Big-planes, boundaries and function algebras

BIG-PLANES, BOUNDARIES AND FUNCTION ALGEBRAS

NORTH-HOLLAND MATHEMATICS STUDIES 172 (Continuation of the Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil

and University of Rochester New York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

BIG-PLANES, BOUNDARIES AND FUNCTION ALGEBRAS

Toma V. TONEV Department of Mathematical Sciences University of Montana Missoula, M7; U.S.A. and Institute of Mathematics Bulgarian Academy of Sciences Sofia, Bulgaria

1992

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 PO.Box 211,1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas NewYork, N.Y. 10010, U.S.A.

Library o f Congress Cataloging-in-Publication

Data

Tonev. Toma V . . 1945Big-planes. noundaries and function algebras ,' Toma, V . Tonev. F. c n . -- (North-Holland mathematics s t u d i e s ; 172) Includes bibliographical references and Index.

ISBN 0-444-89237-0 1 . Analytic functions. 2. Function algebras. functitns. 1. Title. 11. Serles. 0~331 . ~ 5 7 2 '992 515'.9--dC20

3. Almost periodic 9 1-46068

CIP

ISBN: 0 444 89237 0

0 1992 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in The Netherlands

PREFACE

This book treats some of the recent interrelations between theories of analytic functions in one and several complex variables, almost periodic functions and function algebras. The previous stage of this subject matter has been thoroughly treated e.g. in Gamelin’s and Wermer’s books [32], [143].The present book is based on the author’s most recent, some unpublished, results, as well as results of R. Basener, N. Sibony, D. Stankov, S. Grigoryan, Z. Slodkowski, G. Corach and others. In an attempt to make the book self-contained basic facts and results from commutative Banach algebras and in particular from uniform algebras are presented in Chapter I, mostly with proofs. Classical systematic expositions of the subject are given in the books of Browder [22], Gamelin [32] and Stout [107]. An introduction to uniform algebras from the point of view of complex analysis is given in the first chapters of the books of Hormander [54],Gunning and Rossi [48],Garnett [34]. A picture of the latest developments in analytic almost periodic function theory in a contemporary uniform algebra setting is given in Chapter 11. Analytic r-almost-periodic functions in domains of the complex plane are presented as r-analytic functions in corresponding domains of the big-plane. Definitions, various descriptions and general properties of uniform algebras of r-analytic functions on big-plane subsets and especially of the big-disc algebra, and algebras of analytic r-almost-periodic functions are presented in Sections 2.1 through 2.3. Properties of r-analytic functions defined on arbitrary subsets in the bigplane and in particular in the big-disc are presented in Sections 2.4 and 2.6. The important notions of spectral mapping and spectrum of a multiplicative subsemigroup of a commutative Banach algebra are introduced in Section 2.7. The surprising space H g of bounded r-hyper-analytic functions in the big-disc, introduced in Section 2.8, opens new possibilities for investigation of algebras of bounded r-analytic functions in the big-disc, distinct from similar efforts by American authors. Superalgebras of

vi

Preface

the space H g are investigated in Section 2.9. A Bochner-type theorem for analytic measures on certain compact groups with particularly ordered dual groups is proved in Appendix 2.10. Various characterizations (including the original ones due to Sibony and Basener) as well as basic properties of multi-tuple Shilov boundaries of function spaces are presented in Sections 3.1 through 3.3. Sections 3.2, 3.4 and 3.5 are based on the author’s recent results. In particular, there are introduced the so called n-tuple minimal af€ine boundaries of compact convex sets, which are multi-tuple versions of the closure of extreme points. We state here for the first time in a book presentation a general principle, the so called local minimum norm principle for vector-valued functions. Various applications of this principle and related to it n-tuple Shilov boundaries to the study of vector-valued functions are presented in Section 3.2 and 3.3. In Section 3.4 we prove Basener-Slodkowski’s theorem for multituple Shilov boundaries of tensor products of function spaces. A unified approach to a recently arosen generation of important hulls in algebra spectra connected with vector-valued functions is presented in Section 3.5. Chapter IV is devoted to latest developments in the search for n-dimensional analytic structures, analytic r-almost-periodic structures, and r-analytic big-structures in commutative Banach algebra spectra, respectively, including the famous results due to Basener, Sibony, Bear-Hile and others. It depends on several results in Sections 2.5 and 3.1 only. The selection of the topics covered by the book depended to a great extend on the author’s own interests. Many interesting questions, staying somehow aside the mainstream of the book, as well as their authors are not included or even mentioned. At the end of each section we give historical and bibliographical notes. These notes should be regarded only fragmetarily since it was difficult to give always an exact accounting of all contributors to the subject. Though oriented to specialists in analytic functions and commutative Banach algebras, the book is intended for a large au-

vii

Preface

dience of mathematicians. The level of exposition is appropriate for a second year graduate student or mathematician with another specialty who is familiar with the basic results of analytic functions and commutative Banach algebras without necessarily having seen many applications. My work in the field has been greatly stimulated by useful discussions and correspondence with Gustavo Corach, Theodor Gamelin, Suren Grigoryan, Mikihiro Hayashi, Krzysztof Jarosz, Shozo Koshi, Heinz Konig, Bruno Kramm, Donna Kumagai, Guido Luppaciolu, Takahiko Nakazi, Rao Nagisetty, Vlastimil Ptiik, Walter Rudin, Nessim Sibony, Zbigniew Slodkowski, Dimcho Stankov, Edgar Lee Stout, John Wermer, Keith Yale, and Wieslaw Zelazko. I was happy having Mrs. Susan Vayo helping me to prepare the final manuscript. The previous preprint version of this book [127], written for the sake of Bulgarian Committee of Science (through program 386), had served as a basis for several of my courses at the Institute of Mathematics in Sofia (Bulgaria), and series of lectures at Hokkaido University in Sapporo (Japan), Mathematics Division of the International Centre for Theoretical Physics in Tkieste (Italy), the University of Toledo (USA), and the University of Montana (USA), to which I am indebted for their support and hospitality. I am grateful to all friends who silently held my hand when I needed and whose encouragement has kept me writing when the going was hard; and to my family, whose support has meant more than I can say, for helping me get through a very difficult year in my life.

Toledo, Ohio August, 1991

Toma V. Tonev

This Page Intentionally Left Blank

CONTENTS Preface Contents Introduction Chapter I. Uniform Algebras

1.1. Spectrum of an algebra element 1.2. Linear multiplicative functionals 1.3. Maximal ideals 1.4. Some examples 1.5. Shilov boundary

V

ix xi 1 3 12 20 29 43

Chapter 11. r-Analytic Functions in the

Big-Plane 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

Generalized-analytic functions r-analytic functions on the big-disc The big-disc algebra Boundary behavior in the big-disc Algebras of r,-analytic functions r-entire functions Spectral mappings of semigroups The algebra H g

2.9. Algebras between N; and LE 2.10. Appendix. Analytic measures

58 59 71

78 83 93 107 112 120 129 141

X

Contents

Chapter 111. n-Tuple Shilov Boundaries

149

3.1. 3.2. 3.3. 3.4.

n-tuple boundaries of uniform algebras n-tuple boundaries of function spaces Properties of n-tuple Shilov boundaries Shilov boundaries of tensor products 3.5. Multi-tuple hulls

152 163 185 205 212

Chapter IV. Analytic Structures in Uniform Algebra Spectra

231

4.1. n-dimensional manifolds in spectra 4.2. Big-manifolds in algebra spectra 4.3. Almost periodic and r-analytic structures

232 251 262

References Index

280 287

INTRODUCTION

The category of uniform algebras is the natural setting for all those problems in complex analysis which concern a class of holomorphic functions (or their boundary values) which contain sufficiently many elements. One of the main sources for the theory of uniform algebras is the theory of analytic functions in one or more complex variables. It has been utilized intensively to answer many uniform algebraic questions. On the other hand, concepts arising from the study of uniform algebras such as algebra spectrum, Shilov boundary et c. have been increasingly useful in function theory, for instance for offering new problems and involving new methods of proofs. One of the central themes in uniform algebra theory is to locate some kind of analytic structure in algebra spectrum such that the Gelfand extensions of algebra functions are analytic with respect to this analytic structure. Remember that a subset U in the spectrum s p B of a commutative Banach algebra B has an analytic, resp. analytic r-almost-periodic structure if there exists a nonconstant map 1c, from a domain W in the complex plane C into U such that the composition T o 1c, is an analytic, resp. analytic r-almost-periodic function in W for each f E A. The reason for the special interest in the search for analytic structures is based on the fact that the existence of an analytic structure in algebra spectrum provides us with various regularity theorems and integral representations of algebra functions, with theorems for representing measures, derivations, approximations, etc. Roughly speaking, uniform algebras are spaces of continuious functions on a compact HausdoriT set, which possess the most natural properties useful for function theory. Given a uniform algebra A of continuous functions on a compact set X , the algebra spectrum (or maximal ideal space) s p A of A is the most natural superset of X on which all functions in A possess natural extensions, the so called Gelfand extensions, such that the

xii

Introduction

algebra of extended functions remains isometrically isomorphic to the initial algebra A . On the other hand the Shilov boundary a A of a uniform algebra A of functions on a compact set X is the smallest closed subset of X , such that the algebra of restricted on X functions in A is isometrically isomorphic to the original algebra A . The importance of Shilov boundary stems from the fact that it is the minimal among all closed sets E in X (also in s p A ) such that each function in the algebra is completely determined by its values on E. One of the most powerful techniques in the search for analytic structures goes back to a result of Rudin, who has shown that the maximum principle, which is a consequence of analyticity, in certain one-dimensional situations implies existence of (onedimensional) analytic structures. In an attempt to generalize this Rudin’s result, Wermer (see e.g. [143])has proved the following famous theorem:

WERMER’S MAXIMALITY THEOREM. Let A be a uniform algebra on the unit circle S1= { z E C : Izl = l} which contains the identity function id:z c+ z Then either A = C(S1) or every f in A has an andytic extension on the unit disc A . This theorem is very useful in showing that the functions in an algebra are analytic. It has been subject to intensive generalizations, and in particular in multi-dimensional and generalizedanalytic contexts . Generalized-analytic functions, or F-analytic functions in the big-plane in an other terminology, provide us with a convenient and useful language for stating problems and investigating analytic F-almost-periodic functions in C . On the other hand, the theory of analytic r-almost periodic functions helps to answer many questions for F-analytic functions in C G . Generalizedanalytic functions have appeared at a moment when it has been realized that for some specific goals the might of classical analytic function theory was exhausted. Originally, the space of

...

Introduction

Xlll

analytic functions in the big-disc (the so called big-disc algebra) was introduced by Arens and Singer [5] in a rather general setting. In the most useful cases for applications the big disc algebra coincides with the algebra of Gelfand transforms of bounded analytic r-almost-periodic functions on the real line R, where is an additive subgroup of R. Namely, this relation makes possible results for analytic r-almost-periodic functions in C to be associated with corresponding results for r-analytic functions in the big-plane C G and this makes the theory of almost periodic functions become a part of the theory of r-analytic functions in the big plane. The big-plane version of Wermer's maximality theorem, which is presented in this book, says:

r

THEOREM (Tonev 11271). Let Q be the group of rational numbers in R provided with discrete topology, let 0 be its dual group, and let A be a closed subalgebra of C(bQ), where

Q = {Z E C e : 12" - zfl < C, u

E Q+, Z, E

Ce, c > 0)

is a basis neighborhood in the big-plane CG.Suppose in addition that the Shilov boundary d A is identified with the topological boundary bQ of Q. Then either s p A can be identified with bQ, or every function in A possesses a r-analytic extension in Q. As already mentioned, the Shilov boundary d A plays a very important r6le in the search for 1-dimensional analytic structures and analytic r-almost-periodic structures in algebra spectra. For n-dimensional analytic structures, however, the Shilov boundary proves to be too small. It turns out that the appropriate notion for the n-dimensional case is not the Shilov boundary, but the so called n-tuple Shilov boundary, we define below.

DEFINITION. The n-tuple Shilov boundary d(")Aof a uniform algebra A is the smallest among all closed subsets E of algebra

xiv

Introduction

spectrum S P A ,such that n

n

(6,.Fn) of Gelfand extensions

for all nonvanishing n-tuples . ., fj of functions in A, fj, j = 1,. . . ,n. 6

This definition of a(")A, its existence and coincidence with the boundaries discovered by Basener [9] and Sibony [91]are due to Tonev [126]. First non-trivial examples of d(")A were given by Sibony [91]. The importance of n-tuple Shilov boundaries is based on the fact that n-dimensional analytic structures in algebra spectra live mainly outside 8"IA; just as 1-dimensional analytic structure live mainly outside aA. A result of fundamental importance in uniform algebra theory is the Rossi's local maximum modulus principle (e.g. [32]), which generalizes the maximum principle for analytic functions in one variable.

LOCALMAXIMUM MODULUS PRINCIPLE (Rossi). Let K be a closed subset of the spectrum of a uniform dgebra A which does not meet dA. Then for every f in A

This principle is generalized for multi-tuples of functions as follows.

[126]). L f K is a closed set in s p A which does not meet a(")A, then

LOCALMINIMUM

NORM PRINCIPLE (Tonev

n

n

xv

Introd uction h

for all nonvanishing on sp A n-tuples tensions of functions fj in A.

(fi,

. . .,$)

of Gelfand ex-

This principle is presented for the first time in the present book (see also [127]). The result supports the belief that in some reasonable cases the complement sp A \ d(")A of the ntuple Shilov boundary in the spectrum of a uniform algebra A should have some sort of (n-dimensional) analytic structure. A classical result in function theory says that given an analytic function f on the unit disc 2 and a component W of C \ f(Si),then the set f-l(z) = { z E A : f(z) = W ) is finite for all w E W . This result does not hold in several variables. However, as shown by Bishop [17], if K is a compact subset of an n-dimensional complex manifold M with envelope of holomorphy and f1, . . . ,fn are analytic functions on K , then k is, roughly speaking, not too large. More precisely, under above hypotheses the set ( f i , ...,fn)-'Ik( W) = {Z E K : fj(z) = wj,j = I , . . . ,n} is finite for almost all w in cn. The attempt for finding opposite statements to these given above has generated much energy. Bear and Hile have generalized to a great extend a classical theorem of Stoilov [lo51 about analyticity of any light and open continuous mapping f on a complex analytic manifold R, where lightness means that f-'(w) is discrete for all w in f ( R ) . Namely,

k

-

THEOREM (Bear, Hile [15]). I f f E A is light on an open set U in s p A \ dA, then there is an open dense subset R of U which can be given the structure of a one-dimensional complexanalytic manifold such that all functions in A^ are analytic on R with respect to this structure. This subject matter does back to the following already classical result due to Bishop.

xvi

Introduction

THEOREM (Bishop [17]). Let A be a uniform algebra, f E A, and W be a component of C' \ f ( a A ) which meets ?(SPA). Assume that there exists a set of positive planar measure K C W so that the set ?-'(w) = { z E s p A : ?(z) = w } is finite for = {z E SPA : each w E K . Then each point p in f(z) E W } has a neighborhood in s p A which is a finite union of homeomorphic images of A into s p A such that all functions in 2 give rise to andytic functions on A .

T-'(W>

Wermer [143]has given a very instructive proof of this theorem. It has been strengthened afterwards by Aupetit and Wermer [?I. On the other hand Basener [S] has removed the finiteness requirements from the context by replacing it by countability. In 191 Basener has generalized Bishop's theorem for n dimensions. His theorem was the first important application of multi-tuple Shilov boundaries. The next n-tuple generalization of Bear-Hile's result is included in this book. h

(A,. 2)'

THEOREM (Tonev [136]).If the mapping F = .., f j E A , is light on an open set U in s p A which does not meet &")A, then there exists an open dense subset R of U , which can be given the structure of an n-dimensional complex analytic manifold, such that the Gelfmd extensions of all functions in A are holomorphic on R.

In fact, $( m ) is a locally uniformizing variable on R and under is locally holomorphic hypotheses of this theorem every f E in the variable F ( m ) , m E R. B y requiring more specific properties for n-tuple ( f 1 , . . . ,fn) € A", one can recognize even more delicate analytic structures in SPA.

A^

THEOREM (Basener [9]).Let F = ( f l , . . . ,fn), let

w

be a component of the set C"

f j

E A , and

\ F(~(.)A) which

meets

Introduction

xvii

$ ( S P A ) . Suppose that there exists a set K c W with non-zero 2n-dimensional Lebesgue measure such that the set F-l(e) = ( m E s p A : p ( m ) = 2;) is finite for every e E K . Then the set p-'( W ) can be given the structure o f an analytic space of pure dimension n and the Gelfand extensions of all functions in A are holomorphic on 8-1 (w). In fact, under the same hypothesis the set 8-'(W) can be given the structure of an n-dimensional k-sheeted branched analytic cover. Basener's theorem has been strenghtened further by Sibony [91] in effect to recognize a Stein space structure in spA \ d(n)A. While the problem of existence of one-dimensional and multidimensional analytic structures which live in algebra spectra has been given a thorough investigation, comparatively less is known about analytic r-almost-periodic structures (resp. r-analytic structures) in uniform algebra spectra. The following versions of Bishop's and Bear-Hile's theorems concerning analytic r-almostperiodic and P-analytic structures in algebra spectra, respectively, have been proved in [118] and [133] respectively.

THEOREM (Tonev). Let A be a uniform algebra, let the group r possesses property (*) and let f2 = { fa}aEr,be a multiplicative su bsemigroup of functions in A which is algebraically isomorphic t o the semigroup To. Let U be an open subset of s p A \ a A such that r, (U)is an open and connected subset of the big-plane CG.Denote by J ( e o ) the ideal

where y o is a lineax and multiplicative functional, which belongs to U. I f the codimension of J ( a o ) in A is k < 00, then the set U can be given the structure of a kl-sheeted branched ranalytic big-cover over r, (U)for some kl 5 k , and for every open

xviii

Introduction

subset U1 of U on which rn is a homeomorphism there exists a continuous mapping j of an open subset W of the complex plane C into CG whose range is dense in Ul,such that the restrictions offunctions T o j on w are analytic r-almost-periodic functions for every function f in A .

r

THEOREM (Tonev). Let A , 0 and are as above. If the spectral mapping T~ : s p A -+ CG is light on an open subset U of s p A \ dA, then an open dense subset R of U can be given the structure of a r-analytic big-manifold such that the Gelfand extensions of all functions in A are I'-holomorphic in R; for every open subset U1 of U on which rn is a homeomorphism there exists a continuous mapping j of an open subset W of the complex plane C into CG whose range is dense in Ul,such that j on W are analytic r-almostthe restrictions of functions periodic functions for every function f in A .

70

The present book is devoted mainly to described above recent developments in the interaction between the theories of analytic functions, almost periodic functions and commutative Banach algebas, which concern r-analytic functions in arbitrary domains in the big-plane, n-tuple Shilov boundaries of function spaces and their applications to the study of analytic l"-almost-periodic and n-dimensional analytic structures in algebra spectrum.

CHAPTER I

UNIFORM ALGEBRAS

In this chapter we introduce for future use some terminology and basic facts concerning uniform algebras and, more generally, commutative Banach algebras. Let B be a Banach space over the field of complex numbers C . This means that B is a linear space over C (thus there are defined two operations in B - addition and multiplication with complex numbers), which is equipped by a norm , i.e. by a nonnegative function 11 . 11 : B + R+ = [ 0,m) such that: (i) llXall = IXlllall for every a E B and for each complex number X E C ; (ii) Ila bll 5 llall llbll for each a , b E B; (iii) 0 is the only element in B whose norm is zero; (iv) B is a complete space with respect to the norm 11 . 11. Rememeber that completeness means that every Cauchy sequence {an}F=l of elements in B (i.e. given a positive E > 0 there is an integer N > 0 such that the inequality -a,]] < E is satisfied for every pair (a,.,,a,) of elements in B with indeces n, m greater then N ) is convergent.

+

+

1.1. DEFINITION. A Banach space B over the field of complex numbers C is a Banach algebra over C if B is provided with an associative operation (multiplication) which is distributive with respect to the addition and such that the inequality

(4

1 1 4 15 I l ~ l l l l ~ l l Typeset by dMS-TE)(

ChaDter I. Uniform Algebras

2

holds for each a, b E B . A Banach algebra is commutative if its multiplication is commutative, and with unit if there exists an unit element with respect to the multiplication (denoted usually by e , also by 1) such that

(4

llell = 1.

A uniform algebra (or finction algebra) 1.2. DEFINITION. on a compact Rausdodspace X is every commutative Banach algebra A over C which satisfies the following conditions: (i) The elements of A are continuous complex valued functions defined on X , i.e. A c C ( X ) ; (ii) A contains all constant functions on X ; (iii) The operations in A are the pointwise addition and mult iplica tion; (iv) A is closed with respect to the uniform norm in C ( X ) ,

(v) A separates the points of X , i.e. for every two points in X there is a function in A with different values at these points.

Uniform algebras are commutative Banach algebras over C with unit (the constant function 1 on X ) and this fact plays a crucial r d e in their study. The basic results about commutative Banach algebras can be found in many books. The early state of the field is reflected in Gelfand [35] and in Gelfand, Ra,ikov and Shilov [36]. Outstanding contemporary expositions are those of Rickart [82], Gamelin [33], Wermer [143],Browder [22], Stout I1071 and Zelazko [146] among others.

1.1. Svectrum a# an alaebra element

1.1. SPECTRUM OF

AN

3

ALGEBRA ELEMENT

An element f in a commutative Banach algebra B with unit is invertible if there exists an element g in B such that fg = e. Suppose that fgl = e = fg2 for gl and g2 in B. Then g1 = egi = (fg2)gi = (g2f)gi = g2(fg1) = g2e = g2. Consequently, if f E B is invertible, then there is a unique element g for which f g = e. In this case g is said to be the inverse element of f and it is denoted by f - l . Hence e = f f - l for any invertible element f E B. The set of all invertible elements of B is denoted by B-l. It is not hard to see that B-l is a subgroup of B under multiplication.

1.1.1. PROPOSITION. Let B be a commutative Banach d-

c 00

gebra with unit. The series

f" is convergent in B for any

n=O

element f in B with llfll < 1; the sum of this series is an invertible element of B whose inverse element is e - f , i.e.

n=O

Cf", where f" = e by definition. By (ii) n

PROOF.Let g, =

k=O

and (v) from the beginning of this chapter we have that

k=m+l

k=m+l

k=m+l

L Ilfll"+l - Ilfll"+l < llfll"+' 1 - llf II - 1 - Ilf II

Chapter I. Uniform Akebras

4

for every m < n. Hence for a given E > 0 and for rn big enough, - gmli < c for each n > m. Therefore {gn} is a Cauchy sequence and consequently it converges because of the completeness of B. Denote by g the limit of this sequence, i;e. let M ~-

g = lim gn = ndcm

fk.

For the product g(e - f) we have:

k=O k

00

since lim k-m

llfk+'II

5 lim Ilfllk+' k+oo

< 1. Hence

= 0 because of

g = (e - f)-' as desired.

1.1.2. COROLLARY. Let f E B and let s be a complex number with Is1 > Ilfll. Then se - f is an invertible element of B

and its inverse element is the sum of the series

-

O0

Sn+l, f" i.e.

n=O

f"

n=O

PROOF.Denote the element

Ilf II

fs = Af by g. s

By the hypothesis

llgll = - < 1 and Proposition 1.1.1says that e-g is invertible Is/ M ._

and its inverse element is the sum of the convergent series

1.e.

gn, n=O

1.1. Spectrum o f an alqebra element

"

-

n=O

5

-

n=O

Hence s e - f is an invertible element in B and (se - f)-' =

f" Ex , desired.

n=O

as

I

For each s E C the element s e is denoted usually by s. Note that if B # (0) then llfll # 0 for every invertible element f in B. Indeed, llfll = 0 implies f = 0 which certainly is non-invertible, once B # (0). 1.1.3. COROLLARY. Let f E B-' and let g be an element in 1 B with llf - 911 < [If-1 11 *

(1) g is also an invertible element in

c

B;

00

(2) 9-1 =

f-'"+"(f

- 9)";

n=O

PROOF.Since f E B-' then f - l f = e. Thus e - f - l g = f - l f - f - l g = f - l ( e - g) and by the hypothesis we get that ( ( e- f-lgll 5 Ilf-lllllf - gll < 1. By Theorem 1.1.1the element e - (e - f-lg ) = f-'g is invertible and its inverse element h = (e - (e - f-'g))-l = (f-'g)-l possesses the following expansion: M

n=O

00

00

n=O

n=O

Observe that by (f-lg)h = e it follows that g(f-'h)

= e

,

00

i.e. g is invertible and 9-l = f-'h

=

f-'

f-n(f - g)" = n=O

Chapter I. Uniform Algebras

6 Do

f-("+')(f -g)".

Moreover, by the above expression for h we

¶=O

have:

fll

Do

00

n=O

n=l

As Corollary 1.1.3 shows, the neighborhood (9 E B : llg 1 < is contained in B-' for every f E B ; also 9-l

-1 llf

--I

II

tends to f-' as g

t

f . Thus, we have the following

B-' is an open subset of B and the 1.1.4. COROLLARY. f I--+ f-' is a homeomorphism of B-l onto

correspondence it self.

More precisely, B-' is an open group (under multiplication)

in B and the mapping f I-+ f -' which maps B-l into itself is a group authomorphism.

Typical examples for invertible elements are the exponents of the algebra, i.e. the elements ef in B defined by the convergent power series

f ef = 1 + l!

f" + .. +-. + n! *

The usual manipulations show that

7

1.1. Spectrvm of an algebra element

and consequently, the element e-f is the inverse element of e f . The set of all exponents in B is denoted by e B . It is easy to check that the open unit ball {f E B : llf - ell < l} in B centered at the unit element e is contained entirely in the set e B and consequently all its elements are invertible. Note that by examining the power series we can get that (eif)' = ef for each t E R. Thus the exponents of B possess arbitrary real powers. Spectrum of aa element f in a commu1.1.5. DEFINITION. tative Banach algebra B is the set a(f)= { A E C : Xe

(2)

- f 4 B-'}.

- Corollary 1.1.2 shows that a(f)is a subset of the closed disc A(llfl1)= { z E C : Izl 5 llfll} with radius llfll and centered at the origin. Thus the spectrum of each element is a bounded set and therefore C \ a(f)# 0.

1.1.6. PROPOSITION. x f z , @ o(f),then

PROOF.By definition

1% - 4
-lII

z,

- f E B-l

whenever z,

4 a(f). If I

and by Corollary 1.1.3, applied to the elements z, - f and z f , we obtain that z - f E B-l, i.e. z 4 a(f). Hence Iz, 1 for any z E a(f),and the statement follows zI 2 ll(z0 - fl-lll immediately.

8

Chapter I. Uniform Akebras

Proposition 1.1.6 shows that C \ a(f)is an open set, i.e. that a(f)is closed, and being at the same time bounded, it is a compact subset of C. By definition z - f is an invertible element for any z 6 a(f). The B-valued function ~ ( z = ) ( z - f)-', well defined on C \ a(f),is called the resolvent firnction of f . By Corollary 1.1.4 we see that ~ ( zis) a continuous function. Moreover, as the next proposition shows, ~ ( zis) also an analytic function, which means ) a local expansion as a convergent B-valued that ~ ( zpossesses power series (in z ) with coefficients in B nem each point of the open set C \ a(f).

1.1.7. PROPOSITION. UfB # {0}, then the spectrum a(f)of every element in B is a non-empty set and the resolvent function r ( z ) = ( z - f)-' is analytic in its complement c \ a(f).

PROOF.Let 1

ll(z0 - f>-lll

z,

4

a(f) and let

U

= { z E C : Iz

- zol
0. As a continuous function, $ ( z ) is bounded on any disc. By Liouville's theorem 4 is constant. Since + ( z ) tends to 0 as z + 00, this constant can be zero only. Hence $ ( z ) = L ( ( z - f)-') G 0. Since the last equality holds for any bounded linear functional on B, we obtain that ( z - f)-' 0 by the Hahn-Banach theorem, which is absurd because B # (0). The proposition is proved.

Chapter I. Uniform Algebras

10

1.1.8. COROLLARY( Gelfund-Mazur theorem). Any commutative Banach algebra B with B-' = B \ (0) is isometrically isomorphic to the field of complex numbers C and hence is onedimensional.

