)
=
\!
lJ
°YO
- o
.L
B'
,
for any x e: K, whence
I e: A ,
\!
e: M(K). Since
(K) = 0, we reach the contradiction
O. Thus no such
0
can exist, and YO
is a peak point in the weak sense for B'. It is immediate that then K is a peak set in the weak sense for A, and the proof is concluded. As an application, we have the following interpolation theorem, due to Bishop [4]. THEOREM 2.4.10.
Let K be a closed Go subset of X.
Then the following two conditions are equivalent.
i) for every g e: C(K), g f 0, there exists
f e: A such that flK
g, and
I f(x) I
<
"fll for all
x e: X\K; ii) for every
II
I
e: A ,
III I (K) = o.
Suppose i) holds. Taking g = 1, we see that
Proof.
K is a peak set, so by Theorem 2.4.7 XK ll e: Al for every
l.I
e: A1 . Hence, if l.I e: AI
every g e: C(K), so
III I (K)
f K gdl.l =
0 for
O.
Now suppose ii) holds. Then XKl.I
o for every
MEASURES
111
1
v e A , so K is a peak set in the weak sense by Theorem 2.4.9, and since K is a Go' it follows that K is a peak set. Hence AIK is closed (Corollary 2.4.3), and since AIK is evidently dense in C(K), we have AIK
=
C(K). The remaining assertion of i)
now comes from Theorem 2.4.1, and the proof is finished. Sets with property i) are sometimes called peak inteppoZation
sets~
The reader may formulate
the corresponding notion of peak interpolation sets in the weak sense, and prove the corresponding theorem. Theorem 2.4.10 is a generalization of the following "classical" result, found independently by Rudin and Carleson: Let A be the disk
aZgebra~
and K a cZosed
subset of the unit circZe with Lebesgue measure O. Then for every g e C(K) there exists f e A whose restriction to K is g, and which assumes its maximum moduZus onZy on K.
The proof consists of noting that Ivl (K) = 0 for every
V
1
e A , by the F. and M. Riesz theorem,
and applying Theorem 2.4.10. The argument used in the second proof of
FUNCTION ALGEBRAS
112
Corollary 2.4.3 yields a result parallel to Theorem 2.4.10. THEOREM 2.4.11.
Let A be a closed
linea~
subspace
of C(X), K a closed subset of X. Then the following two conditions i)
a~e
fo~ eve~y
equivalent: g e C(K), and every E > 0, there
exists f e A such that flK
~
ii) for every Proof.
e AJ.,
= g, and
~f~ < (1 +
E)Rgl;
I~ I (X\K) ~ I ~ I (K) .
Let T be the restriction map of A into
C(K). Then for any measure A on K, T*A is the linear functional on A given by f
+
from the Hahn-Banach theorem, RT*AR
~ II
f fdA,
and so
inf{~A
+ ~I:
e AJ.}. Thus T* is an isometry if and only if AI <
U
A+
~I
Taking A = then
'xK~1 <
for all measures A on K, all XK~'
~
e AJ. .
we find that if T* is an isometry,
HI -
xK)~1
for all
.L
~
e A, i.e., ii)
holds. On the other hand, if ii) holds, then for any A supported on K and any
~
.L
e A , we have
> n A + XK~ I + I XK~ U
~ n A II
- II XK~ II
+
II XK]1 n
I A II ,
so T* is an isometry. But it is easy to see that
MEASURES
113
T* is an isometry if and only if i) holds. The proof is finished. We conclude this section with the following remark: If K and L are peak sets in the weak sense for the funation aZgebra A, so is K U L. For when1 ever ~ e A , ~(K U L) = ~(K) + ~(L) - ~(K n L) = 0
by Theorem 2.4.7, so K U L is a peak set in the weak sense by Theorem 2.4.9. (Note that for all ~
~
e
A~
if and only if
XK~
e
A~
~(K)
=
0
for all
~
e A ). This argument is easily modified to show
that any closed countable union of peak sets in the weak sense is again a peak set in the weak sense. If K and L are peak sets, Bear has given the following more direct argument that K U L is a peak set: let f e A peak on K, and g e A peak on L. Since (1 - z)1/2 (principal value) is uniformly approximable by polynomials on the closed unit disk, (1 - f)1/2
=
h e A, and (1 - g)1/2
=
k e A.
Since h vanishes only on K, and k only on L, and since \arg h\
<
i
and \arg k\
<
i,
we have that
hk vanishes precisely on K U L, while Re hk > 0 off K U L, so e- hk peaks on K U L. One can interpret this result as asserting: the peak sets in the weak sense are the closed sets of a topology
FUNCTION ALGEBRAS
114
on X. This topology is evidently weaker than the given topology of X, hence is Hausdorff only if it is identical with the given topology. In view of Theorem 2.4.7, this can occur only when A = C(X). 2-5
REPRESENTING MEASURES AND THE JENSEN INEQUALITY Consider our standard example, the disk alge-
bra A. The multiplicative linear functional, evaluation at the origin, is represented, as we saw, by normalized Lebesgue measure a on the circle
r.
This measure a has additional properties with respect to A: 1) if f e A is invertible, then 10glf(0) and
I
=
J loglflda,
2) for any f e A, 10glf(0)
I~
J loglflda.
To see that 1) holds, we just observe that if f e A-I, then log f e A, and 10glf(0)1 Re
J log
fda
=
J loglflda.
=
Re log f(O)
Property 2) is the
classical Jensen inequality of function theory. The importance of 1) and 2) in the general setting of function algebras was first appreciated by Arens and Singer [1]. DEFINITION.
Let A be a function algebra on X,
MEASURES ~
llS
e Spec A,
that
0
is an Arens-Singer measure for
log I ~(f) that
0
a probability measure on X. We say
0
I
=
J loglfldo
if
-1
for all f e A . We say
is a Jensen measure for
10gl~(f)1 ~
~
J loglfldo
~
if
for all f e A.
We remark that any Jensen measure is necessarily also an Arens-Singer measure. For if
0
a
Jensen measure, and f e A-1 ,we have
logl~(f) I
<
J loglfldo
logl~(f-l) I
< -
J loglf-lldo
= -
=
logl~(f) I,
so equality holds. Also, it is easy to see that an Arens-Singer measure for
~
representing measure for
For if f e A, we have
~.
is necessarily a
and similarly, by considering if, 1m
J 1m
fdo, so
~(f)
=
~(f)
=
J fdo.
It can be seen from simple examples that representing measures need not be Arens-Singer measures, and that Arens-Singer measures need not be Jensen measures. For instance, let let X
0 <
r
< R < ~,
= {lsi = r} U {lsi = R}, and let A be the
FUNCTION ALGEBRAS
116
function algebra on X generated by z and
1 z'
or
what is evidently the same, the restriction to X of all functions continuous in the closed annulus {r
~
~
lsi
R} and holomorphic in the interior. It
is easy to see that every real continuous function on X can be uniformly approximated by functions of the form Re f
+
c loglzl, with f e A and creal.
This implies that each point in the annulus admits a unique Arens-Singer measure. If 01'
O2
are the
normalized Lebesgue measure on the inner and outer boundaries, respectively, it is easy to check that 01 -
~
lsi
<
R, it can be shown that
the Arens-Singer measure
~
for s is of the form
~
=
O2
pal
eA. If r
+
<
q02' where p and q are positive func-
tions bounded away from E(OI
-
O 2)
o.
It follows that
is a positive mecsure for
E
~
+
small
enough, hence a representing measure for s which differs from
~
and so is not an Arens-Singer
measure. In the next chapter, we will see an example where the only Arens-Singer measures are point masses, while many points admit representing measures other than the point mass. For an example, where Arens-Singer measures
MEASURES
117
need not be Jensen measures, again take X to be the union of two circles, this time taking r 1 = { (s , 0) e ¢2 : I s I = 1} and r 2 = { (0, s) e ¢2 I s I = 1}, X = r l U r 2 , and A = P (X) , the algebra generated by the coordinate functions zl and z2. There is no difficulty in identifying the maximal ideal space of A with {(s, t) e ¢2 : lsi ~ 1, It I
~
1, st = O}, two disks with their centers
OJ
identified. Let on r j , j
=
be normalized Lebesgue measure
1, 2. Since either 01 or 02 is a Jensen
measure for (0, 0) (the classical fact), we see that 01 - 02 annihilates loglfl for any f e A-I. Now consider evaluation at (s, 0), where 0 < lsi < 1. The Poisson measure on r l ,
~
=
1 -
Is 12
11 - sZ112
is evidently a Jensen measure for (s, 0); it is also the only representing measure for (s, 0) which is supported on r l (this is a restatement of the fact that representing measures for the disk algebra are unique.) Now for £ sufficiently small, ~
+
£(02 - 01) is a positive measure, hence an
Arens-Singer measure for (s, 0). But any Jensen measure v for (s, 0) must be supported on r l , since -
~
<
loglsl
~
f
loglzlldv, and loglzll = -
~
on
FUNCTION ALGEBRAS
118 f
2 . Thus V is the unique Jensen measure for (s, 0). The following basic existence theorem is due
to Bishop [5]. THEOREM 2.5.1. ~
Let A be a funation aZgebra on X,
a muZtipZiaative Zinear funationaZ on
there exists a Jensen measure for
Proof. with
Let P
~(f)
=
=
{u
A. Then
~.
CR(X): there exists f
€
€
A
1, and a positive integer n, such that
nu > loglfl}, and let N
=
{u
CR(X): u < O}. We
€
observe that if u and v belong to P, so does u For if nu
>
loglfl and mv
= 1, then mn(u
~(g)
+
>
loglgl, where
v) > m loglfl
+
+
~(f)
n loglgl =
loglfmgnl, and ~(fmgn)
=
1. Also, if r is a posi-
tive real number and u
€
P, then ru
may assume r is rational, say r
=
€
P. For we
p/q, where p and
q are positive integers; then if nu > loglfl, (qn)(ru) > 10glfPI. Thus P is a convex cone in CR(X). Now P is disjoint from the convex cone N, for if u < 0 and nu > loglfl, then loglfl < 0, so If I
<
1, and hence
1~(f)1
<
1. Since P and N are
open, by the separation theorem there exists a continuous linear functional on CR(X) separating P and N, i.e., a real measure
0
v.
on X such that
MEASURES
119
J vdo ~ f udo
for any v
€.
N, u
P. Evidently,
€.
0
is a positive measure, which may be chosen to be a probability measure, and u
€.
P. Now if f
for all u
€.
€.
f udo > ° whenever
A and cP (f) F 0, we have JUdO > f
CR(X) with u > log cP
, so
(f)
f J log
But if CP(f) Thus
do .:. 0, i.e., loglcp(f)
cp (f)
0
=
°
1
~
J loglfldo.
0, this inequality is automatic.
is a Jensen measure, and the theorem is
proved. One way in which Jensen measures are often employed is to show that certain functions can't vanish too often. Let us illustrate with some classical facts. Let A be the disk algebra on the unit circle. It is easy to see that no f
€.
A, f
F 0,
can vanish on a non-empty open subset of the circle, for then a finite number of translates would have their zero sets covering the circle; their product would vanish identically, so some translate would have uncountably many zeroes in the interior, thus f
=
0. It is less obvious that no f e A, other than
0, can vanish on a subset of r having positive measure. The simplest proof is via the Jensen in-
FUNCTION ALGEBRAS
120
z k g for some
equality. If frO, we can write f k
>
r O.
0, g e A, g(O) -
<
00
10glg(0) I
~
Then
J loglgldo J loglfldo,
so frO a.e. (0). We next give a classical theorem of Rado. The proof which follows is essentially due to Glicksberg [2], with the trick of using Jensen measures supplied by Bishop. Let
be the closed unit disk, r the unit
~
circle, A the disk algebra on THEOREM 2.5.2.
If
f is
~.
Let f e C(6), E
hoZomo~phic
=
{x
€
on int 6\E, then f
6 : f(x)
O}.
A.
€
Proof. Let B be the function algebra on 6 generated by f and A, let X be the Shilov boundary for B. Since every function in B is holomorphic in int in int E, X
c
r
u
and
~\E,
aE. We shall show that X = r.
Since aE C 6\E, it suffices to show that for each x e 6\E, Ig(x) I .::. ~ gllr for all g let
~
€
B. Let x
€
6\E,
be a Jensen measure for x with respect to B,
with supp
~
C X. Since 0
and f = 0 on E, g ex) =
Jr gd~,
~(E)
so
<
If(x) I .::. exp
= O. Thus for all g
f loglfld~, €
B,
I g ex) I .::. II gil r' as we claimed.
MEASURES
121
Thus Blr is a function algebra on r which contains the disk algebra on r, and is clearly not C(r). By Wermer's maximality theorem, 1.2.4, Blr fir
Air. Thus
hlr for some h e A, and since f - h e Band
=
vanishes on r, f - h
= 0
on
i.e., f e A, con-
~,
cluding the proof. Theorem 2.5.2 obviously implies: i f G is an (o~ mo~e gene~atty~
¢
open set in
a Riemann su~-
face), i f f is continuous on G and ze~o
outside its
set~
then f is
hotomo~phic
hotomo~phic
in G.
We next give a generalization, due to Bishop, of Hadamard's "Three Circles Theorem". THEOREM 2.5.3.
Let A be a function
atgeb~a
on X.
Let E and F be ctosed subsets of X with X = E U F. each $ e Spec A,
Then
fo~
~,
< ~ ~
0
1, such that
the~e
exists a constant
fo~ att f
e A:
1$(f)1
~
If~~ I£~~-~. Proof.
Let
~
be a Jensen measure for $. Then for
any f e A, logl$(f)
I ~
Ix
= IE
loglfld~
loglfld~
+
Ip\E loglfld~
FUNCTION ALGEBRAS
122 < ).l(E)
sup{1og(f(x)) : x e. E}
+ ).l (F\E) sup{1og I f (x) I : x e. F} and the theorem, with a = ).l(E) , follows by exponentiating both sides. Applications of this result may be found in Bishop [5], and in Creese [1]. It is often useful to observe that Jensen's inequality persists when we pass to Ll limits. THEOREM 2.5.4.
Let A be a funation algebra on X,
).l a Jensen measure. Let Hl().l) denote the alosure in Ll().l) of A. Then
lOglJ fd).ll
~ J
loglfld).l
for all f e. Hl().l).
f
J
If - f n Id).l + O. We may assume fd).l f 0, else there is nothing to prove. Now for
Proof.
Let fn e. A,
each n, lOglJ fnd).ll
~J
lOglJ fnd ll \
f + f =
or
loglfnld).l log+lfnldll
- J log-Ifnldll
log - I fn I dll ~
J log+lfnldll.
MEASURES
123
log+lfnld~ + J log+lfld~. fnd~1 + loglJ fd~l· Hence
we have f loglJ
lOglf
fd~1
Clearly,
log-Ifnld~ ~.f log+lfld~.
+ lim inf J
Now we may assume, passing to a subsequence if necessary, that fn log-Ifni
+
f a.e., and hence that
log-Ifl a.e. Applying Fatou's lemma,
we have lim inf 10glJ
+
J log-Ifnld~ ~ J log-Ifld~,
fd~1 ~ f log+lfld~
-
f
so
log-Ifld~ = f loglfld~,
as was to be proved. In the classical case, where A is the disk algebra on the circle, cr normalized Lebesgue measure, Hl(cr) , and the analogous spaces HP(cr) formed by taking the closure of A in LP(cr) , are called Hardy spaces. We used H2(cr) (in the general context) already in Section 1 of this chapter, and the spaces HP(cr) will play an important role in Chapter 4. If A is the disk algebra, we can immediately deduce from Theorem 2.5.4 these classical results: i f f e Hl(cr) is not the zero
function~ loglfl is
summabZe (a theorem of Szego); hence, unZess f f
cannot vanish on a set of positive measure
= 0,
(the
"little" F. and M. Riesz theorem). The proof is
124
MEASURES ~
immediate from Theorem 2.5.4 if f(O)
0; in the
general case, we apply this result to inf, where n is appropriately chosen. At this point, we insert some material for readers unfami~iar with LP spaces for 0 < p < 1. Let
~
be a probability measure. For f a measurable
function, 0 < p < even though
~
00,
we write
~f~p
(J
=
IfIPd~)l/p,
Dp is not a norm if p< 1.
From the inequality t P ~ pt
+
1 - P when t ~ 0,
o < p < 1 (easily verified by elementary calculus) it follows that uPd~ ~ 1 whenever u is a nonnegative measurable function with f ud~ = 1, and o < p < 1, and hence
f
(1)
for any non-negative measurable u, 0 < p < 1. If
o< r
<
s, then r
=
sp, where 0 < p < 1,
J urd~
and it follows from (1) that
(J
=
f
(us)Pd~ <
uSd~)P, whence (2)
for any measurable u, 0
<
r
<
s
<
00.
From the inequality log t < t - 1 whenever t
>
0, it follows that if u is a non-negative
FUNCTION ALGEBRAS
125
f ud~ =
measurable function and
1, then
J
log
ud~
<
0, and hence for any non-negative u such that
f log
ud~ exp
exists in the extended sense,
I
log
ud~
.::.
f ud~,
known as the inequaZity of the geometria and arithmetia means. Replacing u by uP, it follows at once
that
J log ud~ any u ~ ° such exp
for
-<
~ul p
(3)
f
that
ud~
log
exists in the
extended sense, and any p > O. If t > 0, it is easily verified by elementary tP - 1 calculus that decreases to log t as p dep
creases to 0. Suppose u is a non-negative measurable function and for
o
<
I urd~
~
<
for some r > 0. Then
p .::. r, we have
logllul
P
=
1 log P
<
-1
P
J uPd~
( J up
d~
- 1)
=
f uP P -
1
d~,
so applying the monotone convergence theorem, we get inf ~u~p ~ exp J log ud~. Combining this with p>O (2) and (3), we find that
FUNCTION ALGEBRAS
126
exp
J
log
ud~
inf p>O
~u~
p
= lim p+o
~uU
(4)
p
whenever u is a non=negative measurable function, and the right-hand side is finite. The arguments of the last two paragraphs are due to F. Riesz [1]. From the fact that the elementary inequalities which we used are equalities if and only if t = 1, it follows that (1), (2), and (3) are equalities if and only if u
const. a.e.
=
(~).
From this characterization of the geometric mean, we have at once: LEMMA 2.5.5.
Let A be a funotion algebra on X,
a probability measure on X. Then measure (for some
for all f e
A, all
~
~
~
is a Jensen
e Spec A) i f and only if
p >
o.
It is obvious that one could also prove Theorem 2.5.4 by using this lemma. Similarly, this lemma implies that whenever
1~(f)1 ~
dense subset of A.
~
exp
is a Jensen measure for
f loglfld~
~
for all f in a
MEASURES 2-6
127
REPRESENTING MEASURES AND SCHWARZ'S LEMMA Let A be the disk algebra on the disk
~.
A
very useful classical fact, known as Schwarz's Lemma, asserts that if f e A and f(O) = 0, then If(s) I ~ lsi ~f~ for all s e ~ (proof: apply the f maximum principle to z)' By composing with linear
fractional transformations, we can give this a formulation not distinguishing the point 0: if lsi < 1, It I < 1, f e A, Ilfll < 1, then If(s) - f(t) I
~
lI s -sttl
11
-
.L~s) -:r:r;:"'\
f(t) I . The first factor being
strictly less than 1, we see that sup{lf(s) - f(t) I: f e A, I f I
~ 1} < 2
whenever I s I < 1, I t I < 1. On
the other hand, if lsi I, It I ~ I, this sup = 2: - z - rs for 0 < r < 1; then for let fr = s 1 - rsz st - r + - 1 as I f r I = 1, f r (s) = 1, and f r (t) 1 - rst r + 1. Identifying points of ~ with the corresponding multiplicative linear functionals, we have shown that I s - t
~ <
2 if and only if both sand tare
interior points of
~;
thus Spec A (A the disk alge-
bra) is divided into equivalence classes by the relation I s - t I
< 2.
In 1956, Gleason [1] made the
remarkable discovery that this holds true for any function algebra, and proposed that the search for
128
FUNCTION ALGEBRAS
analytic structure in the maximal ideal space should proceed via the investigation of the metric topology. This idea has proved successful in the context of Dirichlet algebras (defined by Gleason in the same paper) and their generalizations, as we shall see in Chapter 4. Till further notice, A will be a function algebra on X. LEMMA 2.6.1.
Let
l/J e Spec A. Then I
-
l/J
~
< 2
if and on7"y if sup{Il/J(f)I: f eA, ~fll2.l,
Suppose
~
-
l/J I = 2c < 2. Let f e A,
If 12. 1, O. c - f Put g ; then g eA,lgll< 1, so I
I
1 + c
Now suppose Il/J(f) I < c < 1 whenever f e A,
I f II 2. 1, g
-
(f) = O. If g e A and I g I < 1, put f =
Then f e A, II f I < 1, and 1 - f(g}g Il/J (£) I -< c, i. e. ,
(f) = 0, so
MEASURES
129
11/1 (g) -
<
I .s.
ell - HiJ"1/I(g)
I
<
2c,
2. The lemma is proved.
The next characterization is entirely similar. LEMMA 2.6.2.
Let
1/1 e Spec A, 0 < c
<
1. If there
exists a sequenae {fn} in A, "f n " < 1, with 1
e aH n, and 11/I(fn)
I
-+
1, then
II
-
1/111 = 2.
Passing to a subsequence, we may assume
Proof.
a, lal < 1, and 1/I(fn) -+ b, I b I 1. Let f n -
-+
I
THEOREM 2.6.3.
For
1/1 e Spec A, write
and only i f II
~
1/1 i f
is an equivalenae
relation on Spec A.
Proof. pose
I
e
Reflexivity and symmetry are obvious. Supand
~
1/1. If
e
~
1/1, by Lemma 2.6.1
By the same lemma, there exists c < 1 such that 1
I .s.
c, since e ~
But since
contradicts Lemma 2.6.2. Hence
e
~
~ 1/1, this
1/1, and the the-
orem is proved. DEFINITION.
The Gleason
parts~
or simply
parts~
0,
FUNCTION ALGEBRAS
130
are the equivalence classes defined by the relation
~.
If a subset M of Spec A can be given the structure of connected complex manifold in such a way that the functions in A are holomorphic on M, then M is contained in a single Gleason part. For any two points of M can be connected by a finite chain of analytic disks, and the classical Schwarz lemma and Lemma 2.6.1 applied repeatedly. However, parts can be large without the presence of any analytic structure, as we shall see in the next chapter. Indeed, unless some restrictive assumptions are imposed, essentially nothing can be said about the structure of parts in general. For instance, it was conjectured that parts might have to be connected, or at least could not be finite sets, other than singletons. But Garnett [1] proved the following striking result: given any completely regular, a-compact space P, there exists a function algebra in whose maximal ideal space P can be imbedded as a single Gleason part; further, every bounded continuous function on P is the restriction of some function in the algebra.
(It is clear that
parts must be completely regular, as subsets of a
MEASURES
131
compact Hausdorff space, and a-compact, since for sny ¢ e Spec A,
{~
~¢
e Spec A:
1
- ~ ~ ~ 2 - ~} is
compact by the Banach-Alaoglu theorem.) A conjecture of long standing was that if every part of A reduces to a single point, then A = C(X). This conjecture fell, of course, with the peak point conjecture, since peak points are obviously onepoint parts. We now turn our attention to what can be said about representing measures in relation to parts. We first observe that if
~¢
representing measures for ¢, ~
~ ~ =
~
2, and
~,
so
~ ~
-
p~ ~
pare
respectively, then
and p must be mutually singular (for
sents ¢ -
~,
2, and since
~
- p repre-
~
and pare
probability measures, it easily follows that they are singular.) We can formulate this remark as: a suffiaient aondition that ¢ and
~Zie
in the same
Gleason part is that they admit representing measures whiah are not singular. The converse, and even
more, turns out to be true. The next theorem was first obtained by Gleason in the context of Dirichlet algebras; the general result is due to Bishop ([6] and [7]). We shall follow Bishop's second proof, which yields a good deal more than is stated
FUNCTION ALGEBRAS
132 in the theorem.
THEOREM 2.6.4. ~,
~
e Spec A,
Let A be a funation algebra on X, ~~
~~
-
<
2. Then there exist con-
stants a and b, 0 < a < b <
OO~
a Borel function h
on X with a < h < b, and representing measures A ~
and
~
for
and
~ respectively~
Let M = {f e A:
Proof.
ideal associated with of
~
~(f)
~.
Let
to M. By Lemma 2.6.1, I
such that
~
= O}, the maximal ~'
~'II
be the restriction =
c < 1. Hence
there exists a (complex) measure v on X with c and
~(f)
~ fn~ ~
=
hA.
=
~v ~ =
f fdv for all f e M. Let fn e M,
1, with
~(fn)
points in Loo(lvl)
=
+
c. We regard the fn as
Ll(lvl)*. Since the closed
unit ball of a dual space is weak* compact, {f } n has a weak* cluster point Z e Loo(lvl), I Z I ~ 1. We have then c = lim inf / <
I
IZldlvl
f fndV/ ~
< /
J ZdV!
c,
from which follows that Izi = 1, and in fact that Z = dv thus {fn} actually converges to Z (weak-*). dlvl Let ~ = -c1 Ivl = ~ Zv. Then ~ is a probability meassure on X, and for any g e A, we have (using the
MEASURES
133
fact that gf n e M, all n): c
J gdll
f
f
gZdv = lim
gfndv = lim W(gf n )
= lim W(g)W(f n ) = CW(g). Thus
II
is a representing measure for W. Now for
any geM, W(g) =
f
gdll =
f
gdv =
- cZ)dll = O. In particular,
J gel
0 for all n, hence
and hence
J (1
J Z(l
- cZ) dll
f
gcZdll, so
J f n (1 =
0, so
- CZ)dll
J Zdll
= c,
- cZ) (1 - cZ)dll = 1 - c 2 . Also, for
J gel
- cf n ) (1 - cZ)dll = 11 - cZI 2 2 0, and hence f gil - cZ I dll = O. Le t A = II 1 - c2
g e M, gel - cf n ) e M, so
then A is a probability measure which annihilates M, hence a representing measure for
~.
We have
II
=
1 - c2
11 - cZI 2 1 - c2 (1
+
c)2
A, and the theorem is proved, with a = 1 - c
1 and b
1
+
c
+
c
1 - c
A few remarks are in order concerning this theorem, and its proof. One observes that we have a formal similarity to the classical Poisson formula, from which follows a version of Harnack's inequality in the abstract setting: if f e A, and Re f
>
0 on X, we have
;
FUNCTION ALGEBRAS
134
1 + c
1 - c
Re
1 + c
~(f)
~
~(f)
Re
~
Re
1 - c
~(f).
Another point to notice is this: let H;(~) be the closure of M in L2(~). Then cZ is the projection of 1 on H;(~); for clearly cZ
f
cZ)d~
gel
=
holds for all g
° for €
all g
€
H;(~), and since
€
M, this relation
H;(~), i.e., 1 - cZ
i
H;. The
linear functional ~' on M extends to a continuous linear functional on H2 (in fact ~(g) = (g, cZ) ~
as we saw, where (,) is the L2 inner product). The norm of the extended functional is clearly c. Thus we have the perhaps surprising equality: sup{ I ~ (g) I : g =
€
M, I g II <
sup{I~(g),1
: g
€
l}
M,
f Igl2d~
<
I}.
Harnack"s inequality can be deduced more directly. Suppose
~
and
belong to the same
~
Gleason part, so there exists a constant c that I ~ (g) I
<
c whenever g
°. Now
€
A and Re f > 0, put g = (1 - f) x
if f
€
A, II gil
(1 + f) -1; then g e A, II g ~ < 1. I f ~
(g) = 0, so I ~ (g) I
< C,
or
11 -
< 1,
~
c)-I. It follows that Re ~(f)
Re $(f) whenever Re f
>
O.
and
~
(g)
(f) = 1, then
~(f) I < cll + ~ (f) I ,
whence, by a simple computation, Re (1 -
such
< 1
~(f)
<
(1 +
<
(1 +
c) x
c)(l - c)-l
MEASURES
135
It is interesting to note that the
e~istence
of boundedly mutually absolutely continuous representing measures can be deduced from Harnack's inequality, and thus can be regarded as a theorem about linear subspaces of C(X), rather than function algebras. The next theorem is due to Bishop [6], and evidently contains Theorem 2.6.4. THEOREM 2.6.5.
Let B be a Zinear subspace of
CR(X), containing the constant functions. Let Iji e B*, with
II~II
= 4>(1) =
111ji~
there exists k such that 4>(u)
=
Iji(l) <
~,
1. Suppose
klji(u) and Iji(u) <
k4>(u) whenever u e Band u > O. Then there exist representing measures A for 4> and
A~
k~
Proof.
and
~
<
Let 9 = klji - 4>. We show that there exists
put P = {u e B <
for Iji with
kA.
a positive measure
u
~
0
representing 9. To this end,
9(u) ~ O} and N
OJ. If u e Band u
<
= {u e CR(X):
0, then 9(u)
<
0 by hypoth-
esis, so P and N are disjoint. Clearly, P and N are convex cones, and N is open. Applying the separation theorem, we find a measure that
f vdo ~ f udo
obvious that sup{
0
on X such
whenever v e Nand u e P. It is
J vdo
: v e N} must be 0, so a
is.a positive measure, and JUdO> 0 for all u e P;
136
FUNCTION ALGEBRAS
we may assume that have
f
do
=
k - 1. Now if u e B, we
(u - (k - l)-le(u)) e P, so JUdO
+
Interchanging the roles of $ and
~,
=
e(u).
we find
similarly a positive measure T on X such that
f udT
= k$ (u) -
~
(u) for all u e B. Put A =
(k 2 - l)-l(kT + 0) and
~
= (k 2 - l)-l(ko
+
T). Then
for any u e B, J udA = (k 2 - 1)-1[k 2$(u) -
k~(u)
+
k~(u)
- $ (u)] = $ (u) ,
and similarly, J and 2-7
~
<
ud~
=
kA. The theorem
~ (u).
Clearly 0 < A <
k~
is proved.
ANTISYMMETRIC ALGEBRAS If the real functions in a function algebra
A on X separate the points of X, the StoneWeierstrass theorem tells us that A = C(X). In this section, we obtain a useful refinement of this result. DEFINITION. say that A is
Let A be a function algebra on X. We anti8ymmet~ic
if every real function
in A is constant. A subset E of X is called a set of antisymmet~y (for A) if f e A and fiE real
implies fiE is constant. Evidently, {x} is a set of antisymmetry for
MEASURES
137
each x e X; A is antisymmetric if and only if X is a set of antisymmetry. If ure for some
~
~
is a representing meas-
e Spec A, then supp
~
is a set of
antisymmetry: for if f e A, and f is real on supp
~,
then equality holds in the Schwarz inequality,
J f2d~ eJ
since
=
fd~)2,
hence f
and hence f is constant on supp LEMMA 2.7.1.
n
and E
=
e~),
~.
If E and F are sets of antisymmetry"
F is not empty" then E U F is a set of
antisymmetry. Henae" i f E and F are antisymmetry" either E
Proof.
const. a.e.
=F
maxi~al
sets of
or E and F are disjoint.
Obvious.
LEMMA 2.7.2.
Every set of antisymmetry is con-
tained in a maximal set of antisymmetry.
Proof.
If {Ea} is a collection of sets of anti-
symmetry, linearly ordered by inclusion, it is clear that UEa is a set of antisymmetry. The lemma now follows by Zorn's lemma. It is clear that the closure of a set of antisymmetry is again a set of antisymmetry, hence maximal sets of antisymmetry must be closed. More is true: LEMMA 2.7.3.
Let E be a maximal set of antisymmetry.
FUNCTION ALGEBRAS
138
Then E is a peak set in the weak sense.
Proof.
Let F be the intersection of all peak sets
which contain E; then E
c
F, and F is a peak set
in the weak sense. If E f F, then (since E is a maximal set of antisymmetry) there exists f
A,
€
with flF real and not constant. But fiE is constant, say f = c on E. Put g = 1 - £(f - c)2, where 0 < Th en g IF pea k s on a proper sub set K £ < ~ ~ Ilf"-2. H of F, K
~
E. By Corollary 2.4.4, K is a peak set
for A, contradicting the definition of F. Thus E = F, and the lemma is proved. The following lemma, due to de Branges [1], is the crucial ingredient for the proof of our central result.
LEMMA 2.7.4.
~
Let
be an extreme point of the L
closed unit ball of A . Then supp
~
is a set of
antisymmetry.
Proof.
Suppose f
€
A is real-valued on supp
must show f is constant on supp necessary, by af assume that 0
< f
~.
~.
We
Replacing f, if
b Ca, b real constants) we may
+ <
1 on supp
~.
Since f
€
A and
MEASURES
139 f~
+ A is an algebra, f~ e A~. Now ~ = ~f~~ ~ f~ II (1 - f)~ and ~ f~ I + HI - f) ~ II 11(1 - f)~11 II (1 - f)~ II fdl~1 + (1 - f)dl~1 = 1, since f and 1 - fare
J
J
positive. Since
~
is an extreme point of the unit f~
ball of A~, it follows that ~ = 1/
a.e.
(~),
on supp
~.
f~ ~
so f =
U f~
1/
and since f is continuous, f is constant The lemma is proved.
We have now assembled all that is needed for the following proof (due to Glicksberg [1]) of a theorem due to Bishop [3].
THEOREM 2.7.5.
Let A be a funation aZgebra on X.
Let {X a. } be the aoZZeation of distinat maximaZ sets of antisymmetry for A. Then
i) X = u Xa." and the Xa. are pairwise disjoint; ii) A1Xa. is uniformZy aZosed" and antisymmetria;
iii) A = {f e C(X) : flxa. e A1Xa. for every a.}. Proof.
i) follows from Lemmas 2.7.1 and 2.7.2, and
the observation that singletons are sets of antisymmetry. ii) follows from Lemma 2.7.3, Corollary 2.4.3, and the definition of sets of antisyrnmetry.
FUNCTION ALGEBRAS
140 Suppo~e
f e C(X) and flXa e AIXa for every a. Then
for every extreme point ~
of A , flsupp 2.7.2, hence
~
e Alsupp
f fd~ =
it follows that f
~
in the closed unit ball ~,
by Lemmas 2.7.4 and
O. By the Krein-Milman theorem,
fd~ = 0 for all ~ e A~, hence
f e A. The proof is finished. We observe that Theorem 2.7.5 contains the Stone-Weierstrass theorem, since if the real functions in A separate points, no set of antisymmetry can contain more than one point. This is· de Branges' proof of the Stone-Weierstrass theorem. Often, where Theorem 2.7.5 can be applied, a coarser and more obvious decomposition of X serves as well. For each x e X, let Ex = {y eX: fey) f(x) for all real f e A}. Since for each real f e A, {y e X
fey) = f(x)} is a peak set (as we saw
above, 1 - E(f - f(x))2 peaks there, for E
>
0
small enough), each Ex is a peak set in the weak sense. It is clear that for x, yeA, either Ex = Ey or Ex and Ey are disjoint. If f e A is real on X, then f is constant on each Ex' but AlEx need not be antisymmetric. If f e C(X), and flEx e AlEx for each Ex' then f e A: this follows from Lemma 2.7.4,
141
MEASURES
or can be proved directly. The partitioning of X in this fashion was introduced by Shi1ov, and is the basis for Bishop's proof of Theorem 2.7.5. Theorem 2.7.5 was used by Hoffman and Wermer [1] to obtain the following theorem.
THEOREM 2.7.6.
Let A be a function
and suppose Re A is
Proof.
unifo~mZy
aZgeb~a
on
cZosed. Then A
X~
= CCX).
We shall give first an argument found by
the author [1] which does not use Theorem 2.7.5, but which depends on an interesting lemma from functional analysis, found by H. Reiter [1] . I f S is a subset of a Banach space B, S~ denotes {A e B*: ACf) = 0 for every f e S}.
LEMMA 2.7.7.
Let B be a Banach
space~
Sand T
cZosed subspaces of B. Then S + T is cZosed if and onZy if S~ + T~ is Cno~m) cZosed~ if and onZy if S~
~.
+ T
Proof.
is weak*-cZosed.
Let i be the inclusion map of S into B, and
TI the canonical projection of B onto BIT. Then S is closed if and only if TIoi: S
+
BIT has closed
+
T
FUNCTION ALGEBRAS
142
range. But TIoi has closed range if and only if (TI i) * : (BIT) * 0
+
S* has closed range. Now (BIT) * l.
is canonically identified with T , and S* with B*/Sl.; the inclusion i' of Tl. into B* is (with
this identification) the adjoint of TI, and the projection TI': B*
+
B*/S
J.
is the adjoint of i.
Thus TIoi has closed range if and only if n'oi': T l.
+
B* I S·l. has closed range, i.e., if and only if
Sl.· + Tl. is closed. The range of an adjoint map is
closed if and only if it is weak-* closed. The lemma is proved. We return to the proof of Theorem 2.7.6. To say Re A is closed is evidently equivalent to saying A
+
A is
closed, which by Lemma 2.7.7
l. -.L implies that A + A is weak-* closed, i.e., that
(A
_.L
n
A)
l.
- A
_l.
+
A
For each x e X,
let Ex {y eX: fey) = f(x) for all real f e A}·. We shall show that each Ex = {x}. Suppose Ex contains more than one point. We have seen that Ex is a peak set in the weak sense, hence there exists a peak set K for A whose intersection with Ex is a non-empty proper subset of Ex (see 2.4.5). Let p e Ex
n
K, q e Ex\K. Then op - Oq annihilates
MEASURES
143
every real f e A, hence op - Oq LAn
A.
From the
first sentence of the proof, this implies there exists a real measure v such that ~
L
eA. If f peaks on K, we have
n, but fn
+
op - 0 q J fnd~ =
XK boundedly, whence 1
+
+ iv =
0 all
iv(K) =. O.
This is impossible, since v is real, and thus Ex
=
{x}, and the theorem follows from the StoneWeierstrass theorem. The original proof of Theorem 2.7.6 went as follows. Suppose first that A is antisymmetric. We show that X reduces to a singleton. Fix x e X, and put Ax {f e A : f(x) = a}; we regard Ax as a Banach space over the reals. Since A is antisymmetric, the map f
+
1m f is a one-one, norm-
decreasing linear map of Ax onto Re Ax' If Re A is closed, so is Re Ax' so the open mapping theorem applies, and assures us that there exists a constant k such that I f
~
.::. k Urn f
~
for all f e Ax'
Now if X f {x}, there exists g e Ax with we may assume g(y) with 1 - e
-2k
<
r
=
<
~g~
l',
1 for some y e X. Choose r 1. Put f
=
log(l - rg), where
we take the principal value of log. Since log can be uniformly approximated by polynomials on
FUNCTION ALGEBRAS
144
11 .::. r}, we have f e A. Clearly, f(x)
{s: Is 11m fl
II f X
~ >
=
<
7T 2' and f (y) = log(l - r)
0,
- 2k. Thus
<
kllIm f II. This contradiction shows that
{xl. For the general case, one shows that if Re A
is closed, and Y is a maximal set of antisymmetry for A, then Re AIY is closed (see Hoffman and Wermer [1] for the details). Applying the argument above, we find that each maximal set of antisymmetry reduces to a single point, and hence A = C(X). A companion theorem to Theorem 2.7.6 was found by Wermer: i f Re A is a ring
(i.e.~
u e Re A impZies u 2 e Re A), then A
=
if
C(X). In
this case, the reduction to the antisymmetric case is obvious. The proof again is based on the unboundedness of the map Re f
~
1m f of Re Ax onto
itself. The reference is Wermer [4]. A generalization of Theorem 2.7.6 was found by Sidney and Stout [1]: i f K is a cZosed subset of X, and Re AIK is closed~ then AIK
2-8
= C(K).
THE ESSENTIAL SET. One rather trivial way of making new function
MEASURES
145
algebras out of old ones runs as follows: if A is a function algebra on X, imbed X in the space Y, and let B
=
{f e C(Y) : fix e A}. For example, if
A is the disk algebra on the disk 6, and we take X = 6
x
[0, 1], and B = {f e C (X) : f ( "
0) e A},
we have a so-called "tomato can" algebra. This particular algebra provides an example of analytic structure in the Shilov boundary, but somehow not a satisfying example. In this brief section, we systemize the discounting of such constructions.
DEFINITION:
We say that the function algebra A
on X is essentiaZ if there exists no proper closed subset Y of X such that A contains every continuous function on X which vanishes on Y. (This notion is due to H. S. Bear.) We next show that the study of function algebras can be reduced to the study of essential function algebras. (In a sense, this was already done in the last section, since obviously antisymmetric algebras must be essential.)
THEOREM 2.B.l.
Let A be a funation aZgebra on X.
Then there exists a unique minimaZ aZosed subset
FUNCTION ALGEBRAS
146
E of
x,
called the essential set for A, with the
property that
A contains every continuous function
on X which vanishes on E.