PROOF.Recall that a commutative algebra B with B-l = B \ (0) is called a (commutative) field; a commutative Banach algebra which is a field is called a B a n d field. Consider the subalgebra Ce = {f E B : f = Xe} of a commutative Banach algebra B . Because of IIAell = 1x1, C e is isometrically isomorphic to C. As a simple corollary we observe that the algebra Ce is one-dimensional. We claim that if B-' = B\{O} (i.e. if B is a Banach field) then every one of its elements is of type ze for some z E C. Indeed, since a(f)# 0 for each f E B there exists at least one number z , which belongs t o o(f)c C ,i.e. (z,e - f ) $ B-' = B \ (0). Hence z,e - f = 0 and consequently f = z,e, as claimed.

1.1.9. PROPOSITION. Let f be a fixed element in B. Then w E o(f") if and only ifw = A" for some X E o(f), i.e. a ( f n )=

(4f>I"

*

PROOF.Let XI,. . .,A, be the n-th roots of a fixed complex number w, i.e. let fi = ( A 1 ,...)An). Then A? = w, zn w = (z - XI) (z - A,), f" - w = (f - Xl).--(f - A,), and consequently f" - w is non-invertible if and only if among {f X j } y = l there is some non-invertible element, i.e. 20 E a ( f n )if and only if X j E a(f)for some A j with A? = w, j = 1,.. . ,n.

1.1.10. DEFINITION. The spectral Tadius of an element f E B is said to be the number T,

= max { 121 : z E a(f)}.

11

1.1. Spectrum of an alaebra element

vm]

converges 1.1.11. PROPOSITION. The sequence { for every f E B and tends to the spectral raduis r f off, i.e.

PROOF.Let f E B and X E a(f). By Proposition 1.1.9 we have that An E a(fn)and the previous remark shows that IXnl < r f < Ilfnll, wherefrom 1x1 = = 2 for lim and whence each n = 1,2,. . . Consequently,

vm,

n

vm

the resolvent function r(z) of f possesses the For 1x1 > following convergent series expansion:

Denote z = 1 / X and consider the function g(z) = (l/z)r(l/z) = h ( X ) = X(X

n=O

,.

n=O

,.

- f)-'

n=O

1

Clearly, the last series is convergent for Izl = -

1

( I e + S n * s n c < 2i 4 - 2~ for n big enough,

i.e. p i tend uniformly on

S1to f . The theorem is proved.

1.4.5.THEOREM (Wermer's maximality theorem). Let A be a closed subalgebra of C(S1) which contains the space P(S1). Then either A = P(S1)or A = C(S1). PROOF[Cohen]. Suppose that there exists an f ( z ) E A \ P(S1).Then by the Fejer's theorem (1.4.4)we have that cLn # 0 for some n > 0. Without loss of generality (by replacing, if necessary, f ( z ) with zn-' f(z)/(cf,)) we can assume from the = 1. Hence c;f = cf - 1 and thus co If-1 beginning that ctf - 1 = 0. By the Weierstrass-Stone's theorem the function

.Il

z f ( z ) - 1 can be approximated by functions Qk(Z) =

k

j=-k

cjzj.

Since c;f-l = 0, without loss of generality, we can assume that = 0 for all q k , i.e. that

czk

k

j=-k

k

k

k

E

j=l

j=1

j=l

j=1

i#O

Hence there are polynomials g ( z ) and h ( z ) in z such that

-

+

z f ( z ) - 1= ~ g ( z ) zh(z)

+ k(z), i.e.

ChaDter I. Uniform Algebras

36

z f ( z )= 1

+zg ( z ) + z q z >+ +),

where k(z) E C(S1) and lik(z)ll = maxlk(z)I ZES'

< 1/2. Consider

the function F ( z ) = f ( z ) - g ( z ) - h ( z ) . Clearly, F ( z ) 6 - A and also z F ( z ) - l = Z f ( z ) - z g ( z ) - z h ( z ) - l = z g ( z ) - z g ( z ) + k ( z ) E A. Since a l l numbers of type zg( z ) - zg( z ) axe purely imaginary, we have that

for every r e d t. Hence

+

+

11 t - t z F ( z ) l = 11 t ( z g ( 2 ) - z g ( 2 ) ) - tk(Z)I

+

= 1 t ( t l z g ( 2 ) - zg(z)I2

+

Thus 111 t - tzF(z)ll

+ 5). 1

< 1 + t , i.e. 111 - t z F ( z ) / ( l + t)ll < 1 for

t small enough. Hence t z F ( z ) / ( l

+ t ) E A-l

by Corollary 1.1.1, wherefrom z E A-'. It follows that the function z is invertible in A , i.e. z-" E A for every integer n. By the Weierstrass-Stone's theorem it follows that A = C ( S 1 ) .The theorem is proved. More general versions of Wermer's maximality theorem will be given in Chapters I1 and IV. 1.4.6, DEFINITION. A uniform algebra A on X is m a z i m a l o n X if there are no other algebras between A and C ( X ) except A and C ( X ) . A is a m a x i m a l algebra if its restriction Al,, on the Shilov boundary a A is maximal.

1.4. Some examples

37

The disc algebra A(A) is a maximal 1.4.7. COROLLARY. algebra on

S1.

PROOF.What we shall prove actually is that A(A)lsl = P(S1). The assertion will follow then directly from Theorem 1.4.5. Let f(z) E A ( A ) . By Cauchy theorem for analytic functions

for every k

flsl

5 -1. Hence by Fejer's theorem (1.4.4)we have that

E P(S1). Conversely, as we know, the Gelfand extension ?is analytic on A for each f E P(S1)and therefore P(S') C

44Is1

*

Observe that as we saw above, s p A ( A ) = spP(S') = because of A( A) G A( A) sl = P(S1).

I

2,

According to Corollary 1.3.10, given n analytic functions fl, . . . ,fn in A which are continuous up to the boundary and withthere exist n analytic in A functions 91,. . . , out joint zeros in

z,

n

g, continuous up to the boundary, such that c f j g j

1 on A.

j=l

For n = 1 this is obvious, but the corresponding result for n is not easy verifyable.

>1

1.4.8. EXAMPLE. The algebras P ( E ) and R(E). Let E be a compact subset in C". Let P ( E ) be the closure in C(E)of all polynomials in C" and let R(E)be the closure in C(E)of all bounded on E rational functions in C". P(E) and R ( E ) are uniform algebras on E with respect to the pointwise operations and the uniform norm llfll = maxIf(z)I. The eEE

Chapter I . Uniform Algebras

38

h

spectrum of P(E)is the polynomial convex hull E of E , namely h

E = { z E C" : lp(z)I 5 max Ip(z)l, all polynomials p in .E E

c");

the spectrum of R ( E ) is the rational convex hull r ( E ) of E , namely,

bounded on E rational functions in C " } . Note that the polynomial convex hull of a planar set E C C is the union of E and all its "holes", i.e. bounded components of its complement. The rational convex hull of every planar set E E C coincides with E . For n > 1 both hulls have a more complicated nature, still without a satisfactory description.

EXAMPLE 1.4.9. The algebra A ( R ) . Let 0 be a bounded domain in C". Denote by O(z) the set of all continuous on 3functions which are analytic on some open neighborhood of D.We say that 3 is O ( z ) - c o n v e s if the set { z E C" : \ h ( z ) /5 max lh(z)I for all functions in O(3)) coincides with -

z€3

R. The spectrum of algebra A(S2) of all continuous on 3 and

holomorphic in 0 functions is 3. In particular since - all bounded planar domains D are O(D)-convex, s p A ( D ) = D for any D C C . If the complement of D c C is connected in C , then the famous Mergelyan's theorem asserts that A ( D ) = P(B) = R@). 1.4.10. EXAMPLE. The algebra A K , K c S P A .

Let K be a compact subset of the spectrum spA of a uniform algebra A. Consider the algebra XIK of restrictions of all functions in A^ on K . In general this is not a closed subalgebra of C ( K ) and therefore AIK is not always a uniform algebra. The CI

1.4. Some ezamvles

39

h

closure of AIK in C ( K ) ,which always is a uniform algebra, will be denoted by AK. A-convex hull of K is the set

h ( K ) = { m E spA : If(m)l 5 max If(rn)l for all f E A}. mEK

It consists of all linear multiplicative functionals on A which possess continuous extensions on AK; and therefore, S ~ A K= h(K). K is A-convex if and only if h ( K ) = K , i.e. if and only if

K = ( m E s p A : If(rn)J I max If(m)l for all f E A}. mEK

Simple examples for A-convex sets are the sets of type

{ m E spA : Ifj(rn)l I c,

fj

E a , j = 1,. . . ,n,c

> 0).

In particular the vanishing set V ( S )of any subset S of functions in A is an A-convex set.

1.4.11.EXAMPLE. The algebra H". Let H m be the space of all bounded analytic functions on the open unit disc A. Equipped by the pointwise operations and the sup-norm llfll = SUP If(z>l, %€A

H m is a commutative Banach algebra with identity. The spectrum of H m is a complicated space which still has no satisfactory description. Some linear multiplicative functionals of Hm are obvious, namely the evaluations cp,(f) = f(z) at points z in the open unit disc. There are also others. For instance let I be the set of functions f in H m such that f(z) tends to zero as z approaches 1 along the positive axis. It is clear that I is a proper ideal in Hm; and therefore, I is contained in some maximal ideal, i.e. there is a linear multiplicative functional cp of H m such that y ( f ) = 0 for all f E I. But cp is not a functional of type (pz since there is no z E 2 at which every f in I vanishes. Moreover,

ChaDter I. Uniform Algebras

40

there are a lot of linear homomorphisms of H" which are not of type "point evaluations". The Gelfand transformation A : Hoo -+ : f H f is = f(z) = 0 for each one-to-one. Indeed, i f f = 0, then f(',) z E A. Furthermore, A is an isometry, i.e. llfll = = Indeed, as we know, If(V)I I llfll and,

a"

H"

V ~ S P

h

llfll

Im.

sp

bra H" is isometrically isomorphic to algebra on s p Ha,.

Hm

g", which is a uniform

-

Denote by a the Gelfand transform of the identity mapping z in A, i.e. a ( y ) = y(id), cp E s p H m . id: z

1.4.12.THEOREM. T is a continuous mappingofspH" onto the closed unit disc in the plane; a is one-to-one on the open unit disc A and 7r-l maps A homeomorphically onto an open subset of s p H" .

a

PROOF.By definition a is a continuous map from s p H" into the closed unit disc Because of a ( y Z ) = V,(id) = z , each point z in the open unit disc is in the range of a . Since s p H" is compact, so is the range of a. Therefore, the last range is the entire closed unit disc. Suppose lzol < 1 and a(y) = z., If f

z.

vanishes at z, then f(z) = ( z - z,)g and

4 . f ) = $ 4 2 - zo)cp(g) = 0 - $ 4 7 )

= 0.

Since cp(f) = 0 for every f E H" which vanishes at zo, cp coincides with the evaluation at z, E A. This shows that 7r is one-to-one on ..-'(A). As an one-to-one mapping on a locally compact space into a HausdorfF space a is a homeomorphic correspondence between a-'(A) and A. It is convenient to imagine 7r as a projection of s p H m onto the closed unit disc 2.As we saw above, a maps a-'(A) homeomorphically onto A.

1.4. Some examvles

41

The remainder of the set of linear multiplicative functionals in spH" is mapped via 7r onto the unit circle. The famous Carleson corona theorem says that the set sp H" \ .-'(A) has an empty interior in sp H" , i.e. that .--'(A) is dense in sp H" . However the set s p H" \ 7r-' ( A )has a very complicated nature.

1.4.13. EXAMPLE. The algebra e l . The space l1 of all summable two-sided infinite sequences 00

a=

of complex numbers (i.e. for which

{an}:=

C


5 e(o,ba(f)) mEE EWf) for any f E B ; (3) e(o,f@)) I e ( o , W ) ) for each f E B; (4) The open disc 411 1 .1(min Ilf^(m)II) lies either entirely inmEE side or entirely outside the spectrum a(f)of every element f E B; ( 5 ) min Ilf(m>ll 5 min ~lf^((m>ll of any invertible element

fi~)

mEE

mEsp B

f E B; (6) The Gelfand transform of each element f in B, vanishes within E ; ( 7 ) The disc All.11(min Ilf(m)ll) is contained entirely in the mEE

spectrum a(f) of any non-invertible element f E B . Theorem 1.5.11(5) especially says that the Shilov boundary a B is the smallest among all closed subsets of sp B on which the function Ilf^(m)ll attains its minimum for any invertible element f i n B , where llzll = max { IRezl, IImzl), either llzll = (IRezlPf lImzlp)l’p, p

> 0,etc. attain its minimum on E.

Theorem 1.5.11 implies corresponding characterizations for the points in the Shilov boundary. Namely,

1.5.12. COROLLARY (Local characterization for the points of a B ) . A point m, in spB belongs to the Shilov boundary

54

Chavter I. Uniform Algebras

dB of a commutative Banach algebra B if and only if for each neighborhood U of m, in s p B there exists a function f in B which satisfies one of the following equivalent conditions:

PROOF.The case (1)follows immediately from the definition of the Shilov boundary. Let U be a fixed neighborhood of the point m, in dB. If we suppose that for some f E B-' (2) fails to be fulfilled, by Theorem 1.5.11(5) we get that dB \ U is a boundary of B in contradiction with the choice of m, E dB.

Conversely, let U be a neighborhood of m, such that (2) holds for some f E B-l. Clearly, the subset of s p B on which IlT(m)ll attains its minimum is contained entirely in U. Therefore, U necessarily meets dB and hence m, lies in dB because every one of its neighborhoods contains points of d B . The case (3) can be checked in a similar way.

1.5.13. COROLLARY. The Shilov boundary dB is the smdlest closed subset of sp B which meets every nonempty subset U of s p B of the following type:

(14)

u = { mE S P A

: ~lfi(m)ll< E ,

f E B - ~ E, > o } .

PROOF.Assume that a set U of type (14) does not meet dB. Then Ilf(m)ll 2 E on dB; and therefore, the function Ilf(rn)ll does not attain its minimum on d(n)Bdespite of the invertability o f f , which in the view of Theorem 1.5.11(5) is absurd. If a closed set N is contained properly in aB, then according to Corollary 1.5.10 there exists an f E B-' for which

1.5. Shilov boundary

Hence the set {m E spB : Ilf^m)ll

55

< min Ilf(m)ll) is nonempty, mEN

of type (14)and does not meet N . The corollary is proved. If A is a uniform algebra and K is a compact subset of S P A , then ~ A cKb(spAK)whenever S P A Kdoes not meet dB, which is an immediate corollary of the following

1.5.14. THEOREM (Rossi’s local maximum modulus principle). If U is an open subset of s p A then maxlf(m)l = mE bUu( aA n U)If(m)l mEU

for every function f E A. We can identify easily certain points in the Shilov boundary dA of a uniform algebra A.

1.5.15. DEFINITION. A point x, E X is a peak point of a uniform algebra A if there exists a function f in A such that f (xo)= 1 and If(x)l < 1 for each x E spA \ {xo}. Clearly, each peak point belongs to the Shilov boundary dA. In general the set 6A of all peak points is not a boundary for A. However, for algebras with metrizable spectra it is. Moreover, in this case SA is the minimal (possibly nonclosed) boundary for A, i.e. SA is contained in every boundary of A. Therefore, SA = dA, i.e. the closure of the set of peak points coincides with the smallest closed boundary for A , i.e. with its Shilov boundary.

1.5.16. DEFINITION. A uniform algebra A is a Dirichlet algebra if the space Re(AlaA)is uniformly dense in &(dA), i.e. if every real continuous function on the Shilov boundary d A can be approximated on dA by real parts of algebra functions.

56

ChaDter I. Uniform Algebras

For instance A(A) is a Dirichlet algebra. Indeed, ReA(A) consists of all real continuous functions on A which are harmonic on A and the harmonic conjugates of which are extendable continuously on S1.Consequently, FkA(A) contains all continuously differentaiable functions on S1 and these are dense in CR(S1). 1.5.17. DEFINITION.Let A be a d o r m dgebra on X and let cp E s p A. A representing measure for cp on X is any positive Borel measure dp on X such that the equality

holds for every f E A. Let cp be a point in s p A. Every representing measure dp of cp on X satisfies the equalities

Observe that the set Mv of all representing measures of cp is isomorphic to the set of all norm-preserving extensions of cp E s p A from A c C ( X ) onto C ( X ) . In fact, each dp E Mv obviously determines a norm-preserving extension of cp on C ( X ) , namely by

1

f d p . Conversely, if

$J is a norm-preserving extension of cp

X

on C ( X ) , then $(f) =

f du for some positive Borel measure

1.5. Shilov boundary

where d.1 l

57

is the absolute variation of dv, defined as follows:

where Ej runs over all Bore1 subsets of X } . Note that by Hahn-Banach theorem the sets Mv, cp E s p B , are nonempty. If A is a Dirichlet algebra, then every cp E s p A possesses a unique representing measure on dA, i.e. all MV, cp E s p A , are single-point sets. If not, then the difference of every two representing measures of cp will vanish on A, hence on &A, hence on CR(X); and therefore, it will be identically zero. 1.5.18. NOTESAND REMARKS. The Shilov theorem for algebras of functions was proved originally by Shilov in 1940 by making use of a transfinite induction (see e.g. [22]). Other proofs have been given by Bear [14] and Hormander (e.g. [54])among others. The proof presented here is the Hormander’s one. It does not use the axiom of choice. Shilov boundary is the one-dimensional case of a more general notion, namely of the n-tuple Shilov boundaries (see Chapter 111). The proof of Proposition 1.5.6 presented here is due to Bear and Hile [15]. Corollaries 1.5.8, 1.5.12, 1.5.13 and Theorem 1.5.11 have been proved originally for multi-tuple Shilov boundaries by Tonev (e.g. [126]). Some of the assertions of Theorem 1.5.11 are among well known properties of the Shilov boundary. Thus the case 1.5.11(1) is exactly the property (4). An equivalent version to the case 1.5.11(7) can be found in Rickart’s book [82]. The case (1) of Corollary 1.5.12 is well known (e.g. [32]). For the proof of Rossi’s local maximum modulus principle we refer the reader to the books by Gamelin [32] and Wermer [143].

CHAPTER I1

I'-ANALYTIC FUNCTIONS IN THE BIG-PLANE

We begin this chapter with a review of analytic almost periodic functions in domains of the complex plane C. A quite natural approach of investigation of these functions is to include them in appropriate uniform algebras. It turns out that natural carrier spaces of such algebras are subsets of the so called big-plane and this yields the involvment in the subject of so called r-analytic jbnctions or in an other terminology, generalized-analytic functaons in the big-plane. r-analytic functions in the big-plane came to help uniform algebra theory at a time when it was realized that, for some purposes, the might of classical methods of analytic functions theory (and for analytic almost periodic functions theory in particular) was somehow exhausted. r-analytic functions in the big-plane in fact are Gelfand transforms of certain bounded analytic almost periodic functions in C. This important connection makes possible most of results for r-analytic functions in the big-plane to associate to corresponding results for almost periodic functions and vice versa. On the one hand this connection yields the theory of almost periodic functions to become a part of the theory of F-analytic functions in the big-plane and unexpected properties of analytic almost periodic functions in C to be discovered. On the other hand this helps the study of r-analytic functions and related algebras on certain subsets of the big-plane. As we shall see below, analytic functions in the big-plane inherit many properties of analytic functions in one or several complex variables.

59

2.1. Generalized-analvtic functions

2 .l. GENERALIZED-ANALYTIC FUNCTIONS

A continuous function f on the real line R is called almost periodic if it can be approximated uniformly on R by exponential n

polynomials of type

Cakeisrt,

where

Sk

are real numbers. If

k=l

r

all exponents s k belong to a fixed additive subgroup of R, then f is called r-almost-periodic f u n c t i o n on R. An analytic in a domain D c C function f is said to be analytic r - a l m o s t periodic in D if it can be approximated locally uniformly in D n

by exponential polynomials of type

akeiskz,

where S k belong

k=l

to the semigroup To.A continuous function f in D C C is I'almost-periodic in D if it can be approximated locally uniformly m

n

in

D by functions of type

r,

C a k e i S k z k=O

+X

a l F , where

Sk,tl

1=1

belong to S L 2 0, t i < 0. Let H = { z : Im z > 0) be the upper half-plane. Suppose that is an additive subgroup of the real line R and let v be a fixed nonnegative number. An analytic in H function f is said to be analytic I',,-almost-periodic if it can be approximated uniformly on F? by exponential polynomials of type

r

n k=l

r,

r)

where S k belong to the semigroup = ([v,00) fl U (0). Denote by A,(H) the set of all analytic I',-almost-periodic funcand which possess finite tions in H which are continuous on boundaries as Imz + 00. Under pointwise operations and supnorm llfll = sup If(z)l the set A,(H) is a commutative Banach tEH

algebra with unit. Obviously, the point evaluations at the points

Chapter 11. r-Analytic Functions in the Big-Plane

60

of Tf are linear multiplicative functionals of A,(H). However, there are also other linear multiplicative functionals. The spectrum of the algebra A,(H) is a very complicated space. It will be described below and in the next section. Note that similarly to the case of algebra H-,the Gelfand transformation A : f -+i f is one-to-one and isometric. Consequently, A,(H) is isometrically isomorphic to the space &(H) which is a uniform algebra on spA,(H). Observe that if F is the group Z of integers, then all functions in A,(H) are in fact periodic. In this case the algebra A,(H) coincides with the standard disc algebra h

44.

Let f(sj be a T-almost-periodic function on R. Dirichlet coeficients of f(x) are said to be the numbers u f ( X ) = 7li- 2 3

T

"jf

(x)e(xzdx,

Y

where the limit on the right hand side exists independently from y E R. It turns out that this limit is nonzero for at most countable many A's, which are called Dirichlet exponents of f(x). The fact that Xk are Dirichlet exponents of f(z) and that A{ = u f ( X k ) are the Dirichlet coefficients of f ( E ) for any k = 1,2,. . . symbolically is expressed in the following way: 00

k=l

The series on the right hand side of (16) is called Dirichlet series of f(z). In the sequel we shall often make use of some of the following basic facts about almost periodic functions. The correspondence (16) between almost periodic functions and their Dirichlet series is injective; The red line can be embedded isomorphically and densely into the dual group G = r^

2.1. Generalized-analvtic functions

61

of I' equipped by the discrete topology; every I'-almost-periodic function f on R can be extended as a continuous function on G; Fourier coefficients c[ of the extended in this way function coincide with Dirichlet coefficients A{ = u f ( X k ) of f .

2.1.1. THEOREM. The space of analytic I',-almost-periodic functions in the upper half-plane IT whose Dirichlet coefficients A{ = u f ( ~ k >are zero for X A < o coincides with the class of all bounded analytic I',-almost-periodic functions in H. Each function in this class tends uniformly with respect to Rez to some constant as Im z --$ 00 (namely to the constant term of its Dirichlet series). The proof of this theorem is quite similar, though somehow more sophisticated, to the proof of Theorem 1.4.4.Like in Theorem 1.4.4one can see that the restriction algebra A(H)I, coincides with the algebra of continuous functions on R such that A{ = o for all X, E (-r0) \ (01. The algebra A,(H) is isometrically isomorphic to an algebra over the group G = I'. The relation among these two algebras is crucial for most of what follows. We need some facts about groups with ordered duals and related with them function classes. Let I' be a discrete Abelian group and G = I' be its dual group, i.e. the group of all characters (or, complex homomorphisms x of I' into S'). In other words, G is the collection of all functions x on I' such that x ( z y) = x(z)x(y>and 1x1 f 1. Since I' is discrete, G is a compact Abelian group. As a compact Abelian group G possesses a remarkable measure which is invariant under "motions" in G, the so called Haar measure do for which d o ( g ) = 1 and h

h

+

J

G

also the identity

62

Chapter II. r-Analytic Functions in the Big-Plane

holds for every f E C(G)and every h E G. Each a f gives rise to some continuous character x u of G, namely by x u ( g ) = g(a), g E G. Moreover, Pontryagin's duality t h e o r e m says that every continuous character of G is necessarily of the form x u for some a E T. The group T can be identified naturally with the dual group of G, namely by identifying a with Xu-

Finite linear combinations over C of characters on G form a separating self-conjugate subalgebra of C(G) which is dense in C(G) by the Weierstrass-Stone theorem. = T. The Let To be a semigroup in T for which ToU (-To) relation a + b if and only if a - b E To is an ordering in T ,i.e. it satisfies the following two conditions: (i) If a, b,c E T and if a + b, then a c + b + c ; (ii) If a E T ,then at least one of both relations a + 0, 0 + a holds. Conversely, if '' + " is an ordering in G, (i.e. if (i) and (ii) hold for +) and if Todenotes the set of all "non-negative" elements a in T (i.e. for which a + 0), then by (i) I", is a semigroup and by (ii) ToU (-To) = T. Consequently, there is a bijective correspondence between all orderings in a group T and = T. all semigroups Toin T with the property ToU (-To) If there is an ordering in T which is generated by To, we say that the semigroup Toorders T ,or that G is a group with ordered dual. We say that a semigroup To totally orders T if Toorders T and if in addition ron (-To) = (0). Clearly in this case the relations a + 0 and 0 F a imply that a = 0. Consequently, given a, b E r at least one of the relations a + b, b t a holds. We say that the ordering defined in G by I-', is A r c h i m e d e a n if for each a E Towith -a 4 I", and for each b E Tothere is an integer m > 0 such that m a + b, i.e. for which m a - b E To. This can happen if and only if the semigroup To/ (Ton( -I'o)) induces a total Archimedean ordering in the group T/(Ton( - T o ) ) ,

+

2.1. Generalized- analutic functions

63

i.e. if there is an order-preserving homomorphism of I' into R with kernel Ton (-To). Indeed, it is enough to show that if Ton (-To) = ( 0 ) and if To induces an Archimedean ordering in then there is an order-preserving isomorphism of T onto an additive subgroup of R. To prove this suppose that To induces an Archimedean ordering in T,fix an a, E F and for every a E To denote by E ( a ) the set of all rational numbers rn/n ( m , n > 0) such that nu ma,. If we define cp(u) as the least upper bound of E ( a ) , and cp(-u) = -cp(u), then it is not hard to check that cp is an isomorphism of I' into R and that cp is order-preserving.

r

+

Suppose that 7, orders T and denote by AG the set of all continuous functions on G whose Fourier coefficients

G

are zero whenever a E -F, \ (0). The functions in A are called generalized-analytic f u n c t i o n s on G in the sense of Arens-Singer. A great deal of the classical theory of analytic functions in C is due to the maximality property of the disc algebra A ( A ) (Theorem 1.4.5). The next theorem says that the algebra AG of generalized-analytic functions has the same feature, i.e. is maximal only if the group F = 6 can be equipped by a total Archimedean ordering by some semigroup Toc F , i.e. if only T is a subgroup of the real line R.

2.1.2. THEOREM. If To totally orders T,then AG is a maximal algebra on G if and only if the ordering in G generated by r, is Archmedean.

PROOF.Suppose first that the ordering in T generated by To is total and Archmedean. Let B be a closed subalgebra of C(G)such that AG c B c C(G) and let B # C(G). Since B 2 AG we observe that xa E B for all a E r,. Because of

Chapter 11. r - A n a h t i c hnctions in the Bin-Plane

64

B # C(G)there exists an s E I'o \ {0} such that x - ~4 B. Thus xs has no inverse element in B and it follows that cpo(xs) = 0 for some linear multiplicative functional cpo E s p B . For an a E rothere is an integer m > 0 such that ma + s and if b = ma - s we see that X b E B and c p o ( ~ a ) m = yo(xna) = Vo(Xs6) = yo(Xsxb) = cpo(Xs)vo(xa) = 0. Therefore, cpo(xa)= 0 for every a E ro\ (0). Each representing measure dp of y o vanishes identically on the set {xa : a E ro\ ( 0 ) ) and, at the

{ xa : a E -ro\ {0}}, i.e.

same time, on the set

foralla E F\{O}. Inaddition/X.(g)dp=

Xa(g) dp = 0

J1dp=cpo(l) = 1 G

G

which indicates that

J

G

/ xa dp = / x a do for each xa E G, where h

J

J

G

G

da is the Haar measure on G. By the Weierstrass-Stone theorem

/f

dp =

/ f do

J

J

G

G

for every f E C(G)and thus dp = do since

by the Riesz representation theorem the correspondence between positive Bore1 measures on G and linear positive functionals on

C(G)is one-to-one. Hence c p o ( f ) =

J

f da for every f E B. If

G

f E B and a E ro\ {0}, then f x a E B and

J f(dxa(9) do =

(Po(fXa)

= cpo(f)cpo(Xa) = 0.

G

Hence

cLa =

a E -To

Jf(g)T-.(g)do G

= /f(g)x.(g)da

= 0 for all

G

\ {0}, i.e.