Further~
E is a peak set
in the weak sense~ so AlE is closed~ and A =
{f e C (X) : fiE e A IE}. Proof.
Let rbe the family of all closed subsets
F of X with the property: A Let E =
n
~
{f e C(X) : flF
=
a}.
{F: Fe?}. To establish the theorem, we
must show E e?
Suppose f e C(X) vanishes on E.
If U is any neighborhood of E, there exists (by the finite intersection principle) F e '] such that E
c
F
c
U. Since F e
r,
there exists (Urysohn's
Lemma) g e A, such that g = 0 on F, g = 1 on X\U, and 0 ~ g ~ 1 on X. Then fg e A, since F e ~, and Ifg - fl < If I on U, while fg - f X\U. Choosing U
=
vanishes on
{x: If(x)1 < d, we have
~fg - f~ < E, and fg e A. Hence f e A. Thus E e~.
The remaining assertions are trivial. An alternate proof explains the essential set in terms of annihilating measures. Let E be the closure of the union of the supports of all annihilating measures for A. I.e., x e X\E if and only if there exists a neighborhood U of x such
MEASURES that
I~I
147
(U) = 0 for every
and fiE e AlE, we have
~ ~
f fd~
A. Then if f e C(X) 0 for every
so f e A. On the other hand, if F ~ ~
A such that
f
~ ~
A,
E, there exists
I~I
(X\F) > 0, and hence there I exists f e C(X), van1shing outside a compact sub-
set of X\F, such that
J fd~
r o.
Thus A does not
contain every f e C(X) which vanishes on F: E is minimal, and unique. Evidently, E is empty if and only if A
=
C(X),
and E = X if and only if A is essential. While antisymmetric algebras are essential, essential algebras need not be antisymmetric. The simplest example is P(X), where X is the union of two disjoint closed disks. A connected example is the tomato can algebra: X = algebra on
~,
~
x
[0,1], A the disk
B = {f e C(X) : f(o, t) e A for all
t e [0, I]}. Here B is an essential algebra on X, but the maximal sets of antisymmetry are the disks ~ x
{t}. In general, the points o,f X\E are maximal
sets of antisymmetry, and any closed subset of X\E is a peak set in the weak sense" as is E.
CHAPTER RATIONAL
3-1
THREE
APPROXIMATION
PRELIMINARIES This chapter is devoted to analysis in the
complex plane ¢, with occasional reference to the "-
Riemann sphere ¢. We continue to reserve the symbol
z for the identity function on ¢. If U is an open set in ¢, we denote by CCk)CU) the set of all complex-valued functions on U having continuous partial derivatives up to order k; c~k)CU) {f e CCk)CU): f has compact support in U}. We abbreviate cCk)C¢) by C Ck ). The first order partial c c differential operators a~ af ;; =
1 (
2"
a are defined by
ai
af af) a ax - i ay , a z
- :(~+ia£), 2 ax ay
where we temporarily denote the real coordinates 149
FUNCTION ALGEBRAS
150
on ¢ by x and y, so z
x
=
+
iy. (We shall never
use real coordinates again, and x and y will be available to denote points in
¢.)
The operators
a~' ~ are dual to the complex diferentials dz
az dx + idy, dz
=
af af -- dz + - - dz for az az any f e C(l)(U). A necessary and sufficient condidx - idy, so df
=
=
tion that f be holomorphic in U is that f e C(l)(U) and afjaz
in U (Cauchy-Riemann equations). We
= 0
often denote pf by fz' ~ by f- Lebesgue twoaz az z· dimensional measure in ¢ will be denoted by m. We ~(a;
write
LEMMA 3.1.1. Then pa~
fo~
all
7 · t ~auva~~
Proof.
Is - al
r) = {s e ¢
Let K
s
e
¢,
b~
JK
a
<
measu~able
dm
Is
zl
rL
plane set.
.::. Z/1Tm(K). In
1 Z .~s summa b 7ve on aompaa t se t s.
If m(K)
=
00
(or 0) there is nothing to
prove. If m(K) < 00, let R = ImCK)/1T, so m(K) = 1TRZ. Now put D =
~(s;
R); then Z1TR
so it remains to show that
Z/1Tm (K) ,
RATIONAL APPROXIMATION
fK 1(; dmBut K = [K
n
dm
f ~ D
zl
1(; - zl
D] u [K\D], and m(K) = m(D), so we
have m(K\D)
Ir;
1
dm
on D\K,
R
zl
dm zl
fD\K
<
, and hence that
Ir;
zl
dm
fK
.!.
R
zl
Ir;
>
on K\D, it follows that
< -
Ir;
1
m(D\K). Since
1
and
fK\D
151
f KnD
Ir; - zl
+
f K\D
<
f KnD
+
fD\K
dm Ir; -
zl '
and the lemma is proved. The next lemma supplies the main tool in studying rational approximation.
LEMMA 3.1.2.
Let G be a bounded pZane domain with o~iented
smooth positiveZy
bounda~y
y. Let f
C(l)(U), where U is a neighborhood of aZZ w
€
G.
€
Then for
G,
few)
fdz =
z - w Proof.
Choose £
G. Let G£ =
>
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 <
cuZ-ar~
a.e.
00
Let D =
Proof.
J
dl~
X
(m).
R). Then, by Lemma 3.1.1,
~CO;
dm(s)
I (w) J
D
Iw - sl
< z1TRI~
I CX),
1
and IZI -
is obviously measurable with respect
zzl
to the product measure
I~I
x
m on X x D, so
Fubini's theorem applies, and yields the lemma.
DEFINITION.
such that jlCw) <
o(w)
~
Let X,
J
00,
be as above. For each w e
¢
we define
d~
X z - w
Thus 0 is defined a.e. (m), and surnrnable over bounded sets;
101
~
iJ. We can make a stronger
FUNCTION ALGEBRAS
154
statement if
is boundedly absolutely continuous
~
with respect to m.
LEMMA 3.1.5.
Let X be a compact pZane
set~
g a
bounded BopeZ function on X. Let
few) =
Ix
~
z
w dm,
fop aZZ w e
¢.
Then f is oontinuous on ¢~ hoZomopphic in vanishes at
Proof.
¢\X, and
00
The last two assertions are trivial. That
f is continuous is a special case of a well-known fact: the convolution of a bounded and a summable function is continuous. For completeness, we give the details. We may assume m(X) E >
O. Choose R so X
c
~(O;
= ~
g
~ =
1. Let
R/2). Choose h, a con-
tinuous function with compact support, such that
f
~ (0; R)
Ih - lldm
Ih(s) - h(t) I
It I
<
< E.
Choose 0 > 0 such that
Z
< E
whenever Is - tl
<
o.
Now for
R/2, If(t) -
Ix <
g(s)h(s - t)dm(s)1
Ix
Ig(s)1
\ s I t - h(s - t)\dm(S)
RATIONAL APPROXIMATION
~
I
II g I
Hence, if s, t
155
il(O;R)
I : - h I dm z
< E.
il(O; R/2) and Is - tl
€
0, we have
<
If(s) - f(t)1 < E + E +
Ix
Ig(u) Ilh(u-s) - h(u-t) Idm(u)
Iglldm(X)sup{lh(s) - h(t)l: Is-tl <
< 2E +
o}
< 3E.
Thus f is (uniformly) continuous on il(O; R/2), and ~.
hence on all of
Let X be a compact pZane
LEMMA 3.1.6.
measure on X, f
Proof.
f fz
fd~ 1T
Ix
fd~
set~
a
~
C(l). Then c
€
1
Ix
The proof is finished.
=!
1T
a dm.
IX d~ (w) I~
f-z W -
z dm =
!1T f ~ f-Z adm by Corollary 3.1.3 and Fubini's theorem. COROLLARY 3.1.7.
Let ~ be a measure on
compact support. If U is an open set and a.e.
(m)
in U, then I~I(U)
supp Om). In
particular~
~
o o
(i.e.~
¢
with
a
supp
0 ~
i f and only i f
c
FUNCTION ALGEBRAS
156
a=
0 a. e.
Proof.
(m).
C~l)(U),
If f e
then
f fd~
by Lemma
= 0
3.1.6. If K is a compact subset of U, there exist € C(l)(U) decreasing to XK, so ~eK) = O. It n c follows from regula~ity that I~I eU) = O.
f
In the language of distribution theory, Lemma 3.1.6 says that ~ dZ
a - TI~.
The next result is
suggested by Leibniz's formula.
LEMMA 3.1.8. set X
c
~
Let
be a measure on the compact
~, and let h
v =
€
C~l). Put
1 h~
- - h- Om. z TI
Then the compactly supported measure v has the property:
Proof.
v
hO.
Let w be any point where Dew) hd~
O(w)
Ix
z - w 1
hzCt)
Ix [ I¢ TI
TI
5 -
hzCt)
I¢
Then
00
h-
- ;- f¢
1
<
t -
W
t
z
z - w
Odm 1 d~
dmCt) J
OCt)dmCt)
5
- w
(5)
RATIONAL APPROXIMATION d~
~ I¢
=
[ Ix
1 1T
157
o(t)
(5)
t - w
(s - t) (5 - W) 1
1
f¢ [ t
fx { 5
- W
- t
5
~
]h Z(t)dm(t)
w )
d~
(5)
)1 (t) t
~ I¢
-
w
] hZ(t)dm(t)
h-z (t) CO (t) - O(W) - 0 ( t )) dm ( t ) t - w
f
1
O(w)
hz(t)
¢w
1T
dm(t)
=
h(w)O(w),
- t
as was to be proved. Finally, we observe how to recover the atomic part of
~
from 0 or U.
LEMMA 3.1.9.
~
Let
ure in the plane~ and en
m(~ )-1 n
~({w}) = 1~({w})1
Proof.
be a aompaatZy supported meas-
¢.
let w e
Put ~n = ~(w;
lin),
2
=
~. Then 1T
lim c n
f~n
lim en
Let Fn(s) = en
(w - z)O dm, and
J~n
f~
Iw - z IU dm.
- w dm, and Gn(s) = z n z - 5
158
FUNCTION ALGEBRAS
s
I -
1)-1
~
so Gn(s) (and hence Fn(S)) converges to O. Also, 1
Gn(s) .::.- c n
fA n
dm
Iz -
LIn
< 2
by Lemma 3. 1 . 1. Thus,
sl
Fn and Gn converge boundedly to X{w}' and applying the dominated convergence theorem and Fubini's theorem, we have ].l({w}) = lim
f Fnd].l
lim c n lim c n
Jt - w J d].l (s) t - s dm(t) J (w - t ) 0 (t) dm ( t) ,
and l].ll ({x}) = lim
3-2
lim c n
J
lim c n
J Iz
J Gn
dl].ll (s)
J
dl].ll
I: ~ :
/dm(t)
- wl~dm.
ANNIHILATING MEASURES FOR R(X)
Throughout this section, X will denote a compact plane set. We recall from Chapter 1 that ReX)
RATIONAL APPROXIMATION
159
is the uniform closure on X of the rational functions with no poles on X, P(X) the uniform closure of the polynomials.
Let J.l be a measure on X. Then J.l
THEOREM 3.2.1.
J.
R(X) if and onZy if J.l vanishes off X. A
Proof. J.l
J.
If w
1
f
X,
z - w
R(X). Suppose 0
e R (X) , so O(w) =
a if
a off X, and f is holomorphic
in a neighborhood U of X. We can choose h e c(oo)(U) c Then such that h = 1 in a neighborhood of X.
f fdJ.l
=
I
=
Ix
fhdJ.l =
f¢
(fh)z adm
(fh)z Odm
Ix
f-z adm =
o.
Thus J.l annihilates the restriction of any function holomorphic in a neighborhood of X, in particular, any rational function whose poles lie off X, so J.l
J.
R(X).
As a bonus, we see that any function holomorphic in a neighborhood of X can be approximated uniformly on X by rational functions; even a slightly stronger statement is true:
160
FUNCTION ALGEBRAS
COROLLARY 3.2.2. borhood U of
X~
Let f e C(l)(U) for some neighand suppose
fi
=
a
on X: Then
fix e R(X). Proof.
~ ~
We may assume that f e C2 l )(U). Then for
Jfd~
~
0, since 0 Jx f-z Odm off X by Theorem 3.2.1. Hence fix e R(X).
all
R(X),
=
=
a
We can use measures to prove Runge's theorem, for which we outlined a constructive proof in Chapter 1, Section 4.
THEOREM 3.2.3.
Le t Gl' G2 , .•. .be the bounded aon-
neated aomponents of ¢\X, and let an e Gn for eaah
n. Then any funation holomorphia in a neighborhood of X aan be approximated uniformly on X by rationaZ funations with poles only among the {an}' In partiaular~
if X has aon·neated
aomplement~
any funation
holomorphia in a neighborhood of X is uniformZy ; approximable on X by polynomials.
Proof.
Let B be the function algebra on X gener-
ated by z and by {z 1 = 1, 2, ... }. Clearly, - an : n B c R (X) , and the assertion is that B = R(X). Let ~
be a measure on X which annihilates B. Then il is
RATIONAL APPROXIMATION
161
holomorphic off X, O(a n ) =
O(k) (a ) = k! n
d~
- an =
fz
o,
and
d~
= 0 for every n, k. It (z - a n )k follows that 0 vanishes in a neighborhood of each
f
an' hence throughout Gn for each n. Now for any w, /w/ > max{/s/ : s ~
L
X}, we have z 1- w
€
n ~+l' the series converging uniformly on X, and
ow
f znd~
since
0 for all n, it follows that O(w)
=
=
O. Hence 0 vanishes on the whole unbounded component of ¢\X. Thus
0
=
0 off X, so
~ ~
R(X). The
theorem follows. The next corollary of Theorem 3.2.1 is known as the Hartogs-Rosenthal theorem.
THEOREM 3.2.4. Proof.
Let
X, so 0
=
~
If m(X) ~
= 0,
then R(X)
= C(X).
R(X). By Theorem 3.2.1, 0
=
0 off
0 a.e. (m), and by Corollary 3.1.7 it
follows that
~
= O. Hence R(X) = C(X).
Constructive proofs of Theorems 3.2.2 and 3.2.4 are easy to obtain. If f e C(l)(U), where U is a neighborhood of X, choose an open set G with smooth boundary y such that X
c
G
cITe
U. If
FUNCTION ALGEBRAS
162
=
either f-z
0 on X or m(X)
that for w e X, IIG z
~Zw
=
0, we can choose G so
dml is arbitrarily small
(it suffices to take m(G\X) small, by Lemma 3.1.1). Now if g(w) =
I
fdz, then g e ReX) by an arguY z - w ment given in 1.4 (the approximating sums are rational functions of w), and 3.2.2 and 3.2.4 then follow from Lemma 3.1.2. Theorem 3.2.4 is totally unimpressive unless one realizes that there are compact sets X with empty interior for which R(X)
+ C(X).
crucial example, known as the swiss
Here is the
cheese~
for
reasons which will be apparent. Start with a closed disk DO' and choose a countable family {Dn} of open disks, such that: i) Un
c
for n
+ m;
iii) DO\~ D has empty interior; and
rn
00,
iv)
~
int DO' for each n; ii) Un
<
1
n
Urn is empty
n
where rn is the radius of Dn' (It is
1
easy to define such Dn inductively, with centers chosen from any dense subset {an} of int DO' and radii from any given sequence {En} with En > 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
<
~
on X, which we may
J fd~ Iyo =
~ + 0,
in fact
fdz -
~~~
=
2TI
~ 0
rn'
If f is a rational function whose poles lie off X, N
then ali the poles of f are contained in
u
1
D , for n
some N. From the Cauchy theorem, it follo",s that
JYO
fdz
-
I
fd~
that
J
nL 1
=
0
for all n > N, and hence
O. Hence
~
.L
Yk
=
fdz
R(X), and hence R(X) f
C(X) . Evidently, in this construction one may replace disks by Jordan regions with rectifiable boundary; then iv) becomes L tn < 00, where tn is the length of Yn . A consequence of the Hartogs-Rosenthal theorem is that the Swiss Cheese has positive plane measure. William Allard has given the following direct argument for this unobvious fact. We can assume DO is the unit disk. For each x e [ -1 , 1] , let In(x) be the number of points on the circle Yn which meet the line {s : Re s = x}; thus I ex) = o , 1, or 2. n
FUNCTION ALGEBRAS
164
It is trivial that for each n, Hence,.
~ 1
.fl
-1
I n (x)dx <
fl
-1
I (x)dx n
=
4r n .
so by the monotone con-
00,
vergence (or Beppo Levi) theorem, Jl ~ I (x)dx < -lIn 00 00, and in particular, L I (x) < 00 for almost all 1
n
x e [-1, 1]. Thus, for almost all x, In(x) = 0 for all but finitely many n, so Xx = {s eX: Re s = x} is a finite union of non-degenerate intervals. In particular, Xx has positive one-dimensional measure for almost all x, and hence m(X) > 0 by Fubini's theorem. We now begin an examination of R(X) as a function algebra, and see how the various definitions and theorems of Chapter 2 fit in.
LEMMA 3.2.5. R(x)IK Proof.
=
Let K be a peak set for R(X) . .Then
R(K). Let G be a bounded component of ¢\K. Then
G meets ¢\X: for if G
c
X, and f peaks on K, f e
R(X), then f is holomorphic in G and equals 1 on the boundary of G, hence f = 1 in G, a contradiction. It follows from Theorem 3.2.3 that R(X)IK is dense in R(K). Since K is a peak set, R(X)IK is
RATIONAL APPROXIMATION
165
closed (Theorem 2.4.3), and the proof is finished.
COROLLARY 3.2.6.
Let X = uX
ex be the deaomposition of X into maximal sets of antisymmet~y fo~ R(X). Then R(X) IX
ex
fo~
to a singleton
Proof.
R(X ex )
=
fo~ eve~y
ex, and Xex
~eduaes
all but aountably many ex.
Since each Xex is a peak set, the first
assertion is a consequence of Lemma 3.2.5. Since ReX) Ix ex is antisymmetric, the second now follows from the Hartogs-Rosentha1 theorem.
COROLLARY 3.2.7. ReX). Then ReX) Proof.
Let E be the essential set
fo~
{f e CeX) : fiE e -R(E)}.
=
Immediate from Theorem 2.8.1 and Lemma 3.2.5.
The next simple observation can be very useful.
LEMMA 3.2.8. exists v
the~e
supp v Proof.
c
X
n
~ ~ ReX), and h e C(l) c . Then
Let ~
ReX) suah that
v
=
hO, and
supp h.
Let v =
h~
1
- -7T h-Z
hP, and by Lemma 3.2.1, v
Om. ~
By Lemma 3.1.8,
~
R(X).
The last two results allow us to strengthen
=
FUNCTION ALGEBRAS
166
Corollary 3.2.2.
THEOREM 3.2.9.
Let E be the essentiaZ set for
R(X). Let f e C(l)(U) for some neighborhood U of X. Then fiX e R(X) i f and onZy i f fz
=
0 on E.
Proof.
Suppose f-z = 0 on E. For any ~ ~ R(X), we have supp ~ c E and 0 = 0 off E, by Lemma 3.2.5 and Theorem 3.2.1, so
J fd~
J f-
!TI
=
Z
Odm
= O.
It
follows that f e R(X). Now suppose f e R(X). Let K
a}.
For any
~ ~
J fzhOdm
R(X), all h e C~l). Thus fzO ~ ~
f
R(X), we have
O. Hence, by Lemma 3.2.8,
R(X), and hence 0
=
=
~
=
0 for all
~ ~
0 a.e. em), for any ~ ~
0 a.e. (m) on ¢\K, so
R(K). Since this holds for any that R(X)
{w eX: f-(w) = z fz Odm = TI f fd~ = =
~
~
R(X), it follows
{f e C(X) : flK e R(K)} and so K
~
E,
and the theorem is proved. Theorem 3.2.9 enables us to put some limitations on the real functions in R(X).
THEOREM 3.2.10.
Let f e C2 (U), where U is a neigh-
borhood of X, and suppose that fix e ReX) and fix is real. Then f is constant on each component of
RATIONAL APPROXIMATION
167
the essentiaZ set E for R(X). In is connected and R(X) is
particuZar~
essentiaZ~
if X
there exist no
non-constant smooth reaZ-vaZued functions in R(X).
Proof.