B = AG. Suppose now that the ordering in G is not Archimedean. Let xal, x a z ( a l , a z E r0)be two characters of G such that a1 4 Po and m a 1 - a2 4 To for all m >_ 1. Let B be the uniform algebra generated by AG and Clearly B # AG. We claim that all

xal.

2.1. G e n e r a l i z e d - a n a l v t i c f u n c t i o n s

65

characters { x u : a # 0) in AG are contained in a maximal ideal of AG. If not, then there are characters x a l , .. . , x a , E n AG m

and functions f i , . . . ,fn E AG such that

C

fjxaj

= 1. Let

k be

j=1

the smallest (with respect to the To-ordering in G) amongst all numbers a j , j = 1,.. . ,m. We have

j=l

i.e. both k and -k belong to Toin contradiction with the identity To n (-To) = { 0) which holds because To totally orders T. If cpo is a linear multiplicative functional of AG such that Kercp, 3 { x a : a E To \ {0)}, then cpo(xU)= 0 for each a E To\{O). As in the first part of the proof we observe that cpo( f) =

J

f do for every f E AG. Since ma1 - a2 E T \ To,then

a2

-

f z z X a 2 da

=

G

ma1

E To \ (0) for all m 2 0. We obtain that

J

G

G

Hence

Jgxa2

da = O for each g E

B;and therefore, B # C(G).

G

Consequently, AG is not a maximal algebra on G. The theorem is proved. Theorem 2.1.2 motivates our further interest in big-disc algebras AG related with compact groups G whose duals T = 8 are subgroups of the real line R. If T is an additive subgroup of the discrete real line and if G = ?, then by Theorem 2.1.2 AG is a maximal algebra. Consequently, if a continuous function h on G does not belong to AG,

66

Chapter II. r-AnaJytic Functions in the Big-Plane

then every continuous function on G can be approximated unin

fk(g)hk(g), where fk E AG.

formly on G by functions of type k=l

Note that by replacing xa by elaz, the integral with respect to the Haar measure do by the integral from (15) and by using the above arguments one can prove that the algebra A,(H) from the beginning of this section is also a maximal algebra on R. In the sequel, unless especially required, the I',-ordering in G will be supposed to be total and Archimedean. In other words we will assume that r is (isomorphic to) some additive subgroup of real numbers which is equipped with discrete topology. We will assume also that in the case when F # Z the group I' is dense in R. A

Let s be a real number. Define the character qs E F = G in the following way:

The induced mapping j , : s H q s is an isomorphism of the real n

line onto a dense subgroup of G (see e.g. [53]).I f f =

C cjxaj is j=1

a finite linear combination of characters on F , then the function n

f ( j e ( s ) )= f ( q , ) = ~ c j e z a j 3which , obviously belongs to the j=l

algebra A,(H), can be extended analytically on the upper halfplane H as n

n

If f E A G , then a j 2 0 for all j and the extension f i s bounded. From the discussion on the algebra A,(H) from the beginning

2.1. Generalized-anahtic

functions

67

FE

of this section we see that Ao(H)IR. The function ?can be expressed by the integral formula 00

-00

which holds for each f E AG. More generally, if g E G and n

if

f

=

Ccjxaj then the function f(jg(.s))= f ( g

je(s)) =

j=1 n

f(g

qs) =

*

C c j X a j ( g ) e i a j s belongs to the algebra A,,(H)I, j=1

and can be extended analytically on the upper half-plane H by setting

1 tds ,O 0, ~ " ( gr ) 1, where g E T 2 (i.e. to the big-disc algebra A(Ap,(za)) of p(Z2)-analytic functions on ApP(z2)). Observe that if the restriction on T 2 of a function in A&

+

-

a2,

possesses an analytic continuation in the unit bidisc then this continuation in general does not coincide with the original function. The mapping p : Z2 4 R maps the set of all generating A& monomials of type unvm homeomorphically into the additive group of real numbers. It is easy to see that instead of & we can take any positive irrational number a in the above example in order to obtain, in the same way, a corresponding algebra A, isometric to some big-disc algebra. The algebra A ( A 2 ) of continuous functions on T2 which possess holomorphic continuations on the bidisc in fact coincides with the intersection of all algebras of type A,, a > 0.

IT2

a2

r

Note that if a subgroup is not dense in R then the algebra A ( A p r ) is isomorphic to the classical disc algebra A ( A ) . That is why in the sequel we will assume that F is dense in R (and thus is dense in Ro). By its definition the big-disc algebra A ( A G )is a uniform algebra. Because of A ( A G )= AG, we have that:

r

h

(i) s p A ( n ~=) &;

2.2. r-analvtic functions on the Lipdisc

75

(ii) aA(Ac) = G; (iii) A(AG)satisfies a local maximum principle, namely for every T-analytic function f(z) on & and for every compact neighborhood U C AG the inequality

holds for each zo E AG. (iv) Every T-analytic function which vanishes on a nonvoid open subset of AG vanishes identically on &; (v) Any real-valued T-analytic function is constant; (vi) A(AG)is a Dirichlet algebra; (vii) A(AG)is a maximal algebra. Indeed, properties (i), (vi) and (vii) follow directly from the corresponding properties of the algebra AG stated in Section 2.1; (iv) and (v) are direct consequences from the fact that the functions f(j,(H)) are analytic in H for each f E A(AG)and any g E G; (iii) follows directly from Rossi’s maximum modulus principle (Theorem 1.5.14)for the big-disc algebra. From Section 2.1 and because of A(&) = & f AG we get that ~ A ( A GC)G. In Section 2.5 we shall prove (ii) (and also all the results from above) for more general domains in C G . r-analytic functions possess many properties of analytic functions of one and many complex variables. Below we state some of them. Let f be a fixed r-analytic function on the big-disc AG,where G = and let Pf = {{P,},a E Ep}pEl be the set of all possible sequences of polynomials Pa in C G , where a E Cp, which tend uniformly on &. For a given sequence {P,}aE~B define a , = lim a,, where a , is the smallest index for which xu aECp

takes part in

Pa. Let a f

= supap. The next theorem extends BEI

the classical Schwarz lemma for T-analytic functions.

76

Chapter 11. r-Analytic finctions in the Bin-Plane

2.2.5. THEOREM. Let f be a fixed r-analytic function on the big disc &, where G = such that max lf(g)l 5 1 and

/3r,

f(0)= 0. I f a f > 0 then lf(z)I

OEG

5 1zI"f for every z E &.

PROOF.Because of f(0) = 0 we can consider without loss of 1 is not involved in the set generality that the function xo Pf.If a E To is such that 0 < a < af,then we can assume also that all indeces b for which xb is included in some polynomial in Pj are not smaller than a. Since the function f(z)/xa(z) is r-analytic on & we have that

-

wherefrom JF(r g)1 I 121" because G is a boundary for A(&). We conclude that If(z)l I IzI"f since the last inequality holds for every a < a f . Q.E.D. The following result extends for the case of r-analytic functions a particular case of Glicksberg's uniform algebra version of Rouchk's classical theorem.

2.2.6. PROPOSITION. Let f = (x")"cp and h = (x")"$ be two r-analytic functions on &, G = /3r,such that cp and 1c, do not vanish on &. If the inequality h( < if1 + Ihj holds on G, then m = n.

If +

PROOF.Suppose that m 1 n. Then on G we have:

According to the Glicksberg's result mentioned above, (x")"-"cp and $ are simultaneously either invertible, or non-invertible. But this is possible only if m = n.

2.2. r-analutic functions on the baa-disc

77

As we saw in Section 2.1, the big-disc algebra A ( A G )= 2~ is tightly connected with certain almost periodic functions on the upper half-plane H = { z E C : Imz 2 0). Denote by j e the standard embedding of the upper half-plane H in the big-disc & with dense range (and thus with je(R) dense in G as well) which contains the unit element e E G. Recall that for every r-analytic function f the composition f o j c is analytic in H. Moreover, f o j e is bounded and analytic r-almost-periodic function in H and its restriction on R is approximable by linear combinations of functions x'(je(t)) = eidt, s E F,. In other words, the restriction of f o j e on R is a r-almost-periodic function whose Dirichlet coefficients coincide with the corresponding Fourier coefficients of the function f E A ( A G ) . RRcall that the embedding j e is defined in the following way:

r

r.

where qz is the character of defined as qz(a) = eiza, u E Obviously, e = je(0). If f is a polynomial in C G , say, f(z) = ckxak(z), where uk E ro, E &, then for the corresponding analytic F-almost-periodic function f o j e we have f o je(z) = x c k e i a r z . If g ) - f(0)l< E for some E > 0, where r , > 0, 0 5 r < r,, then o je(r g ) - f(0)J < E for each y > yo = - In r,, i.e. f o je is a bounded function in H. The embeddings j , = je g of H into & possess dense ranges in & as well and pass through the points g E G correspondingly, i.e. j,(O) = j e ( O ) g = eg = 9.

.(fI -

If

-

-

The next theorem extends for the case of r-analytic functions classical Rudin-Carleson theorem about interpolation sets on the unit circle for analytic functions in one complex variable.

2.2.7. THEOREM. Let E be a closed subset of G and let the sets j,'(E n j,(R)), g E G, have zero Lebesgue measures for

Chapter II. T-Analytic Functions in the Bin-Plane

78

every g E G. Then every continuous on E function f can be extended as a r-analytic function on the big-disc & in such a way that If(z)l = max If(s)l*

7

%€do

SEE

PROOF.Let L b be a measure on G which is orthogonal to the big-disc algebra A ( A G ) ,i.e.

J

f ( g ) d v ( g ) = 0 for any r-analytic

G

function f f A(Ac). According to de Leeuw-Glicksberg's characterization of subsets of G with zero measure, d.1 l = 0 (and therefore dvlE = 0), because the Lebesgue measure of the set jil(Enjg(R))is zero for any g E G. Hence A ( A G )is a Dirichlet algebra and d v l E = 0 for every orthogonal to A ( A ) measure d v , because E is a closed subset of G. Therefore, as shown by Glicksberg [38], C ( E )= A(AG)IE.Q.E.D.

2.2.8. NOTES AND REMARKS. De Leeuw-Glicksberg's characterization of the subsets of G with zero Lebesgue measures can be found e.g. in [31]. See also [33]. An other proof of the result has been given by Mandrekar and Nadkarni "751. Theorem 2.2.5, Proposition 2.2.6 are from Tonev [113], Theorem 2.2.7 is from Tonev 11191. A general form of Rudin-Carleson's interpolation theorem for algebras for which F. and M. Ftiesz theorem holds was given by Bishop [17]. Other properties of r-analytic functions were discovered by Grigoryan P31, [441, [46l.

2.3. THEBIG-DISCALGEBRA In this section we will concentrate on the big-disc algebra

A ( A c ) for which we give various characterizations useful for the sequel.

2.3. The bia-disc algebra

79

The maximality property of A(AG)implies the following

2.3.1. PROPOSITION. The big-disc algebra A(AG)is isometrically isomorphic to the algebra B of continuous functions f on the closed unit big-disc & whose compositions f o j , with the embeddings j g , g E G, axe analytic functions in the upper half-plane H.

PROOF.B is a uniform algebra which contains the functions xa, a E F,,; and consequently, A(AG)C BG c C(G).Because of BIG # C(G)we conclude that BG = A(AG)IGby the maximality of the big-disc algebra.

2.3.2. PROPOSITION. The algebra A(AG)IGof restrictions of the big-disc algebra on G is isometrically isomorphic to the algebra of all r-almost-periodic functions on R which possess bounded analytic continuations in H, provided with the supnorm.

Clearly, the restrictions of compositions f o j, of F-analytic functions with the embedding j , on R are F-almost-periodic functions on R which possess bounded analytic continuations in H. Conversely, the algebra Br of F-almost-periodic functions on R which possess bounded analytic continuations in H is isometrically isomorphic to to the algebra grlGof continuous functions and the latter algebra obviously differs on the group G = from C(G).Since grlG is a uniform algebra which is situated between A(AG)IGand C(G), we conclude that Br S A(AG)IG as claimed.

pr,

In the same way we obtain

80

Chapter II. r-Analytic h c t i o n s in the Bin-Plane

2.3.3. PROPOSITION. The big-disc algebra A ( A c ) is isometrically isomorphic to the algebra of analytic r-almost-periodic functions on the upper half-plane H. By making use of Proposition 2.3.3 and the correspondence between Fourier coefficients of T-analytic functions f and Dirichlet coefficients of compositions f o j , , discussed in Section 2.1, we obtain that the big-disc algebra is isometrically isomorphic to the algebra of analytic T-almost-periodic functions in H whose negative indexed Dirichlet coefficients are zero.

The big-disc algebra characterizations from above imply the following results about F-almost-periodic functions.

2.3.4. COROLLARY. The algebra of analytic r-almost-periodic functions in H whose negative indexed Dirichlet coefficients are zero coincides with the algebra of bounded analytic F-almostperiodic functions on H, i.e. with the algebra of r-almostperiodic functions. The functions in either of these algebras tend uniformly with respect to Re z to some h i t as h z + +CQ. These limits coincide with the corresponding zero indexed Dirichlet coefficients.

As another immediate consequence from the above characterizations and the maximality of the big-disc algebra, we obtain the following approximation result.

2.3.5. COROLLARY. Let f be a r-almost-periodic function on R which does not possess a bounded analytic continuation in H. Every I'-almost-periodic function on R can be approximated m

uniformly on R by polynomials of type

gi f i= 1

where 9, are

bounded analytic r-almost-periodic functions on H.

2.3. T h e baa-disc algebra

81

The result is clear because every T-almost-periodic function on R possesses a unique continuous extension on G.

r

More specific subgroups c R naturally generate more interesting, in properties, big-disc algebras A ( A c ) . The following theorem gives a characterization of Q-analytic functions, where Q is the group of rationals in R. In the sequel by 0 we will denote the group PQ = which is dual to the group Q d of discrete rationals. Note that aa a group of type PT,T c R,the group 0 is compact and connected.

2.3.6. THEOREM. The unit disc & is homeomorphic to an infinite-dimensionalsubset V of the infinite-dimensional polydisc -m A and the big-disc algebra A(&) of &-analytic functions on A s is isometrically isomorphic to the algebra P ( V ) . -W

PROOF.Consider the set V c A of sequences { ~ j }zj~E ~ C for which Z k n = zlm whenever n/k = m/l, where k,E,m,n are positive integers. Observe that the subset F of V which consists of sequences { z j } g 1 with lzll = 1 forms a compact -W group under coordinatewise multiplication in A . Because of { w j } y = {rl/j%}W, the set V admits the following descripr1/J 1 tion:

v = {{rl"zj}gl

: 1' = 1211, {~j}? E F}.

Observe that because of the hypothesis on z j , all zj are zero whenever z1 = 0. Let { z j } r be a fixed element in V \ ( 0 ) . The function g defined as g ( l / n )= z,/Iz~J'/~can be extended as a multiplicative function on Q in the following way:

g(m/n)= g m ( l / n )= Znm/lZ1

Im/ " .

By the hypothesis on zj, this extension is well defined. The following calculations:

ml+ kn

ml+kn

,

82

Chapter II. r-Analvtic h c t i o n s in the Big-Plane

show that g is multiplicative. Since Igl = 1, it is clear that g is a character of Q, i.e. g is a point in the dual group 0 = PQ. Note that different points in F correspond to different characters in PQ. Conversely, if go is a character of Q then clearly (go(l/n)}~=l E F c V . Consequently, there exists an isomorphism between groups F and 0,which can be extended on the set V (and onto &), by {r'ljzj} H( r 9). For every integer k we define

-

V , the class of Q-analytic functions on 36 coincides with the class of continuous functions on V , which can be approximated by linear combinations over C of nonnegative integer powers of coordinate functions z,, j = 1,2,. . . , i.e. by polynomials in --oo A . The theorem is proved. 2.3.7. NOTESAND REMARKS. Theorem 2.3.6 is from Tonev [119]. Tonev [113] has shown also that the big-disc algebra A ( A 6 ) coincides with the uniform closure in C(0)of an inductive limit of algebras of type Pa(@)GZ P(S'),and that A ( A e ) is the smallest uniform-algebraic extension of the classical disc algebra A ( A ) , which is closed under taking arbitrary Q-powers of its generators.

2.4. Boundaru behavior an the baa-disc

83

2.4. BOUNDARY BEHAVIOR IN THE BIG-DISC

Rememeber that a point in the unit circle S1is called a Futou point for a bounded analytic function in the open unit disc A in the complex plane C if the function under consideration possesses a radial limit at this point. It is well known that the set of Fatou points of a bounded analytic function in A is: (a) a BoreZ set of type Fm6, i.e. it can be presented as countable intersection of countable many unions of closed subsets of S1, and (b) of fiZZ measure in S1, i.e. the measure of its complement in S1is zero. As we shall see in this section, the notions of Fatou points and radial limits can be extended quite naturally for bounded T-analytic functions in the open unit big-disc AG over a group G whose dual group is ordered. It turns out that results similar to (a) and (b) from above hold for the set of points in the boundary of the big-disc which are Fatou points of bounded r-analytic functions in AG. By definition a point g E G is a Fatou point for f if the radial limit limf(r g ) of f at the point g exists.

-

r-1

2.4.1. PROPOSITION. Let G be a compact group whose dual group = 8 is an additive subgroup of the real line R and let f ( z ) be a continuous function on the open big-disc AG. The set of all Fatou points off is a Bore1 F,b-subset of G .

r

PROOF.The set E of all Fatou points of f can be described as

E

=

nu

O<e 0 0 and S > 0. Let { T ~ } : ? ~be an increasing sequence of positive numbers which tends to 1. The set

Ee=

u

Ec,6

cG

01 00

) 5 -1n (1 - t

= - ln(1- a t ) ,

n=l

1

w h e r e a = x n 2 " , Ol" = r;, as claimed. For every xa E I-', we have

r

X"(Zv>= xa (l'vg,) = C X a ( g , ) = I4x

l

+

>Ia x a (gv)

= Icp(xa)ls,(4= cp(X")*

Hence cp(p) = p ( z , ) for each F,-polynomial p(z) in CG.Consequent ly, z, E p , ( K ) and cp( f ) = 0 ) = cps, (f) by the continuity argument. We conclude that every linear multiplicative functional cp E spP,(K) can be identified with the point evaluation at some point of p , ( K ) ; and, moreover, the natural extensions f of all functions in P ( K ) coincide with their Gelfand extensions. The theorem is proved.

f(

h

r

Note that Theorem 2.5.2 holds for groups which are not dense in R as well. In these cases there is no necessity to extend the function g, from above on the whole line R. Observe that the big-discs &(R) are r,-polynomially convex subsets in the big-plane CG for each u E [0,1]. Consequently, the I-',-polynomial convex hull p , ( K ) of any subset K in &(R) which contains its topological boundary {R} x G coincides with the big-disc &(R) because of p , ( { R } x G) c p , ( K ) C &(R).

98

Chapter 11. r-Analytic finctions in the Bin-Plane

r'

In the sequel we will denote by the set (0,1] f l F . In many to situations in the future we will require the group F = possess the following property: (i) 2 c r

(*>

c Q, and

r'

(ii) for any a , b E there exists a c E r' and integers k , l , such that ck = a and cl = b.

Clearly, the group R of re& satisfies neither one of these conditions. The group Q of rationals, the group 2 of integers, the group of dyadic rationals are examples of groups F which possess property (*). If I' possesses property (*), then the neighborhoods of type

Q = &(Zo,€,a)

=

z, E

(2

E CG : (X"(2) - X"(%)I

cG,a E

rl,

< €,

> 0)

form a basis of the topology in C G . Indeed, the topology in CG is generated by neighborhoods of type

{z E CG : Ix"'(z) - X"J(z0)l < e, aj E r', E > 0, j = I,.. . ,n].

ZCJE

CG,

By (*) it follows that there exists a number c E P and positive integers k l , . . .,k , such that ckj = a j , j = 1,.. . ,n. It is clear that the neighborhood (a E CG : IxC(z)- xc(zo)l < S} is contained in the set

{z E CG : Ix"'(z) - x"'(zo)l < € 7 zo E CG, aj E P ,j = 1 ,..., n, E > O ) for some 6

numbers S j

> 0 because for every j > 0 such that

Ix"(4 - x"(zo)1

= 1 , . . . ,n, one c a n find

= )(Xc)k'(Z) - (x")"(zo)~ < E

whenever Ixc(z)- xc(z,)l < Sj.

2.5. Algebras o f r,-anahtic functions

r

99

h

2.5.3. PROPOSITION. Let the group = G possess property (*) and let v be a fixed number in [0,1].Each proper compact subset K of G is I',,-polynomially convex.

PROOF.Let zo be a fixed point in G \ K and let

Q = Q ( z o , & , a )= (1: E CG : Ixa(z)I:-.

<eo,

a d , eo>O}

be a fixed neighborhood of zo in C G which does not meet K. Clearly, x4(K)is a proper subset of S1.Moreover, xd(K)is a proper subset of S' for each d E l" such that drn = a for some positive integer m. Choose an arbitrary point z1 in CG \ K and let Q1 = {z E C G : Ix*(z) < e, b E I", e > 0 } C C G \ K be a neighborhood of z1 in CG which does not meet K . According to (*) there is a number c E and integers Ic and I such that clc = a , cl = b. By the above remark we have that xc(zl) 6 xC(K)C S1.Being a connected subset of S1,the set x c ( K )is polynomially convex in C ;and therefore, there exists a polynomial p ( z ) in C for which

z~I

r'

Hence the inequality

holds for the r-polynomial q = p o xc and thus z1 6 int p u ( K ) . We conclude that K = intp,(K), i.e. K is a I'-polynomially convex set in CG. Since the polynomial q o xu belongs to P,(K), Y > 0, and satisfies the inequality

IQ for every 21 ity

O

XU(Z1)I

> zy

Iq 0 x"(z)l

6 K whenever the polynomial q satisfies the inequalIq(zdl

> zy Iq(z)I,

we see that K = p , ( K ) as well. The proposition is proved.

Chapter 11. r-Analvtic Functions in the Big-Plane

100

A

2.5.4. PROPOSITION. Let the group I' = G possesses property (*) and let v be a fixed number in [0, I]. If I< is a r,polynomially convex compact set in the big-plane CG, then the Shilov boundary of the algebra P,(K) coincides with b K . PROOF. It is easy to verify that a P v ( K ) c b K . We claim that dP,(K) is not a proper subset of bh'. Indeed, if z, E b K , then for every neighborhood

Q

= {Z E

C G : 1x"(z) - Z z l < E , a E r',&

> o}

of z, in CG we can find a point, say z1 in V \ K. Hence there is a b E I" such that Ixb(z)- Xb(zl)l 2 E for any z E K . Let c E I" and let the numbers k, 1 E Z, be such that ck = a , cZ = b. Clearly, the function

attains its maximum within V . Because of xc(zl) 4 which follows from Xb(zl)4 xb(K), we see that

x"(K),

wherefrom z, E dP,(h'). The proposition is proved.

Let Q = Q(z,,&,a) = {z E C G : Ix"(z) - zE1 < E , z, E CG, a E (v,11 n r, E > 0 ) be a basic neighborhood in CG, which does not contain the origin 0 of CG. Then spP,(Q) = and aP,(G) = bQ for every v E [O, 11, as it follows from Theorem 2.5.2 and Proposition 2.5.4, since Q is I',-polynomially convex. 2 . 5 . 5 . PROPOSITION. If the group (*), then P(&) is a Dirichlet algebra.

T = i? possesses property

2.5. Algebras o f r,-analvtic functions

101

PROOF.Because of 0 $! Q we have that 0 4 A(x"(z,),e) = x"(Q), and therefore X b ( r . g) = (xa )b/a (r . g) in Q for any a , b E r,, a # 0. It follows that A(&) = [{f E C(Q) : f = h 0 xa for some h E A(xa(Q)))]. If f E C R ( ~ Q )then , IbQ = limfno(Xa)*n/aIbQ n = lim(jnozbn/a)oXalbQ n f = limfnoXbn n

for some bn E To and fn E C(xbn(Q)). Note that fn o zbnla E C(bx"(g)). Since A(x"(&)) = A@(x"(zo,e))) is a Dirichlet algebra on ba(xa(zo,e))= bx"(Q), we see that fn o tbn/albQE [ReA(xa(Q))lbQ]for every n = 1,2,. . . . Hence fn o x b n l b Q = (fn 0 t b n / " )o xalbQE ReA(Q)lbQ for each n, and consequently f E [ReA(Q)lbQ].The proposition is proved.

As we shall see further, P(&) is also a maximal algebra. Note that in the case when v > 0 Pu(Q)is neither a Dirichlet nor a maximal algebra. 2.5.6. DEFINITION. Let K be a bounded set of the big-plane CG and let v E [0,1]. The set

for every r,,-rationd function

T )

is called r,,-rational hull of K . A set K c C G is r,,-rationally convex if it coincides with its r,,-rational hull r,,(K).

Following the same line as in Theorem 2.5.2 we can get the following 2.5.7. PROPOSITION. Let K be a compact set in the bigplane C G and let u be a number in [0,1]. The maximal ideal space of the algebra R,,(K) coincides with the I',,-rationd convex, hull r,(K) of K .

102

Char>terI I . f -Analytic Functions in the Big-Plane

Instead of Proposition 2.5.4 now we have the following weaker result. 2.5.8. PROPOSITION. The Shilov boundary of the algebra R,(h') is contained in the topological boundary bK of K .

PROOF.Suppose that z, E int K n aR,(K) and let U be a neighborhood of zo in int K . Then there exists a function f E R,( K ) for which

The composition g = fojz0-loff with the standard embedding j e o of C into C G through z, is an analytic function in C which attains the maximum of its modulus at certain inner point (namely at the point je0-l(z)E jeOA1(U)) of the set jz0-'(K) c C. Consequently, g is constant on the connected component N of the set j e 0 - l ( K )which contains the point j e o-'(z0); and therefore, f attains the maximum of its modulus somewhere in the set j e o ( b N ) c bh' in contradiction with the choice of f . We conclude that int K n aR,(K) = 0; and therefore, aR,,(K) c bK. Q.E.D. Observe that in the case when the group T satisfies the condition (*) and K c C G ,from the proof of Proposition 2.5.4 one can derive that aR,(K) = bh' for every v E [0,1]. As shown by Grigoryan [30] both these sets do not coincide in general. The next result is a F-version of classical Hartogs-Rosenthal theorem. Let K be a compact set in the big2.5.9. PROPOSITION. plane CG, G = /3T.If the Lebesgue measure of the set x " ( K ) is zero for each a E To, then R ( K ) = C ( K ) .

2.5. Alqebrus o f I',-unalutic

functions

103

PROOF.Let f E C ( K ) and e > 0. By the Weierstrass-Stone theorem we can find an a E I-', and an h E C ( x " ( K ) )such that

The function f can be approximated uniformly on x " ( K ) by rational functions in z by Hartogs-Rosenthal's theorem, because of dzdy(x"(K)) = 0. If

for some h' E R ( x " ( K ) )then , Ih'(x"(z))- f ( z ) l < E ; and hence f E R ( K ) , as desired. In other words Proposition 2.5.9 claims that every continuous function on a compact set K c C G which satisfies its hypothesis can be approximated uniformly on K by T-rational functions.

2.5.10. PROPOSITION [Grigoryan-Stankov]. If the group r = G possesses property (*), then every compact set K in the big-plane C G is r-rationally convex. h

E CG\ K . Choose c E To and E > 0 such that the neighborhood Q = { z E C G : l x c ( z ) - x c ( z l ) l< E } does not meet K . Thus the point x"(z1)does not belong to the

PROOF.Assume that

zl

compact set f ( K ) c C and hence I1/(XC(Z) -XC(Zl>)

I > max I l / ( x " ( z )- X C ( Z l ) > I. 5EK

Consequently z1 4 r,(K). We conclude that K is T-rationally convex because the neighborhoods Q from above form a basis of the topology in C G .

2.5.11. COROLLARY. If I-' =

s p R ( K ) = IE xd(bQ) then we will have x b ( z o )= ( ~ ~ ( 2 , E) )( ~x ~ ( ~ Q =) (xb(bQ)) )~ which is absurd). By the maximality of the algebra P(bK,) we obtain that P(bK,) = Bd,, = R(bK,) in contradiction with the choice of g. We conclude that B n Rd(bQ) = Pd(bQ), and therefore,

The theorem is proved. In the sequel we shall assume, without mentioning it especially, that the group possesses property (*). If Q = Q ( z , , E ,u ) is a basis neighborhood in C G ,we will assume that 0 6 Q whenever z, # 0.

r

As an immediate consequence from Theorem 2.5.12 we get that P(G) = R(Q) = A(&) for any neighborhood Q in CG.In particular, any function in R(G)and A(&) can be approximated uniformly on c CG by polynomials in C G .

I06

Chapter II. r-Analvtic finctions in the Bin-Plane

2.5.13. DEFINITION. A continuous function on an open subset U of the big-plane C G is r - a n a l y t i c (or generalized-analytic) in U if it can be locally approximated in U by polynomials in CG.

The set of all analytic functions in in the sequel by Hol (U).

U c CG will be denoted

A famous theorem from classical function theory, due to Rad6, says that if a continuous function is analytic outside its vanishing set, then it is analytic everywhere. The next result generalizes this theorem for analytic functions in the big-plane C G . 2.5.14. PROPOSITION. Let the group T C R possess property (*) and let Q(z,,e,a) be a fixed basis neighborhood in and r-analytic in Q\ C G . If a function f is continuous on int f-'(o), then f is an analytic function on the whole &, i.e. f E 4Q) = P(g>-

PROOF.The function F ( z ) = f o j e ( z )is continuous on the set je-l(Q) C C G ,where j e is, as before, the standard embedding of C into C G through the point e E G. Suppose that j e ( C ~f l ) f-'(O) = 0. Then F is analytic in je-l(&); and consequently, f is a r-analytic function in Q, because of the maximality of the algebra P(&). If K = j e ( C )n f-l(O) # 0, then F is analytic in j e - l ( Q) \ K and vanishes identically on K . According to Rad6's classical theorem in C , the function F is analytic in j e - l ( Q ) . Proceeding as before we conclude that f E P(&). Q.E.D. 2.5.15. NOTESAND REMARKS. The results in this section are from Tonev [113], [114], [127]. For the case v = 0 Theorem 2.5.2 is proved in [113]. Various

2.6. r-entire functions

107

conditions for F-polynomial convexity have been found by Grigoryan [43] and Stankov [99]. As shown in Tonev [113], the algebra P(K) for K c C e can be presented as the uniform closure of an inductive limit of algebras of type P ( K j ) on certain compact subsets Kj of some Ftiemann surfaces. Moreover, also there it is shown that for any compact K C C there is a unique uniform algebra extension of the algebra P ( K ) in which the image of the identity mapping possesses arbitrary rational powers. For the original Hartogs-Rosenthd theorem see e.g. in Gamelin’s book [32]. Its proof for the case = Q is given in Tonev [113]. Proposition 2.5.9 was announced originally by Grigoryan [43], but proven completely by Stankov [97]. As shown by Kramm [32], A y ( A ~are ) Schwarz algebras for each u > 0. Theorem 2.5.14 is from [113].

r

2.6. r - E N T I R E FUNCTIONS

In this section we extend some results about entire functions in one variable for entire functions in big-planes.

2.6.1. DEFINITION.F-entire finetion in the big-plane CG we call any function which is holomorphic in C G . The class of F-entire functions in the big-plane will be denoted by Hol ( C G ) . Let f be a r-entire function in the big-plane CG which is bounded. The restriction of f on the embedded plane j e ( C )c CG is a bounded entire function in C ;and therefore, it is constant according to the classical Liouville’s theorem for analytic functions in one complex variable. Being constant on a dense subset of the big-plane CG,the function f is constant on C G . Thus we obtain the big-plane version of the classical Liouville’s

Chapter II. F-Analytic finctions in the Big-Plane

108

theorem, namely: Every bounded r-entire function in the bigplane CG i s constant. The so called "real part of Liouville's theorem" can be extended also in the big-plane. Namely, 2 . 6 . 2 . THEOREM. Let F be a r-entire function in the bigplane CG and C,N axe real constants such that

for every z in C G . Then the Fourier coefficients

J G

J G

of F (where da is the Haar measure on G ) axe zero for every a >N.

PROOF.Without loss of generality, we can assume that F ( 0 ) = 0. Let Em > 0, E m --+ 0, 1 < Am -+ 00 and let P m ( Z ) = k

I3 C $ ~ ) Z ' J

be a polynomial in CG such that Pm(0) = 0 and

j= 1

-max IF(2) - P m ( Z ) ( < e m . By the hypothesis it follows that A,(&)

Re(P,(z)) that

G

5 C1zlN + Em on &(Am).

G

On the other hand the identity

fiom one hand we have

2.6. r - e n t i r e functions

which holds for every fixed a,

J Re

pm

109

> 0 gives:

(r- 9) Re Xa' (9) ds

G

k

(22)

=

xaj

j= 1

J

(Recjm) Rexaj(g)ReXa'(g)dg G

and also

/Rexa1(g)Imxa2(g)dg and

J

ImXa1(g)ImXa2(g)dg,where

Chapter II. r-Analytic finctions in the Big-Plane

110

and also

By (24) we have that /Rexa1(g)Rexa2(g)= 0, and

J

ImXa1(g)ImXa2(g)dg= 0.

G

G

Correspondingly, (25) implies that

I

G

0. Let now

and

a1

1

= a2 = a

#

0. The identities

X"(g)y(g)dg= 1 imply:

G

and therefore,

Re xal(9) Im Xa2(g)dg =

/ G

(x"(q))'dg = 0

111

2.6. r-entire functions

The identities (22) and (23) now take the forms 1 /Re Pm(r g) ReXal(g)dg = -ralRR crm; 2

-

G

G

correspondingly. According to (21) we have: f R e c p " = A/RePm(r-g)(lkRexal(g))dg 5 -(Crf+cm); 4 fa1 rat G

2 TImcFm = r /Re ~m(r.g)(lrfImx"'(9)) G

4

44 L --$(Cr,N+sm),

+

because of RePm(s) 5 ClzlN cm and 0 5 lrfRexal(g) 5 2; 0 5 lrfImXa'(g) 5 2. Since crm + c ~ fas m + 00, IFkcfI = IIrncfl = 0 (and thus cf = 0) for every number QI > N. The theorem is proved.

An entire function f in C is said to 2.6.3. DEFINITION. be entire r-almost-periodic f u n c t i o n i f f can be approximated uniformly by exponential polynomials of type c k e i a k r with k

To on each halfplane { z E C : Imz > a, a E R}. Theorem 2.6.2 implies the following

ak E

2.6.4. COROLLARY. Let f be an entire T-almost-periodic function in C and let z A k e i A k t be its Din'chlet expansion. If k

there exist constants C and N such that Re f(z) every z E c, then A k = 0 whenever x k > N .

5 Ce-NY for

2.6.5. NOTESAND REMARKS. The results in this Section are from Tonev [131]. See also [125].

Chapter 11. r-Analytic Fhctions in the Big-Plane

112

2.7.

SPECTRAL

MAPPINGSOF

SEMIGROUPS

The theories of analytic functions in one and several complex variables are closely connected with commutative B anach dgebras generated either by one or several of their elements respectively. In a similar way the theory of analytic r-almost-periodic functions in C (and respectively, of F-analytic functions in the big-plane C c ) is tightly related with commutative Banach algebras which are generated by specific multiplicative subsemigroups I

Let B be a commutative Banach algebra over C with unit. be an additive Fix a positive number v, 0 < v 5 1, and let subsemigroup of R such that r,\{O} is dense in [v,00) and which contains the semigroup 2,. Let R = { ba j }”3 7 1 be a multiplicative semigroup of elements b,, in B which contams the unit of B and which is algebraically isomorphic to the semigroup I‘,. The set R 0-’ is a subgroup of B which is algebraically isomorphic to the set r = F, (-F,,). As it is not hard to see, r is a dense subgroup of R which contains 2. Let G = /?F = F d . Fix a linear multiplicative functional cp in s p B such that cp(b,) # 0 for some (and thus for any) element ba E 0. Consider the function

r,

-

+

4 b a >/I d b a 1I

for a E F,,

g+,( - a ) = l/g+,(- a )

otherwise.

Clearly, g, is a well defined function on the set I‘, U (-I-‘”). Following the same lines as in Theorem 2.5.2 we can extend gv on F as a character, which we shall keep denoting by g+,. 2.7.1. DEFINITION. Let R = { b a } a G r , be a multiplicative subsemigroup of a commutative Banach algebra B . Spectral

2.7.Spectral mappinas

113

of semiaroups

mapping of 52 is said to be the continuous mapping r, : s p B CG which is defined in the following way: r,(cp> =

{

if cp(b,) = 0 for some a E r, \ {0},

0 Icp(b1)l

I-+

- g$9 ifcp(ba) # 0 for each

The range a ( a ) = r n ( s p B ) C

aE

r, \ (0).

CG of r, we c d spectrum of 0.

Observe that xa(ro(cP>> = Ka(Y)

(28)

for each a E

n

b O ( d

r, and for every cp E s p B.

= 1 and X " ( T , ( V ) )

Indeed,

x"(rn(9)) =

= X"(lCP(~l>l * gv) = lcp(~l)lag,(a) = n

l c p ( b a~ ) (l m~ a cp(ba) ) = ba(cp) for every a

+ 0. Consequently,

xa(r, ( s p B ) ) = %a ( s p B ), wherefrom (29)

x"(a(.n>> = a(ba).

The continuity of r, is an immediate consequence from (28). Obviously, a(52)is a compact subset of the big-plane CG. 2.7.2. EXAMPLES.

1.) Observe that one can consider also spectral mappings and spectra of multiplicative semigroups 52 C B which are nondense in R. Let b E B and 52 = {b"}F=, 2 Z,. We can easily ver(-2,) = Z and G = ,f32 = 2 = S1. ify that now I' = Z, The spectral mapping r, of 52 maps sp B into C z = C ,namely: ~ , ( c p ) = 0 whenever cp(b) = b(cp) = 0 and rn(cp) = lcp(b)leiep for some number eiep E S1. In fact eiep is determined by e i e p = cp(b)/lcp(b)l. Therefore r,(cp) = cp(b) = b(cp). We conZ, of clude that the spectral mapping r, of 52 = nonnegative powers of a fixed element b E B coincides with the Gelfand transform of b. Consequently a( 52) = a(b). The iden) ~a(f"),i.e. it coincides with tity (29) now states that ( ~ ( f ) = the statement of Proposition 1.1.8.

+

n

n

h

{b"}z=o =

114

Chapter I I . r-Analvtic finctions in the Big-Plane

2.) Let b E e B be a fixed exponent in B and let F be a dense additive subgroup of R which contains 2. Consider the semigroup 52 = {ba}aEr, c B and let G = p r . The spectral mapping T~ : s p B + C G of 52 now can be evaluated as T, (9) = Icp(b)lgV for some g, E p r = G, where g,(a) = cp(ba)/lcp(ba)l = (cp(b)/lv(b)l)a for each a E F,. Therefore, in this case I ~ ~ ( c p )=l h

b(v)and g d l ) = ~ ( c p ) / l ~ ( c p ) l for every cp E SPB.

3.) Let T be as before, G = p r , and let 52 = { X ' } ~ E ~ where ~ , 0 < v 5 1. Let A be a uniform algebra on a compact subset K in the big-plane C G which contains In this case the spectral mapping T~ is the identity on K , because according to (28)

n.

for each a E

r, and every cp E s p A .

As it follows from (28), a point z E CG belongs to the spectrum u( 0 ) of a semigroup G?c B if and only if all elements of tYPe (2" - b a ) a E r u = ( ~ " ( 2 )- ba)a,=ru belong together to a maximal ideal of B. Consequently

2.7.3. PROPOSITION. The spectrum ~ ( 0of)a sem'group $2 = { b a } a E r , in a commutative Banach algebra B (where I', is as above) consists of all points z in the big-plane CG such that (Z'l,.

. . ,zan) E u ( b o 1 ,... , b a n )

for all finitely many collections of numbers a j , j = 1,. . . ,n , in

r v.

In the case when the group obtain the following

r satisfies the condition (*) we

2.7. Spectral mappings of semaqroups

115

r

2.7.4. PROPOSITION. If possesses property (*), then the spectrum a(52)of every multiplicative semigroup 52 = { b a } a c r o in 52 consists of all points z in the big-plane C G such that za E a(b,) for each a in To. PROOF.The identity (29) implies that za E a(ba) for all z E a(52) and all a E To.Suppose that z: E a ( b a ) for each a E though z, 4 a(52). For some E > 0 and a E To the neighborhood Q = { z E C G : Ixa(z)- zzl < E } of z, does not meet a(52). Because of z: E a(ba),the neighborhood {z E C :IZ-Z:~ < E } meets the set a ( b a ) = x"(a(52))according to (29); and consequently, there is a point z1 in a(52)for which Ix"(zl) - zzl < E , i.e. z1 E Q n a(52) in contradiction with the choice of Q.

r,,

2.7.5. THEOREM. Let Q = Q(z,, E , a ) = {Z E C G : Ixa(e)z: < E } be a basis neighborhood of CG,and let A be a uniform algebra on bQ which contains the semigroup 52 = { X " } a E r , . Then s p A = bQ whenever a(52)= bQ, and A E P(&) whenever a(52)= T ~ ( As) = ~ &.

I

PROOF.According to Proposition 2.5.4 we have that aP(&) = bQ. On the other hand

& = spP(&) = s p P ( b Q ) = p,(bQ).

Therefore, P(&) = P(p,(bQ)) = P(bQ) c A and ~ ~ ( s p cA ) T ~ ( & ) = &. Since, according to Proposition 2.5.5, P(bQ) is a Dirichlet algebra, then for each z, E bQ the set T;'(z,) c s p A contains only the point evaluation at z,. Hence T~ is an oneto-one correspondence between the sets ~ i ' ( b Q and ) bQ. If in ) then evidently s p A coincides addition a ( 0 ) = ~ ~ ( s p =A bQ, with bQ and this proves the first part of the theorem. Let ~ ( 5 2 = ) &, z1 E Q, and let y q , ( p 2 be two functionals in the set T ; ' ( z ~ ) with representing measures dp1 and dpz on bQ respectively. Because of y l ( p ) = c p ~ ( p )= p ( z o ) for each polynomial p and because P(&) is a Dirichlet algebra, we observe

116

Chapter I I . T-Analvtic Fhctions in the Big-Plane

that both measures coincide. Thus 91 = 9 2 ; and therefore, Since the spectral mapping rn is injective on the set a(R) = we can identify s p A = r;l(a(R)) = rL1(g)with Hence A IIP(&), s p A = s p P ( Q ) = and a A = aP(Q) = bQ wherefrom A Z A1 bQ. Because of s p C(bQ) = bQ # we see that A # C(bQ). By the maximality of P(bQ) (Theorem 2.5.12), we obtain finally that A Z Al,, P(bQ).

0,

r;'(g).

g

g.

g,

Theorem 2.7.5 implies the following result for r-almost-periodic functions in C . Let U be either a half-plane { z E C : I m z > p, p > 0} in C or an open subset of type { z E C : l e i a z - e i a z , I < E } in some half-plane { z E C : Imz < a } . Denote by the set j e ( U ) c C G , where G = /3r and let A be a sup-algebra of continuous r-almost-periodic functions on U which contains all functions e i a z , a E Then s p A = bQ if a ( { e i a " } a E r = o ) bQ and A is isometrically isomorphic to the algebra of all bounded analytic r-almost-periodic functions on U whenever , ( { , i O z } a ~ = r o&. )

r,.

Recall that a commutative Banach algebra B is Zinearly generated by its subset S c B if and only if the linear combinations over C of elements in S are dense in B.

2.7.6. THEOREM. Let v be a positive number, 0 < v _< 1, and let r, be an additive subsemigroup of R such that r, \ (0) is dense in [v,oo) and which contains the semigroup Z,. If a commutative Banach algebra is linearly generated by a multiplicative su bsemigroup R = { ba}=Erv, 11 b, 11 5 1, which is algebraically isomorphic to r,, then the spectrum a(0 ) of 0 is r,polynornially convex and the spectral mapping r, : s p B -+ a( 0) of R is a homeomorphism. Observe that according to the hypothesis the semigroup R contains the unit of B. The spectral mapping rn of R is one-to-

2.7. Spectral mappinqs of semigroups

117

one because each point cp E s p B is uniquely determined by its values {cp(b,)},Erv. Being an one-to-one and continuous mapping from a compact set (namely s p B ) into a Hausdorff space (namely C G ), T~ is a homeomorphism. Let z, be a point in the big-plane C G which belongs to the T,-polynomial convex hull p,(a(R)) of ~ ( 0 For ) . every r,-polynomial we have:

By the continuity we observe that for each b E B there exists a unique function p in the algebra P(pv(a(12))) such that Ip( z,) 5 llbll and, moreover, the point z, gives rise to a linear multiplicative functional in B,say ye,, defined as cpzo(b) = p(zo). Let us )) = evaluate the point ~ , ( c p ~ , ) . Because of x ~ ( T ~ ( ( P ~=, cpz,(b,) xa(zo) for every a E I', (and thus for each a E ro)we obtain that rn((pz,)= z,, i.e. z, E a ( 0 ) = ~ ~ ( s p AWe ) .conclude that p,(a(R)) = a ( 0 ) . The theorem is proved.

I

2.7.7. THEOREM. Let v be a positive number, 0 < v I 1, and let T, be an additive subsemigroup of R such that I',\(O} is dense in [v, m) and which contains the semigroup Z,. A commutative Banach algebra B is linearly generated by a multiplicative subsemigroup 0 c B(0,l) which is algebraically isomorphic to the semigroup if and only if B is isometrically isomorphic to an algebra of type P,(K) for some I',-polynomially convex set K in the big-plane CG,where G = p r .

r,

The "if" part of the statement follows immediately from the previous Theorem 2.7.6, and the "only if " part follows from Theorem 2.5.2.

118

Chapter I I . r-Analytic Functions in the Big-Plane

2.7.8. COROLLARY. Let A be a uniform algebra on a compact set I< which is linearly generated by some multiplicative subsemigroup R c B(0,l) as in Theorem 2.7.6. If a(R) 3 G, where G = PI', then A is isometrically isomorphic to the algebra

PV(AG). PROOF.According to Theorem 2.7.7 the spectrum a ( 0 ) is a

r,-polynomially convex subset of the big-disc & and 7;, maps

A isometrically and isomorphicaly onto the algebra Pv(a(R)). Thus s p A = spP,(a(R)) = p , ( a ( R ) ) . By the hypothesis, we have that G c a(R) c &, because of R c B(0,l). Consequently, A E Pu(&) because the r,-polynomial convex hull of G coincides with &. 2.7.9. LEMMA. Let A and R be as in Theorem 2.7.7and let the group (-rv)satisfies condition (*). If a A # s p A and l f a l l a A = 1 for some (and thus for each) function fa E R, then a ( R ) 3 G.

r, +

PROOF.In the same way as in Corollary 2.7.7 we see that p , ( a ( R ) ) = a(R), A E P,(a(R)) and s p A = s p P , ( a ( R ) ) = a(0).By the hypothesis we see that d A c I-1( 1) = Ixa 1) = G. Assume that d A # G. According to Proposition 2.5.4 we

lf?L

have that

SPA = a ( Q ) = s p C ( a ( Q ) ) = p u (ba(R)) = pu(dP,(a(R))) = dP,(a(R>) c d A , because, according to Proposition 2.5.3, the proper compact subsets of C G are r,,-polynomially convex. But this contradicts the hypothesis d A # S P A . We conclude that G = a A c s p A = 40). Corollary 2.7.8 and Lemma 2.7.9 imply the following characterization of the algebra Pu(&):

2.7. Spectral m a p p i n q s of semaqroups

119

2.7.10. THEOREM. Let u be a positive number, 0 < u 5 1, and let I',, be an additive subsemigroup of R for which the set r,,\ (0) is dense in [v,00) and such that the group = (-r,,) possesses property (*). Let A be a uniform algebra on a compact set I{ which is linearly generated by a multiplicative subsemigroup 52 c B(0,l) which is algebraically isomorphic to r,,.I f a A # s p A and if If a \ = 1 for some (and thus for each) function f a E 52, then: (1) The spectrum s p A of A is homeomorphic to the unit big-disc AG, where G = p r ; ( 2 ) The Shilov boundary d A of A can be equipped by the structure o f a compact Abelian group isomorphic to G; (3) A is isometrically isomorphic to the algebra P,,(&).

r r,,+

laA

Theorem 2.7.10 implies the following

2.7.11. COROLLARY. Under hypotheses of Theorem 2.7.10 there exists a continuous embedding j of the upper half-plane H into s p A with a dense range, such that all functions of type j , f E A , are analytic T,-almost-periodic functions in H.

70

Indeed, for j one can take the composition r;' o j , of any standard embedding j , of H into & with the inverse of the spectral mapping rsI.

2.7.12. NOTESAND REMARKS. The notion of spectral mapping and of spectrum of a multiplicative subsemigroup of a commutative Banach algebra are due to Tonev [114]. In the case when u = 0 Propositions 2.7.3, 2.7.4 and Theorems 2.7.6, 2.7.7 are proved in Tonev [114], [117]. Theorem 2.7.5 is from [127]. Theorem 2.7.10 was proved originally by Grigoryan and Tonev in [47] for the case of antisymmetric algebras A , a condition which is somehow more restrictive than

Chapter II. r-Anal ytic finctions in the Big-Plane

120

the one that is required above, namely, d A # s p A . For other applications of spectral mappings and spectra of multiplicative semigroups in commutative Banach algebras and, in particular, for a "big-plane" version of holomorphic calculus see Stankov 1951, PSI, WI.

2.8. THEALGEBRA Hz

As we shall see in this section, most of the delicate properties of bounded analytic functions in the unit disc can be extended for the class of bounded r-hyper-analytic functions in the open big-disc A G ,introduced below. 2.8.1. DEFINITION. Let G be a compact abelian group whose dual = 6 is a subset of the group of rationals Q which satisfies condition (*). r-hyper-analytic finction in the open unit bigdisc AG we call every continuous function on AG which can be approximated uniformly on AG by functions of type h(z") = h o x a ( r .g ) where h E H o l ( A ) and a E

r

r,.

r,

Observe that each function z', a E \ {0}, is r-hyperanalytic on AG and maps the big-disc A c onto the unit disc A . Moreover, every r-analytic function on AG is a bounded T-hyper-analytic function on AG. This section is devoted to the study of algebra H z of bounded T-hyper-analyticfunctions on the open big-disc AG. Equipped with the sup-norm Ilfllm = sup If(z)l and the pointwise operBEAG

ations H Z is a commutative Banach algebra. It is clear that a function f(z) belongs to H Z if and only if f can be approxim mated on & by function of type h(z') with h E H" = H s l .

2.8. The alqebra H Z -

121

Note that the restrictions of T-hyper-analytic functions on each of the big-discs r < 1, are approximable uniformly on A G ( Tby ) T-polynomials.

a,(,),

Note that the Gelfand transformation A : H Z + gg of algebra H g is one-to-one and isometric. Indeed, f(cp5) = f(z) = 0 for each z E AG (and thus f z 0) whenever f E 0. On the other hand we have that h

h

since A does not increase the norm; and therefore,

Consequently, being the range of an isometric and one-to-one transformation, H l is a uniform algebra on its spectrum. What can be said about the spectrum sp HG of the algebra H l ? It is clear that each point z in AG generates some linear multiplicative functional, say pz,of H g , namely the point evaluation cp5 at z.

r

2.8.2. THEOREM. Suppose that the group possesses property (*) and let G = PI’. Then the spectral mapping r, of the standard multiplicative semigroup R = { x a } a E r o is a continuous mapping from sp H l onto the closed unit big-disc & which is a homeomorphism between r i l ( A c ) and the open big-disc AG.

PROOF.The spectral mapping r, is continuous by definition. Obviously, each point z E AG gives rise to some linear multiplicative functional, namely, the point evaluation cp5 E s p H ; at z . Let cp E s p H Z and let rn(cp) = z, E AG. Clearly, cp(x”) = ~ “ ( z ,for ) each number a E To. If a function f E H E vanishes at z , , then without loss of generality we can assume

122

Chapter II. F-Analvtic finctions in the Bin-Plane

that the function h o xu also vanishes at zo for every function hox" which approximates (with respect to the sup-norm on A G ) f because the point = x " ( z o )belongs to the open unit disc A. From h ( x " ( z o ) )= h ( z z ) = 0 we obtain

where h l ( z ) is some function in Hm. Consequently, ' p ( f ) = P(X"

- Z,")'p(hl

0

x") = 0

*

V(hl

0

x") = 0.

Because of ' p ( f ) = 0 for every function f E H Z which vanishes at z,, we see that 9 is the point evaluation at zo, i.e. 'p = pe,. This shows that T ; ~ ( A = ~ ){ ( p a : z E A c } . According to (30) we see that x"(T,((P,)) = cpe(x") = ~ " ( z for ) each a E To; and consequently, ~,('p,) = z for every z E AG. Since s p HG is a compact set, the range of the spectral mapping T, is also compact; and therefore, it must be the entire closed unit big-disc &. As an one-to-one and continuous mapping from a locally compact set in a Hausdorff space T~ is a homeomorphism between T ; ~ ( A Gand ) A c . Q.E.D. Example 2.7.2(1)shows that in the classical case (when F = Z and G = ,f?Z = S') the spectral mapping T~ coincides with the Gelfand mapping 2 of the identity id ( z ) = z in A. Proposition 2.8.2 allows some of the results for the classical algebra H - = HYl to be extended for the case of the algebra Hg of bounded F-hyper-analytic functions in the big-disc AG. First of all, it indicates that the mapping 7;' : z 'pe embeds the open big-disc AG homeomorphically into s p H z . Equivalently, the set T;'( A c ) is maped homeomorphically onto A c via T ~ The complement of T;'(AG) in s p H g is mapped via T, onto the group G. The set S, = Ti1(g) c s p H E is called the fibre of s p H i over the point g E G. Namely,

-

8

Sg = TG1(g) = {'p E s p H G : ~,('p) = g } .

.

2.8. The alaebra H 2 -

123

The fibre S, is a compact subset of sp H l . Since each linear multiplicative functional in s p H l \TL'(AG)belongs to the fibre S, over some point g E G, we see that sp H g = T;'(AG) U U S,, where all the entries in the last union are disjoint sets. SEG

Equivalently, SPH;

(31)

\T;~(AG)

=

U sg. SEG

Note that since Ip(x")l = Ip(x')l = 1for every p E S,, g E G, we have that cp(x") = 1 for each p E s p H g \ T ~ ~ ( A G ) .

2.8.3. PROPOSITION. The fibres S,, g E G,are mutually homeomorphic.

PROOF.Let g and go be two points in G and let R, : AG

AG be the "rotation" Rg(z)= zg in AG by g. We claim that the adjoint mapping Ri : sp Hi + sp H: defined by

(R,*cp)(f) = cp(f 0 R,) maps homeomorphically the fibre S,, onto the fibre Sggo . Indeed, for each a E roand every p E S,, we have

x"(.,(R;cp))

4 x " 0 R,) = xU(g)(P(xU)

= (Rf4(XU) =

= Xa(g)Xa(7n(9)) = x"(s)x"(so) =x"(ggo);

and thus, r,,(R,*p) = ggo. Hence R i p E S,,,,. As a continuous and one-to-one correspondence between the fibres Sgoand S,,, (with inverse Ri), the restriction R*l is a homesollo g s,, omorphism between Sgoand S,, and this justifies the claimed result.

I

Similarly to the classical situation, there are no reasons to expect the mapping T,, to be a homeomorphism between the whole

124

Chapter 11. f -Analytic Functions in the Big-Plane

spectrum s p H Z and &. A question which arises naturally is if the embedded open big-disc T;' (AG)is dense in the surrounding space s p H z or not? In other words, are the fibres S,, g E G, fat enough or not? Carleson has solved the classical setting of this problem by his famous proof of the Corona theorem for the unit disc in the complex plane C . What he has proved actually is that the "corona" int ( s p H g \ ~,-'(AG))of the algebra H Z is empty in the case when G = S1 and = Z. Below we prove a similar corona-type theorem for groups G other than S1 whose dual groups possesses property (*). Though the result obtained seems somehow too abstract, similarly to the classical situation, it can be easily reformulated into a very explicit theorem for bounded T-hyper-analytic functions on the upper half-plane H.

r

2.8.4. LEMMA.The range T;'(A,) of the open unit bigdisc is dense in s p H E if and only if the following condition is fulfilled: If f l , . . . , f n is a collection of bounded r-hyper-analytic functions in the open unit big-disc for which the inequality

is fulfilled on AG, then there exist an other collection of bounded r-hyper-analytic functions 91,. . . ,g, on AG such that the sum fi(z)gl(~) - .+fn(z)gn(z) is identically equal to 1 on the open unit big-disc AG.