We can assume f is real in U. By Theorem
3.2.9, f-z = 0 on E, and hence (since f is real) fz = 0 on E, as well. Thus feE) is a set of critical values of f. It follows from the Morse-Sard theorem (see, e.g., Sternberg, "Lectures on Differential Geometry", Prentice-Hall, 1964) that feE) has linear measure zero. In particular, f(K) is a connected null set of reals for each connected subset K of E; and the theorem is proved. One might conjecture that if X is connected and R(X) essential, then R(X) is antisymmetric. However, Steen [1] has constructed a Swiss cheese X such that R(X) is not antisymmetric (it is clear that any Swiss cheese is connected, and R(X) is essential). The non-constant real f Steen displays is given by f(s)
€
R(X) which
F(Re s), where F
is the familiar Cantor function on the line; thus f is differentiable a.e. (m), which shows the limitations of Theorem 3.2.10. From Lemma 3.2.8 we can also deduce a local
FUNCTION ALGEBRAS
168
maximum modulus principle for R(X).
THEOREM 3.2.11.
Let seX, U a
neighbo~hood
of s,
and suppose the~e exists f e R(X) suoh that If(s)1 >
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
<
n, such that X
c
u
1
U., and V. J
J
=
X has the property: fiV. e ReV.). For any J J n .L ReX), write II = L II . as in Lemma 3.2.12. Then 1 J fdll . = o , so f e ReX). fdll = L n
J
J
This last theorem answers for ReX) a question that can be raised for any function algebra. We make the definition: A function algebra A on X is called ZocaZ if whenever Vl"",V n is an open cover of X, f e CeX) and flv. e Alv. for J J 1 < j ~ n, then f e A. The question is: if X = Spec A, does it follow that A is local? For ReX), as we have just seen, the answer is yes; in fact, ReX) has the stronger property of being "approximately local", the definition of which we leave to the reader. However, for the general case, the answer is no. The counter-example was found by Eva Kallin [1]; it is peX) for a certain polynomially convex X c ¢4.
3-3
REPRESENTING MEASURES FOR ReX) Throughout this section, X denotes a compact
plane set.
RATIONAL APPROXIMATION
171
Let x e X. If v is a (complex) measure which represents x for R(X), then (z - x)v obviously annihilates R(X). The very useful converse of this remark is due to Bishop, and we formulate it as a theorem.
THEOREM 3. 3 • 1.
R(X), If x e X, i}(x) < "" " lJ = _1_ is a compZex and 11(x) f o , then v 11 (x) z - x measure which represents x. Proof.
Let lJ
.L
Since il (x) < "", the measure v is well-
defined. Evidently,
f dv
=
1. If f is a rational
f - f(x) function whose poles lie off X, so is z - x f - f(x) so O. Thus, for such f, dlJ z - x
f
f(x)
=
f f(x)dv J fdv
and hence f(x)
=
J
- _1_ 11 (x)
f fdv
[f - (f - f(x))]dv
Jf
- f_ (Xx) dlJ
=
f fdv,
z
for all f e R(X). The theo-
rem is proved.
COROLLARY 3.3.2.
Let x e X. Then x is a peak
point for R(X) i f and only i f O(x) lJ
.L
R(X) and ilCx)
<
00
=
0 whenever
172
FUNCTION ALGEBRAS
Proof.
~
If there exists
i
R(X) with
~(x)
< ~
and
a(x) f 0, Theorem 3.3.1 shows that there exists a measure v
rep~esenting
x with v({x}) =
o.
As we
saw in Chapter 2, this implies that x is not a peak point (recall: if f peaks at x, then fn ~ x{x} boundedly, so 1 = fn(x) =
J
fndv
~
v({x})
whenever v represents x). On the other hand, if x is not a peak point, there exists a representing measure A for x with
A({X}) = 0 (Theorem 2.3.4). Then R(X), and a(x) =
~(x)
=
~
=
(z - X)A
i
1. The proof is finished.
The "only if" part of the above immediately yields a theorem of Bishop [2]:
THEOREM 3.3.3.
R(X) = C(X) if and only if almost
all (m) points of X are peak points for R(X).
Proof.
If almost all (m) points of X are peak
points, Corollary 3.3.2 shows that for all R(X), 0
o
a.e. on X, and hence (3.2.1)
a.e., and hence
~
= O. Thus R(X)
a
~
i
o
C(X). The other
direction is obvious, so the proof is complete. This argument of Bishop's also has the fol-
173
RATIONAL APPROXIMATION lowing formulation:
THEOREM 3.3.4.
Let E be the essentiaZ set for
R(X), B the minimaZ boundary (set of aZZ peak points). Then E is the cZosure of X\B.
Proof.
Let F be the closure of X\B. Since X\E
consists of peak points, X\E hence E have 0 ~ ~
=
~
c
B, so E
F. On the other hand, if
~
~
X\B, and
~
R(X), we
0 a.e. on X\F by Corollary 3.3.2, so ~
R(F) by Theorem 3.2.1. Thus R(X)
flF e R(F)}, so F
~
{f e C(X)
E. The theorem is proved.
With Theorem 3.3.4, we can obtain another proof of the "only if" part of Theorem 3.2.9, in fact, a slightly stronger result.
LEMMA 3.3.5.
Suppose f e R(X) is differentiabZe at
x e X. Then either x is a peak Proof.
=
f(x)
+
Iy - xl-llg(y)1 h
=
or fz(x) =
o.
Recall the definition of differentiable:
there exist a, b e that f
point~
~
(a = fz(x), b = fz(x)) such
a(z - x)
+
b(z - x)
0 as y
+
x in X. If b
+
+
g, where
r
0, put
b-1(z - x) [£ - f(x) - a(z - x)]; then h e: ReX).
FUNCTION ALGEBRAS
174 Since h = Iz - xl 2
+
b -1 (z - x)g, we see that
Re h > 0 in some punctured neighborhood of x, while hex)
= O.
Hence e- h e ReX) peaks at x, re1a-
tive to a neighborhood of x. The lemma now follows from Theorem 3.2.11. The argument used here was employed by Wermer [6] to obtain a much deeper theorem.
COROLLARY 3.3.6.
Let f e R(X), and suppose f dif-
ferentiabZe at each point of X. Then the cZosure of
{x: fz(x) = O} contains the essentiaZ set for R(X). Proof. Immediate from Lemma 3.3.5 and Theorem 3.3.4. We can apply Theorem 3.3.1 to the study of the Gleason parts, and more generally, the norm topology on X induced by R(X). The next result was found by Wilken [1].
THEOREM 3.3.7. If x is not a peak point for R(X), the CZeason part of x has positive measure.
Proof. If x is not a peak point, there exists a positive representing measure A for x with A({X})
RATIONAL APPROXIMATION o~
Let V = (z - X)A. Then V
Y v =
{y e X:
~(y)
<
00
175 ~
and O(y)
R(X), and V
r OJ.
r
O. Let
If Y e Y,
1
is a complex measure which reprez - y sents y; by Theorem 2.1.1, there exists a positive }ley)
representing measure for y absolutely continuous with respect to v, and therefore absolutely continuous with respect to A. It follows that y belongs to the Gleason part of x. Since m(Y) > 0 by Lemma 3.1.4 and Corollary 3.1.7, the theorem is proved. Wilken has extended this argument to show that if x is not a peak point for R(X), then x is
~n
fact a point of density for its Gleason part. Melnikov [2] has a proof of this fact using the tools of analytic capacity. The author [2] found (independently) a slightly more general result, which has some interesting corollaries. This resuIt (Theorem 3.3.9 below) can also be obtained by Melnikov's arguments. We denote by II x - y I (for x, y e X) the distance between the associated functionals on R(X), i.e.,llx-YII
sup{lf(x) - f(y)l: nfft,::.l,
f e R(X)}. Then the Gleason part of x is
FUNCTION ALGEBRAS
176 {y:
Ix -
yl < 2}.
THEOREM 3.3.8.
Let x
€
X, let
~
be a measupe whiah £
peppesents x, let £ > 0, and let 0 = II~II +
Then
I y - xI
Proof. Since
< £ whenevep
(z-x)~
I y - x IP (y)
1 +
£
o.
<
R(X), the measure 1c zz -- xy
~
~,
z - x d~ , represents y whenever c :f 0, z - y z - y - (x - y) But C 1 and P (y) < d~ Z - Y (x - y)O(y), so I c I -> 1 - Ix - yIP(y) > o when y
with c
=
f
f
00.
satisfies the hypothesis of the lemma, since 0 Thus I x - yll.::. =
fI
-1c zz -- xy - 11
(1 -
f I(x Ic I Ix - yl
<
!c I
(IO(y)1 II
Ix - yIP(y) 1 - Ix - rIP(y)
<.
0 1 - 0
(II~II
+ 1)
~
I
CII~II
£.
z - ,y +
P(y))
+ 1)
dl~1
y -
+
1
- y) [0 (y)
_1
<
- y + y
+
1
1.
dl~1
I z - x - c(zcz - y)cy c)(z - y) z - y lei J I
J
<
]
xldl~1
Idl~1
RATIONAL APPROXIMATION THEOREM 3.3.9. Let P E let ~n = {y e ¢ peak
point~
and
177 =
{y e X: ~y - x~ < E}~ and
Iy - xl
<
~}. If x e X is not a
0, then
E >
n ~n)
m(P E
---=.--==- -
1
as
n
+
00.
m(~n)
Proof. Let that
~
~({x})
be a measure which represents x such E
= O. Let 0 =
According
11~II+l+E
to the last Lemma, P E :;, {y
e ¢ : Ix - ylii(y) Ix - ylil(y) < o}
< o},
so
But according to Lemma 3.1.9,
as n
+
00.
This proves the theorem.
An immediate consequence is
COROLLARY 3.3.10. The isolated points of X, in the nopm fop R(X).
topology~
·ape ppeaisely the peak points
FUNCTION ALGEBRAS
178
Using results from Chapter 1, we get
COROLLARY 3.3.11. Let x e X. There exists a nonzero point derivation on ReX) at x if and onZy if
x is not a peak point for ReX). Proof. Apply Corollary 3.3.10, and Theorem 1.6.5 with Corollary 1.6.4. Thus, by Bishop's Theorem 3.3.3, ReX) admits non-zero point derivations at many points whenever R(X) C(X). For bounded point derivations, the situation is different. Wermer [7] has constructed a Swiss cheese X such that ReX) admits no non-zero bounded point derivations. In the same paper, he constructs Swiss cheeses with non-zero bounded point .derivations at almost all points. Hallstrom [1] has given a necessary and sufficient condition for a point x e X to admit a non-zero bounded point derivation. This condition uses the notion of analytic capacity, and is analogous to a condition, due to Melnikov [1], which is necessary and sufficient for x to be a peak point. For Melnikov's work, we refer the reader to Zalcman [1], and Curtis [1], where a significant simplification is made. Hallstrom's paper also contains
r
179
RATIONAL APPROXIMATION many other interesting results concerning bounded point derivations, and derivations of higher order, on R(X) and A(X).
3-4
HARMONIC FUNCTIONS In this section, we use the symbol
note the Laplace operator: of class C(2) is harmonia
to de-
~
= 4f zz . A function if ~f = O. We begin with ~f
some analogues of the results obtained in Section 3-1.
LEMMA 3.4.1. Let f e C~2). Then for every w e ¢, few) =
~ 27T
f 10glz
-
wl~f
dm.
Proof. Choose R large enough so that supp f ~(w;
R). Let Ge:
c
{s : e: < Is - wi < R}, let Ye:
be the positively oriented circle of radius e: and center w. By Green's theorem, we have - i lYe: f-z 10glz - wldz d
2
fGe:
dZ
(f Z
log I z - wi) dm f-
Z
f-
zz
Now I:::.f
4f
zZ '
Z
-
dm. w
and the line integral above is of
FUNCTION ALGEBRAS
180
the order of
log
E:
so letting
E:,
1
2:
J
J fz
wldm =
loglz
Since loglz
-
dm
w - z
7T
E: +
0 we have
few).
wi is summable em) over compact
sets for any w, an application of Fubini's theorem yields:
LEMMA 3.4.2. Let on
¢.
~
be a compactZy supported measure
em) w, loglz - wi e
Then for aZmost aZZ
Ll(I~[),
and w
wldl~1
+
J loglz
~
is a compactly supported measure
-
is summabZe over
compact sets.
DEFINITION. If on ¢, we put new) =
J loglz
-
wld~
for each w such that the integral is absolutely convergent. Thus
~
is defined a.e. em) and summable
over compact sets.
LEMMA 3.4.3. Let and f e C2
2).
~
Then
be a compactZy supported
J fd~ =
2;
measure~
J ~f ~ dm.
Proof. Immediate from Lemma 3.4.1 and Fubini's theorem.
RATIONAL APPROXIMATION
181
COROLLARY 3.4.4. Let
~
be a oompaotly supported
~
c
supp
= 0 i f and only i f
~
= 0 a.e.
measure. Then supp ~
v
~m;
in partioular 3
em).
Proof. Immediate from Lemma 3.4.3.
DEFINITION. If X is a compact plane set, we denote by HeX) the set of all real-valued continuous functions on X which are uniformly approximable on X by functions harmonic in a neighborhood of X. We put HbeX)
=
H(X) lax.
Since each f e H(X) is harmonic in int X, the maximum principle for harmonic functions shows that HeX) and Hb(X) are isometrically isomorphic. We now look at annihilating and representing measures for HeX).
LEMMA 3.4.5. Let
~
be a real measure on the oompaot
plane set X. Then ~ ~ HeX) i f and only i f ~
= 0
off X. Proof. If
~
since loglz X. If
~
~
H(X), then w(w) = 0 for every w
~
X,
wi is harmonic in a neighborhood of
= 0 off X, and f is harmonic in a neighbor-
hood U of X, choose ¢ e C(2)(U) with ¢ c
= 1 in a
FUNCTION ALGEBRAS
182
neighborhood of X. Then
J fd]..l J f~d]..l since
o
~(f~)
= 2:
J ~~(f~)dm o
on X and ~
=0
off X.
LEMMA 3.4.6. Let X be a compact pZane
set~
Zet
Gl , G2 , ... , be the bounded compZementary components of X, Zet a. € G. for each j. Then the reaZ J J measure ]..I on X annihiZates H(X) i f and onZy i f ]..I
i
~(a.) J
R(X) and
Proof. If f
€
= 0
for every j.
R(X), then Re f and 1m f
H(X), so
€
"only if" is immediate. Suppose]..l
i
R(X) and ~(a.) J
=
0 for every j.
Then 0 = 0 off X by Theorem 3.2.1. Differentiating under the integral sign, we find jJ = 2. a a~ off X; V
since ~ is real, it follows that ~ is constant on each component of ¢\X. Since
~(a.)
J
o for each j,
00
~ vanishes in u G .• Since log(z - w) € ReX) if Iwl 1 J
is sufficiently large, ~ vanishes in the unbounded component of ¢\X. Thus
o
~
off X, and the lemma
follows from Lemma 3.4.5.
COROLLARY 3.4.7. Let X, {a.} be as in Lemma 3.4.6. J
183
RATIONAL APPROXIMATION Then the set of all finite
lineap oombinations
I, with f € ReX) and c J' peal J oonstants, is dense in HeX). In paptioulap, i f
Re f + L c. 10glz - a. J
¢\X is oonneoted, then Re P(X) is dense in HeX). Proof. Immediate from Lemma 3.4.6.
COROLLARY 3.4.8. Let X be a oompaot plane set, ppobability measupe on X. Then measupe fop x ~
if
€
~
~
a
is a peppesenting
X with pespeot to HeX) i f and only
is an Apens-Singep measupe fop x with pespeot
to R(X).
Proof. If f and f- 1 belong to R(X), then loglfl
€
H(X), so a representing measure with respect to H(X) must be an Arens-Singer measure with respect to ReX). The other direction follows from Corollary 3.4.7. We next take up the question of finding sufficient conditions in order that Hb(X) = CRCaX). It is clear that a necessary condition is that for each X
€
ax, Ox is the only Arens-Singer measure for X
with respect to RCX), and a foptiopi, the only Jensen measure. We shall prove the converse of this fact. The proof uses the solvability of the
FUNCTION ALGEBRAS
184
Dirichlet problem for smoothly bounded sets, and we first give a proof of this well-known result by methods in the spirit of this chapter.
LEMMA 3.4.9. Let X be a smoothZy bounded compact pZane set~
wn
+
Wo
Zet
e
ax.
Then there exist wn e ¢\X,
wO' and a constant k such that loglw n - zl >
loglw o
-
zl
k on X.
+
Proof. There exists a neighborhood U of
Wo
such
that U\X contains a sector, and hence a sequence wn e U\X, wn
for all w e U w
n
wo ' such that larg
+
n
w~ ~ :~
I~
8 > 0
X. It follows that
- w - w
11
wn w
- Wo - Wo
I ->
sin 8 > 0,
and hence loglw - wi > loglw o - wi + log sin 8 n for all w e U n X. Since loglw - zl converges n uniformly to loglw o - zl on X\U, the lemma is proved.
THEOREM 3.4.10. Let X be a smoothZy bounded compact
Proof. Let
~
be a real measure on
ax
which anni-
RATIONAL APPROXIMATION
185
hilates H(X). We must show
~
=
3.4.4 is equivalent to showing ~
Lemma 3.4.5, we know that
0, which by Corollary ~
= 0 a.e. From
= 0 outside X, and
ax
has measure 0 since X is smoothly bounded, so it suffices to show that ~(~) point
~
of X. Let
0 for any interior
=
be such a point, and let A be
~
a representing measure for s with respect to H(X). Then for any Wo e
ax,
choosing {w n } by Lemma 3.4.9, wi, and n
f loglz - woldA
> lim
J
loglz - wnldA
logls - wol > -
00
by Fatou's lemma, so loglz - wol e Ll(A), so logl~
- wol by dominated convergence (Lemma 3.4.9).
Then
J dl~1 (w) J loglz
J logl~
- wldA >
-
00
-
wldl~1
(w)
,
so by Fubini's theorem, ~(w) e Ll(A). Again applying Lemma 3.4.9, for any
J loglz
-
O. Thus
~
o
woldl~1 =
ax
e
> - 00,
0 a.e.
f ~d\
Wo
such that
we have
~(wo)
(A). so
I
d\(w)
f loglz
-
wld~
lim
~(w
n
)
186
FUNCTION ALGEBRAS
=
J d~(w) f
loglz - wldA
J logl~
wld~(w)
-
~(~).
=
The proof is finished. This proof follows Carleson [2]. Instead of the trivial Lemma 3.4.9, Carleson proves a more difficult lemma, and uses the argument above to obtain Theorem 3.4.14 at once. We have chosen a slower, more scenic route.
THEOREM 3.4.11. Let X be a oompaot pZane
set~
and
suppose that for eaoh x e ax, Ox is the onZy Jensen measure for x with respeot to R(X). Then Hb(X)
=
CR(aX). Proof. Let Xl' XZ ' ... be a sequence of smoothly bounded compact sets such that Xn+ 1
int Xn for
c
each n, and n X = X. Let u e CR(aX); by Tietze's 1 n extension theorem, there exists U e CR(X l ) with u
= ulax. By Theorem
3.4.10, for each n there exists
un e H(X n ) such that un1aXn = ulax n . Let x e ax.
°
We shall show that u n (x) + u(x). Let n be the representing measure for x on ax n with respect to H(X n ). Identifying each on with an element of
RATIONAL APPROXIMATION
187
C(X l )*, the weak-* compactness of the unit ball leads us to the existence of a measure a on Xl which is a weak-* cluster point of {an}. If v is any continuous function with compact support disjoint from ax, then v vanishes on ax n for n sufficiently large, so f vda n = 0 for large n, so
f vda
= O. Thus a is supported on ax. We next ob-
serve that a is a Jensen measure for x, with respect to R(X). If f is holomorphic in a neighborhood of X, and has no zeroes on ax, then f has no zeroes on ax n for n sufficiently large; applying Theorem 3.4.10, we find vn e H(X n ) such that vnlaXn = loglfl lax n . Since loglfl is subharmonic in int Xn ' we have loglfl ~ vn in Xn ' in particular, 10glf(x)1 -< v n (x) =
f v n da n = f loglflda n
for all large n, whence loglf(x)
I
<
f
loglflda.
Since the set of f e R(X) with no zeroes on ax is dense in R(X), it follows that a is a Jensen measure for x (see Section 2 - 5) . By hypothesis, we have then a = 0 x· Since a was an arbitrary cluster point of {an}' we conclude that a (weak-*). I t follows that u (x) = n
f Udo n
converges to
f
Udo x
f u n do n
= u(x). Since
n
188
FUNCTION ALGEBRAS
~un~aX ~ ~U~,
the convergence is bounded. If ~ is
any measure on ax which annihilates Hb(X), we have
J ud~ =
lim
J und~ =
0 by bounded convergence, so
u e Hb(X). The proof is finished. There are various ways of showing that if X has connected complement, each boundary point of X is a peak point for P(X). The argument that we shall give uses a function-theoretic idea which has formed the basis of a long series of successes in the theory of rational approximation by the Soviet school. If f is a function holomorphic in a neighborhood of
00
in the Riemann sphere, we denote by
f'(oo) the coefficient of liz in the Laurent expansion of f; thus, f'(oo) g' (0), where g
1 im 5 [ f (s) - f (ex»] s+oo
=
= f(l/z).