+

PROOF.Suppose that there is a linear multiplicative functional 9 in s p H g which does not belong to the closure of the set T;'(AG).Having in mind the definition of Gelfand topology on s p H ; this means that there exist finitely many functions f l , . . . ,f n in H Z and a positive number S > 0 such that the open set

2.8. T h e alaebra H Z -

does not intersect the open big-disc AG, though every j = 1,.. . ,n. In particular,

125 Q(fj)

= 0 for

on the open big-disc AG and at the same time f 1 , . .. ,f n lie simultaneously in some proper ideal of H i , namely, in the kernel of Q. The fact that the functions f 1 , . . . ,fn lie in a proper ideal is equivalent to the statement that 1 does not belong to the ideal generated by these functions, i.e. there are no functions W g l , . . . ,gn in HG such that

on A c . It is not difficult to verify that the above argument is easily reversible.

2.8.5. THEOREM (Corona theorem for r-hyper-analytic functions). Let the group r possesses property (*) and let G = p r . I f f1,. . . ,fn are n bounded r-hyper-analytic functions in the open unit big-disc AG for which

then there exist other n bounded r-hyper-analytic functions g l , . . . , gn on AG such that the equality

is fulfilled for every point z in the open unit big-disc AG.

PROOF.The strong version of Carleson’s corona theorem ass e r t s t h a t i f f l , . . . , f k E H o J , llfjllOoI1 and I f i l + . . * + l f k l > > 0 on A , where 0 < Q < 1/2, then there exist g l , . . . ,gk E H w and a constant C(lc,a) > 0 such that llgjll 5 C(lc,a) and fig1 . - . f k g k = 1 on A . Without loss of generality we can 0

+ +

Chapter 11. r-Analvtic Functions in the Big-Plane

126

assume that IlfjII 5 1/2 for all functions f j from (32) and that 6 < 1. Let C(n,S/2) be the corresponding Carleson’s constant and let c 2 max(1, C(n,S / 2 ) ) . For each j = 1,.. . ,n there exist functions f: E Hm and numbers a j E r‘ such that the inequality

-

holds for the functions f j = f j o xaj, j = 1,.. . n. Without loss of generality we can assume that the indices u j do not depend on j (by replacing them, if necessary, by some b E T’for which bm; = u j , j = 1,..., n , and fj(z) by fj(zmj)). NOW If11

-+

Note that 6/2 < 1/2. Consequently, for f:, . . . ,f:, which are I1 bounded analytic functions on A , the inequality I +. 6 / 2 > 0 holds on A ; in addition = 5 llfjllm

-

llfJllw

+ S/(2nc) 5

If; llflillw

.+If:

+

1. According to the strong version of Carleson’s theorem there exist functions g; , . . . ,g; E H w for which f;gi + . . - + f A g h = 1 on A and such that llgillw 5 C(n,6/2) 5 c. Denote by Fj the functions 9: o xb, j = 1, . . . ,n. llfj

fjIlm

=

llfjllw

- + +fncn -

+ + fk

We have f1 TI - = (f;0 x6)(gi 0 x b) . . - ( 0 xb)(g; 0 xb) = 1 on AG and also 11gj;.11m5 c. Denote by $ the bounded - .. fnijn. We have F-hyper-analytic function .1c, = f l & +

+ +

Consequently, $ is invertible in H Z and hence the identity fig1 + fngn 1 holds for the bounded r-hyper-analytic functions g j = $;/$, j = 1,... n, on the open unit big-disc AG. The theorem is proved. + .

*

+

2.8. The alaebra H Z -

127

As a corollary from Theorem 2.8.5 we obtain that the set \ r, ( A G )is a disjoint union of mutually homeomorphic fibres Sg,g E G, with empty interior. sp HZ

over a point g E G Roughly speaking, the fibre Sg= r;'(g) consists of all linear multiplicative functionals of the algebra H g which resemble evaluation at g. More precisely,

r possess property (*), let G = p r and let f be a bounded rhyper-analytic function in the big-disc AG. The range f ( S g o )of the fibre Sgoconsists of all limits of all sequences of type {f(zn)), where z, E AG, z, + go. 2.8.6. THEOREM. Let the group

go be a point in the group

PROOF.Let 6 = limf(zn) where zn E AG, zn + go Without n loss of generality (by passing to some subsequence, if necessary) we can assume that p(,, = r;'(zn) -+ $ E s p H Z . On the one and on the other hand f(zn) -+ 6. hand f(zn) = cp,,(f) -+ $(f), Consequently, [ = +(f) = ?($). If rn($) = z, then 1x1 = I$(x')l = lirnlcp,,(x1)I = limIx'(zn)l = limlznl = 1. Hence n n n r, ($) = z = go because, according to (30), ~ " ( 2 )= x " ( T ~ ( + ) ) = lim~"(,n((~sn)) n

= limcp,,(f) n

= limf(Zn) = f ( g 0 ) . n

Therefore, $ E Sgo;and hence, the point 6 = f($)belongs to T(Sgo). Suppose conversely that 6 E f^(Sgo), and let = f^(cp) for some cp E SsoC s p H Z \ r, (AG).By the Corona theorem for r-hyper-analytic functions there exists a sequence {zn} c AG which tends to cp within s p H Z . Because of cp E Sgowe obtain that Z n = :,(ps,) + r,(cp) = go and at the same time 6 = f^(cp) = lim f(cp,,) = lim f(zn). The theorem is proved. n

n

Theorem 2.8.6 implies the following

Chapter II. r-Analvtic Functions in the Birr-Plane

128

2.8.7. COROLLARY. Let go be a fixed point in the group G. A bounded r-hyper-analytic function f on the open unit bigdisc A c is continuously extendable on the set AG U {go} if and only i f its Gelfand transform is constant on the fibre Sgo. Every analytic r-almost-periodic function on the upper halfplane H which can be approximated uniformly on H by functions of type h(eiar),where s E roand h E Hol ( A ) ,we will call hyperanalytic in H. The functions of type h ( e i s z ) are well defined because each of the functions eisz maps H onto the open unit disc A. The Corona theorem for r-hyper-analytic functions in AG implies the following

COROLLARY. Let the group r c R possess property (*) and let fi, . . . ,fn be bounded hyper-analytic I'-almostperiodic functions in H for which the inequality 2.8.8.

n

j=1

holds on H for some 6 > 0. Then there exist bounded hyperanalytic r-almost-periodic functions 91,. . . ,gn in H, such that efjgj

3

1 on N.

j= 1

The result follows immediately from Theorem 2.8.5 by restricting all functions involved on the range of the half-plane je(H) by j,, which is dense in AG, and by the easily verified fact that every hyper-analytic r-almost-periodic function in H coincides with the restriction on je(H) of some r-hyper-analytic function in the open unit big-disc AG.

2.8.9. NOTESAND REMARKS. A brief picture of the maximal ideal space of algebra H w (without the Corona theorem) is given in Hoffman's book [51].

129

2.9. Alaebras between H Z and ~5,"-

F-hyper-analytic functions were introduced by Tonev [112]. In [loo] Stankov has investigated automorphisms and isometric isomorphisms of the algebra H - by utilising the spectral mapping of semigroup L? = { x " } ~ E ~ , . Carleson has proved his famous Corona theorem in [24]. Another and shorter proof was given afterwards by Wolff (see e.g. in [SO]). Lemma 2.8.4 is from Hoffman's book [51]. It was thought one time that a corona for the algebra of bounded analytic functions might appear in the big-disc setting. The proof of Corona theorem for Q-analytic functions on A e , due to Tonev [112], has shattered these expectations. Proposition 2.8.3 and Theorem 2.8.6 are from Tonev [116]. For other boundary behavior properties of r-analytic and F-hyper-analytic functions in the open unit big-disc see Tonev [115], [116] and [127]. There it is proved, in particular, the connectedness of the set s p H z \ ~;l(Ae). For the classical case of this last result see e.g. in Hoffman [51].

2.9. ALGEBRAS BETWEEN H l

AND

Ll

Throughout this section r will stand for an additive subsemigroup of the group of rationals which satisfies condition (*), G will be the group /3F and du will be the Haar measure on G. Let Loo(G) be the space of all bounded measurable complexvalued functions on G. Lm(G) is a Banach algebra under the norm llflloo = esssup If(g)l. For each number a E F, consider g€G

the algebras

which are isometrically isomorphic to algebras L- = L m ( S 1 ) and Hm = H m ( S 1 )respectively. It is easy to check that L r c LF and H r c H r whenever a = kb for some k E Z,

130

Chapter 11. r-Analvtic finctions in the Bin-Plane

2.9.1. DEFINITION. The space L l is the closure in L"(G) of the union of d spaces of type L z , a E r,. The space Lz is a commutative Banach algebra if we identify every two functions in L l which coincide do-everywhere on G. Moreover, Lz is a B*-algebra with involution (namely f * = 7). Consequently, by Gelfand-Neimark theorem the algebra L z is isometrically isomorphic to the algebra C ( s pL z ) .

As we saw in the previous section, every bounded analytic function on the open unit big-disc AG possesses radial limits almost everywhere on G. 2.9.2. DEFINITION. A r-analytic function f in the unit bigdisc A c whose radial limits on G are of absolute value 1 doalmost everywhere on G we call inner function in A G . For each a E r, the mapping f H f o x" is an isometric isomorphism from H w onto H", and its conjugate 9 H 1c, : s p Ha" --+ s p H w : $(fox") = 'p(f) is a homeomorphism between the spectrums s p Ha" and s p H w . Consequently, all results which are valid for algebras Hm are valid for algebras Haw , a E To,as well. Let Tdenote the function on G which is defined by the radial limits of a given function f E HZ. Clearly, is defined doalmost everywhere on G, f E L z and llflloo = llfll. Hence we can consider the algebra H Z as a closed subalgebra in L z . The mapping T" : s p La s p H," defined by

-

7

00

--f

(TaV>(f)=

~(7)for each

f E Ha"

is a homeomorphism between the sets s p L r and aHaW, because of L r E L", H" S H and because of the corresponding classical result for algebras L" and H w . 00

2.9. Aiaebrcls between HZ&Z-

131

In this section we investigate some subalgebras of L z which contain the algebra HG .

2.9.3. THEOREM [Stankov]. The Shilov boundary algebra H l is homeomorphic to the space s p Ll.

PROOF.Consider the mapping r : s p L l as

aHl

of

+ s p H g defined

( v ) ( f )= v(7h f E H l .

Clearly, r is continuous. Observe that because of

-

- SUP

V € 5 P L;

Id)l+€7(5P =SUPL; 1 If(+>I,

the set r ( s p L l ) is a closed boundary for H l . The mappings m

r," : s p L z + s p La : (r,"p)(f)= p(f) for every f E L r , and W

rf : ~p H l -+ s p H a : (rfp)(f) = p(f) for every

f

E Haw

are continuous, though not one-to-one in general. Note that

where ru : sp L r + s p Ham is the corresponding mapping relative to G Z S1. We claim that r is one-to-one. Let 91 # 972 be two points in s p LG, and let pl(F) # p2(F) for some F E L z . By Definition 2.9.1 there exists a sequence {Fn)Fz1 of functions F, E Ly' in Lg such that llF, - Film + 0 as n -+ 00. Consequently, { Fn>,tends to ? in C(spL l ) . Therefore, we can find an a E and an Fa E L r such that W

r'

Chapter 11. r-Analytic Functions in the BipPlane

132

i.e. ra L cpi

(r:

# 7rtcp2.

Now

( 7 r f o ~ ) ( p 1= ( T a o r Q L ) q 1

#

( T ~ o TL, ,)(p2

=

because the mapping Ta is homeomorphic. Therefore, ~ ( ( p 1 )# ~ ( 9 2 ) and ; this shows that the mapping T is one-to-one. p and let y o E ~ ( s Lz)\ p Let S be a closed subset of ~ ( s Lz), S. Let a , b E To. Since Ham c H r whenever b = Ica for some Ic E Z, and since by definition the set UH," is dense in H,", we o T)(PZ

0

can find an m E 2, and a neighborhood U of p0, for instance u = { $ E SP H," : I p o ( < € 7 fj E Ha , j = 1, * k} 3 which does not meet the set S. Clearly, for each $ E S one can find a j = 1,.. .,k, such that (p,(fi) # +(&). Therefore,

A) $($)I

-

+(A)

(w0)(fj)= cpo(&) # = (r$)(fj); and hence, the mapNote that according to (33) 7rF o ping T separates cp, and ~(spL;= ) 7, o nQL(spLZ)c T , ( S P Lr). As it is not difficult to be seen 7raH 9 , E 7 r F o ~ ( s p L ; ) \ r f ( S ) = Ta(SpL;)\.?r,H(S). Ac-

s.

cording to the corresponding classical result ~ , ( s pL:) = aHam. Consequently, .?rFcp,E a H a \ rf(S) and

for some f E Ham.Since

we obtain that sup Icp(f)l

(7rfpo)(f) = cp,(f)

< Icp,(f)l;

and because of

and consequently, S is not

Cp€S

a boundary for H,". The theorem is proved.

2.9.4. COROLLARY. The space H Z is a logmodular algebra on its Shilov boundary aH,".

Let f E L z C ( s p L z ) be a real-valued function on G and fn E La , be an approximating sequence of f, i.e. let {fn}:=l, 0 Ilfn-flloo + 0 as + 00. It follows that Il(fn+Tn)/2-fllw 00

2.9. Alaebras between H E and L z

133

+

as n + 00 as well. Since every function of type (fa, J , )/2 is real and belongs to the space Ly, E L" E C ( s p L z ) and since H" is a logmodular algebra, we can find a function h, E C (HE)-1 such that (fa, +7,,)/2 = In Ihnl. It follows that H g is a logmodular algebra on its Shilov boundary a H g E s p L; . 2.9.5. LEMMA.r f ( a H g ) = aHaWfor each a E

To.

According to Theorem 2.9.3, r f ( a H , " ) = ~f(.r(spL;)) C T, ( r t ( s pL; )) c ~ , ( s L p r ) S aHaW. Assume for a moment that aHaW\ rF(aH,") # 0 and choose a t,b0 E aHaW\rF(aH,"). It follows that there is an f E Haw such that

for every

+ E rF(aH,").

Because of aHaWE dHw

c U S,, aESf

it is clear that +o belongs to the fiber S, c s p H g over a point a E S1.Therefore, (f^oXa)(Sg) = f^(S,) whenever ~'(1-g) = a, i.e. there exists a vo E Sg such that v 0 ( f o x a ) = ~ ) ~ ( f By ) . (34) we get that

A F-hyper-analytic function on the open unit big-disc AG which is a limit C(&) of functions of type /? = b o xa,where a E Toand b is a Blaschke product on A we call T-hyper-Blaschke product on AG.

Chapter 11. r-Analytic finctions in the Bin-Plane

134

2.9.6. THEOREM [Stankov]. Let cp E s p H Z . The following conditions are equivalent:

EBHE; E aHaw for every a E r,; (3) l.^('p)l = 1 for every inner r-hyper-analytic function u on AG; (4) Ip('p)l = 1for every r-hyper-Blaschke product ,B on A c .

(1)

'p

(2)

7f'p

PROOF.First we show that (1) implies (3). Let u be an inner I'-hyper-analytic function on AG, i.e. u E H g and liil = 1 daalmost everywhere on G. Since according to the Corollary 2.9.4 H z is a logmodular algebra on aHZ,the functional cp has a m unique representation measure on aHc s p L z , say dp. The measure dp gives rise to a linear multiplicative functional of L; , which we will denote again by cp. Therefore,

=

/

= SP

L;

1

Z(x)dp

1

.^(x)$(z)dp = SP

c(x)dp

J

ldp = 1.

L;

We show now that (3) implies (1). Fix an a E r, and let u1 be an inner function on the unit disc S', i.e. u1 E H w and I = 1 almost everywhere on S'. Since u1 o xa is an inner rhyper-analytic function on the big-disc aG,by (3) we have that I(r:cp)(G1 o xa)l = 1. According to the corresponding classical result for inner functions on A we obtain that 7r;p E dHaw. To show that (2) implies (1) suppose that 'p E a H g . By (2) 7rFp E a H a for each a E I', and according to Lemma 2.9.5

2.9. Alqebras between €€:

and L z -

135

T ~ E Q ?r:(aH,"). If we assume that Q $ aH,", then as in the course of the proof of Theorem 2.9.3 one can find a number u E r0such that T ~ 4QrF(aH,"). If b is a Blaschke product in A and if TFQ E aH;, then (cp(bo xa)( = I(~fv)(box")I = 1 according to the corresponding result for classical Blaschke products. By the continuity argument we see that for every I'-hyper-Blaschke product p (2) implies (4). Finally, following the same line as in the course of proving that (2) follows from (3), one can show that (2) follows from (4). As it is naturally to expect, inner F-hyper-analytic functions and r-hyper-Blaschke products on the big-disc AG possess properties similar to the properties of their classical versions on the unit disc A . In particular, the following result can be proved by applying its classical 2-version.

2.9.7. THEOREM [Stankov]. (1) The unit ball of the algebra H g coincides with the closed convex hull of the set of all I'hyper-Blaschke products in the big-disc AG; (2) The algebra Hg is generated by the set of all r-hyperBlaschke products in AG; ( 3 ) For every t > 0 and for every function f E H i , f f 0, one can find a number u E To,an inner r-analytic function u E Ham and a function v E Ha" which does not vanish on AG and such that SUP If(? * s) - ( 4 r * s>l< E r,gEAc

(&-factorizationof functions in H:). Let A c B be two algebras and let S be a subset in B. By [A,S] will be denoted the subalgebra of B which is generated by the sets A and S. Namely, [ A , B ]is the closure in B of sums of products of type ucrsc*,where ua E A and s, E S. It is well known that the algebra [Hm,Z] is the smallest closed subalgebra of Lm(S1) which contains the algebra H" (actually

Chapter IZ. r-Analytic finctions in the Bin-Plane

136

the algebra of radial limits of functions in Hm. It can be preC(S'). It turns out that in the case of sented also as H w compact groups G other than the unit circle S' the spaces of type H (G) C(G) are not always closed subalgebras and even not always closed subspaces of L" (G). Below we show that if G is a compact group with rational dual group which possesses property (*) and if H Z is the algebra C(G) of bounded r-hyper-analytic functions on G, then H," is a closed subalgebra of L" (G).

+

O3

+

r

+

Denote by Ca the subalgebra Ca = {f o x a : f E C(S')} of C(G) which naturally is isometrically isomorphic to C(S' ). According to Weierstrass-Stone's theorem the space C(G) coincides with the closure in C(G) (and also in L w ( G ) )of the union of algebras of type Ca, a E r,. It is clear that if a = kb then

ca c cb-

2.9.8. THEOREM [Stankov]. The space H g +C(G)is a closed subalgebra of L z .

U (Haw + Ca) in Lm(G)is a closed subalgebra of LE, We claim that H l + C(G)= [ U ( H r + a€ro Call. The inclusion H Z + C(G)C [ U {H," + Ca}] follows The closure of the set

a€ro

a€ro

from the definition of H Z and from the discussions preceding the statement of the theorem. The opposite inclusion we shall prove following the the proof ( H a Ca)] we of its classical 2-version. For each t~ E [

u

+

+

a€ro

can find a sequence {fn W n } T = I , where fn E H a l and W n E Can,such that )l(fn ton) - vll, < 2-" for each n E 2,. We can assume in addition that an = Ic,an+l for some kn E Z, wherefrom fn E wn E Ca,+l. Let fn = f; o x a w + 1 and

+

2.9. Alaebras between H Z and L;-

137

w, = U I o~ xan+l,where f; E H", wk E C(S1). Denote A, = { h o xa : h E A(A)l,, }, where A(A) is the disc algebra. Since dist (9,H " ) = dist (9,A(A)lsl) for each g E C(S1) we have

I II(Wn - %+l)

- (fn+l - f n ) J (

I 1/2"-'.

Consequently, for each n there is a g, E Aan+lsuch that II(wn w,+l)-gnl( 5 2-,+'. Let bl = 0 and b, = a l + . . . + ~ , - ~ E A,,, for n > 1. Now {s"}, s, = w, b, E Can+l is a Cauchy sequence. Let s, .--) w E C(G). We have v - UI E H Z because of ( f n + ~ n ) - S n = fn-bn E Han+, c H," and ~~((hl+wn)-S,) (v -W)I(.p + 00. Consequently, o E H Z w C H; C(G). The theorem is proved.

+

00

+

+

2.9.9. COROLLARY. If 52 is the restriction on G of standard multiplicative subsemigroup { x a } a E r o in A ( A G) , then H," +

C(G)= [ H z , a ] .

+

PROOF.It is clear that H," C(G) c [H,",C(G)]. On the other hand H," - C(G)C H g C(G) because as we saw H," C(G)is an algebra. It follows that [H," ,C(G)]= H," +C(G)and thus H," +C(G) = [H,", G^] since C(G) = [C, G] by WeierstrassStone theorem. Consequently, H," + C(G)= [H," , = [H," ,R Ua]= [HG,521 because of 52 c H g . The corollary is proved.

+

00-

+

(3

2.9.10. COROLLARY. Let U be the set of functions u E ( H l + C(G))-l for which IuI = 1 do-almost everywhere on G. T h e n H z +C(G)=[H,",U].

138

ChaDter 11. r-Analytic finctions in the Big-Plane

+

This is true because U 3 3. Note that HZ C(G) is a logmodular algebra on s p Lz . Indeed, if f E H i and g E C(G), then there are functions hl E H: and h2 E C(G)such that Ref = log JhlI and fteg = log lh21 because both H E and C(G) are logmodular algebras on s p Lz . Therefore, Re (f+g) = Re f+ Reg = log Ihl]+log lh21 = log IhlhZI. Since the space HZ+C(G) is an algebra, hl h2 E H," - C(G)c H: C(G).

+

2.9.11. THEOREM [Stankov]. The spectrum s p ( H Z +C(G)) is homeomorphic to the set s p H g \ AG.

+ C ( G ) )-+

sp H Z defined as (v(p))(f) = p ( f ) for each f E H Z . q is a continuous : . We claim that q is mapping from s p (HZ C( G ) ) into s p H one-to one, and its range coincides with s p H Z \ AG. Let 9 1 , (p2 be two elements in s p (H: C(G))and assume that &(f) = &(f) for each f E H," where $ j = ~ ( p j )j , = 1 , ~ . Since vj(x-') = pj((X')-') = l/vj(x') = ~/$j(x'), we have that pl(x-') = cpz(x-'>. Consequently, (pl(fx-') = cpl(f)cpl(X-') = 972(f)v2(x-') = p2(fx-l) M for each f E H i , and whence 9 1 = 9 2 because of [HG ,a] = [HZ, x-'1. Let t,b = t,br,.g, for some r,, 0 5 ro < 1, and let q(p) = 1c, for some cp E s p ( H Z C(G)). Then ~ ( f=) f ( r , - go) for every f E HE. Clearly, r , # 0 because of Icp(x-')l = l/lp(xl)l = l/lxl(ro.go)l = l/ro. In addition, cp(x-')-l/x~(r,~g,) = l/r,1/xl(ro .go) = 0 in contradiction with the invertability of the function x-'-l/xl(ro-go) E C( G ) c H," +C(G).Consequently, no point in A , belongs to the range of q; and therefore, is an one-to-one mapping of s p H g C(G ) into s p H g \ A c . It remains to show that q is an "onto" mapping. If t,b E s p H Z \ AG, then according to (31) $ E Sg for some g E G. Hence T,($) = 1 - g; and consequently, g(a) = $(xa) whenever a E Q o and g(a) = g(-a) whenever a E -Qo by the definition of spectral mapping 7, of the standard subgroup 0 =

PROOF.Consider the mapping '7 : s p ( H ;

+

+

+

+

2.9. Algebras between. HZ and

LZ-

139

Consider the functional cp : A --$ C defined on the algebra A of functions of type fx-", where f E H l and n E 2, U {0}, as follows:

cp(fX--")= $(f>x--"(l 9). It is clear that cp(f) = cp(f(x-l)") = $(f)for each f E H Z . We *

claim that cp is a linear and multiplicative functional on A. Indeed, for n > k we have:

funcWe claim that cp can be extended as a linear multiplicative tional on the space H l C(G) = [H," ,x-'] = A. If fx-" is a fixed element in A, then

+

i.e. the functional cp is bounded on d;and therefore, it can be extended continuously on 2 = [ H z,x-'] = C(G)as a linear and multiplicative functional. The theorem is proved.

+

2.9.12. COROLLARY. The Shilov boundary a(H," of algebra H Z C(G)coinsides with s p LE .

+

+ C(G))

140

Chapter II. f -Analytic finctions in the Bin-Plane

SUP

ll~('p)ll=

QELZ

SUP

Ilf^(Oll =

(PEL;

(37)

-

SUP

VESP

w:

SUP

VE~H P g \AG

Ilf^(Cp)II

ll%)ll~

+C(G))

+

The first inequality shows that the space H Z C(G)is isometrically isomorphic to the space its Gelfand transforms. The second inequality implies that s p L ; is a boundary for the algebra H Z C(G). We conclude that a(H," C(G))= s p L z because of H Z C(G ) 3 HZ and i3HZ = s p L z .

+

+

+

Denote by H a ( & ) the weak*-closurein Lw (G) of the algebra AG, i.e. of the restricted on G big-disc algebra A ( A c ) , where da is the Haar (i.e. Lebesgue) measure on G. Let G be a compact abelian group 2.9.13. COROLLARY. whose dual group r possesses property (*). If in addition G # S', then H Z is a proper subalgebra of H m ( d a ) .

+ +

Indeed, as shown by Rudin the space H m (do) C(G) is closed in L w ( G ) if and only if G E S1 (and therefore, F = G 2 Z). According to Theorem 2.9.11, the algebra H Z C(G)is closed in the space Lz c Lw (G); and we conclude that H Z # H m ( d a ) whenever G # S'. h

141

2.10. Azwendix. Analutic measures

2.9.14. NOTESAND REMARKS. The case r = Q of the results in this section are due to Stankov [104], [105]. Corollary 2.9.13 is due to S. Grigoryan (personal communication). The proof presented here was hinted by K. Yale. For a survey picture on classical versions of these results (i.e. when I' = Z) see e.g. Hoffman [51],or Garnett [34]. Note that most of the classical theory of HP-spaces related with the unit circle S1 possess r-hyper-analytic analogues relative to the spaces HG = [ U HP o x"]. If the set E, runs over all aEr,

measurable subsets of G, then as in the classical Z-case one can show that the sets of type V = {'p E s p L l : 2,('p) = 0, s = 1,. . . ,n } ,where K , are the characteristic functions of sets of type ( x " ~ ) - ~ ( E form , ) , a basis of the topology in s p L l . As shown by Grigoryan [46], Lz = [AG,L r ] for every a E It turns out that in the case when the group F is distinct from Z but still possesses property (*), there arise three natural algebras of type H" related with the group G = p r . One of them is the algebra H l studied above, and the other two are: (1) The algebra H"(AG) of all bounded analytic functions on the open unit big-disc, and (2) the algebra H"(d0). Corollary 2.9.13 asserts that H l # H-(do). As shown by Grigoryan [46], H"(AG) # H Z . Little is known about properties of the space H" (AG) C(G).In particular, if it is a closed subalgebra of the space L" (G) or not is an open question.

r,.

+

2.10. APPENDIX.ANALYTIC MEASURES Let G be a compact Abelian group and let To be a fixed subsemigroup of the dual group 'I = 8 of G. Recall that the classical F. and M. Riesz theorem about analytic measures on the unit circle says that if G is the unit circle S1 and F, = Z, = Z+, then every complex Bore1 measure dp on G with zero

Chapter II. r-Analytic finctions in the Big-Plane

142

non-positive Fourier-Stieltjes coeficients crt”, =

J xn(g) dp(g) G

/

2?r

1 27r

=-

eintd p ( t ) , n E To \ {0} = Z+

\ {0}, is absolutely con-

0

tinuous with respect to the Haar (i.e. Lebesgue) memure do on

G. A pair ( G , K ) of a compact Abelian 2.10.1. DEFINITION. group G and a subset K of its dual group = we call a Riesz pair if every finite Bore1 measure dp which is orthogonal to K

r

(i.e. for which

J

x ( z ) d p ( s ) = 0 for every

x

E K ) is absolutely

G

continuous with respect to the Haar measure da on G. The F. and M. Riesz theorem cited above actually says that (S1, 2), is a Riesz pair. According t o Bochner’s generalization of K ) is a Riesz pair whenever K this F. and M. Riesz theorem, (T2, is a subset of the complement in Z2 = T2of a plane angle edged at the origin and less than 27r, where T2is the two-dimensional torus. In this appendix we give a general construction for Riesz pairs which leads to a generalization of Bochner’s theorem. Let the group T is provided with the ordering generated by the semigroup Toas discussed in Section 2.1. If this ordering is total, or complete, i.e. if ron (-To) = {0}, then (S1, 2), and (S1,-2,) are the only Riesz pairs of type ( G ,To),To c G. Here we study Ftiesz pairs related with incompletely ordered groups h

G. 2.10.2. DEFINITION. Let 2 be a partially ordered set and let R be a subset of 2. We say that 0 is low-complete with respect to the ordering in 2 if and only i f for any subset Y c 2, which is bounded from below by an element of 0, there exists

143

2.1 0. A p p e n d i x . A n a l u t i c m e a m r e s

the greatest among all lower bounds of Y which belong to the set

n\Y.