The next lemma can be quickly deduced (with K =
1/4) from a celebrated theorem of Koebe and
Bieberbach (see, e.g., Titchmarsh, Theory of Functions~
p. 209). In this book, we wish to avoid
appealing to such deep results of function theory, so we include a direct proof.
LEMMA 3.4.12. There exists an absolute constant
RATIONAL APPROXIMATION K > 0 with the
189
following property: for any compact
connected subset J of
¢,
with diameter d, there A
If I
exists f ho lomorphic in ¢\J, with
A
<
1 in ¢\J,
f(oo) = O~ and If' (00)1 = Kd. Proof. The principal value of
Iz
maps
~(1;
1) onto
a neighborhood G of 1-, choose c > 0 so that G 2 -1 1 ~ (1 ; c) . Take K = '2 c(4 - c) . Let U be the
::>
A
component of 00 in ¢\J. Choose a, b e J such that la - bl = d. Since U is simply connected, there z - b exists g ho10morphic in U such that g 2 z - a , g (00) = 1. We note that g' (00) = 1 (a - b) . Since '2 z b la - b l < 1 outJ c t; (a; d) , and I 11 z - a z a z b side ~(a; d), we see that z _ a maps U onto a set containing
~(1;
I), and hence g(U)
follows that g (U) does not meet
~
::>
~(1;
c). It
(- 1; c) (else
there would exist s, t e U such that g(s) = - get), whence g 2 (s) = g 2 (t), so s = t, an impossibili ty) . c(z - 1) Let h = we observe that h maps 2 ; 2 (z + 1) + c {s: Is + 11 > c} onto ~ (0 ; 1) (for h is the composition of
c Z +
2Z) . Put f 1 an d c2 _- cz
= h 0 in gU ,
A
f = 0 in the bounded components of ¢\J. Then f maps U into f' (00)
I), f(oo) = 0, and s(g(s) - 1)) lim c s-+oo 2g(s) + 2 - c 2
~(O;
FUNCTION ALGEBRAS
190
c
--""'2 g' (00) = 4 - c
K
(b - a),
and the proof is finished.
THEOREM 3.4.13. Let X be a oompaot pZane
G,
oomponent of ¢\X. If x e X n
Bet~
Ga
then x e Ch(R(X)).
Proof. For simplicity of notation, we assume x = O. We shall use the with
=
0.
"0.
-
S" criterion of Theorem 2.2.1,
2 ~ K' and any S, 0 < S <
0..
Given any
neighborhood V of 0, choose r > 0 so that
~(O; S-lr)
V. Now ~(O; r)
c
G has a component of
n
diameter > r (any component whose closure contains
o
will do) if r does not exceed the diameter of G,
so there exists an arc J >
i
c
~(O;
r)\X with diameter
r. Applying Lemma 3.4.12, we find f holomorphic r-
and bounded by 1 in ¢\J, with f(oo) = 0, f'(oo) > f·' (00) - zf Then g is holomorphic f'(oo) + r in ¢\J, and g(oo) = O. If s e ~(O; r)\J, then
21 Kr. Put g
=
r-
Ig(s) I
< 1,
and hence by the maximum modulus princir-
pIe, Igl r g (z)'
< 1
in ¢\J. Applying Schwarz's lemma to
we find Ig(s) 1< _r_ if 151 > r. Let h = glX.
We have h(O)
- 151
f' (00) f' (00) + r
>
K 2 +
K
0.,
and
RATIONAL APPROXIMATION Ih(s)1
~
e
191
for all s • V, while IhU
1. By Theo-
<
rem 2.2.1,0 e Ch(R(X)), and the proof is finished. The idea behind this proof is due to Gonchar [1]. We have followed a proof of Curtis [1]. The reader will observe that the hypothesis can be weakened considerably, without changing the conclusion or the proof. It suffices to assume that for arbitrarily small r > 0, ~(x; r)\X contains a connected set of diameter ~ kr, where k is independent of r.
THEOREM 3.4.14. Let X be a compact pZane set hlith a finite
numbe~
of bounded
compZementa~y
components
Gl, ... ,G . Let a. e G.
fo~ j = 1, ... ,no Then eve~y n J J reaZ continuous function on ax can be unifo~mZy
approximated by functions of the form Re f + n I: 1
ajl~ hlhere f
c. loglz J
is a rationaZ function
hlith poles onZy among the a. ' s ~ and c. are J
constants. In complement~
J
~eaZ
if X has connected
particuZar~
then Re P(X) is dense in CR(aX).
Proof. By Theorem 3.4.13, each point of ax admits a unique representing measure for R(X), and hence by Theorern3.4.11, Hb(X)
=
CR(X). The proof is
FUNCTION ALGEBRAS
192
finished by applying Corollary 3.4.7. This theorem is usually referred to as Walsh's theorem.
It is also attributed
to Lebesgue. See the bibliography. Another corollary of Theorem 3.4.13 is a theorem of Lavrentiev:
THEOREM 3.4.15. Let X be a compact pZane set with connected compZement and empty interior. Then
peX) = eeX). Proof. By Theorem 3.4.13, each point of X is a peak
=
point for ReX) rem 3.3.3, peX)
peX). By Bishop's theorem, Theo-
=
CeX).
This result contains a theorem of Walsh: if
J is an arc in
¢~
then
P(J) = C(J).
It is possible to have H(X) = CReX) without having ReX)
=
C(X)
(i.e., in view of Theorems
3.3.1 and 3.4.11, to have for each point of X only one Arens-Singer measure, but for many points of X, more than one representing measure). One example is the Swiss cheese X discovered by McKissick [1]. McKissick's example has the following amazing
193
RATIONAL APPROXIMATION
property: ReX) , CeX), but for any disjoint closed subsets F and K of X, there exists f e ReX) such that f = 0 on F, f
1 on K. It follows immediately
that the only Jensen measures are the point masses, so HeX) = CReX) by Theorem 3.4.11. Another example was found by Huber [1]. We present here what is probably the simplest example. It is a square Swiss cheese with square holes, constructed as follows: For s e ~, we denote by Isl oo the "sup norm" of s, i. e. , Isl oo = max{ I Res I , 11m s I L and r > o , we put Q(c; r) = {s: Is Qec; r) = {s: Is - cl oo
<
-
For c e ~
cl oo
<
d, and
rL We define the nested
sequence of compact sets {X } inductively, as n - 1 1 follows. Let Xo = QC the closed unit square.
z; z)'
Baving defined Xn- l' let Sn be the set of all Gaussian integers p such that p2- n e int Xn- l' and put G
n 00
Let X
n
sists of less than 22n squares, each with perimeter 8. Z- 3n , the sum of the lengths of all the deleted squares is finite, and so ReX) f
c eX) as we saw
FUNCTION ALGEBRAS
194
in Section 3-Z. The key fact about X is this: Let x e X. Then for infinitely many n there exists p e Sn such that Ix - pz-nl < Zl-n.
Assuming this for the moment, we show that each x e X admits only the point mass at x as an Arens-Singer measure. Put u
=
n
log Z - loglz - qn l log Z - 10glx - qn l
where qn
=
pZ -n ,with p e Sn and
Ix
- qn I
<
Zl-n.
Then un e H(X), and (since diam X < 2 and Iz - qn l > Z-3n on X for every n) we have
o
log Z < u
n
while un(x) qn
+
<
=
log Z
+
log Z3n
+
log Zn-l
1, and un(y)
x). Thus, if
0
+
=
3n
+
1 <
4,
n
0 for all y f x (since
is any Arens-Singer measure
for x, 1
= un(x) =
f undo
+
o({x})
by bounded convergence, and so
0
=
0 . x
It remains to verify the italicized assertion above. We show that if x e X, and n is not divisible by 3, there exists peS
n
such that x e Q(pZ-n; Z-n).
If Q(x; Z-n) is disjoint from Q(qZ-m; Z-Zm) for all
RATIONAL APPROXIMATION
195
m < n, q e Sm' then any Gaussian integer p with pZ-n e Q(x; Z-n) will do. Otherwise, there exists p such that Z-3m < Ipz-n _ qZ-ml oo < Z-3m
+
Z-n
for some m < n, q e Sm' and IpZ-n - xloo < Z-n. I claim that peS . If not, there exists k < nand n -n -k -3k r e Sk such that I pZ - rZ l 00 ~ Z . Then
o
I r2 -k
<
- qZ -ml 00
with r e Sk' q e Sm' k < n, m < n. We finish the argument by showing this is impossible. We may assume k < m. Then IqZ-m - rz-kl oo > Z-3k since q e Sm' so Iq - rzm-kl oo > Zm-3k. If m ~ 3k, it follows that Iq - rzm-kl oo > 1
+
Zm-3k, so 1
Zm-n, a contradiction since m I q - rZ m-kl 00 ~ 1, so 1 < Z-Zm
<
n. If m < 3k, then 2m-3k + Zm-n . This
+
<
Z-Zm
+
is possible only if 3k - m = 1 = n - m, ruled out by our hypothesis that n is not divisible by 3, or if m
=
k
3-5
THE ALGEBRAS A(X) AND AX
=
1, ruled out since Sl is a singleton.
Let X be a compact plane set. We recall from Chapter I that A(X) is the algebra of all continuous
FUNCTION ALGEBRAS
196
functions on X which are holomorphic in the interior of X. The results of the last section enable us to give the following generalization of Wermer's maximality theorem:
THEOREM 3.5.1. Let X be a compact ptane set, with interior G and boundary Y. Suppose that G is dense in X, that G is connected, and that each point of
Y admits a unique Jensen measure for R(X). Then A(X) I Y is a maximat ctosed subatgehra of C (Y); in fact,
i f B is a function atgehra on Y and B
R(X) IY, then either B
c
A(X) IY
or B
~
= C(Y).
Proof. Suppose that there exists w e G such that ¢(z)
rw
for all ¢ e Spec B. Then (z - w)-l e B
(Corollary 1.2.13). Since G. is connected, and B R(X) IY, it follows that B ~ R(Y)
~
(Theorem 3.2.3).
Since each point of Y lies in the closure of G, each point of Y is a peak point for R(Y) by rem 3.4.13, and hence R(Y)
=
Theo-
C(Y) by Theorem 3.3.3.
Thus B = C(Y). Suppose on the other hand that for every w e G there exists ¢ w e Spec B such that ¢ w (z) ~
w
=
w. Let
be an Arens-Singer measure on Y for ¢ . Then w
~
w
197
RATIONAL APPROXIMATION
is, a fortiori, an Arens-Singer measure for w with respect to R(X), i.e., a representing measure for w with respect to H(X). Let feB. By Theorem A
3.4.11, there exists f e H(X) such that fly For each w e G, we have few) ~w(f). ~
=
f.
J fd~w = J fd~w
It follows that (Zf)A(W) =
w (z)~ w (f)
=
~w(zf)
=
wf(w) , for each w e G, whence zf -
=
(Zf)A is harmonic in G, as well as f. Then f is holomorphic in G, since 0
=
(zf)zz
=
(zfz)z
=
A
zfzz + fz
=
fz· Thus f e A(X), so f e A(X) Iy. The
proof is finished. We refer the reader to Gamelin and Rossi [1] for more general theorems of this type, and a deeper study of the space H(X) and related questions. The main result in this section will be a theorem of Arens which identifies the maximal ideal space of A(X) with X. The theorem actually works for a larger class of algebras.
DEFINITION. Let X be a compact subset of the RieA
A
mann sphere ~, G an open set in ~, G
c
X. We define
A(X; G) to be the set of all functions continuous on X and holomorphic in G. We define A(X) to be
FUNCTION ALGEBRAS
198 "-
"-
A(X; int X), and AX to be A(¢, ¢\X). In general, AX need not contain any nonconstant functions. Our first result is due to Wermer [1].
LEMMA 3.5.2. Let X be a aompaat
p~ane
set, and
suppose that AX aontains a non-aonstant funation. Then there exist f, g, h e AX suah that {f, g, h} "-
separates the points of
¢.
Proof. Suppose there exists f e AX which is not constant on ¢\X. Choose points s, t in ¢\X such that f(s) g
=
l
f(t). Let
f - f(s)
f
- f(t)
h z -
s
z - t
It is clear that g, h e AX' If P and q are distinct "-
points of ¢, and f(p)
f(q), then g(p) = g(q) im-
plies that h(p) 1 h(q). Now if every f e AX is constant on ¢\X, then m(X) = 0, by Lemma 3.1.5. In particular, X must "-
have empty interior, so ¢\X is dense in ¢, and hence AX reduces to the constants. The lemma is proved.
COROLLARY 3.5.3. Unless A(X; G) reduaes to the
199
RATIONAL APPROXIMATION constants~
A(X; G) is a function atgebra on X.
Proof. Clearly, A(X; G) is a uniformly closed subalgebra of C(X). Using linear fractional transforma~
tions, we can assume that either X = ¢ and or that X
c
¢. In the first case, A(X; G)
~
=
e G, Ay '
~
where Y
=
¢\G is a compact plane set, and Lemma ~
3.5.2 shows that Ay separates the points of ¢. In
the second case, z separates the points of X. The algebras AX have an interesting property, again first pointed out by Wermer.
LEMMA 3.5.4. Suppose that X has empty interior. Then ~
for aZt f e Ax' f(X) = f(¢).
Proof. It suffices to show that if f e AX has no ~
zeroes on X, then f has no zeroes in¢. Now if f has no zeroes on X, then f has at most a finite ~
number of zeroes in ¢. Let al, ... ,a n be the zeroes of f in ¢ (a zero of order k being listed k times). Put g
=
(z - a l )
-1
... (z - an)
-1
f. Then g e AX' Bnd
~
if f has any zeroes in ¢, zeroes in ¢, so g
=
g(~)
=
O. But g has no
exp h for some continuous func-
tion h on ¢, necessarily holomorphic in ¢\X. Since
ZOO
FUNCTION ALGEBRAS
g(~)
=
0, this is incompatible with the argument
principle. The lemma is proved. According to Theorem 3.4.15, if J is an arc in ¢, then P(J) = R(J) = C(J). The preceding results of Wermer show that the higher-dimensional analog of this theorem fails to hold: there exists an aro J in ¢3 suoh that R(J) f C(J). For let X be an arc in ¢ such that AX does not reduce to the constants, for instance, an arc X with m(X) f O. (The existence of such arcs was first shown by Osgood [1]. For further examples of arcs X such that AX is nontrivial, see Denjoy [1].) Let f, g, h be the functions given by Lemma,3.5.Z, and let J
=
{(f(s),
g(s), h(s)) : seX}. According to Lemma 3.5.Z, J is an arc in ¢3, homeomorphic to X. If F is a polynomial in 3 variables, then F(f, g, h) e Ax; if F(zl' zZ' z3) has no zeroes on J, then F(f, g, h) has no zeroes on X, and hence by Lemma 3.5.4, none A
on ¢, so F(f, g, h)-l e Ax. Thus R(J) is imbedded as a suba1gebra of AX (is it a proper sub algebra? A
no one knows.) If w e ¢\X, the map F
+
F(f(w),
g(w), hew)) is a homomorphism of R(J) onto ¢, not evaluation at any point of J (since f, g, h separate
201
RATIONAL APPROXIMATION A
the points of ¢ by Lemma 3.5.2), so R(J) f C(J). (In other words, J is not rationally convex.) Rudin [1] has modified this construction to obtain an arc J in ¢2 such that P(J) f C(J). The next theorem is due to Arens [1]. The proof we give is a slight modification of Arens' proof, probably due to Vitushkin (see Gamelin and Garnett [1] and Zalcman [1] for further uses of the construction).
THEOREM 3.5.5. Let f e A(X; G), let p
X, and
€
A
let
O. Then there exists g, continuous on
>
€
¢~
holomorphic in G and in a neighborhood of p, such that ~f - g~x < E.
Proof. We may assume that p = 0, and that f(O) = O. A
If X
o
~
A
¢, extend f to be continuous on ¢. Choose
> 0 so that
If(s) I
< E/9
whenever lsi
00
Choose ¢ e Cc such that ¢ = 0 in ~(O;
outside
!TI
J few) W
_
20), and
0<1>-11 <
z
~(O;
<
20.
0), ¢ = 1
2/0. Put g(w) =
z- f z dm. By Corollary 3.1.3, g(w) 1
f(w)(w) - TI
J wf¢_zz
dm. By Lemma 3.1.5, it follows
FUNCTION ALGEBRAS
202 '"
that g is continuous on ¢, holomorphic in 6(0; 8). We next observe that g is holomorphic in G. Fix w e: G. Then g (w + h) - g (w) h 1
J [f (w
hlT
~J
+ h)
+
w - z
h) - few)
cP-z
h
- : J few)
- f
w -
IT
Now (w - z)-l(f(w)
few) - f]
- z
W + h
few
f
-
+
-
z
cP-z W +
Z
w
h
h - z
CPz dm
dm
dm.
f) is continuous, since f is
holomorphic at w; therefore, by Lemma 3.1.5, the second integral above is a continuous function of h. Evaluating the first integral by Corollary 3.1. 3, we have g (w + h) - g (w)
lim h+O
h
few + h) - few) lim - - - - - - - CP(w h h+O
+ h)
few)
cP-z
+
lim h+O
~J
f I (w)
+
-
f (z)
dm w + h - z w - z ¢few) - f (z) z dm. w - z w - z
~J
203
RATIONAL APPROXIMATION Thus g'Cw) exists. Finally, we estimate
~f
-
g~.
For any w e ¢, If (w) - g (w) I
= If(W)(1 -
~ ~fllL1
~(w))
+
~J
IlfllL1ll~zll ~
+
t
(w -
z)-lf~z
dml
Iw - zl-ldm
(where we write L1 for L1(0; 20)), whence by Lemma ~f
3.1.1 and our choice of 0 it follows that ~f~L1(1 +
(2/0)
-
g~
• 40) < E. The theorem is proved.
COROLLARY 3.5.6. Let f e A(X; G),
~.e
X, p f
00.
For each E > 0, there exists g e A(X; G) such that ~f
- f(p) - (z -
p)g~
< E.
Proof. Immediate from Theorem 3.5.5.
THEOREM 3.5.7. Suppose A(X; G) contains a nonconstant function.
Then the maximaL ideaL space of
A(X; G) is X (with the usuaL identification). In A
particuLar~
Spec A(X) = X, and Spec AX =
¢ unLess
AX reduces to the constants. Proof. By Corollary 3.5.3, A(X; G) is a function algebra on X, so it suffices to show that if
~
is
<
204
FUNCTION ALGEBRAS
a multiplicative linear functional on A(X; G), there exists p e X such that
~(f)
= f(p) for all
f in A(X; G). By using a linear fractional transformation if necessary, we may assume that either X
C
¢.
A
¢ and
or X =
00
z e A(X; G). Put p =
e G. In the first case, ~(z).
(z - p)-l e A(X; G) if P
f
Clearly. p e X, since X. Let f e A(X; G). By
Corollary 3.5.6, there exist gn e A(X; G) such that (z - p)gn ~(z
-
p)~(gn)
+
f - f(p). But
= O. so
~(f)
~((z
- p)gn) =
= f(p). Now suppose
A
X = ¢, and
00
e G. If ¢ is not evaluation at
00,
there exists h e A(X; G) with ¢(h) f h(oo); replacing h by h - h(oo) , we may assume h(oo) = 0 f Since h(oo) = 0 and 00 e G, zh e A(X; G). Put _ ¢(zh) P ~(h) Let f e A(X; G). By Corollary 3.5.6, ~(h).
there exist gn e A(X; G) such that (z - p)gn f - f(p). Now 1
-- • ¢(h)
~(h)¢((z
- p)g ) n
1
- - ¢((z - p)hg) ~ (h)
1
n
+
205
RATIONAL APPROXIMATION Thus
(f)
f(p). The proof is complete.