2.10.3. EXAMPLES.

1.) Let 2 be the standard Z-lattice Z2 in R2equipped with the partial ordering generated by the semigroup To = { ( n ,m ) E z2 : n 2 o}. In this case T,n (-T,)= {(o,n): n E Z} # 0. The set 52 = {(n,O) : n 0) is low-complete with respect to the To-ordering in Z2. Indeed, suppose that Y is a subset of Z2 which is bounded from below by an element ( n o ,0) of 52. This means simply that Y c { ( n , m )E Z2 : n 2 no, no 5 0} and it is clear that there exists a greatest among all lower boundaries of Y which belong to 52\Y,namely this is the point (n1,O)with n1 = max { n : (n,O)4 Y } .


o ) , where zoruns in X , form a basis of the topology of X . A minimizing s et for B is said to be each closed subset E of X such that f

j

~

2;I f ( 4 = 2;I f ( 4

for every nonvanishing on X function f in B. The next theorem is an analogue to the Shilov theorem for minimizing sets of function spaces.

3.1. THEOREM. Let X be a compact Hausdorff space, and let B be a function space on X . Then there exists a minimizing set for B which is contained in every minimizing set for B.

PROOF.Consider the set of all minimizing sets for B ordered ~ a linearly ordered system of under inclusion. If € = { E , } a E is {E,}, then F is a closed minimizing sets for B and if F =

n

aEI

and nonempty subset of X . For a fixed f E B the compact set M ( f ) = { z E X : f(z) = minIf(z)l) meets each set E, ZEX

in €, so it meets any finite number of sets in €. By the finite intersection property, M ( f ) meets also F ; and therefore, F is a minimizing set for B. According to Zorn's lemma there exists a maximal element E, (with respect to the inclusion) of the set of all minimizing sets for B , which, clearly, is a minimizing set for B as well. Fix an arbitrary minimizing set E c X for B . Suppose that E, \ E # 0, and let 5, E E, \ E. Let

V

= {z E

x : Igj(z,,)l

< E,

j = 1,. .. , n ,

E

> 01

be a basis neighborhood in X such that z, E V c X\E. Since E, is the maximal minimizing set for B , E, \ V is not a minimizing set for B. Hence there exists an f E B which does not vanish on X and such that

Chapter 111. n-Tuple Shilov Boundaries

151

Let k be a big enough positive number such that

on E, \ V and €or which If(z)l > E on X . Hence €or the function g = k f E B we have:

Let cy be a complex number with Icy1 = 1, and let j be an integer, 1 5 j 5 n. For every x E V we have

= min 19(X>I- ( xEE,\V

zy&I s d z > l + - - + xEE,\V m= *

> min Ig(z)I > min Ig(z)lzEX

Hence 1g(z)

x EX

+ agj(s)I > min Ig(z)l XEX

E

Isn(x)l)

e.

for every z E X and for

+

each j = 1,. .. ,n, because the functions g Lygj do not vanish on X . Let now z1 be a point in X such that

152

Chapter III. n-Tude Shilov Boundaries

For every j = 1, . . .,n we can choose complex numbers aj) lcyj I = 1 such that

wherefrom Igj(zl)l < e for any j = 1 , . . . ,n. We conclude that g ( z ) does not assume the minimum of its modulus outside V c X \ E in contradiction with the choice of E. The proposition is proved. 3.2. DEFINITION. The minimal minimizing set of a function space B on X is said to be the intersection a(')B of all minimizing sets of B.

3.1. n-TUPLE BOUNDARIES OF UNIFORM ALGEBRAS

Let A be a uniform algebra of complex-valued continuous functions over a compact Hausdorff space X . Clearly, A^ is a function space on spA. For the sake of simplicity we shall assume below that spA is identified with X ; and therefore, A^ 2 A and f^ = f for every f in A. As it is not difficult to be checked, the minimal minimizing set 8 ' ) A coincides with the Shilov boundary d A of A . Let F = (fl, . . . fn) be an n-tuple over a commutative Banach algebra B. There is a natural projection oF = a(fi,...,fn) of s p B into C",the so called spectral mapping of F , defined by a&> = O ( f l )...,fn,(4 = ( A W ,* * , K ( m ) ) = F ( m ) , m E sp B. )

153

3.1. n-tuple boundaries of uniform alaebras

3.1.1. DEFINITION. The range of the spectral mapping a, in C" is called the joint spectrum of the n-tuple F = (fl,. . . ,fn). The joint spectrum of the n-tuple F = (f1, . . . ,fn) is denoted by a ( F ) or a(f1,.. . ,fn). As a continuous image of a compact set, a ( F ) = .,(spB) = F ( s p B ) is a compact subset of C". In the case n = 1 the joint spectrum becomes the usual spectrum ~ ( fo)f f E B. It is clear that if n = 1 the spectral mapping a, coincides with the Gelfand transform ?of F = f E B. In the sequel B", n 2 1, will denote the set of all n-tuples of elements in B , i.e. the set of all n-tuples over B. Multi-tuple boundaries of Shilov type are connected with a multi-tuple version of inclusion (7) from Chapter I. h

3.1.2. DEFINITION. A subset E of s p A is an n-tuple boundary of a uniform algebra A i f the inclusion

F ( E ) 3 ba(F) = bF(spA)

(39)

holds for every n-tuple F = (f1,

.. . ,fn) of functions in A .

Observe that being a continuous image of a compact set, F( E ) is a closed subset of C" whenever E is a closed set in s p A . Therefore, a ( F ) \ F ( E ) is open in C" for every F E A" and for each closed subset E of spA. Let

11 . 11 be the Euclidean norm in C",namely

(x n

(40)

IIZII

= ll(z1,**.,zn)II =

IZjI2)',

zE

c".

j=l

An n-tuple F = (f1,. . . ,fn) over a commutative Banach algebra B is regular if the functions fi , . . . ,fn E B have no joint zeros in s p B , i.e. if Il$(rn)ll # 0 for each m E s p B . The set of all regular n-tuples over B will be denoted by B,". h

h

h

154

Chapter I l l . n-Tuple Shilov Boundaries

3.1.3. THEOREM. A closed set E in s p A is an n-tuple boundary of a uniform algebra A if' and only if

holds for every regular n-tuple F = (fl,. . . ,f n ) of functions in the algebra A.

PROOF.Let E be an n-tuple boundary of A. Then F ( E ) 3 ba(F) for every n-tuple F = (fl, . . . ,fn) of functions in A. Since min llzll = min llzll for any regular F E A: we have that

sEbdF)

EEU(F)

Consequently, the identity (41) holds for every regular n-tuple F E A:. Assume conversely that E is not an n-tuple boundary for A . By definition there exists an n-tuple F = (fi, . . . ,fn) E An such that F ( E ) ba(E). If z is a point in ba(F)\ F ( E ) c C", we define an n-tuple H E An by

If rn, E F-'(z), we have that H(rn,) = 0 = (0, ..., 0) E ba(F) since F( m,) = z E ba(H) and since a(H) can be obtained from a ( F ) by a translation with (-z) in C". Moreover, 0 4 H ( E ) since z $! F ( E ) . Hence

Because of z E ba(F) we can find a point ii outside a ( F ) and close enough to z such that the n-tuple = F - 2 also satisfies (43). Consequently, H is a regular n-tuple over A which violates (41). This completes the proof of the theorem.

-

3.1. n-tuple boundaries o f uniform alqebras

155

In other words, Theorem 3.1.3 asserts that every regular ntuple over A attains the minimum of its norm within any n-tuple boundary of A. Note that every multi-tuple boundary E is a boundary. Indeed, as it is not hard to be seen, every multi-tuple boundary is a 1-tuple boundary. Therefore, f ( E ) 3 ba(F); and consequently, The next theorem is a multi-tuple version of Shilov theorem for uniform algebras.

3.1.4. THEOREM [Tonev]. The intersection of all closed ntuple boundaries of A is a closed n-tuple boundary of A. We prove Theorem 3.1.4 by following the idea of the proof of Shilov theorem from Section 1.5. First we need the following:

3.1.5. LEMMA.Let A be a uniform algebra on X and let (44)

V = { m E s p A : I g j ( m ) ( > l , g j E e A , j = l ,...,k}

be a fixed basis neighborhood of Gelfand topology in s p A, generated by elements 91,. . . ,gk E eA. Then either V meets every n-tuple boundary of A or its complement E \ V in every closed n-tuple boundary E of A is d s o a closed n-tuple boundary of A .

PROOF.Let E be a closed n-tuple boundary of A for which E \ V is not an n-tuple boundary. If E \ V = 0, then V 3 E. Therefore, Igj(m)l > 1 on spA for every j = 1,.. .,k since Igj(m)l > 1 on the boundary E of A. Hence V = spA and of course V meets each n-tuple boundary of A. Let now E \ V # 0. According to the initial supposition E \ V is not an n-tuple boundary and therefore

156

ChaDter III. n - f i d e Shilov Boundaries

for some regular n-tuple F E A:. Observe that for every j = 1,.. . ,k the sequence { (gj(m))'/'} tends uniformly on s p A to 1 as t -+ 00. Henceforth there will be a number t o > 0 such that the sets { lgj(m)ll"o, m E s p A ) , j = 1,. . . ,k, are so close to 1 that the inequalities min llF(m)lllgj(m)l"'o

mEE\V

> 1, j = 1,.. . ,k

hold on E \ V. For each j = 1,.. . ,k the inequality Igj(m)l > 1, and together the inequality

(45) IIF(m)IIIgj(m)I"'O > 1 holds on V. Consequently, the inequality (45) holds on E as well. Because of gj E e A , by one remark in Chapter I there are functions hj E e A such that hfo = gj; and hence lhj(m)I = /gj(m)/l/'ofor each j = 1,. . . ,k. We conclude that the inequalities 1 < IIF(m)IIIgj(m)I1/'o = IIF(m)IIIhj(m)I = Ilhj(m>F(m>ll hold on the n-tuple boundary E for every j = 1,.. . ,k. According to Theorem 3.1.3, these inequalities hold on s p A as well, i.e.

min Ilhj(m)F(m)l( > 1, j = 1,.. . ,k,

mEsp A

because of the regularity of n-tuples h j F for each j = 1,. . . ,k. Let mo be a point in s p A such that IIF(mo)ll = 1. Inequalities (46)show that 1 < llhj(mo)F(mo)ll = Ihj(mo>l, .i= 1, * -

k, wherefrom 1gj(mo)l = Ihj(mo)lto > 1 for each j = 1,.. . , k. By 7

(44)we get that mo E V . Therefore, the positive function z t--) 11Ffrn)llattains its minimum only within V. We conclude that every n-tuple boundary E of A necessarily meets V because in the opposite case the point 1 E b F ( s p A ) will not belong to the range of E via F , i.e. E will not be an n-tuple boundary of A . Consequently, V necessarily meets every n-tuple boundary of A as desired.

3.1. n-tuple boundaries o f uniform alaebras

157

PROOFOF THEOREM 3.1.4. Let F E A: be a fixed regular n-tuple of functions in A, such that IIF(rn)ll > 1 on the intersection E, of all closed n-tuple boundaries of A. We shall prove that IIF(rn)ll > 1 on s p A under this assumption. Suppose that the set K = { m E s p A : IIF(m)ll 5 l} is nonempty and let m, E K . Since IIF(rn)ll > 1 on E,, there is a closed n-tuple boundary E of A which does not contain m,. Hence there is a basis neighborhood VEof m, of type (44) which does not meet E. According to Lemma 3.1.5, SPA\VE is dso an n-tuple boundary of A. Because of the compactness of K there are finitely many . .,.,V E in~ s p A , such that s p A\Ej are n-tuple open subsets V E ~ ~ Ej = 0 and whose union covers K . By boundaries of A , V E n Lemma 3.1.5 and by the induction argument it follows that the set sp A\

n

U V E is~ also an n-tuple boundary of A. Being fulfilled

j=1

on an n-tuple boundary, the inequality IIF(m)II > 1 holds everywhere in s p A because of the regularity of F . This contradicts our supposition on K ; and consequently, the function IIF(rn)ll attains its minimum within E,. This proves that the set E, is a (closed) n-tuple boundary of A, as claimed. 3.1.6. DEFINITION. The n-tuple Shilov boundary of a uniform algebra A is the smallest closed n-tuple boundary of A. The n-tuple Shilov boundary of a uniform algebra A will be denoted by d(")A.According to one of the remarks which follow Definition 3.1.2, we conclude that the 1-tuple Shilov boundary d(')A is exactly the usual Shilov boundary d A of A. The next theorem establishes an useful description of multituple Shilov boundaries. Given an n-tuple F = (fi,. . . ,fn) over A , denote by V ( f 1 , .. . ,fn) the vanishing set of functions fi,...,fn, i.e. V(fi,..., fn) = { m E spA : IIF(m)ll = 0 } = { m E spA : f ~ ( m=)f 2 ( m )= = f,(m) = 0 } = F-'(O). The set of all constant elements {Ae} in B will be denoted by B".

158

Chapter I l l . n-Tuple Shilov Boundaries

3.1.7. T H E O R E[Tonev]. M For every n 2 1 the set

coincides with the n-tuple Shilov boundary d(")Aof algebra A. The proof of this theorem is not less important than the theorem itself since it shows a very efficient way for introducing and investigating boundaries of Shilov type.

PROOF.Let F = (fi ,...,fn) be a regular n-tuple over A, and let rn, be an arbitrary point in S P A . Without loss of generality we can assume that fj(m0) = 0 for all j # 1. Indeed, by applying, if necessary, an orthogonal transformation, n

say

u = { C a i j f i ( m o ): aij E C,j

= I,. . . , n ) , in C" we can

k l

get U(F(rn,))j = 0 for all j f 1. Note that since U is an isometry in C " , the distance from 0 to U ( F ( m ) )in C" equals the distance between 0 and F(m). Therefore, U(F(rn))is also a regn

ular n-tuple over A and by replacing

fj

with x a i j f i for every i=l

j = 1,.. . ,n we can assume from the beginning that f j ( m o )= 0 for each j # 1. Clearly, IIF(mo)ll = If1(rn0)l. Consider the set V = V ( f 2 , .. . ,fn). The function f1 does not vanish on

Iv

V , which because of the A-convexity of V coincides with spAv. is invertible in A v . Since m, E V = sp A v , we have Hence f1 where f - l is the inverse elethat IfF'(rn,)l 5 max If;'(rn)l,

I

ment of

f1

I

mEaAv

in Av. Therefore,

3.1. n-tuple boundaries o f u n i f o n alqebras

= fin

{ IIF(rn)ll: E [

u

159

aAV(G)] }-

GEAn-l

Observe that, as a continuous function, IIF(rn)ll attains its minid A v ( ~ )for ] every regular n-tuple F mum within the set [ GEAn ~ A v ( G )is] an n-tuple boundary over A. Hence the set [

u

u

GEAn-l

of A; and consequently, it contains the n-tuple Shilov boundary d(")A of A because of the minimality property of d(")A. Conversely, let G = (gl, . . . ,gn-l) E A" \ A t . Suppose that V(G) n d(")A = 0. There exists a positive constant c > 0 such that IIG(rn)ll 2 c on d(")A. For each positive number d < c the n-tuple (G,d) = ( g l , . . . ,gn-l,d) is regular and ll(G(rn),d)ll = JIIG(rn)l12 8 > on d(")A. Hence ll(G(rn),d)ll 2 d m ' on s p A by the Theorem 3.1.3; and therefore, on V(G) we have d 2 in contradiction with the positivity of c. We conclude that V (G) n d(")A # 0 for every irregular G E A". Suppose that If(rn)l 2 r > 0 on V(G) n d(")Afor a function f in A whose restriction is invertible. Note that elements V(G) of this type are dense in (Av(G))*. For every positive c < r we can find a neighborhood V. of the set V(G) n d(")A such that the inequality If(rn)l > r - c holds on V,. Consequently, for any m E 8 " ) A we have

+

d

d

w

w

fl

for some positive constant C, large enough. Because, according to our hypothesis on f , the n-tuple (C,gl,. . . ,C,g,-l, f ) is regular, Theorem 3.1.3 implies that (48) holds on s p A . In particular on V(G) = V(g1,. . .,gn-l) we get If(rn)l > r - c ; and moreover, If(m)l r because of the liberty of the choice

160

Chapter III. n-Tuple Shilov Boundaries

We obtain that each invertible funtion in A v attains the minimum of its modulus on V ( G )n d(")A. According to Theorem 3.1.3, this means that V ( G )n a(")Ais an n-tuple boundary (i.e. a closed boundary) of A v . Hence V(G) n acn)A 3 aAv(G) for every ( n - 1)-tuple G of functions in A. Consequently, @")A 3 U ( V ( G )n a(")A)3 ~ A v ( G )and ; of

E.

u

GEAn - 1

therefore, 8 " ) A 3

[ U

GEAn-'

aAV(G)].The theorem is proved.

GEAn-'

Note that for every n-tuple F = (fi, .. . ,f n ) of functions in a uniform algebra A , the equality F ( a ( k ) A = ) F ( s p A ) holds for each k > n. Indeed, obviously F(O(k)A)c F ( s p A ) for each integer k. On the other hand

=F([

(J

21)

= F(SPA)

eEF(spA)

+

whenever k 2 n 1, as claimed. In particular f ( a c k ) A )= a ( ' ) for every function f in A and for each k > 1.

3.1.8. EXAMPLE. ~(")A(A") Let A = A ( A n ) be the algebra of all continuous functions on the closed unit n-dimensional polydisc A n in C" which are analytic in its interior. We claim that a ( k ) A= for each k > n and d("A consists of these points z = ( 2 1 , . . . ,z,) in A n , whose n - k 1 coordinates z j are of modulus 1 for k 5 n. Indeed, the first claim follows immediately from the last remark from above.

zn

+

3.1. n-tuple boundaries of uniform aloebras

161

Let k 5 n and, to be specific, assume that n = 3. If k = 1, then obviously 8 ' ) A ( A 3 ) = a A ( A 3 ) = T 3 = {(z1,22,t3) E

z3:

= 1221 = lz31 = l}, i.e. a(l)A(A3)coincides with the distinguished boundary of A3. If k = 2, then according to Theorem 3.1.7 1211

8 2 ) A ( A 3= )

U

[

BAv(q].

F € A 2( A s )

We claim that d ( 2 ) A ( A 3coincides ) with the set

Indeed, fix a number 230 in 2, and consider the function A ( A 3 )defined as f 3 ( 2 1 , 2 2 , 2 3 ) = 2 3 - 2:. Now

=

{(21,22,z;)

€a3:

(21,Zz)

€a2) d;

thus AV(f3)= A(V(F3))Z A ( A 2 ) ,and d A v ( f 3 = )

I=

E

f3

{ (z1,22, 2 : )

:

I = l} Z T 2 . Consequently, the set { ( 2 1 , 2 2 , 2 3 ) E z3: 1.~11= 1221 = l} is contained in the boundary U dAv(f), 121

122

f EA(A3) and therefore in the set d ( 2 ) A ( A 3 )In . a similar way one can show that all points in (49) belong to d ( 2 ) A ( A 3 )We . claim now that . an f in A ( A 3 ) .Since V ( f ) the set (49) contains d ( 2 ) A ( A 3 )Fix is a (two-dimensional) analytic subset of b( V (f)nZ3)c bA3 and If(z)l 5 max I f ( . ) [ 5 maxIf(z)l, according to the b( V (F)nx3) LA3 maximum modulus principle for analytic varieties, applied to the set V ( G )nz3. Consequently, d A v ( f )= a A ( V ( f ) )c bA3. Let ( z f , z ~ , z : ) E d A v ( f ) c bA3 and to be specific let 1 . ~ ~ =1

z3,

1. If V ( f ) coincides with the set

( ( Z ~ , Z ~ , Z J )E

Z3 :

23

=

ChaDter 111. n-Tuple Shilov Boundaries

162

n z3,then

A v ( f ) = A ( V ( F ) ) ;and hence a A v ( f ) is contained in the set { ( z ~ , z ~ , z ;€) : ( ~ 1 ~ 2 2 E) A 2 } which of course is a subset of (49). If V(f) f l {(z1,22,23) E A3 : 2 3 = 2 : ) is an one-dimensional analytic subvariety of the set n { ( Z l t Z 2 , Z 3 ) : 23 = z;}, then b V ( f )n { ( z l , z 2 , z ~ E) : ( 2 1 , ~ ~ E )A 2 } c b(-d3 n { ( z ~ , z z , z ~E~-21" ) : ( 2 1 , ~ ~E )A 2 } and according to the maximum modulus principle applied to the va2:)

x3

z3

z3

in the set

i.e. in the set (49). Finally, if k = 3, then according to TheU d A v ( f l , f 2 )We ] . claim orem 3.1.7, a ( 3 ) A ( A 3 )= [ (fl , f d € A 2 ( A 3 )

that a ( 3 ) A ( A 3= ) bA3. Indeed, by applying (39) to the identity mapping in C3 we get a ( 3 ) A ( A 33) bA3. Let ( f 1 , f Z ) E A 2 ( A 3 ) . Because V (f l , f 2 ) n z 3is an one-dimensional analytic subvariety , then b( V (f1 ,f2) n C bA3 and according to the maxin imum modulus principle for the variety V (f1, f 2 ) n , we have that a A v ( f i , f 2C) bA3. We conclude that 8 3 ) A ( A 3 c ) bA3, as desired.

z3

z3)

z3

Further we shall consider other examples of multi-tuple Shilov boundaries.

3.1.9. NOTESA N D REMARKS. The proof of Theorem 3.1 follows the same lines as Bear's proof for the existence of Shilov boundaries for function spaces from [14].The n-tuple Shilov boundaries Ckn)Awere discovered

3.2. n-tuple boundaries of function spaces

163

independently by Basener [9] (boundaries dn-lA) and Sibony I911 (boundaries En-lA) in 1975. Basener has reached them in the course of his investigation of A-variaties of finite codimensions and multi-dimensional analytic structures in the spectra of uniform algebras. Sibony has reached multi-tuple Shilov boundaries in the course of his study of plurisubharmonic functions. Description (47) of d(")A, which is due to Basener, is very useful for investigating multi-dimensional analytic structures in algebra spectra (see e.g. Chapter IV). Definition 3.1.2, Theorems 3.1.3, 3.1.4, 3.1.7 and Definition 3.1.6 are from Tonev [127]. In a more general context Theorem 3.1.4 is proved in Tonev [123]. More results about joint spectra of several algebra elements may be found in any book on commutative Banach algebras.

3.2.

n-TUPLE

BOUNDARIES OF FUNCTION SPACES

In this section we introduce multi-tuple boundaries and multituple Shilov boundaries for a function space B on a Hausdorff compact space X and investigate some of their properties. The definition is quite similar to n-tuple Shilov boundaries' description (47) for a uniform algebra, but instead of uniform algebra functions we use elements of an arbitrary function space B. Let B be a function space on X over C or R, consisting of C or R-valued functions. In what follows K will stand either for C or for R . As for the case of uniform algebras, given a compact subset K of X , by BK we shall denote the function space BK on K . 3.2.1. DEFINITION. The n-tuple Shilov boundary o f B we call

the set (50) d(")(B)=

[ Ud(l)BIV(F): F

where V ( F )= (fl,.

. . ,fn-1)-l(0).

= (fl,.

.. ,fn-l)

E

Bn-'],

ChaDter III. n-Tur>te Shilov Boundaries

164

3.2.2. THEOREM. d ( " ) ( B )is the smallest among all closed sets E in X such that the inclusion

holds for every n-tuple F E B".

PROOF.Note that the proof of Theorem 3.1.2 can be adjusted for the case of uniform algebras, considered in this section. We prefer to give below an independent proof, which in addition is more general. Let F = (f1, . . . ,fn) be an n-tuple over B and suppose that (51) is false. Let z, = F(s,) E b F ( X ) \ F ( B ( " ) ( B ) )for some z, E X. We shall make use of the following geometrical fact: Given a compact set K in K", then

where zik are the ik-th coordinate functions in K". Consequently, without loss of generality we can assume that z, E b(K n ( z 2 , . . . ,z~)-~(z,)), where K = F ( X ) . Hence

We conclude that (zo)l = fl(z,> E bfl(V), where V = - fn(Z0)). On the other hand z,

- f 2 ( 4 ,

* *

7

fn

4 F ( B ( " ) ( B ) )implies that

V(f2

E & ( V ) \ f1 (V n B(")(B))c bfl(V) \ fl (B(l)BlxnV).Without loss of generality we can assume (by adding some constant, if necessary) that fl # 0 on X n V . The last inclusion shows that (z,)~

3.2. n-tuple boundaries of function spaces

165

we can find a constant c E K \ f1 (X n V) such that the function g = fl -c attains the minimum of its modulus out of a(l)BlxnV, in contradiction with the definition of @")BI,,,. We conclude that (51) holds for E = a(")(B). Let now E be a closed subset of X for which (51) holds for each F E B". Take an arbitrary (n-1)-tuple G = (gl,. . . ,gn-l) of elements in B , and suppose that bf(V(G)) $ f ( E n V ( G ) ) for an f E B. There exists an 5, E X n G-l(O) such that f(zO)E b(f(V(G)) \ f(E n V(G)). BY bf(V(G)) c b(f, g1, * * * 9 9"-1 )(X) we have that (f(4 0, * * ' ,0) E b(f, 91 ,* * * ,gn-l)(X) \ (f,g1, gn-l)(E) in contradiction with (51). Hence bf(X n V(G)) c f(E n V(G)) for each f E B. Consequently, for each f E B which does not vanish on V(G) the function f l v ( G ) attains the minimum of its modulus within E n V(G). We conclude that E n V(G) 2 i3(1)Blv(G) for every ( n - 1)-tuple G of functions in B . Consequently, "')

By taking closures in both sides we get

The theorem is proved.

A thorough examination shows that the above proof holds for spaces of functions B which satisfy the following two conditions: (1) B is a uniformly closed in C(X) point separating space of K-valued continuous functions on a compact Hausdoflspace X (K is either R or C),which contains all translations with scalars in K of its elements; (2) Foreverysubset K o f X oftypeV(f1, ...,f,) = ( f i , ..., fn)-l(0), where fj E B , there exists a smdlest among

Chapter ZIZ. n-TuDIe Shilov Boundaries

166

all closed subsets of X (say a ( l ) B ~on) which dl nonvanishing functions in B assume the minimums of their moduli. By # F will be denoted in the sequel the cardinality of a given set F c A. Sometimes we shall prefer to write (47) in the following form:

In particular, (52) implies the following chain of inclusions:

(53)

aB

c

c a(2)Bc - - - c &")B c - - . c X .

3.2.3. DEFINITION. A subset E of X is an n-tuple boundary o f B if (54)

b F ( X ) C F ( E )f o r every F

ere F

cB

= {fj>Tz ~ ( s=) (fi(z), .

-

with # F 5 n.

and F ( X ) c K#F. Theorem 3.2.2 asserts that 8 " ) B is the smallest closed n boundary of B. Inclusion (54) shows that every n-tuple boundary is also a k-tuple boundary for each k = 1 , 2 , . . . ,n - 1. *

f#F(s))

Unless otherwise stated, 11 . 11 will denote in the sequel a fixed but arbitrary norm in C" which is equivalent to the Euclidean norm. Proceeding as in Theorem 3.1.2 we can obtain several rather general characterizations of multi-tuple boundaries of a function space.

3.2.4. PROPOSITION. A closed subset E of X is m n-tuple boundary of B if and only if the inequality (55)

3.2. n-tuple boundaries of function spaces

holds for every subset F of B with # F F ( E ) n b F ( X ) # 0 and the equality

is fulfilled for every F C B with # F

167

5 n, or equivalently, if

5 n.