By virtue of Theorems 3.5.5 and 3.5.7, various facts that we have learned about R(X) carryover for A(X). For instance, if ~
i
o
R(X), so 0'
{x e X: 0(x)
<
measure (unless
~
i
A(X), then a fortiori
off X. It follows that E
and O'(x) f O} has positive plane
00
~
Theorem 3.5.5 that
= 0). If x e E, it follows from 1
~
z ~_ x is a complex measure
which represents x. The proof of Theorem 3.3.9 carries over word for word, and hence so does its corollary: there exists a non-zero point derivation on A(X) at x i f and only i f x is not a peak point for A(X).
The most natural question about A(X) is, of course, when does R(X)
= A(X)? The best answer to
date is due to Vitushkin [1], who gives necessary and sufficient conditions in terms of "continuous analytic capacity." An exposition of Vitushkin's work is to be found in Zalcman [1]; see also Vitushkin [2]. We shall give in the next chapter a proof along function algebra lines, due to Glicksberg and Wermer, of a theorem of Mergelyan which gives a sufficient condition for R(X)
= A(X). An
206
FUNCTION ALGEBRAS
outstanding problem is this: if each boundary
point of X is a peak point for R(X), does it foZZow that ReX) = A(X)?
CHAPTER DIRICHLET
4-7
FOUR ALGEBRAS
DIRICHLET ALGEBRAS In his 1956 paper [1], Gleason singled out
one of the crucial properties of the disk algebra on the circle as offering a defining property for a class of algebras that seemed amenable to analysis. In the years that followed, his prediction was amply confirmed: Dirichlet algebras, and their generalization introduced by Hoffman, logmodular algebras, provide a setting in which an unexpectedly large amount of classical function theory on the disk finds generalization.
DEFINITION. The function algebra A on X is said to be a Dirichlet algebra if Re A is dense in CReX).
FUNCTION ALGEBRAS
208
We have observed that the disk algebra on the circle is a Dirichlet algebra. The big disk algebra is a Dirichlet algebra on the torus, and for the same reason: trigonometric polynomials are dense. The bicylinder algebra on the torus is very far from being Dirichlet. The Lebesgue-Walsh theorem (Theorem 3.4.14) shows that if X is a compact plane set, then P(X)
IY
is a Dirichlet algebra on
Y, where Y is the Shilov boundary for P(X)
(the
boundary of X, or the boundary of the unbounded component of ¢\X). It has been shown (Browder and Wermer [1], [2]) that there exist Dirichlet algebras on the unit interval, and proper subalgebras of the disk algebra which are Dirichlet algebras on the circle. One of the outstanding open problems in the subject is: does there exist'a non-triviaZ
ViriahZet aZgebra A on X, with Spec A
=
X?
The giant step in the discovery that Dirichlet algebras were a natural setting for function theory was taken by Helson and Lowdenslager [1], who worked in the setting of the big disk. Bochner's note [1] called attention to the generality of their arguments, and Wermer's theorem (Theorem
209
DIRICHLET ALGEBRAS
4.4.1 below) established the power of their method in the context of function algebras. The next major breakthrough was the work of Hoffman [1], who extended the results to logmodular algebras.
DEFINITION. The function algebra A on X is said to be logmodular if loglA-ll = {loglfl : f e A-I} is dense in CR(X). Evidently, a Dirichlet algebra is necessarily logmodular (Re f
loglefl). In terms of annihi-
lating measures, a function algebra A on X is Dirichlet if and only if 0 is the only real measure in A~. This immediately implies that each ~
e Spec A (indeed, each
~
e A* with
~ ~~
=
~(l)
1) admits a unique representing measure. The corresponding fact for logmodular algebras is due to Hoffman.
LEMMA 4.1.1. Let A be a logmodular algebra on X, ~
e Spec A. Then
~
admits a unique representing
measure.
Proof. Suppose A and
~
are representing measures
for ~. Then for every f e A-I,
210
FUNCTION ALGEBRAS 1
= ~(l) = ~(f)~(f-l)
J fdA J f-ld~ whence 1 <
J
eUdA
2
J e-ud~
J
IfidA
J If-lld~,
for all u e log lA-II ,
and thus for all u e CR(X). Fix u e CR(X). Put Fet) =
J etudA
J e-tud~,
for t real. It is easy
to see (expand in power series) that
J
etudA is a
differentiable function of t, with derivative
J uetudA, Je
and a similar statement holds for
-tu d~. Hence F is a differentiable functlon, .
and for all t, Fet)
o=
F'I(O)
=
J
~
1 = F(O). Thus
udA -
J ud~,
and since this holds for all u e CR(X) , we have A =
~,
and the lemma is proved.
It was shown by Lumer [1] that all the function theory which was generalized to the context of Dirichlet algebras and then logmodular algebras depended essentially only on the conclusion of Lemma 4.1.1: the uniqueness of representing measures. This phenomenon is explained by Theorem 4.1.2 below. We have made some reference already in Chapter
211
FUNCTION ALGEBRAS
2 to the Hardy spaces associated with a function algebra. In this chapter, they become of fundamental importance. We recall that if A is a function algebra on X, and
a probability measure on
0
X, HP(o) is defined to be the closure in LP(o) of A, for 0 < p <
We define Hoo(o) to be the weak-*
00.
closure of A in Loo(o) (= Ll(o)*). It is obvious that Hoo(o) is a uniformly closed subalgebra of the Banach algebra LOO(o) , which, as we saw in Chapter 1, is isometrically isomorphic to C(Y), where Y = Spec Loo is a totally disconnected compact Hausdorff 00
space. It follows that H (0) may be regarded as a function algebra on some yr (Y' = Y if Hoo(o) separates the points of Y, otherwise Y' is obtained from Y by identifying certain points). In this chapter, we will always take
0
to be a representing
)
measure for some
~
e Spec A. In this case, one
sees without difficulty that on Hoo(o): 00
H (0).
J fgdo = J fdo
(Thus
functional
~
~
0
is multiplicative
J gdo for all f, g e
extends to a multiplicative linear 00
on H (0); but note that 0 is not a
representing measure for
~:
it is not a measure on
the space on which HooCo) is a function algebra.) More generally, if f e HP(o) and g e Hq(o), where
FUNCTION ALGEBRAS
212
(i
p and q are conjugate exponents
J fgdo
=
J fda J gdo;
+
~ = 1), then
this follows easily from
Holder's inequality. There is a quasi-converse to Lemma 4.1.1, due to Hoffman and Rossi [1], [2].
THEOREM 4.1.2. Let A be a function aZgebra on X, ~
e Spec A, and suppose that
~
admits a unique
representing measure a. Then Hoo(o) is a ZogmoduZar aZgebra on Spec Loo(o); in fact~ more is true: 00
every u e LR~ there exists f
00
e (H (a))
-1
for
such that
u = loglfl. Proof. Without loss of generality, we assume that
J udo
= O. There exists a sequence {u n }, u n e
CR(X), such that un
-+
u a.e. (0),' and II Un II ,::.llull.
By Corollary 2.2.4, there exist g
J
n
e A such that
Re g n -< u (u n - Re g n )do -+ O', without loss n' and of generality, we may assume 1m gn do = O. Put gn un then f n e A, and Ifni = exp Re gn < e < fn = e e llull . Let f be any weak-* cluster point of {f } n in Loo(o). Since f e A, f e Hoo . Since fndo n
J
J
J exp have
gn do = exp
J fda
=
J gn do
1. Since
exp
J
Re gn do
-+
1,
we
un Ifni -< e , we have I f I -< e U
DIRICHLET ALGEBRAS a. e. (0).
IE
213
(For if E is any Borel subset of X,
Ifldo =
I
fkdo, where k = XE sgn
is a cluster point of
IE
Ifldo =
I
{I
fkdo
~
I
t. But
fkdo
fnkdo}, so lim sup
II
fnkdol
< lim sup <
lim sup
<
IE
eUdo
by Fatou's lemma.
Thus If I ~eu a.e. (0).) We have found f e HOO(O) , loglfl
~
fdo
=
1, and
u. Similarly, we find g e HOO , ~
and loglgl Ifgl and
I
J fgdo
(0), so f- l
gdo
1,
- u. Then
exp(loglfl
=
I
+
J fdo J gdo
= =
loglgl) < 1, =
1, whence fg
g e Hoo , and loglfl
=
=
1 a.e.
u. The theorem
is proved. In connection with Theorem 4.1.2, the following theorem of Gorin [1] is interesting.
THEOREM 4.1.3. Let X be a metrizeable compact space~
and A a function algebra on X. Suppose that
FUNCTION ALGEBRAS
214
for every u e CR(X) there exists f e A-I such that
u
=
loglfl. Then A
=
C(X).
Proof. Let u e CR(X). Then for each real t, there exists f t e A-I such that tu
=
loglftl. Now since
X is metrizeable, A satisfies the second axiom of -1 countability, and hence A has at most countably many connected components. Since the reals are uncountable, it follows that there exist s, t, s f t, such that fs and f t lie in the same compo-1 -1 nent of A . Then fsft lies in the component of 1 in A
-1
; by Theorem 1.4.3, there exists g e A such
that f f- l s t
=
exp g. Thus (s - t)u
= loglfsf~ll
Re g, and so u e Re A. Thus Re A = CR(X). It follows from the theorem of Hoffman and Wermer (2.7.6) that
A = C(X). It would be interesting to know if there exist function algebras A (other than CeX)) on a metrizeable X with the property that for every positive continuous function u on X there exists f e A with u
= I fl· Gorin's theorem and 4.1.2 assure us that if
o is a unique representing measure, then the com00
pact space on which L (0) is realized as an algebra
215
DIRICHLET ALGEBRAS
of continuous functions is not metrizeab1e-a conelusion which should astonish nobody.
REMARK: If A is a Dirichlet algebra on X, and X is totally
disconnected~
then A
=
C(X). For if K is
an open and closed subset of X, there exists f with ~Re f - XK~ < disjoi~t
1 z.
€
A
Then f(K) and f(X\K) lie in
half-planes, hence in disjoint closed disks,
so Xf(K) e P(f(X)), and hence XK
€
A. Thus A con-
tains the characteristic function of every open and closed subset of X, and since linear combinations of these are dense in C(X), the conclusion follows. Since Spec Loo(cr) is totally disconnected (1.3.4), Theorem 4.1.2 provides us with examples of logmodular algebras which are not Dirichlet. For another example (the "big annulus" algebra), see Hoffman [1].
4-2
ANNIHILATING MEASURES Throughout this section, A will be a function
algebra on X,
~
a mUltiplicative linear functional
on A which admits a unique representing measure cr.
FUNCTION ALGEBRAS
216
We denote by M the maximal ideal associated with ~:
M = {f e A:
~(f)
2 P 2
HP(a) by HP. For 1 {f e HP : 1
2 P
<
00,
I
= OJ. We shall abbreviate 00,
we put Hb
=
fda = a}. It is easy to see that for
Hb is the closure of M in LP, and H~ 00
is the weak-* closure of M in L . Our objective in this section is to describe the annihilating measures for A in terms of the Lebesgue decomposition and the Radon-Nikodym theorem.
LEMMA 4.2.1. Let E be an Fa subset of X such that
aCE) = O. Then there exist fn e A, with
~fn~
such that fn
+
0 pointwise on E, and f
+
n
2 1,
1 a. e.
00
Proof. We may write E = u K , where each K is n n 1 closed and Kn
Kn+ 1 for all n. There exists, by
c
the regularity of a, an open neighborhood G of E n such that a(G n )
<
1 -Z.
By Urysohn's lemma, there
n
exists un e CR(X) with un on Kn' and - n
<
un
<
-
=
0 on X\G n , un
=
-
n
0 on X. By Corollary 2.2.4,
there exist g n e A such that Re g n -< u n and udo -! We may assume (adding an Re J gn do > n n
J
(
217
DIRICHLET ALGEBRAS
f gnda
imaginary constant if necessary) that 1m
o.
1 undo> - na(G n ) > - -n' we gn e . Then fn e A, and gnda > -2/n. Let fn
I
Since
f
have
undo =
IG n
exp Re gn ~ exp un' so ~fn~ ~ 1 and ~fn~K e- n
Since
J fnda
<
n
= ~(egn) = exp ~(gn) =
J gn da
> exp (- 2/n) and ~ f n II -< 1, the sequence {fn} converges to 1 in Ll(a), and hence a subse-
exp
quence converges to 1 a.e. (a) • If x e E, then x e Kn for all sufficiently large n, so Ifn(x) I = exp Re gn(x) < e -n for all sufficiently large n, and the proof is finished. If
~
is any measure on X, by Lebesgue's de-
composition theorem we may write where
~a
~
=
~a
is absolutely continuous, and
+ ~s' ~s
is singu-
lar with respect to a. The next theorem is referred to as the F. and M. Riesz theorem.
THEOREM 4.2.2. Let ~ e A~. Then ~~ e A~ (and hence ~
s
e A~).
Proof. Since
~s
is singular with respect to a,
there exists a Borel set E such that aCE) = 0 =
FUNCTION ALGEBRAS
218
Iv 5 I (X\E).
By regularity, we may assume that E is
an Fa. Applying Lemma 4.2.1, we find f n € A, with ~fn~ ~ 1, such that fn + 1 a.e. (0) and fn + 0 on E. Then fn for any g
+
a.e. (P a ) and f n + 0 a. e. CPs) , so A, we have 0 = gfndp = gfndPa + 1
J gfndp s
+
J gdP a
f gdj.1 a
0
for all g
=
Proof. Let c P
s
€
J
J
€
=
by bounded convergence. Thus
f dp.
A, which was to be proved.
€
Then P - co
€
Ai , so (v - co) = s
Ai by Theorem 4.2.2. From the general F. and M. Riesz theorem, we
may quickly deduce the classical one:
COROLLARY 4.2.4. Let a be normalized Lebesgue r~
measure on the circle
r suoh that
J zndp
=
and let p be a measure on
0 for every positive integer
n. Then j.1 is absolutely oontinuous with respeot to o. Proof. Let A be the disk algebra on
r;
then A is a
Dirichlet algebra, and the considerations of this section apply, with ¢
evaluation at the origin.
=
The hypothesis is that p this implies that Ps
€
€
Mi. By Corollary 4.2.3,
Ai. Since M = zA, it follows
DIRICHLET ALGEBRAS
219
that z~s e M~, and again applying Corollary 4.2.3, that z~s e A~. By the obvious induction, we find that z-n ~s e A~ for every non-negative integer n. Thus
~s
annihilates A and ft., and since A + A is
dense in C(n, i t follows that
~s
=
O. The proof
is finished. There are a number of different proofs of the F. and M. Riesz theorem in an abstract setting, going back to the ground-breaking paper of Helson and Lowdenslager [1]. We have given the proof of Forelli [1]. This argument was improved by Ahern [1] to estaolish the following result: Let A be a function algebra on X, ~ e Spec A, o a representing measure for ~. T h en ~ e A ~·~mp l'~es
~a e A~ i f and only i f every representing measure for ¢ is absolutely continuous with respect to 0. This situation was studied in detail, notably in connection with problems of rational approximation, by Glicksberg [4]. Glicksberg [3] has also found other versions of the F. and M. Riesz theorem. Our next order of business is to describe the absolutely continuous annihilating measures. By the Radon-Nikodym theorem, if ~ is a measure absolutely 0,
then ~
= fo for some
f e Ll(o). Let H = {f e Lleo)
: fo e
A~}. It is
continuous with respect to
FUNCTION ALGEBRAS
220 I
clear that HO I
c
-
H; we shall prove that in fact
-
HO = H. We begin with a theorem due to Hoffman and Wermer (unpublished; but see Wermer [5]). THEOREM 4.2.5. Let f e HI, and suppose that If I ~ 1 a.e.
f n e A, with Ilf n I -< 1,
(0). Then there exist
such that fn
f a. e. (0).
+
Proof. Since f e H1 , there exist gn e A such that
J
If - gnldo
O. Passing to a subsequence, we
+
may assume that gn
f a.e. (0). Let un
+
=
then un e CR(X). Let En
=
log+lgnl;
{x e X: Ign(x)1 > I}.
Then
Ix
undo
<
IE n
log+lgnldo
IE
I g n- ,f I do -<
n
IX
Ig n - fldo
since log+lgnl ~ Ign l on En' Thus
I
may
e A such that Re h < - u , Re h n do > n n and 1m I hndo = O. Since un > 0, we
- 2
undo
O. By Corollary 2.2.4, we
+
J
-u
have
exp Re h ~(e
exp
hn
f
<
n -
e
) = exp hndo
n
<
1. Also,
~(hn)
221
DIRICHLET ALGEBRAS
= exp Re so
f
f hndo
h
>
exp(- 2
J
undo),
h
e ndo
1. Hence, e n
+
+
1 in Ll(o), so passing h
to a subsequence, we may assume e n h
Let fn = gne n
Then fn
+
+
1 a.e. (0).
f a.e. (0), and Ifni =
I g n lexp Re h n -< Ig n lexp(- 10g+lg n I) -< 1. The theorem is proved.
COROLLARY 4.2.6. For any p ~ 1, Hoo = HP Proof. We observe that HP
c
n
Loo •
HI for p ~ 1 (an im-
mediate consequence of Holder's inequality), and 00
since the weak-* topology of L
is
stronger than the weak LP topology. Thus Hoo Loo
c
HI
n
HP n
c
Loo ; but it is clear from Theorem 4.2.5
10000
that H
n
L
c
H , so all is proved.
We shall need two lemmas from the general theory of measure and integration. The first is due to Hoffman [1].
LEMMA 4.2.7. Let A be a probabiZity w e Ll(A), w > O. Then exp
flOg
wdA
measure~
and
FUNCTION ALGEBRAS
222
{J eUwdA:
inf Proof. If u e
L~(A)
u e
L~(A),
J udA
and
J udA = O}.
= 0, then applying
the inequality of the geometric and arithmetic means, we find
J
exp
= exp
log wdA
J
(log w + u)dA
~
f eUwdA;
which establishes the inequality in one direction. If w were bounded above, and bounded away from 0,
= flOg wdA - log w would yield the
choosing u
opposite inequality. In general, we proceed as follows. Let v
+
n
min{n, log+w} and v
n
min{n, log-w} for each positive integer n. Put
cn
J vndA.
vn+ - v n ' and c n
vn +
Then vn e L00 R , and
flOg wdA by the monotone convergence theorem
(note that this is true even if flOg wdA
e
-v
n
n
J
=
n -
+
1 . . W pOlntwlse,
ence that lim
f
e -v nwdA
f e unwdA
we have by dominated converg+
l. Hence
= lim e
cn
lim e cn Thus
00).
c n - v n . Then u n e Loo u n dA = O. Now R and < v < log-w, so e-vnw _< max{l, w}. Since
Put un - v
= -
f
e
-v
exp
nwdA
J log
wdA.
223
DIRICHLET ALGEBRAS
J
inf { JeUwdA: u e L;,
udA =
exp
<
o}
flOg
~
wdA,
and the proof is finished. The next lemma is due to Arens.
LEMMA 4.2.8. Let A be a finite positive
measupe~
let g e L~(A). Suppose there exists 8 > 0 such that
J logll o
- tgldA > 0 for all t e (-8, 8). Then g =
a.e. (A).
Proof. We may assume 8 = 1, since it suffices to show 8g
f
logll
O. For +
5
i(l - s)gldA. Then F is harmonic in the
open unit disk. If we have logll ~
F(s)
e ¢, lsi < 1, define F(s) =
+
5
=a
i(l - s)gl
D for aIls, 151
inequality, we have F(s)
151
<
f
>
logll
+
tgl, so
1. Hence, by Harnack's
<
~
i : I!J
F(O) for aIls,
1, and in particular, F(D)
<
i
t
<
1. But F(l - t) =
log(l
+
t Zg Z)dA. By elementary calculus, one
for 0 lz
it, with a and t real,
+
<
F(l - t)
easily verifies thatt-llog(l
f logll +
+
itgldA =
tZx Z) ~ 0 as t ~ 0,
FUNCTION ALGEBRAS
224
real number x. Thus
by dominated convergence. Hence F(O) so g
i
=
J
log(l
+
g2)dA
0,
=
0 a.e. (A), which was to be proved.
=
2 If we assume g € LR , Lemma 4.2.8 can be proved in a much more direct and elementary fashion (see Hoffman [1]), and this result would suffice, as we shall see, for most of what follows.
LEMMA 4.2.9.
FoX' each
g
€
H,
J
logll - glda
> O.
Proof. According to Lemm'a 4.2.7,
J
exp =
logll - glda
inf{
J eUll
- glda : u
00
€
LR ,
But as we saw in Theorem 4.1.2, if u
J udcr
=
0, there exists f
loglfl and exp
J
If
€
fdal = 1. Thus
logll - gldcr
J uda
=
o}.
00
€
LR and
(Hoo)-l such that u
=
DIRICHLET ALGEBRAS
= inf{
J If I
11 - g I dO'
J fda
00
But for f e H
J If I
225
=
=
(H)
-1
,
1, we have
I J (f
11 - glda >
00
(f e
I
- fg)dal
f fdal
=
1
since g e H, and the lemma follows.
THEOREM 4.2.10. For eaah p, 1 ~ P <
A is
dense in LP; A +
00,
A +
A is
weak-* dense in Loo •
Proof. Suppose that g e Hand g is real-valued. Then
J logll
- tglda > 0 for all real t by Lemma
4.2.9, and hence g
If g e Ll and
= 0 a.e.
(a) by Lemma 4.2.8.
J fgda J ~gda
=
0 for all f e A,
then Re g and 1m g belong to H, and thus g = 0 a.e. (a). Thus A
+
A is
weak-* dense in Loo • Since
. h . P p_ 1 1S t e conJugate exponent
to p, the same argument shows that A
+
A is
dense
in LP. COROLLARY 4.2.11. .
-
2
sum); thus H n L .
$
H2 (orthogonal direat
226
FUNCTION ALGEBRAS
2 . d . L2 ; Proof. By Theorem 4 .2. 10 , H02 + -H 1S ense 1n since HO2 and -2 H are orthogonal closed subspaces, the direct sum is closed, and the corollary is proved. THEOREM 4.2.12. H 1
Proof. We have already observed that HO pose g e Loo and g -
f
f fgd~
gda e H n Loo
c
0 for all f e
H n L2, so g -
-
H. Sup-
c
H~.
f gda
Then e
H~
by
Corollary 4.2.11, and thus g e H2 n Loo • Hence 00
g e H
by Theorem 4.2.5 and hence
f fgda
= 0
whenever f e H. Thus HI0 is dense in H, and of course HI0 is closed, so the proof is finished. Combining Theorems 4.2.2 and 4.2.12, we have a description of annihilating measures in terms of the representing measure a: if ~ e A~, then ~ = fa + v, where f e HO1 and v is singular with respect to a. The next theorem, due to Glicksberg and Wermer [1], extends this result to a more global description of annihilating measures for Dirichlet or logmodular algebras.
THEOREM 4.2.13. Suppose that A is a function alge-
DIRICHLET ALGEBRAS bra on
227
X with the property that every multiplica-
tive linear functional on senting measure. Let ~ e
A admits a unique repreAi. Then there exists a
sequence {an} of representing measures~ a sequence {f } with f
n
n
e HOI(O )~ and an annihilating measure n
v which is singular with respect to every representing measure for A, such that
~
= v
fno n ,
+ L
the series converging in norm.
Proof. For each
~
e Spec A, let
senting measure for 2-6, if ~~ - ~ ~
=
~.
o~
be the repre-
As we observed in Section
2 then o~ and o~ are singular,
while if ~~ - ~~ < 2, o~ and o~ are boundedly equivalent (mutually absolutely continuous) by Theorem 2.6.4 and uniqueness. Let I be the set of all the Gleason parts of Spec A. For each choose ~~ e ~; let o~
=
o~~.
~
e I,
Then {oa : ~ e I} is
a pairwise singular collection of representing measures, and for each
~
o~
e Spec A,
is equiva-
lent to a a for some a e I. For each a e I, let ~
=
~
a + v a , where ~ a is absolutely continuous with respect to a and v is singular. For each finite a a subset F of I , let v F L ~ Then v F is ~ aeF a singular with respect to aS for each S e F, since
.
FUNCTION ALGEBRAS
228
VF = Vs -
~~,
L
and
~fS
Vs
and each ~~ (~ f S) is
singular with respect to aS. Hence v F is singular ~S'
with respect to each
S e F, and hence singular
with respect to L ~ . Thus ~ L ~ ~ F ~ F ~
II ~ II. But I
L ~~ ~
L
F
F
I i-t II, since ~
=
{~
~~~
-
~vF~ ~
: ~ e F} is a
~
pairwise singular collection. Thus L ~~~~ ~ ~ ~~ F
for every finite subset F of I. It follows that
=
J
{~
e I :
~
f O} is countable, and that
~
converges in norm. We put v
=
~
- L
~~
L ~
~
(what else?)
and observe that v is singular with respect to aS for every S e I, and hence with respect to ~
any
Vs
e Spec A. For v
and each
~~
with
~
Vs
L
~fS
~
~
o~
for
, and since
f S is singular with respect
to aS, and the series converges in norm, it follows that v is singular with respect to aS. Now for each
~
e I,
~~
e Al. by the F. and M. Riesz theorem
(Theorem 4.2.2), and hence by Theorem 4.2.12, ~~ 1 f a for some f e HO(o~). The proof is finished ~
~
=
~
by taking an enumeration o£ J.
4-3
APPLICATIONS In this section, we apply the results of pre-
vious sections to obtain some theorems of classical
DIRICHLET ALGEBRAS
229
analysis, construct some interesting function algebras, and generalize some classical functiontheoretic results to an abstract setting. We begin by using the analysis of annihilating measures obtained in the last section to obtain a theorem in polynomial approximation. The theorem is due to Mergelyan, the proof we give to Glicksberg and Wermer [1].
THEOREM 4.3.1. Let X be a compact plane set with connected complement. Then P(X)
A(X).
Proof. Let Y be the boundary of X. Then P(X) is a Dirichlet algebra on Y, by Theorem 3.4.14.
(We
may, and shall, regard P(X) and A(X) as function algebras on Y.) Suppose
~
is a measure on Y which
annihilates P(X). By Theorem 4.2.13, we can write
where each a
n
is a representing measure on Y for 1
P(X), and fn e HO(on)
(the Hardy class formed from
the algebra P(X)), and where v is singular with respect to every representing measure for P(X). We show first that v = O. By Theorem 3.3.1,
230
FUNCTION ALGEBRAS
for each x e X such that vex) <
00
and vex) f 0,
there exists a complex representing measure A for x, with A absolutely continuous with respect to v. By Theorem 2.1.1, there exists a (positive) representing measure a for x with a absolutely continuous with respect to A, hence with respect to v. Since v is singular with respect to every representing measure a, it follows that vex) = 0 whenever vex) v
<
00,
and hence by Corollary 3.1.7, that
= o. (An alternate argument runs as follows. For
each x e X such that
J dlvl
Iz - xl
<
00
, the measure
z v- x ,singular with respect to the representing measure a x for x, annihilates the maximal ideal at x. By Corollary 4.2.3,
J z dv-
x
= O. Thus v
vanishes a.e. on X, as well as everywhere off X, so v = 0 by Corollary 3.1.7.) We next show that each fno n annihilates A(X). We observed in Chapter 1 that Spec P(X)
= X when
X has connected complement. Thus each an represents a point x n e X, for P(X). If Tn is a measure on Y which represents xn for A(X), Tn a fortiori represents xn for P(X). so Tn
an by uniqueness. Since
231
DIRICHLET ALGEBRAS
fn is the limit in Ll of a sequence of functions in P(X) each vanishing at x n ' it follows that
J gfndo n
= 0
for all g e A(X).
Thus each measure on Y which annihilates P(X) also annihilates A(X), so P(X)
=
A(X).
Mergelyan's original proof of this theorem (see Mergelyan [1], or for a more readable exposition, the book of Rudin [2]) has the advantage of being constructive in character, and also of yielding, simultaneously, Theorem 4.3.2 below. A proof of Theorem 4.3.1 on functional analysis lines was first given by Bishop [8]. The Glicksberg-Wermer proof which we gave is also to be found in Carleson [2], stripped of all references to the general theory of Dirichlet algebras.
THEOREM 4.3.2. Let X be a compact plane
set~
and
suppose that the diameters of the components of
¢\X are bounded away from
o.
Then R(X)
=
A(X).
Proof. If 26 > 0 is a lower bound for the diameters of the complementary components, then for each x e X, X n 6(x; 0) is a compact set with connected complement. Let f e A(X). By Theorem 4.3.1, the restriction of f to X n 6(x; 0) is in P(X n 6(x; 0)).
FUNCTION ALGEBRAS
232
The theorem now follows from Bishop's localization theorem (Theorem 3.2.13). This derivation of Theorem 4.3.2 from Theorem 4.3.1 was found by Kodama [1], and rediscovered by Garnett [2]. As an application of the (classical) F. and
M. Riesz theorem, we give a construction of a Dirichlet subalgebra of the disk algebra, and some other Dirichlet algebras. Let A be the disk algebra on the circle r, let 0 be normalized Lebesgue measure on r. Let S be a strictly monotonic continuous map of [0, 2n] onto an interval [a, a + 2n] such that S' (t) = 0 almost everywhere (see, e.g., Riesz and Nagy [1], ·t ·S(t) p. 48). Put a(e 1 ) = e 1 • Then a is a homeomorphism of r, which transforms 0 into a measure singular with respect to 0; i.e., the measure 0 , a
defined by 0a(E) = o(a
-1
(E)) for Borel subsets E
ofr, or equivalently, by
J
fdo a
f foado for
f e C(r), is singular with respect to o. Let A
a
=
C (r) : foa e A}. Then A is a Dirichlet algebra a on r, and AJ.a = {J.l : J.l e AJ.}. By the F. and M. Riesz a theorem (Corollary 4.2.4), each J.l e AJ. is absolutely {£ e
DIRICHLET ALGEBRAS
233
continuous with respect to 0, and hence
~a
is ab-
solutely continuous with respect to 0a' Thus, if ~
I~
e A~ and v e Aa~ , +
and v are
~
.
so
s~ngular,
v I = ~ ~ U + II v II. By Lemma 2.4.8, it follows
closed. Since A~ + A~a is a is weak-* weak-* dense in (A n A a )~ ' it follows that that
A~
+ A~
(A n Aa)~ = A~
+ A~.
If ~
+
v is real, with ~ e A~
and v e A~, then the absolutely continuous and a
singular parts of are real, so
~
~
+
v are real, i.e.,
= v = O.
~
and v
Thus A n Aa is a Dirichlet
algebra on the circle, properly contained in A. Now suppose that aoa is the identity map of r. Let B = {f e A: foa = fl. Let X be the space obtained from r by identifying the orbits of a to points. One may verify that there are only two possibilities: either a has no fixed points, when a is orientation-preserving, and X is homeomorphic to r; or a has two fixed points, when a reverses orientation, in which case X is homeomorphic to
[0, 1]. Since C(X) may be identified with {f e C(r): f
=
foal, B may be regarded as a closed
subalgebra of C(X). We show that B is a Dirichlet algebra on X by showing that if A is a real measure on r, annihilating B, then A
~
{f e C(r): f = foal,
234
FUNCTION ALGEBRAS
i.e., that ACE) E of f. Let
~
= - A(a(E)) for every Borel subset
be the set of all such "odd" measures
A. Clearly, ~ is weak-* closed, and BL is the weak-* closure of AL
1:.2 II~
+
v
+
~. Now if ~ e AL and v e ~, then
+
~a
+
va I
1:.2 (II ~
<
~
where we used the fact that - v since v e and v a again, we find that AL
hence BL = AL A =
~
+
and hence
A
~
a ~
=
~
I ~a
v a II)
+
are singular,
is weak-* closed, and
e AL , v e
~
+
~a
+
Suppose A e BL is real. Then
~.
v, where
+
A
+
and
v II
Applying Lemma 2.4.8
~.
+
+
~
a
+
~.
~
v + va
O. Thus A
='V
e
Hence
~,
+ ~
a
is real,
and B is a Dir-
ichlet algebra on X. We next show that B is a maximal subalgebra of C(X). Let BI be a closed subalgebra of C(f), with BI
~
B and with f
=
foa
for every f e BI . Then every A e B~ is of the form L C(X) if and ~ + v, with ~ e A , v e ~, and BI f only if there exists ~ + v e BL with ~ f O. Now I for any f e B1 , f~ + fv e BL I , so f~ + fv ~1 + vI' L f oa, so with ].11 to A , vI e ~. But fv to ~ since f
DIRICHLET ALGEBRAS f~
-
~l
e Q. Since
235 f~
-
it follows that f~ - ~l f e Bl . Thus
~
~l
=
is absolutely continuous, O. Thus f~ e A~ for all
annihilates the algebra generated
by A and Bl ; since
~
f 0, this algebra is proper,
and we conclude from Wermer's maximality theorem that Bl
c
A, and hence Bl
c
B, concluding the
argument. it can be shown that Spec (A n Aa)' and Spec B when a reverses orientation, are homeomorphic to the 2-sphere. When a preserves orientation, Spec B is homeomorphic to the real projective plane. See Browder and Wermer [2] for the details. Since A~ +
A~a is closed, it follows from Lemma 2.7.7 that A
+
Aa is closed, and hence A
+
Aa
=
C(r).
(If a
is any orientation-reversing homeomorphism of r, it is shown in the above paper that A + in C(r), though A +
A need
A is
dense
not be closed, nor an
algebra, when a is not singular.) Let J be an arc in the complex plane. Let
~
A
be the Riemann map of the unit disk
~
onto ¢\J. It
is known that ~ extends continuously to
X.
For
each s e r, there exists a unique a(s) e r such that ~(s)
=
~(a(s)),
and one may verify that a is
a homeomorphism of r, and aoa is the identity. The
FUNCTION ALGEBRAS
236 map f
~
f·~
is an isomorphism of AJ onto B. Thus
AJ is a Dirichlet algebra on J if the boundary identification a induced by
~
is singular. Arcs J
with this property were constructed in Browder and Wermer [1].
For the remainder of this section, we return to the setting of Section 2: A is a function algebra on X, ¢ e Spec A admits a unique representing measure
0,
M is the maximal ideal associated with ¢.
We shall next obtain, in this context, a theorem due to Szego when A is the disk algebra. It is of fundamental importance in the prediction theory of stationary stochastic processes (see, e.g., the book of Grenander and Szego [1]).
THEOREM 4.3.3. Let
~
be a positive measure on X;
by the Lebesgue deaomposition and Radon-Nikodym theorem~
~
= wo
+ v, where
w e L1 (0) and v is
singuZar.with respeat to o. Let 0 < p <
inf {
f 11 - fIPd~:
- fldo
~
Then
f e M} = exp flOg wdo.
Proof. Let f e M. Then
J logll
~.
logl4>(l - f)1
0
DIRICHLET ALGEBRAS
237
since a is a Jensen measure for
I 11 -
~
flPwda
exp
I
Since
~.
(log w
+
P logll - fl)da
(inequality of the geometric and arithmetic means) it follows that
J
exp
log wda
~
inf{
J 11 -
flPwda: f
flOg wda
On the other hand, by Lemma 4.2.7, exp inf {
J eUwda:
u e L;(a),
eM}. =
f uda o}, =
while by Theorem 4.1.2, for each u e L;(a) with
I
= 0, loglfl = u uda
P
exp
I
there exists f e (Hco)-l such that and
I
fda
=
1. Thus
log wda
=
inf
> inf
=
inf
{I {J
IflPwda: f e (Hco ) -1, co IflPwda: f e H ,
{ I 11 -
flPwda: f e
I
J fda = I}
fda
=
I}
H~}.
co From Theorem 4.2.5, we see that each f e HO is the pointwise a.e. (a) limit of a bounded sequence in
M; it follows from bounded convergence that exp
J
log wda
~
inf {
J 11 -
flPwda: f
eM}.
FUNCTION ALGEBRAS
238
Thus
f
exp flOg wdo = inf {
11 - flPwdo: f eM},
and it remains to show only that inf {
f \1
-
fIPd~:
f e M}
J
inf {
11 - flPwdo: f eM}.
Let f e M. By Lemma 4.2.1 (the heart of the F. and M. Riesz theorem), there exist gn e A such that gn
1 a.e.
+
(0) , gn
+
0 a.e.
and II gn I < 2 for a11 n. Then
(v) , with
J
((1 - f)g )
n
gn do = 1
=
fH(gn) = 1, so (1 - f)g n = 1 - h n for some h n e M. Applying bounded convergence, we have
-
J
hnlPd~
11 -
=
J
1(1 - f)gnIP(wdo + dv) +
J Jl
-
f IPwdo .
Thus inf {
J 11
inf {
J
- hIPd~:
heM}
fIPd~:
f e M}
so 11 <
inf {
J 11 -
<
J 11
-
flPwdo,
flPwdo: f eM}.
Since the opposite inequality is obvious, the proof is concluded.
DIRICHLET ALGEBRAS
239
Szego proved this theorem (for A the disk algebra) with the stronger hypothesis that
~
is
absolutely continuous; the general case is due to Kolmogorov and Krein. The most interesting case is, naturally, p = 2; the theorem for any p can in fact be deduced from the theorem for p = 2, though there seems no good reason to do so. An interesting corollary of Szego's theorem is the following, taken from Hoffman's book [2].
THEOREM 4.3.4. Let on the aircle r~
0
be normalized Lebesgue measure
let 1 < P < oo~
let g e LP(o). Then
the linear span of {zng : n a non-negative integer} is dense in LP(o) i f and only i f i) g and ii)
J loglgldo
= -
f
°
a. e.
(0),
00
Proof. Let S be the closed linear span of {zng ; n
>
o}. Suppose ii) holds. Then, taking A
to be the disk algebra and applying Theorem 4.3.3, we have inf
{ J 11
or. since M inf
{J
- fjP IglPdo: f e M}
0,
zA, Ig - zfg!Pdo: f e
A}
O.
FUNCTION ALGEBRAS
240
Since fg e S whenever f e A, and since Izl
=
1, we
have zg e S. By an obvious inductive argument, we conclude that zng e S for every n. Now if h e Lq(o), where q = p P l' and it follows that
J
J fhdo
znghdcr
=
= 0 for every f e S, 0 for every integer n,
positive or negative, so gh = 0 a.e. (0). If i) holds, it follows that h = 0 a.e. (0), and hence S = LP(o). On the other hand, if S = LP(o), then i) obviously holds, and since zg e S, we have 0
so
= inf
{J
= inf
{ J 11 -
zfl P IglPdo: f e A}
= inf
{ J 11 -
fl P IglPdo: f eM},
f loglgldo
Izg - fglPdo: f e A}
= - ~ by Theorem 4.3.3. The proof is
finished. Theorem 4.3.4 leads naturally to the question: which closed subspaces S of L2(0) (or of H2(0)) are invariant under multiplication by z? There are some obvious candidates: if E e L~(o) (or H~(o)) and lEI
=
1 a.e., then S
EH2(0) = {Ef: f e H2 (0)}
is evidently such a space. In 1949, Beur1ing [1]
DIRICHLET ALGEBRAS
241
proved that every invariant subspace of H2(0) is of this form. Beur1ing's proof used deep results from function theory, but he observed that the function E was essentially the orthogonal projection of 1 on S. It was the inspired contribution of Helson and Lowdenslager [1] to make this observation the basis of the proof; they thus not only obtained Beurling's theorem in a more
gen~ral
setting, but a much simpler proof for the classical case. There is now a large literature on various generalizations of Beurling's
theorem;~
see the
books of Hoffman [2] and Helson [1] for a start, and for further references. We shall next give the natural generalization of Beurling's theorem in the context of this chapter.
THEOREM 4.3.5. Let S be a olosed subspaoe of L 2 (0), with the properties that
i) i f f e A and g e S, then fg e S; ii) the olosed linear span of {fg: f e M, g e S} is not dense in S. Then there exists suoh that S
Ee
LOO(o) , with
lEI
=
1 a.e.
= EH 2 (cr) = {Ef: f e H2(cr)}; E is
uniquely determined (up to multiplioation by a
(0),
242
FUNCTION ALGEBRAS
oonstant of modulus 1).
Pr90f. Let T be the closed linear span of {fg: f e M, g e S}. Since T is a proper closed subspace of S, there exists E e S,
J
IE 12 do = 1,
fgEda f we have in
such that E is orthogonal to T, i . e . , for all f e M, g e S. Taking g
f
particular that
=
E,
= 0
flEI 2da = 0 for all f e M. Thus
IEI 2 a is a probability measure annihilating M, hence a representing measure for
~.
By uniqueness,
we conclude that IEI 20 = a, i.e., that lEI = 1 a.e. (0). It follows that multiplication by E is an isometry of L2(0) into itself, so EH2(0) is a closed subspace of S. Suppose that g e S is orthogonal to EH 2 (0), Then
J
fE.g-do
=
0
for all f e A; but.for f e M, fg e T, so
J
fgE do = 0;
thus Eg is orthogonal to A 4.2.11, we conclude that
+
M. By Corollary
Eg =
0, and since lEI
a.e. (0), that g
= O. Thus EH 2 (0) = S. If also
S
IFI
=
FH2(a) where
=
1 a.e. (0), then E
=
=1
Ff for
DIRICHLET ALGEBRAS
243 f e H2 (0), and similarly,
some 2
-
2
EF e H (0). Hence EF is orthogonal to HOeo) + -2 HO(o), and hence EF = const. a.e. by Corollary 4.2.11. The proof is concluded.