PROOF.Let E be a closed n-tuple boundary of B . Then F(E) 3 b F ( X ) for every subset F of B with #F 5 n; and consequently,

i.e. (55) is fulfilled. The equality (55) now follows immediately because of F(E)f l b F ( X ) = b F ( X ) . Assume that E is not an n-tuple boundary of B. By definition there exists a subset F = {fi>j”=”1c B, # F 5 n such that b F ( X ) F(E).If x o is a point in X for which F ( z o )E b F ( X )\ F ( E ) , then we have T ( z o )= 0 E b T ( X ) for the following subset T of B: (57) because of T ( X ) = F ( X ) - F ( s o ) . Moreover, 0 of F ( z o )# F(E),and thus

Consequently, the set T c B with #T = satisfy (55). According to (25)

4 T ( E )because

#F 5

n does not

i.e. T does not satisfy (56) as well. The proposition is proved.

Chapter ZIZ. n-Tuple Shilov Boundaries

168

Denote by e(zl,az) the metric which is generated by a fixed norm I[ . 11 in Kn, i.e. e(zl,s2)= 11z1-22II. We can rewrite (55) and (56) respectively as follows: e ( 0 ,F ( E ) ) 5 e(0, b F ( X ) ) for every

F

cB

with # F 5 n; (59)

e ( 0 ,F ( E ) nb F ( X ) ) = e ( 0 , b F ( X ) ) for every

F c B with #B 5 n. For a given positive number c > 0 and integer k 2 1 we will denote the open ball { z E Kk: llzll < c] with radius c which is centered at the origin 0 of Kkby Bk(0,c). 3.2.5. COROLLARY. A closed subset E ofX is an n-boundary of B if and only if for every subset S c B with # F 5 n the balls

are either entirely inside or entirely outside the set F ( X ) .

PROOF.If 0

4 F ( X ) , then

since F ( E ) c F ( X ) ; and moreover, e(O,bF(X)) = e(O,F(X)) 5 ~ ( 0F , ( E )f~b F ( X ) ) . In this case (59) takes the form

169

3.2. n-tuple boundaries o f function spaces

Consequently, in this case E is an n-boundary of B if and only if both balls from above are outside the set F ( X ) . If 0 E F ( X ) , then

e ( o , b W ) ) = e(o,K# F \ F ( x ) ) I e(0, F ( E )n W X ) ) and (59) now takes the form

e(0, V

)5)e(0,F ( E )n W X ) ) = e(o, b F ( X ) )

= Q(o, K # \~F ( x ) ) , i.e. B#F(o, = B#F(O,

min

Ilzlr>

BEbF(X)

min 11~11) c F ( X ) . n b F (X )

B E F (E )

Since B # F ( o ,

min

EEF( E)nbF(X )

11~11) 2 B#F(O,minIIF(x)II), xEE

we see

that B # ~ ( O , m i nllF(x)11) C F ( X ) as well. Consequently, in this GEE

case E is an n-boundary of B if and only if both balls from above are inside the set F ( X ) . The corollary is proved. For a given n 2 1 consider the following classes of n-tuples over B: (1) B," = { F C B : # F = n, 0 $! F ( X ) } - the class of all reguhr n-tuples over B. Equivalently, F E B,"if and only if F is not contained in any maximal ideal of B; (2) B,"= { S c B : #F = n, 0 E b F ( X ) } - the class of these n-tuples over B for which the origin of K#F is contained in the topological boundary of the set F ( X ) ; (3) B" \ B,"= { F C B : 0 E F ( X ) } - the class of all irregular n-tuples over B.

3.2.6. COROLLARY. A closed subset E of X is an n-tuple boundary of B if and only if E satisfies one of the following equivalent conditions: (1) min llF(x)11 = min llF(x)11 for every regular F c B with zEE

# F I n;

XEX

170

Chapter III. n-Tuple Shilov Boundaries

(2) F vanishes on E for every F E BH" with # F 5 n; (3) B#F(O,minllF(5)11) C F ( X ) for every irregular F C B ZEE with # F 5 n.

PROOF.If E is an n-tuple boundary, then according to Proposition 3.2.4 the equality (55) is fulfilled for every subset F of B with # F 5 n. In particular, (55) is fulfilled for each F c B , # F 5 n which belongs to every one of the classes B?", B f " or Bf" \ BF". As it is not hard to see, applied to each of these classes the equality (55) takes the forms (l), (2) or (3) respectively. Assume that 33 is not an n-tuple boundary of B. Then there is a set F in B with #F 5 n such that bF(X) $ F ( E ) . Proceeding m in Proposition 3.2.4 we can find a set T E B f " which satisfies (25), i.e. for which min llT(z)11 > min ilzll = 0. This ZEE

5EbT(X)

contradicts with the property (2) from above and also the property (3) as well, because of B,#" c B#" \ B?". Indeed, applied to the class B#" \ B,#" the above inequality shows that no ball with nonzero radius are contained in T ( X ) . By applying a translation with a suitably chosen point z in K#" \ T ( X ) we can get a regular subset T - z of B with

i.e. for which min llT(x) ZEE

zII

>

min

sE( T - s ) ( X )

llzll in contradiction

with (53). Observe that, similarly to the case n = 1, B," is a dense subset of the topological boundary bBF of BF with respect to the coordinate convergence. Indeed, if F E B,", then 0 f b F ( X ) and F = lim F f z k for some suitable chosen Z k E K" \ F ( X ) which k+m

+

tend to 0 . Since F z k E B," and F E B" \ B,", it is clear that F E bB,". Hence B," C bB2. Let now F E bB,",let Gk + F for

171

3.2. n-tuvle boundaries o f fiinction spaces

some Gk E B," as k + 00 and let rk = ,o(O,Gk(X))> 0. Clearly rk + 0. If zk are points in bGk(X) such that e(0, G k ( X ) ) = rk, then the functions Hk = Gk - zk belong to B: and H k -+ F, i.e. bB," c as claimed.

xr,

Since, by definition, the n-tuple Shilov boundary a(")Bis the smallest closed n-tuple boundary of B , we get the following

3.2.7. THEOREM (Global characterizations of n-tuple Shilov boundaries). Let 11 . 11 be an arbitrary norm in K" which is equivalent to the Euclidean norm. The n-tuple Shilov boundary 8 " ) B of a function space B coincides with the intersection of all closed subsets E of X which satisfy one of the following equivalent conditions: (1) bF(X) C F(E) (equivalently, F ( X ) \ F(E) is open in K") for any F c B with #F I n; (2) minIIF(x)II 5 min llzll for eacb F C B such that zEE

zEbF( X )

#F I n; (3) F(E)nbF(X) # 0 and

min

s € F ( E ) n b F.( X,)

for every F C B with #F

i k;

llzll

llzll = .

I

(4) e( 0,F(E)) I e(0, b F ( X ) ) for any F c B with #F I n; ( 5 ) F(E) n bF(X) # 0 and e(O,F(E) n bF(X)) = e(0, b F ( X ) )for every F C B with # F 5 n; (6) Both open (in C") balls B#F(O, min llzll) and zEF(E)nbF(X)

B#p(O, min llF(z)11)lie either entirely in, or entirely out Z€E

of the set F(X) for each F c B with #F 5 n; (7) min l l ~ ( x ) 1 1 =min ll~(x>11 for every F E B,#" such that zEE

zEX

#F I n; ( 8 ) Every F E Bf" with #F 5 n vanishes within E; (9) B#F(O,min ll~(z>11)c F ( X ) for every F E B#F\ ~ ZEE with #F 5 n.

f "

172

Chapter 111. n-TuDle Shilov Boundaries

The case (7) from Theorem 3.2.7 in particular says that the ntuple Shilov boundary a(")Bis the smallest among all closed subsets o f X on which the function llF(z)11 = Il(fl(x), . . . ,fk(x))ll assumes its minimum for every regular subset F c B with #F = k 5 n. Note that for uniform algebras we know this result from Section 3.1, but for the Euclidean norm only. Now we conclude that &")B is the smallest among all closed subsets of n ..

X on which every one of the functions mnaX { lfj(z)I), j=l

lfj(x)l, j=1

Observe that the case n = 1 of Theorem 3.2.4 applied to the algebra of Gelfand transforms of a commutative Banach algebra B over s p B coincides exactly with Theorem 1.5.11 for the Shilov boundary dB.

3.2.8. THEOREM [Tonev]. Let n, k, 1 5 k 5 n , be two fixed numbers. Then

PROOF.Let

11 . 11

be the Euclidean norm in K", let F = ( f ~. .,. ,fn) be a regular n-tuple of functions in B and let zo be a.n arbitrary point in X . Without loss of generality (applying, if necessary, an orthogonal transformation in K") we can assume that fj(50) = 0 for all j > k SO that (lF(~,)ll= ll(fl(5,),...,f~(~o))II. Consider the set V = V ( f i c + l , . . . , f n ) . Because of x, E V = X and by the Theorem 3.2.7(7), the regular k-tuple (fll,, . . . ,fkIv) E ( B v ) satisfies ~ the inequality

3.2. n-tuple boundaries of function spaces

173

Hence

j=l

j= 1

= min { ll~(x>11 :x E

[

U

d(")BV(G)]).

GEBn-k

We obtain that every regular n-tuple F ( x ) over B attains the minimum of its absolute vdue within the set [ d(k)BV(G)].

u

GEBn-k

Hence this set is an n-tuple boundary of B; and consequently, it contains the n-tuple Shilov boundary a(")Bof B because of the minimdity property of 8 " ) ~ . Conversely, let G = ( 9 1 , .. . , g n - k ) E Bn-k \ Bc-k. Suppose that V ( G )n a(")B = 0. There exists a positive constant c > 0 such that 11G(x)11 2 c on a(")B. For any positive number d < c the n-tuple (G,d ) = (91,.. . , g n - k , d,O,. . . , 0 ) is regular and ( I ( g l ( x ) , . . . , g n - k ( t ) , d , O , - - * ~ O ) l l J]lG(Z)I12 + d 2 v'Zi-3 on 8 " ) Hence ~ . ( g l ( x ) , . . ., g n - k ( x ) , d , ~ .,. . ,o) 2 on X , according to Theorem 3.2.7(7). Therefore, on V ( G )we have d 2 d m in contradiction with the positivity of c. We conclude that V(G) n 8 " ) A # 0 for every irregular G E B"-'. Suppose that ll(fl(z),. . . , f k ( z ) ) 2 r > O on V ( G )n 3(n)Bfor some Ic-tuple (fl,. . . ,f k ) E B k , whose restriction (fi . . ,f k l v ) on V = V ( G ) ,G E B"-k\Bz-k, is regular. For every positive E < r we can find a neighborhood V, of the set V ( G )f l a(")B,such that the inequality If(x)l > r - E holds

11

4 -

>

11

11

lv,.

>

174

Chapter III. n-Tuple Shilov Boundaries

for each m E d(")B. Therefore, the inequality l l ( f i ( z ).,. . , f k ( z ) , C , g i ( Z ) , . . . ,c,gn-k(x))II > r - & holds on X because the n-tuple ( f i , . . . ,fk, Cegl, . . . ,Ccgn-k) is regular. In particu1 x 7 On V ( G )= V(g1,- - * 9 g n - k ) we have ( . f l ( x ) , 7 f k ( x ) ) > r - E ; and moreover, ( f l ( z ) , . . . ,fk(z))II 2 T because of the liberty of choice of e. We obtain that every regular k-tuple Over the 'Pace BV(gl,...,gn-k) attains the minimum of its norm on the set V ( G )n d ( n ) B . According to Theorem 3.2.7(7) the (closed) set V ( G )n 8 " ) B is a k-tuple boundary of the space Bv. Hence V ( G )n &")B 3 d ( k ) B V ( G ) for every (n - k)-tuple G = ( 9 1 , . . . ,g n - k ) over B. Finally, from

11

11

8 " )3~

U

(v(G) na(")B) 2

GEBn-k

we have that

a(n)B3

---

11

U a(k)~V(G) GEBn-k

[ U

a(k)BV(G)].

The theorem is

GEEn-'

proved. Observe that if k = 1 the equality (60) restricts to (50). From the proof of Theorem 3.2.8 we can extract even stronger results. Namely,

3.2.9. PROPOSITION. The set

is a (possibly non-closed) n-tuple boundary of B.

3.2. n-tuvle boundaries of function spaces

175

PROOF.Subject to an easy verification, one sees that E c X is a (non closed in general) n-tuple boundary of B if and only if for every regular n-tuple F E B,",there exists an x, E E such that llF(xo)ll 5 llF(x)11 for every z E X , where 11 . 11 is the Euclidean norm in K". Following the same line of proof as in the first part of the proof of Theorem 3.2.8 (with k = 1) we observe that for every point x E X and for every regular n-tuple F of functions in B with F ( z ) = (fl(z),0,. . . ,0) the inequality n

xi E dBv(f2,...,fn) C

u

G E Bn- 1

dBv(G). Therefore, the set (62) is

an n-tuple boundary of B.

The next characterizations of the points in the multi-tuple Shilov boundaries of a uniform algebra follow immediately from Theorem 3.2.7.

3.2.10. COROLLARY (Local characterizations of points in d(")B). Denote by 11 . 11 an arbitrary norm in K" which is equivalent to the Euclidean norm I . I in K", 1 5 m 5 n. A point x, in X belongs to the n-tuple Shilov boundary d(")B of a function space B if and only i f for each neighborhood U of x,, there exists a subset F of functions in B with #F 5 n which satisfies one of the following equivalent conditions: (I)F ( U ) n b F ( X ) # 0 , F ( X \ U )n b F ( X ) # 0 and min < s € F ( X \min U)nbF(X) 1 141; E E F ( v ) n bF( X ) (2) F is regular and min llF(x)11 < min llF(x)11;

Z€B

(3) F E B, and F ( x ) # 0 on X

zEX\U

\ U.

PROOF.Let U be a fixed neighborhood of the point x, E a(")B. Assuming that for some subset F c B with #F 5 n the condition (1) fails to be fulfilled, by Theorem 3.2.4(3) we obtain

176

ChaDter III. n-Tude Shilov Boundaries

that a(")B\ U is also an n-tuple boundary of B in contradiction with z, E d(")B. Conversely, let U be an arbitrary neighborhood of z,. Denote by F a subset of functions in B for which # F 5 n and for which the condition (1)holds. Clearly, the set of points in F-'(bF(X)),where llF(z)11assumes its minimum, is contained ; consequently, entirely in U. Thus U necessarily meets d ( n ) B and z, lies in d(*)B because every its neighborhood intersects the closed set d(")B. The other cases can be proved in a similar way. Observe that the case n = 1 of Corollary 3.2.10 applied to the algebra of Gelfand transforms of a commutative Banach algebra coincides exactly with Corollary 1.5.12 for the usual Shilov boundary dB. 3.2.11. EXAMPLES . . 1.) d ( k ) B where , B c C ( K ) ,K C K".

Let B be a function space on K c K" which contains all coordinate functions z I-+ z j , j = 1 , . . . ,n. Because the identity mapping id : K + K now belongs to B", the inclusion (39) applied to it implies that 8 " ) B 2 bK. Moreover, the remark which preceedes Example 3.1.8 (applied again to the identity mapping id) shows that d ( k ) B= K for each k > n. If in addition B c A ( K ) , then d(")B = bK. Indeed, assume for a while that d(")B # bK, and let zo E d(")B\ bK. Let U be a neighborhood of z, in d(")B\ bK. By Corollary 3.2.10 there exists a regular n-tuple F = (fi,. . . ,fn) of functions in B for which ZEU

llF(z>Il < z?jyu llF(z>lI.

Therefore, on V = V ( f 2 - fi(z,), . . . ,fn - fn(zo)) we have

3.2. n-tuvle boundaries o f function svaces

177

According to the maximum modulus principle for the variety K n V the set Vr-7V meets b(K n V ) c bK in contradiction with the choice of U . We conclude that a(")B = bK. According to (52) a("B c bK for every k 5 n; and similarly to the remark which precedes Example 3.1.8, one can see that d("B = K for any k > n. 2.) a(")P(A") Let X be the Hilbert cube, i.e. the infinite-dimensional polydisc

Am = {Z = ( ~ 1... , , ~ j , . .) . E C" : maxIzjI < 1). 00

j=l

-"

Equipped with the product topology A is a compact subset of CaY.Let B = P(Am) be the space of continuous functions on -aY -00 A which can be approximated by polynomials in C" on A . As it is not hard to see, aB ="'2 = { z E 2" : Izjl = 1 for every j E Z). Also d(")B = { z E 2" : lzjl # 1for at most n - 1 coordinates zj ) . 3.2.12. COROLLARY [Tonev]. The n-tuple Shilov boundary d(")B is the smallest closed subset of X which meets every nonempty set U c X of the following type:

(63) U = {m E X : llF(x)11 < e, F E B,#',

#F 5 n, E > 0).

PROOF.Assume that a set U of type (63) does not meet d(")B. Then llF(x)11 2 e on d(")B;and therefore, the function llF(x)11 does not attain its minimum on a(")B despite of the regularity of F, which is absurd (e.g. by the Theorem 3.2.7(7)). If a closed set N is contained properly in 8(")B,then according

178

ChaDter III. n-Tuple Shilov Boundaries

to Corollary 3.2.10(2) there exists an F which

c B,

with # F 5 n for

Hence the set { z E X : liF(z)11 < min llF(z)11},which is of type zE N

(30) and nonempty, does not meet N . The corollary is proved. Recall that according to a uniform algebra version of classical RouchC's theorem if f , g E A and if the inequality

I f W - d m ) l < If(m ) + s(m>l holds on the Shilov boundary dA, then both f and g are simultaneously invertible or non-invertible elements of A . Next theorem shows that vector-valued functions over a function space B behave in a similar way. Namely,

3.2.13. THEOREM (Multi-tuple Rouche"s theorem). Let F and G be two n-tuples over a function space B on X . If the inequality

(64)

IIF(4- GWll < llF(z)+ G(.>ll

holds on the n-tuple Shilov boundary a(")B,then F and G simultaneously vanish or do not vanish on

x.

PROOF.Because of the compactness of X we can find an integer k such that

Assume that the statement is false. Then the first and the last member of the finite sequence

2kF, (2k - l ) F

+ G, (Zk - 2 ) F + 2G,. .. ,F + (2k - 1)G, 2kG

3.2. n - t u p l e boundaries of f u n c t i o n spaces

179

are not simultaneously regular or irregular n-tuples over B. Consequently, there are two neighboring members in this sequence, one of which vanishes and the other one does not vanish on X . To be specific let (k - Z)F (k Z)G does not vanish and let (k- Z + l ) F + ( k + Z - l)G vanish on X for some k. We have

+ +

Is

Since (k-Z)F+(k+l)G is a regular n-tuple over B, this inequality holds everywhere on X by the Theorem 3.2.7(7). Therefore, if z o is a point in X for which (k - Z l)F(zo) (k Z - l)G(zo)= 0, we obtain:

+

+ +

i.e. an absurd.

3.2.14. PROPOSITION. The function min { Ifj(rn)l}? 3=1 as3

sumes its maximum on the n-tuple Shilov boundary d(")B for every n-tuple (f1, . . . ,f n ) over B.

Chapter III. n-Tuple Shilov Boundaries

180

PROOF.Let F = ( f i , . . . ,f n ) E B", and suppose for a while that min (Ifj(m)l};=l 5 T on acn)Bfor some r > 0. Let x, E I

X . Without loss of generality we can assume that Ifi(x,)l 5 min {/fj(z)i};=2.Denote V = V ( { f j- fj(zo)}yii). On V n 3

acn)Bwe have: Ifi(z)l = min { l f j ( ~ ) l } ~ = I l r ; and therefore, I

If~(x)l5 r on V because dBv c V n O(")B. Since z, E V, we have r > Jfi(xo)l = minj { Ifj(zo)l);=,. Hence min { lfj(z)I)" 3-1

5

r on X, and this completes the proof.

3

3.2.15.EXAMPLE. Minimal n-tuple f i e boundaxies. Let M be a compact set in a real locally convex linear topological space V , and denote by A ( M ) the set of restrictions on M of all real &e continuous functions in V, i.e. f E A ( M ) iff f is continuous and f ( t z (1 - t)y) = tf(z) (I - t)f(y) for every t E R. Remember that a subset N of M is called an end subset of M iff it consists of points z which satisfy the following condition: z can not be represented as z = Ax p y with X > 0, p > 0, X p = 1, unless z and y belong to N . The extreme points of M are the points which are end subsets of M . Let EM stand for the closure of extreme points of M . Obviously, A ( M ) is a function space on M . We claim that if M is convex, then E M is the smallest closed subset of M on which every nonvanishing convex function in A ( M ) attains the minimum of its modulus, i.e. that &')A(M) = E M . Indeed, let f E A ( M ) , f > 0 and let m i n f ( z ) = a < b = min f(z).

+

+

+

+

zEM

=€EM

Since f is f i n e , ( 5 E M : f(z) 2 b } is a compact convex set which contains E M ;and consequently, by the famous KreinMilman's theorem, it contains also the closed convex hull of EM. In particular, the set { z E M : f(z) 2 b ) contains the whole set M . Hence f(z) 2 b > a on M , which is a contradiction. So every positive convex function in A ( M ) attains its minimum within E M . If a closed subset N of M possesses the same property,

3.2. n - t u v l e boundaries of function spaces

181

-

then its closed convex hull ( N ) coinsides with M . In fact, if ( N ) # M , we can find a positivecontinuous f f i e function f in V for which f(z) 2 a > 0 on ( N ) but which is less than a for some point inM in contradiction with our supposition on N . The equality ( N ) = M implies N 3 E M because, according to the Milman's theorem, the latter is the smallest closed subset whose closed convex hull coincides with M . For nonconvex connected sets M we define EM to be the set E ( g ) . It is possible to describe also the smallest closed subset EM of M on which all nonvanishing functions in A(M) assume the minimum of their modulus for disconnected compacts M . Note that EM is also the smallest closed subset of M on which every function in A ( M ) attains the maximum of its modulus, i.e. EM = a(l)A(M)= aA(M). The n-tuple Shilov boundary of A ( M ) is called also minimal n-tuple afine boundary of M . By definition this is the set

where AF-'(M) is the set of all non-vanishing on M (n - 1)tuples of real af€ine functionals on M , and Z(g1,. . . ,gn-l) is the set of common zeros of functions gl, . .. ,gn-l in M . Note that E,(M) may differ largely from E M . All previous results in this section hold for the function space

B = A ( M ) c C ( M ) with a(")A(M)= E,(M). Below we state some of them.

3.2.16. THEOREM [Tonev]. Let f 1 , . . . ,fk be k &e functions in A(M) without joint zeros on M , k 5 n. Then each one of the functions

(1) (2) (3)

fl(.)2 + ... + fk(.)2; Ifl(4l+ - + l f k ( 4 I ; 2%(Ifl(4L lfd.>I); *

*

(4) Ifi(z>Ip+ * * -

* * * 7

+ Ifk(s>lp

for any p 2 I,

182

ChaDter III. n-TuDle Shilov Boundaries

attains its minimum within the minimal n-tuple affie boundary E n ( M )of M . En(M ) is the smallest closed subset of M on which these minimums are attained for each k-tuple ( f l , . . . ,fk) E A k ( M ) , k 5 n, without common zeros on M . From (53) we have

EM = E l ( M )C E z ( M ) C

. . .E , ( M )

C

bM c E,+1(M) = M ,

and Theorem 3.2.8implies that

3.2.17. THEOREM (Global characterizations of minimal n tuple afine boundaries). Let 11 . 11 be an arbitrary norm in R" which is equivalent to the Euclidean norm. The minimal affine n-tuple boundary E , ( B ) of M is the smallest among all closed subsets E of M which satisfy one of the following equivalent conditions: (1) The inclusion b ( f 1 , . . . ,f n ) ( M )c ( f l , . . . ,fn)(E)holds for every n-tuple ( f 1 , . . . ,f n ) of affine functions in A ( M ) ; (2) Every n-tuple of &e functions (fi,. . . ,fn) in A ( M ) with 0 E b ( f 1 , . . . ,f n ) ( M )has a joint zero on E ; ( 3 ) Each point in the ball Bn(O,min I l ( f l ( z )., . . , ZEE

fn(+))l l )

in C" is a vdue for the n-tuple (fl(z), . . . ,fn(z))for some z E M for any n-tuple (fl ,. . . ,fn) of affine functions in A ( M ) with common zeros in M . (4) E meets every nonempty open subset of type U = { z E M : II(fl(.),...,fk(.))Il < e, 5 72, E > 0) of M , where fi, . . . ,fk are &e functions in A ( M ) .

3.2.18. COROLLARY (Local characterizations of points in E n ( M ) ) . The minimal n-tuple &e boundary E , ( M ) of A4

183

3.2. n-tuple boundaries o f function spaces

consists of points x, E bM such that for every neighborhood U of x,, there exist: (1) k a d h e functions fi,. . . ,fk in A(M), 1 5 k 5 n, without common zeros, such that min (f;(x) * fi(x))

+ +

Z€iJ

+ --+

fl(x) for every 5 E M \ U ; < f;(x) (2) k a d h e functions f ~. .,. , f k in A(M), 1 5 k 5 n, with 0 E b(f1,. . . ,fk) and without common zeros on M \ U . 3.2.19. THEOREM (Rouche' theorem for multi-tuples of afine functions). Let f 1 , . . . ,fn and 91,. . . ,gn be two n-tuples of f i e functions in A(M). If the inequality

holds on the minimal n-tuple a d h e boundary E,(M), then both ( f i , . . . ,f n ) and (gl, . .. ,gn) have or have not common zeros in M simult meously. 3.2.20. EXAMPLES. 1.) The smallest among all closed subsets E of M that the equality

cV

such

holds for every complex-valued real-afine functional f E A( M )8 C which does not vanish on M is neither E ( M ) nor M , but

E'(M).

More generally, the smallest among the closed subsets E of M such that

184

Chapter ISI. n-Tude Shilov Boundaries

for every non-vanishing on M n-tuple (f1,. . . ,f n ) of complex valued real-&ne functionals fj E A ( M ) 8 C is the set E2n(M). Indeed, applying (50) and (40) to the space A ( M ) 8 C, we have:

~(")(A(M 8)C ) =

[ IJ{ ~ ( " A ( M )B cIclnz(gl,...,gn-l) :

2.) The n-tuple Shilov boundary 8 , ) A C ( M ) of the space of complex-afhe functionals on M is the set

3.3. Properties of n-tuple Shilov boundaries

185

3.) Let A be a uniform algebra. Then s p A is a weak*compact subset of A*, the dual of A. Let A' c A(A*ISpA) be the natural image of A into A** (i.e. A + A' C A** : u H d j a , where Ga(cp) = y ( u ) for every cp E s p A ) . The set B = A'lSpAC A C ( s p A ) is a function space of complex-affhe functionals on s p A c A*. According to the previous example, d(")(A'IspA)c E z ( M ) . Since AllspA 3 A^, it holds that $")A = &")A' = 8 n ) ( A ' l { A * , )) c 8 " ) ( A C ( s p A ) ) ;and *PA

therefore, d(")A c E2,,-1(spA), since every minimizing set for A'I, G XIK, K c s p A c A*, is a boundary for A'lK. 3.2.21. NOTESAND

REMARKS.

For the case of uniform algebras Theorems 3.2.2, 3.2.7, 3.2.8, Proposition 3.2.4, Corollaries 3.2.5, 3.2.6 and 3.2.10 are proved in [126], [127]. For more details about algebra P ( A m )we refer the reader to Stout's book [107]. The applications to convex analysis, presented in this section, are based mainly on Tonev [128], [134]. For Krein-Milman's theorem we refer the reader e.g. to Yosida [145]. Milman's theorem can be found e.g. in Phelps [79]. Subject to an easy verification is the fact that for convex sets M the boundaries E,(M) are the closures of all end subsets in M which are contained in (n- 1)-dimensional a&e subspaces of V . A uniform algebra version of multi-tuple Rouch6's theorem was proved by Corach and Maestripieri [26], and a multi-tuple a f h e version of it was proved in [128].

3.3.

PROPERTIES OF

TUPLE

SHILOV

BOUNDARIES

The purpose of this section is to establish properties of multituple boundaries which are connected with the boundary behaviour of vector-valued functions over a uniform algebra. We

186

Chapter IZI. n-Tuple Shilov Boundaries

give also some properties of multi-tuple boundaries penetrating subideals and superalgebras of a given uniform algebra. As mentioned before, $')A = a A . For the sake of simplicity throughout this section 11 . 11 will denote the Euclidean norm in C". 3.3.1. PROPOSITION.Let F be an ( n - 1)-tuple over A , n 2 1, and let g be an arbitrary function in A. Then the inequality

holds on s p A whenever it holds on d(")A.