COROLLARY 4.3.6. Let S be a olosed subspaoe of H2 (0), suoh that
i) i f f e A and g e S, then fg e S·, i i) 1 is not orthogonal to S. Then there exists E e Hoo(o), suoh that S
=
lEI
=
1 a.e.
(0) ,
EH 2 (0).
Proof. Since 1 is orthogonal to {fg: f e M, g e S}, ii) implies that the second hypothesis of Theorem 4.3.5 is satisfied. The function E produced by Theorem 4.3.5 belongs to H2(0) n Loo(o), so E e 00
H (0) by Corollary 4.2.6. We note that when A is the disk algebra on the circle, hypothesis ii) of Corollary 4.3.6 is unnecessary; replace S by znS if necessary.
4-4
ANALYTIC STRUCTURE IN THE MAXIMAL IDEAL SPACE Again in this section we keep fixed a function
244
FUNCTION ALGEBRAS
algebra A on X with a multiplicative linear functional
which admits a unique representing measure
~
o. We denote by M the maximal ideal associated with
~,
and we denote by P the Gleason part of
thus P =
{~
e Spec A:
~~
-
~~
~:
2}. This section
<
is devoted to proving the theorem (due to Wermer [3] when A is a Dirichlet algebra, extended by Hoffman [1] to the case of logmodular A, and proved by Lumer [1] in the present generality) that unless P reduces to a singleton, it is an analytic disk.
THEOREM 4.4.1. Suppose P f a map
~
{~}.
Then there exists
of the open unit disk U
such that for every f e A,
fo~
~(O;
1) onto P
is a bounded holo-
morphic function on U (where f is the Gelfand transform of f). The map ~ is a homeomorphism i f P is given the metric topology~
and thus a one-one
continuous map when P is given the weak (Gelfand) topology.
Proof. We shall use several lemmas.
LEMMA 4.4.2. Let
~
e P. Then
representing measure lent to a
(i.e.~
~~
and
~ ~
admits a unique is boundedly equiva-
there exist constants a, b with
245
DIRICHLET ALGEBRAS
o
<
a
< b
<
such that
00
a~
< b~).
a
<
Proof. By Theorem 2.6.4 (or 2.6.5) there exist representing measures A for ¢ and
~
constants a, b with 0
such that
A <
b~.
<
a
<
b
<
00
for
~,
and a~
<
Since a is the only representing measure
for ¢, we have A = a. Suppose v is any representing measure for since a -
a~
~.
~
Then a(v -
~)
annihilates A, and
0, it follows that a + a(v -
~)
is
a representing measure for ¢. From the uniqueness of a, it follows that v
=
~.
The lemma is proved.
In view of Lemma 4.4.2, if
~
e P and
~
is a
representing measure for ~, the spaces H2(a) and
H2(~) are identical as sets of (equivalence classes of) functions; as Hilbert spaces, they have distinct but equivalent norms. Thus H~ is the closure of M in H2, where H2
H2 (a)
= H2(~),
and closure
may be taken in either the L2(a) or L2(~) sense Without altering the meaning. Our next lemma tells us that, loosely speaking, M is a principal ideal; more precisely, it does Contain the assertion that the associated maximal ideal in Hoo is principal.
FUNCTION ALGEBRAS
Z46 LEMMA 4.4.3. There exists Z e
=
a. e. , and suah that HZo
H~ suah that IZI = 1
ZHZ.
Proof. The function Z may be constructed either as in the proof of Theorem Z.6.4 or as in Beurling's theorem of the last section. Let us repeat the ~
latter argument. Choose
~
e P,
be the representing measure for proj ection of 1 on
i ~.
~,
~
and let
Let E be the
H~ in HZ (~). Thus E e H~,' and
1 - E is orthogonal to H~ in LZ(~). Let c Z
J IEI2d~.
Then c 2 > 0, for otherwise
J fd~ =
0
for all f eM, contradicting the assumption that ~
so
f
We put Z = c -1 E. If f e A, then fE e
J fE (1
- E) d~
particular, on H2
,
= 0,
f Ed~
c
or Z
.
J
fEd~
Since
~
H~,
J fIEIZd~;
in
is multiplicative
it follows that
C2~(f)
=
f fEd~
J flEl2d~
for all f e A, i.e., c-2IEI2~ = IZI2~ is a representing measure for
~.
Hence, by Lemma 4.4.2,
Izi = 1 a.e. From Izi = 1 a.e. it is obvious that ZH 2 is a closed subspace of HZ, and from the multiplicativity of a that ZH 2 ZH 2 =
c
H~. To show that
H~, it suffices then to show that no non-
DIRICHLET ALGEBRAS
247
zero element of H~ is orthogonal to ZH2. Suppose g e
H~
then gf e c- l
J gZfd~
and
H~,
f gfd~.
we have
so
= 0 for all f e A. If f e A,
J gf(l
-
E)d~
Hence, for all f e A with
J gZfd~
J gZfd~
0, or
J £d~ = 0,
= O. Thus gZ is orthogonal to
as well as A. But A
+
=
A
A is dense in L2(~) by Theo-
rem 4.2.10 (and Lemma 4.4.2). Hence gZ = 0 a.e., and hence g
=
0 a.e. The proof is concluded.
The next lemma is a simple corollary.
LEMMA 4.4.4. For eaah s e U, the orthogonaZ aompZement of (Z - s)H 2 in H2(o) aonsists of the aonstant -1 muZtiples of (1 - sZ) .
Proof. For any f e HZ,
J feZ
- -1 do - s) (1 - sZ) =
J fZdo
= 0,
Z Thus (1 1 a.e. and fZ e HO' is orthogonal to (Z - s)H 2 .
since
IZ I =
-
- -1 sZ)
Suppose g e H2 and g is orthogonal to (Z - s)H Z. Then
o
f
g(Z - s)fdo =
f
gel - sZ)Zfdo
for all f e HZ. Thus g(l - 5Z) is orthogonal to
FUNCTION ALGEBRAS
248
H~ in view of Lemma 4.4.3, and hence g(l - 5Z) = const. The lemma is proved. We now proceed to the proof of Theorem 4.4.1. For each 5 e U, we define the linear functional ci>s on A by ci>s(f) =
J
f(l - sZ)-lda
for each f e A; since (1 - sZ)-l is bounded, the definition makes sense and ci> e A*. Since 00 5 - -1 n-n (1 - sZ) = L 5 Z , the series converging uni-
o
formly, we have ci>s(l) = 1. Since ci>s(f) is the inner product of f with (1 - 5Z)-1, Lemma 4.4.4 tells us that ci> 5 (f) = 0 if and only if f e (Z - s)H2. Hence the kernel of ci> 5 is an ideal, and thus ci>s e Spec A for each 5 e U. For f e A and 5 e U, let f(s) = ci>s(f). Then 00 fZnda]5 n , so f is holomorphic in U. f(s) = L 0
J
-
1 (Lemma 1.2.9), we have If(s)1 .:=..llfll for aIls e U, f e A. By Schwarz's lemma it follows that
-
If(s) - f(t)1 < 215 - t i l l - stl- l whenever 5, t e U, f e A, HfH
<
1. Thus
249
DIRICHLET ALGEBRAS
for all s, t e U. In particular,
so
~
e P for all s e U.
s
Now by Theorem 4.2.5, there exist fn e A, ~fn~ ~
JfnCl J
1, such that fn - sZ) -Ida
+
f Z (1 00
L
Z (1
o
~s(fn)
Z a.e. (0). Then
+
- sZ) -Ida. But
J s n ZI-n do
=
s. Thus
~
s
(f ) n
From the inequalities Is - t I -< II ~ s - ~ tII <
I st I
21s - t
h-
it is clear that
~
is a homeomorphism of
U into P, when P is given the metric topology. We have already observed that f =
f·~
is holomorphic
for every f e A. It remains to show that
~
is
onto. Let 8 e P, let v be the representing measure for 8. Let s
=
J Zdv;
s is well-defined by Lemma
4.4. 2, I s I < 1 since I Z I
1 a. e., and I s I = 1 is
ruled out since Z is not constant a. e. (0) (for
J
Zdo
= 0) and hence not constant a.e. (v). Thus
s e U. If f e A, by Lemma 4.4.4 we can write
FUNCTION ALGEBRAS
250
c
f
Using the multiplicativity of v
for some g e H2 on H2
,
(Z - s)g
+
sZ
1 -
we find 8(f)
= f
fdv
=
~
c
=
sn
c L:
I
f
c
J
~Vsz
1
J
Zndv +
+
J
(Z - s)gdv
(2 - s)dv
J
gdv
s 1 2n
On the other hand, cP (f)
s
f
J1 c c
da
J
- sZ da
J
- 12 + 11 - sZ L:
J
snsm
fZ da Z - s
J fZda
Znzmda
m,n c L: Isl 2n Thus CPs
=
=
cel -
IsI2)-1.
8. We have shown that cP maps U onto P,
and the proof of Theorem 4.4.1 is finished. Special cases of Theorem 4.4.1 are worth examining. If Y is a compact plane set with connected complement, then Y can be identified as we have seen with the maximal ideal space of P(X), where X is the boundary of Y, and P(X) is a
251
DIRICHLET ALGEBRAS Dirichlet algebra. In this case, the map
~
pro-
vided by Theorem 4.4.1 is the Riemann map of U onto a component of the interior of Y. It is easy to see that the map f
+
f developed in the proof
of Theorem 4.4.1 extends to a map of
Hoo(cr) onto
Hoo(U), the algebra of all bounded holomorphic functions on U. As a consequence, one obtains the theorem of Farrell: every bounded hoZomorphia
funation on the interior of Y aan be uniformZy on aompaat
sets~
approximated~
by a uniformZy bounded
(on Y) sequenae of poZynomiaZs. See Wermer [5] for more details. Another example to look at is the big disk algebra described in Chapter I. In that chapter, we described a one-one continuous mapping of the half-plane into the maximal ideal space which we nowesee is (modulo the equivalence of disk and half-plane) the mapping
~
promised by Theorem
4.4.1. This example show that
~
need not be a
homeomorphism when P is taken with the Gelfand topology; for as we saw in Chapter I, the functions in A are taken into almost-periodic functions on the half-plane, which define a topology quite distinct from the ordinary topology on the half-plane.
FUNCTION ALGEBRAS
252
One more example must be mentioned. Let cr be Lebesgue measure on the circle. Then Hoo(cr) is a logmodular algebra, as we have seen, and one of the Gleason parts is rather obvious. But there exist many other parts in Spec Hoo(o), as was first pointed out by Gleason. The full story is quite complicated, and is to be found in Hoffman [3]. We must not conclude this section without mentioning some generalizations. A function algebra A on X is called hypo-Diriahlet if there exist f l , ... ,fn e A-I such that the linear combinations n
Re g + L c. loglf. 1 J J
I
(where g e A and c. are real J
constants) is dense in CR(X). For example, if X is a compact plane set with finitely many complementary components, then R(X) is a hypo-Dirichlet algebra on the boundary of X (Theorem 3.4.14). Wermer [8] showed that if A is a hypo-Dirichlet algebra, and
~
e Spec A belongs to a non-trivial
Gleason part, then there exists an analytic disk through
~.
O'Neill [1] showed that in this situa-
tion, the part of
~
could be given the structure
of analytic space; he also showed, using the argument of Bishop which we saw in Theorem 2.6.4, that
DIRICHLET ALGEBRAS
253
Wermer's conclusion could be obtained with the weaker hypothesis that the uniform closure of Re A has finite codimension in CR(X). Abstract function theory in the context of hypo-Dirichlet algebras was developed by Ahern and Sarason [1]. Using their work, O'Neill and Wermer [1] showed that each nontrivial part in a hypo-Dirichlet algebra could be regarded as a finite-sheeted covering of the unit disk. Finally, Gamelin [1] showed that each such part is in fact a finite open Riemann surface. Finally, we bring up an open problem. The results of this chapter, and the theory of hypoDirichlet algebras, exhibit a close relation between certain phenomena in function algebras and results from the theory of functions of one complex variable. Is there a bigher-dimensional analog? For instance, are there any hypotheses of a general character which would yield the existence, in the maximal ideal space of a function algebra, of complex analytic structure of dimension greater than one?
APPENDIX
COLE'S COUNTEREXAMPLE TO THE PEAK POINT CONJECTURE
THEOREM. Thepe exists a function aZgebpa A on a compact metpizeabZe space
i) Spec A
=
X such that
X;
i i) A f C (X) ;
iii) evepy point of X is a peak point fop A. Proof. Let X be a compact subset of
~
with the
properties: a) R(X) f CeX), b) for each x
€
X, 0
X
is the only Jensen measure for x with respect to R(X). (For an example of such an X, see pp. 193195.) Let A
R(X). Choose a countable dense sub-
set {f } of A- l ; since X has no interior, {f } is m m then dense in A. We shall co~stru~t the space X, and the algebEa A on X, with a continuous map TI of X onto X, in such a way 255
FUNCTION ALGEBRAS
256
that the functions fon (f e A) belong to A, and such that each f on admits an m n-th root in A for every positive integer n. (In Cole's thesis [1], a more general construction is_carried out, so that every function in A admits n-th roots; in this way, Cole obtains (besides the example given here) algebras where every Gleason part is trivial, and no non-zero point derivations exist, while the Shilov boundary is a proper subset of the maximal ideal space.) Let I be the set of all ordered pairs (m, n) of positive integers, with n > 2. For each (m, n) e I, let Ymn
=
{l, ... ,n}, and let Y be the Cartesian
product IT Y ; for y e Y, we denote by y(m, n) the I mn coordinate of y in Ymn . We define X (as a set, not as a topological space) by X = X x Y. Let n be the natural projection of X onto X
nex, y)
= x.
For each em, n) e, I, let f;/n denote the principal determination of the n-th root of f m; thus each f;/n is a bounded (riot necessarily continuous) function on X. For each n, let wn be a primitive n-th root of unity. For each em, n) e I, we define the fUnction gmn on'X by
257
APPENDIX and we put f e: A} We give X the
~
u
{gmn : (m, n) e: I}.
topology, i.e., the weakest topology
which makes each F e: CJ- con tinuous . We observe that the countable subset {fm o~} u {g
mn
} of
a
tT
separates the points of X: indeed, if
= (x, y) and x,
x
(x', y') are distinct points
r x',
of X, then either x
f (x') for some m, so m
x
in which case g
mn
(fmo~)(x)
r y'(m,
= x' and y(m, n)
(x)
in which case fm(x)
r
(fmo~)(xl),
r or
n) for some (m, n) e: I,
= fl/n(x)wy(m,n) f
m n Y' fl/n(x)w (m,n) = g (x'). It follows that X is a mn m n Hausdorff space, satisfying the second. axiom of countabili ty. Let A be the uniformly closed algebra generated by
fJ..
A is
Clearly,
a commutative Banach algebra
with unit; we shall show that X is compact (and hence that A is a function algebra on X) by showing that Spec A
= X. Since A is a separating algebra
of continuous functions on X, it suffices to show that for each that ~(F)
=
~
e: Spec A there
F(x~)
Then the map f
+
e~ists x~
e: X such
for every F e: A. Let ~ e: Spec A. ~(fo~)
is a mUltiplicative linear
FUNCTION ALGEBRAS
258
functional on A; since Spec A = X, there exists x~
e A such that
~(fon)
=
f(x~)
for every f e A.
For each (m, n) e I, we have g~n
so there exists
y~(m,
n) e Ymn such that
f;/n(x~)w~~(m,n). Let y~ (x~, y~);
we have then
~(gmn)
{y~(m, n)} and x~
~(F)
=
F(X~)
=
=
for every
F e~, and_hence for every F e A. Thus A is a function algebra on X, and Spec A = X. Since X satisfies the second axiom of countability, it is metrizeable. To show that A f C(X), it suffices (since
A f C(X)) to show that if f e C(X) and fon e A, then f e A. We shall show even more: there exists a continuous linear map P of C(X) onto C(X), such that P(fon) = f for every f e C(X), and peA)
c
A.
We define P by "averaging over the fibers of TI". For each positive integer N, let SN = {y e Y : y(m, n)
=
I if max{m, n} > N}. Let c N be
the cardinality of SN (c N is clearly finite). For F e C(X), define F(x, y).
APPENDIX
259
Thus, for each N, PN maps C(X) into B(X), the space of bounded functions on X; clearly, PN is linear,
~PN~ =
1, and PN(fon)
f for every f e
C(X). Let;z.= {fon : f e C(X)}u {gmn: (m, n) e {gmn : (m, n) e I}. Let F be a polynomial in
I}
u
7.
We shall show: PNF e C(X) and is independent of
N, fop N sufficiently lapgej if F is a polynomial inp~
then PNF e A fop N sufficiently lapge. We
map assume, since PN is linear, that F is a monomial: F
where f e C(X), (m., n.) e I, and k., ,Q,. are nonJ
J
J
negative integers, for 1 r
g
= II j =1
I f m. I
~
< j
J
r. Let
(k.+,Q,.)/n. J
J
J,
J
k. -,Q,. so g e C(X), and let w.
w J n. J
J
Then r
<
r.
J
y(m. ,n.)
w. SN 1 J
and if wI = ..• = wr
< j
w.
j =1
r
for 1
y (m. , n.) J J
F(x, y) = f(x)g(x) II
L II
J
J
J
• Thus PNF e C (X) ,
1, then clearly PNF
=
fg for
all N. On the other hand, if say wI f 1, then PNF = 0 for all large N. In fact, if N
>
max{ml,n l },
FUNCTION ALGEBRAS
260
then putting SN' ,= {y e: SN ,J I
<
y
j } for
mInI
j .::.. n l , we have nl
y (m. , n.) w. II LI J J J SN r
L
j=l
r
L ye:S
II
N ,J. I
y(mk,n k ) wk .'
nl
r j II w I L j=l I yeS N . 2
y(mk,n k )
Wk
o,
,J
~
since wI is an nl-th root of unity, and wI
l~
Thus PNF = 0 for N > max{m l , nl}' If F is a polynomial in
9,
then f e A, £j = 0 for I .::.. j < r, and
we may assume 0
<
k.
J
<
n. for I .::.. j J
<
r. Then for
large N, either PNF = f or PNF = 0; in either case, PNF e: A. We have proved the italicized assertion. We define PF, when F is a polynomial in~, as lim PNF. We have: PF e: C(X), P(fon) = f whenN+oo
ever f e: C (X) , PF e: A whenever F is a polynomial in
9,
and II P I
l. Since polynomials in 7are
dense in C(X) by the Stone-Weierstrass theorem, we may extend P to C(X) by continuity. Then P maps C(X) into C(X), peA) C(X), and
~p~
c
A, P(fon) = f for all f e:
= 1. We have proved assertion ii)
of the theorem. It remains to show that each point of X is a peak point for A. Let
x
= (x, y) e: X. Let
~
be a
APPENDIX
261
representing measure for X. Let v be the. measure
~(TI-l(E));for Borel sets
on X defined by veE) E
c
X; equivalently,
f e C(X). Since that supp
~
TI
c
~
-1
J
fdv =
J
fOTI
d~
for
all
is a positive measure, we see (supp v).
For any (m, n) e I, we have
J
Ifmll/ndV =
>
J IfmoTIII/nd~
I J gmnd~1
J Igmnld~
= Igmn(x) I
Ifm(x) Il/n. Thus Ifm(x) I
~
[
J Ifmll/ndV]n for
all n, m. Since ~
{fm} is dense in A, it follows that If(x) I
[ J Ifl l/ndv]n
f or every f · . e A , every posltlve
integer n, and hence If(x) I
~
exp f loglfldv for
every f e A (see p. 125). Thus v is a Jensen measure for x with respect to A, and hence, by our choice of A, v = ox' It follows that supp TI
-1
-
({x}) = Xx . Now for every (m, n) e I,
If m(x)l l / n
~
c
FUNCTION ALGEBRAS
262
since stant on supp
~
for every (m, n) e I. Since
{gmn : (m, n) e I} separates the points of Xx' it follows that supp
~
= {x}, i.e., that
We have shown that for each only representing measure for
x,
xe
X,
~
Ox
i.e., that
= ox' is the
xe
Ch(A). Since X is metrizeable, it follows that each
xe
complete.
X is a peak point for A, and the proof is
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