PROOF.Let F = ( f l , . . . , f n - l ) E A"-l and suppose that ]g(m)l 5 IIF(m)ll on a(")A. For afixed point m, in s p A consider the set

v = V ( F - F(m0)) = v({ f j - fj(mo)>j=l 1 ). n-

Obviously, V is nonempty and m, E V. On V n a(")A (which according to ( 4 3 ) contains d A v and therefore is nonempty) we have: (67)

Ig(m>l I IIF(4ll =

IIFhJ>ll.

Since (67) holds on d A v , it holds also on V = s p A v and in particular at mo,i.e.

Is(mo>l5

IIF(mo>ll.

We conclude that (66) holds everywhere in s p A .

Note that Proposition 3.3.1 does not hold with 8 " ) A changed by d(')A for some k < n. For instance both functions g ( z ) = 1/2 and f(z) = z belong to the disc algebra A(A); and Ig(z)I 5 If(z)l on S1 = d A = d(')A, but nevertheless g ( 0 ) = 1/2 > f ( 0 ) = 0. Observe that d ( 2 ) A ( A = ) ;? # S'.

3.3. Properties o f n-tuple Shilov boundaries

187

Using a similar argument as above we obtain the following

3.3.2. PROPOSITION. For a given F E An-1, n 2 1, and g E A the inequdty

holds on sp A whenever it holds on d(")A.

PROOF.Proceeding as in the proof of Proposition 3.3.1 we get: Reg(m) 5

IIF(m)ll = I I F ( ~ o > l l

on V n @")A IIdAv for an arbitrary point m, E V = V ( F F(rn,)). Consequently, Eteg(rn) 5 IIF(rno)ll on V. Indeed, assuming in the contrary that RRg(rn1) > IIF(rno)ll for some rnl E V, we get I,g(ml)l = eReg(m1) > ell'(mo)ll in contradiction with the inequality 5 e ~ ~ F ( mwhich o ) ~ ~holds on dAv and therefore on V. Hence &g(rn,) 5 IIF(rno)ll as desired. The next corollary generalizes Proposition 3.3.1.

3.3.3. COROLLARY [Tonev]. For a given k-tuple F, 15 k 5 n - 1 , and a given ( n - k)-tuple G = ( g l , . . . , g n - k ) over A the equdity

holds on s p A whenever it holds on d(")A.

PROOF.The case k = n - 1 is exactly the statement of Proposition 3.3.1 proved above. Suppose that k < n - 1. Let G = (gi, . . . ,gn-k) E An-k, and let (69) holds on d(")Afor an F E A k . Consider the set V = V({gj - gj(rno)]!-k-l), 3=1 where

188

Chapter HI. n-Tuple ShiJov Boundaries

m, is a fixed point in SPA. On d ( k + l ) A ~which , by Theorem 3.2.8 is contained in the set V n d(")A, we have: n-k-1

j=1 By Proposition 3.3.1 this equality holds on s p A v = V . In particular it holds at rn, E V , i.e. we have

IIG(mo>ll5 lI~(m0)lL as claimed.

In fact acn)A is the minimal closed subset of s p A for which the first two propositions in this section hold true. Namely, 3.3.4. THEOREM. If E is a closed subset of s p A such that either (66) or (68) holds on s p A whenever they hold on E for every g E A and every F E An-1, n 2 1, then E is an n-tuple boundary of A.

PROOF.Let first (66) holds on s p A whenever it holds on E for every F E A"-', g E A. Fix an F = (fi ,...,fn-1) E A"-' \ A,"". If we assume for a moment that V ( F )n E = 0, then we can find an E > 0 such that 11 F(rn)ll 2 c on E. According to the hypothesis on E, the equality IIF(m)ll 2 E , being of type (66), holds on s p A as well, and hence V ( F )= 0 in contradiction with the choice of F . Suppose now that Ih(rn)I 5 1 on V ( F )n E for some h E A. For each e > 0 there is a neighborhood V, of the set V ( F )n E on which Ih(rn)l < 1 E . Therefore, if C, is a big enough positive constant, the inequality

+

3.3. Proverties o f n-tuple Shilov boundaries

189

holds on E. Being of type (66), inequality (70) holds on S P A , according to the hypothesis on E ; and therefore, on V ( F ) we have Ih(m)l < 1+ e . Moreover, Ih(m)l 5 1 on V ( F ) because of the liberty of choice for t . Hence V(F)n E is a boundary for A v ( F ) , and thus V ( F )n E 3 ~ A v ( F )Therefore, . by E 3 U ( V ( F )n E ) 3 FEAn-l

U

FEAn-l

d A v ( ~we ) get that E 3 a(")Aby taking the closures in

the both sides of the last inclusion. We conclude that E is an n-tuple boundary of A. Let now (68) holds on s p A whenever it holds on E for every F E An-', g E A. Proceeding as above we obtain that (70) holds on E. Since IzI = max Re{zeie}, we see that the inequality @ €[O ,2 n]

also holds on E for each 6 E [0,27r]. Since (71) is of type (68), it holds on s p A for any 6 E [0,27r] according to the hypothesis on E. Therefore, on V ( F )we have Re {e"h(m)) < 1 t for each 6 E [0,27r]; and hence Ih(rn)l = max Re {eieh(m)} < 1 c .

+

BE[O,S~I

+

Continuing as before we obtain that E 3 a(")A. The theorem is proved. Now we conclude that

3.3.5. COROLLARY. A closed subset E of s p A is an n-tuple boundary of A if and only if either of inequalities (66) or (68) hold on sp A whenever it holds on E for each F E An-', n 2 1, andgEA.

190

ChaDter 1x1. n-Tude Shilov Boundaries

Since, by definition, the n-tuple Shilov boundary d(")Ais the smallest closed n-tuple boundary of A , we get the following 3.3.6. THEOREM. The n-tuple Shilov boundary d(")A, n > 1, of a uniform algebra A coincides with the intersection of all closed subsets E of s p A which satisfy one of the following equivdent conditions: (1) The inequality 1g(m)l 5 IIS(m)ll holds on s p A whenever it holds on E for any subset S of A with #S 5 n - 1 and any g E A; ( 2 ) The inequality Reg(m) 5 IIS(m)ll holds on s p A wheneveritholdsonEforanysubsetSofA with # S S n - l and any g E A.

Using the same arguments as in Corollary 3.2.11 we obtain

3.3.7. COROLLARY [Tonev]. A point m, in s p A belongs to the n-tuple Shilov boundary d(")A, n 2 1, of A if and only if for each neighborhood U of rn, there exist a function g E A and a subset S of functions in A with #S 5 n - 1 such that one of the following conditions holds:

3.3.8. COROLLARY [Tonev]. Let F be a Ic-tuple, 1 5 k 5 n1, and let G be an (n - Ic)-tuple of functions in A . If IIF(m)ll = 116'(m)11holds on d(")A,then IIF(m)II E IIG(m)ll on S P A .

This can be seen for instance by switching the places of F and

G in Corollary 3.3.3,

3.3. Provedtes of n-tuple Shilov boundaries

191

3.3.9. EXAMPLES. 1.) As a subset of A*, the dual of A, s p A can be given two topologies: the Gelfand one (i.e. the weak*-topology) and the topology induced by the d u d norm. Let HP(E) denote the p t h HausdorfF measure on s p A associated with the metric induced by the dual norm. Denote by N the set of points m E s p A at which the norm topology coincides with the Gelfand one. As shown by Sibony [91],d(")A = { m E N : H2"({rn E s p A : llpn - m, 11 5 r ) ) = 00 for some r < 00. Also if R is an 0-convex domain in C" and if A contains the polynomials, then

d ( k ) Ac {z E bR : H2kpfl B ( z , r ) )= 00 for any r > 0}, where B 2 k ( E )is the 2k-th Hausdorff measure in R2".

2.) Let R be a bounded domain in C" with a C2-boundary bR. There exists a real C2-function e(z) (defining function) in a neighborhood U of 3 such that grade # 0 on bR and

R = { z E u : e(z) < 0). The Levi form L,(z) of R at z acts on the complex tangent space T,(R) to bS2 at z as follows:

where w =

..., w,) E TB(R) with " aae(z) w

(w1,

j,k=l

j = 0. A

zj

point z E bR is strictly k-pseudoconvex, 0 5 k 5 n - 1, if L J z ) has at least n - k - 1 positive eigenvalues. As shown by Basener, the closure of the set

Fk(R) = {z E bS2 : R is strictly k-pseudoconvex at z )

Chanter 111. n-Tuple Shilov Boundaries

192

is a ( I c

+ 1)-tuple boundary for the algebra A(52),i.e. @+')A(S2)

c

m.

Near a strictly E-pseudoconvex boundary point z in &R there exists an analytic variety of codimension E 1 which touches 0 precisely at z. To determine if such a point z belongs to $ k + l ) A seems to be a difficult question as this involves global existence questions, such as the existence of a globally defined analytic variety of codimension k 1 which touches 3 precisely at z. Recall that according to Corollary 3.2.14 this is enough to conclude that z E &k+')A(R). Below we show that the set Fk(Q) from above coincides with the ( I c 1)-tuple Shilov boundary a(k+l)A whenever R is an intersection of Stein domains and a pseudoconvex domain with a C"-boundary.

+

+

+

3.) Let 52 be a bounded pseudoconvex domain in C" which is an intersection of Stein domains and whose boundary is C"smooth. Recall that 52 is pseudoconvex if and only if

(Lg(Z)W,W)2 0 for every z E bf2 and every w E T,(R). For every k 2 0 consider the sets Nk(52) = { z E bf2 : dimKerL,(z) 5 I c } . Under hypothesis Nk(R) = F k ( 0 ) . As an intersection of Stein domains is O(D)-convex and so s p A( 52) = 3.Example 3.2.11 asserts that a(k)A(R)= 3 for E > n and d(")A(Q)= b 0 . We claim that for k 5 n the closure of the set Nk(R2) coincides with the ( I c 1)-tuple Shilov boundary a('+')A(52) of algebra A(O), i.e. - d(k+')A(R) = Nk(52) = Fk(52).

+

Indeed, the set N o ( 0 )of the points in bR where the Levi form is injective coincides with the set of strictly pseudoconvex points; and therefore, a(')A(R) = dA(f2) = N,(R), as shown by Rossi.

193

3.3. Properties of n-tuple Shilov boundaries

The case k = 1 [Sibony]. Since by the supposition 52 is an = where intersection of Stein domains, we can write

nop

QP are domains of holomorphy. Let g

P

E Hol(Op),

f

E A(S2)

and suppose that Ref(z) I Ig(z)l on Nl(52). We claim that Ref(z) 5 Ig(z)l on 3. If g is a constant, then the assertion is obvious as it follows easily from the case k = 0. Assume that g is nonconstant. By Sad's theorem the set of critical values of restriction g(z)lbQof gl(z) on b ~ 2is of measure zero in cl. The set of critical values of g, considered as a mapping from OP to C1, is also of measure zero. Since by the Hartog's theorem g(52) c g(b52) = gl(b52), almost every point in g(52) is a regular value for g and 91. Denote by W the set of all points in Cf which are regular values for g and g1 simultaneously. Fix a 20 in W and denote by V, the preimage of w, i.e. V, = g-l(w). Since w is a regular value for g, the set V, is a manifold in OP;and clearly, V, is a Stein manifold. Moreover, since w is a regular value for gl as well, V, n b52 is a submanifold of b52; and thus V, n 0 is an open set in V, with a smooth boundary. Every strictly pseudoconvex point a, in b52 n V, belongs clearly to the set Nl(52). Since Ref(z) 5 Ig(z)l on N1(52), we have that

The case k = 0 implies that the set N,(V, n 52) of all strictly pseudoconvex points in V, n bS2 is dense in i3A(Vwno). Therefore, Ref(z) 5 IwI on aA(V, n 52). The case k = 1 of Proposition 3.3.6 implies that Ref(z) on V, n 52. Consequently, Ref(z)

5 IwI

I Ig(4l on g - l w ) .

Since g ( 0 ) \ W is of measure zero and g is an open set, it is not hard to see that g-'(W) is dense in 52. Therefore,

Chavter 111. n-Tuvle Shilov Boundaries

194

for every f E A(R) and g E Hol(Rp). Since UHol(Rp) is P

dense in A(R), it is clear that Ref(z) 5 1g(z)1 on 3 whenever Ref(z) 5 1g(z)1 on Nl(R) for every f , g E A(R). Theorem 3.3.6(2) implies that N1(R) is a 2-tuple boundary for A ( 0 ) ;and therefore, Nl(0) 3 ac2)A(0). Let now z, E Nl(R) so that dimKerLp(z,) 5 1. If it happens that dim Ker L p ( z o )= 0, then zo is a strictly pseudoconvex point and hence z, E dA(R) c 6(2)A(R) by the case k = 1. Let dimKerL,(z,) = 1. Without loss of generality we can assume that zo = 0 and that R c { z f Cn+' : Rezl > O}. The tangent space Ts0of $2 at z, is { z E C"+l : z1 = 0} and KerL,(z,) = R n ( 2 2 = -.. = zn = 0). Denote by V ( z o )the set R n { zn+1 = 0). z , is a strictly pseudoconvex point of V( 2,) since V ( z o )is an open subset of 0 with a smooth boundary and the restriction of Levi form on it is injective. Therefore, z, E N , ( V ( z , ) ) C d A ( V ( z , ) ) = ~ A V ( ~ c ,d)( 2 ) A ( R )by Theorem 3.2.1. Hence Nl(R) c d(2)(R)wherefrom we conclude that N1 (Q) = a(2)A(0). For arbitrary k 2 2 the proof follows the same line as for the k = 1 by using the inductive argument.

case

Applying Theorem 3.3.6 to Example 3.3.9(3) we get the following

3.3.10. COROLLARY. Let R = { z E C" : e(x) < 0} be a bounded pseudoconvex domain with a COO-smooth boundary. If f2 is an intersection of Stein domains, then the smallest closed subset of 3 OR which for each ( f l , . .. , f k ) E Ak either of functions

j=I

3.3. Provedies of n-tuple Shilou boundaries

j=l j=l attains its minimum is the closure of the set Nk-l(S2) = { z E bS2 : dim KerLp(z) 5 k - l}. Since B(')A(M) = BA(M) for the space of all real valued affine functions over a compact subset of a vector space, all previous results hold for the space set A ( M ) as well. Namely,

3.3.11. THEOREM [Tonev]. Let f 1 , . . . ,fk be k affine functions in A ( M ) , k 5 n. I f the inequality fl(.)2

I f a x ) + .-.+ fm2

holds on the minimal n-tuple f i n e boundary E n ( M ) , then it holds on M . E,(M) is the smallest closed subset of M for which this property holds for every k-tuple ( f l , . . . ,fk) E A k ( M ) .

3.3.12. COROLLARY. Let f 1 , . . . ,fn be n f i e functions in A( M ) and let k be a fixed integer, 1 5 k 5 n- 1. I f the inequality

f?(.)

+ - + f,"(.) * *

I f;+,(.)

+ + f%> * * *

holds on the minimal n-tuple &ne boundary E,(M), then it holds on M . I f the equality fi"(x)+-+f,"(x)

=f,"+1(x)+-+f:(x)

holds on E,(M), then it holds also on M .

3.3.13. COROLLARY (Local characterizations of points in E,(M)). Minimal n-tuple &ne boundary E,(M) of M consists of these points x o E bM such that for every neighborhood U of 2, there exist k f i e functions f 1 , . . . ,fk in A ( M ) , 2 5 Ic I n , with m e (f;(x) - fi(x) - ..-- f,"(.)) > 0 and f,"(.> I XEU

f,"(x)

+ - - - + f , " ( x ) for every x in M \ U .

196

Chapter III. n-Tuple Shilov Boundaries

Let Ic 5 n. Since all the sets U = { m E s p A : Igj(m)l < E , 15 j 5 k, gj E A-'} = { m E s p A : lgT1(m)l > l/e, j 2 k} are of type (63) (with respect to the norm llzll = mkclzjl),

1s

j= 1

Corollary 3.2.12 implies the following

3.3.14. COROLLARY. The n-tuple Shilov boundary d(")A meets each neighborhood U C s p A in the basis (4) of the Gelfand topology of s p A with k 5 n. The next result follows immediately from Corollary 3.3.14. [Corach-Suhez]. The union of all n3.3.15. COROLLARY tuple Shilov boundaries of a uniform algebra A is dense in sp A. 00

Indeed, assuming that

[ U d(")A] # S P A , we

can find a

n=l

neighbourhood U = {m E s p A : Igj(m)l

< E , 1 5 j 5 k, g j

E

u 8 " ) A and 00

A - l } in the basis (4) which does not meet the set

n=l

moreover - the k-tuple Shilov boundary $')A in contradiction with Corollary 3.3.14.

u d ( " ) B = A" 00

For instance

n= 1

00

coordinate zj} and

= {z E

zm: lzjl = 1 for some

[ U d(")B]= [A=] =A . -00

n= 1

Applied to the space A(R), where R is an O(z)-convex domain in c",Corollary 3.2.11 implies the following

3.3.16. COROLLARY. The k-tuple Shilov boundary d ( k ) A ( R ) consists of these points z E bR such that for every neighborhood U of z there exist k functions fi, . . . ,f k which are holomorphic

197

3.3. Properties o f n-tuple Shilov boundaries

on 3 and for which the analytic variety V( f 1 , . only in points belonging to U.

..,fk)

touches

3.3.17. PROPOSITION. If the spectrum s p A of a uniform algebra A is a metrizable space, then the set

FEAn-l

is an n-tuple boundary of A.

PROOF.If s p A is metrizable, so are all of its subsets V(G) = s p A v ( ~ )G , E A"-1. Thus the minimal boundary (Chapter I) SAV(G)of the algebra Av(G) exists for all G E A"-'. For each point m E s p A and for every regular n-tuple F with F ( m ) = (fl(m), 0 , . . . , 0 ) we have

for some point ml in SAv, where V = V(f2,. . . ,fn). Hence the point 5 1 which appeared in the the proof of Proposition 3.2.9 can be choosen to be inside 6Av; and therefore, the set (72) is an n-tuple boundary of A as well.

3.3.18. DEFINITION.A point m, E s p A is an (n - 1)-peak point for A if there exist n functions f 1 , . . . ,fn E A with fj(m0) = 0, j = 1,.. . , n - 1, fn(m,) = max Ifn(m)l = 1 and V(f1,. ..,fn- 1)

Ifn(m)l < 1 (or, equivalently, Refn(m)

< I) for each m

E

V(f1, - - fn-1). If we interprete ( f l y . .. ,f n - l ) as "coordinates" near the point ( ~ ~ ( T T Z ,.).,. , f , - l ( m,)) E C", then fn serves as a "barrier"

198

ChaDter 111. n-Tuple Shilov Boundaries

function to a($'). According to Definition 3.3.18 the usual peak points are 0-peak points of A. As mentioned in Chapter I, the closure of the set of all peak points for A coincides with the Shilov boundary d A whenever s p A is a metrizable space. Since in that case all subsets V(f1,. . . ,fn-l) are metrizable, by the Proposition 3.3.17 we get the following 3.3.19. COROLLARY. H s p A is a metrizable space, then the n-tuple Shilov boundary ll(n)A coincides with the closure of the set of all (n - 1)-peak points for A , The next theorem shows how the n-tuple boundaries of A are related to the Shilov boundaries of the extensions of A by conjugates of some functions in A. Given an S c C ( s p A ) denote by A ( S ) the function algebra on sp.4 which is generated by A and S. Namely,

3.3.20. THEOREM [Basener]. For each n > 1 the n-tuple Shilov boundary 8 " ) A is the smallest among all closed subsets E of s p A which axe boundaries for all algebras A@'), where S c A, #S 5 n - 1.

PROOF.We claim that a closed subset E of s p A is an n-tuple boundary of A if and only if it is a boundary of the algebra A @ ) where S C A , #S 5 n - 1. Suppose first that E is an n-tuple boundary of A. Let S = { f l , . . . ,fn-1) c A and let cp E A ( S ) -*n-1 so that cp = c g , J f ' .. . f n - l for some g, E A . Choose an I

3.3. Properties of n-tuple Shilov boundaries

199

m, E s p A with max Icp(m)I = Icp(m,)l and let mEsp A

f

=~g,~f'(m,)...~ E~ A ,~ ~ ( m , ) I

T = {hl, * * - 9 hn-l} C A, where hj = fj - fj(mo), j = 1,.. . ,n - 1. Then m, E V(T), and thus V(T) # 0. Since E is an n-tuple boundary of A , max

mEV(T)

If(m)l = mEF(yjnE If(4l.

Observe that m, E V(T) and that f cp on V(T). Hence Ip(m)l = maxIp(m)l, i.e. E is a I ~ ( m o > l= If(mo)l = mEsp A

mEE

boundary for the algebra A@), as claimed. Suppose now that E is a boundary of the algebra A@) for any S c A with #S 5 n - 1 and let S c A with #S 5 n - 1 and V ( S )# 0. We claim that rnax If(m)l= max If(m)l for every

f

mEV(S)

m € V (. S,) n E

E A . Let S = ( f i , . . . ,fn-l). Define n-1

n-1

Clearly, cp 1 on V( S) while 0 < 'p < 1 on s p A \ V ( S ) . For each k 2 0 we have fcpk E A @ ) , so that

Because cp peaks on V(S), it follows that

n E is a boundary of A and therefore V(S) n E IId A v ( s ) . Since this inclusion holds for any S c A with #S 5 n - 1, (20) implies that E 3 d(")A. The theorem is proved. as claimed. Consequently, V(S)

Chapter H I . n-Tuple Shilov Boundaries

200

The following result we give without proof. 3.3.21. PROPOSITION [Basener]. acn)A =

[

u

{ d ( A / I ) ],

1

where I runs over all closed ideals in A of codimension n - 1. The next theorem establishes multi-tuple versions of some Shilov boundary properties, related to subideals of A. Let I be a closed ideal in A, and let V denote the vanishing set of I , i.e. V = V ( I )= { m E s p A : f ( m ) = 0 for every f E I } . Then A / I can be identified with A v ( l ) ,s p A / I can be identified with V ( I )and s p I can be identified naturally with the set spA with all points in V identified. Denote the last set by s p A { q . 3.3.22. THEOREM [Tonev]. Let I be a closed ideal of a uniform algebra A and let V = V (I ) . The the n-tuple Shilov boundary of 1 can be obtained from the n-tuple Shilov boundary of A by identifying the points in V , i.e.

PROOF.Consider first the case n = 1. Clearly, aA{v} is a closed boundary for the algebra I. Thus d I c d A { v ) . Let m, E dA \ V , and let U be a neighborhood of m, which does not meet V. Denote by f a function in A such that 1 = f ( m 1 ) = max (f(m)l for some point ml in U and If(rn)l < e < 1 on mEsp A

spA \ U. Since m,

4

V , there exists a function g E I such that gl V 0 and g(m,) # 0. For every integer k > 0 the function g f k belongs to the ideal I c A. By choosing k big enough we can get that gfk assumes the maximum of its modulus only within U. It is clear that m, E a I ; and consequently, a A \ V C d I C d A { v } .We conclude that dI = d A { q because of ~ A { V=) [dA\ v ] s p A i V i = [dA\ V]S,IC C~A{v)*

3.3. Proverties of n-tuple Shilov boundaries

201

Let now n > 1. For each ( n - 1)-tuple F E A"-l the algebra Iv(F) is a closed ideal in Av(F). By the case k = 1 from above we get that dIv(F) is the set ~ ( A v ( Fwith ) ) the points in { V f l V ( F ) )identified, i.e. ~ ( A v ( Fwith ) ) the points in V identified. Therefore

The theorem is proved. Denote by w the point V { V }in s p I = s p A { v ) which is obtained by identification of the points of V . As shown above d(")I\ w = d(")A\ V ; and therefore, we have the following

3.3.23. COROLLARY. Under the hypothesis of the previous theorem the following inclusions hold:

Remember that Rossi's local maximum modulus principle for a uniform algebra A asserts that if U is an open subset of s p A then

(73)

a),

for every f E A. If is A-convex (i.e. if SPAT = this equality implies that d A r C ( d A n U ) U bU; and in particular, d A r C bU whenever U does not meet the Shilov boundary dA. The next theorem extends Rossi's principle for vector-valued functions over A .

Chapter I l l . n-TupIe Shilov Boundaries

202

3.3.24. THEOREM (Local minimum norm principle for vector-valued functions) [Tonev]. I f U is an open subset o f s p A and 11 . 11 is the Euclidean norm in C", then

for every regular n-tuple F E A".

< min IIF(m)ll for a point mEbU U and for some regular n-tuple F = (f1 ,. . . ,f n ) f A:.

PROOF.

Assume that IIF(rno)ll

mo E We claim that mo belongs to 8 " ) A . Applying, if necessary, an orthogonal transformation in C" we can assume from the beginning that F(rn,) = (fl(m,),O,. . . ,O) and thus m, E V = V ( f 2 , .. . ,fn) = sp A v . Then

because of b(U n V ) c bU element of A v and hence

n V. Clearly,

f1

Iv

is an invertible

in contradiction with the Etossi's maximum modulus principle, since U n V is an open subset of V and m, # b(U n V ) . We C conclude that mo necessarily belongs to a A v = aAv(fz,...,fn)

d(")A. Q.E.D. Note that (74) holds also for n-tuples F E A" whose restrictions Flu are regular n-tuples in A;. Indeed, assuming that

203

3.3. Properties of n-tuple Shilov boundaries

for some m, E U and some F E An with FIv E A,; we obtain, as above, that If1(m0)l< min Ifl(m)lfor some fl E A such meb( U U V ) that fi is invertible in Avnv. Then again as before

lvuv

I(fi-l lUnv(m0)) I > m

I(fi-'

E ~ ~ v ,

I

Ivnv(m>>

and hence there exists a function g E A such that glvUv E AGiv

I

I

I,

and for which (glnnv(mo))I > mEmb( UuU V ) (glvnv(m)) as well. We conclude as before that m, E d(")A.

3.3.25. COROLLARY [Basener]. Let K be a compact A-convex subset of SPA. Then

8 " )c b~ ~ u~( a ( " )n~K ) . PROOF.The set of n-tuples over A whose restrictions on K = S ~ A are K regular, obviously is dense in ( A K ) ~ .By the above remark we observe that the equality (74) is satisfied (with K instead of for every n-tuple F E ( A K ) ~ . Therefore, bK U ( 8 " ) A n K )is an n-tuple boundary of AK by the Theorem 3.1.3. This completes the proof of the statement because ~ ( " ) A Kis the smallest closed n-tuple boundary of A K .

a)

a

As a direct corollary we observe that if is A-convex, i.e. if h ( r ) = 21,then (74) holds for any norm in C". 3.3.26. COROLLARY. The n-tuple Shilov boundary d(")Ais the smallest among dl closed sets E in spA such that (75) for m y open subset

F E An.

U of s p A \ E and for every regular n-tuple

Chapter III. n-Tuple Shilov Boundaries

204

Theorem 3.3.24 and Corollary 3.3.26 show that the local manimum principle for regular mappings is as natural in function theory as the usual local maximum principle. The only reason which have made the local maximum principle to prevail in classical function theory is its relative simplicity in the onedimensional case. Note that with the exception of the single case, d(")A does not coincide in principle with the smallest closed subset E of sp A for which the inequality

holds for every F E A" and for each m E SPA. Actually the condition (76)

max IIF(rn)(l = max IIF(m)))for every mEE

mEspA

F

E A",

where [I . 11 is the Euclidean norm in C",is not sufficient to identify the set E as an n-tuple boundary of A . Indeed, a point m, which belongs to the intersection E of all closed subsets E of s p A satisfying (76) can be characterized as follows: For every neighborhood U of m, there exists an n-tuple F E A" for which (77)

Suppose that IIF(m)ll assumes its maximal value at ml E U. Applying, if necessary, an orthogonal transformation in C" we can assume from the beginning that f2(ml) = = fn(ml) = 0, i.e. that IIF(m1)ll = If~(rnl)l. By (77) we see that

- -

Consequently for each neighborhood U of m, there exists a function in A which attains the maximum of its modulus within U only. By Corollary 3.2.10(2) we conclude that m, E d(')A = 6'A which, of course, differs in principle from d(")A.

3.4. Shilov boundaries of t e n s o r products

205

3.3.27. NOTES AND REMARKS. The case (2) of Theorem 3.3.6 is precisely Sibony's characterizing property for multi-tuple Shilov boundaries from [91]. For pairs of functions Corollary 3.3.8 was proved by Sibony [91). In a more general context Corollaries 3.3.3 and 3.3.8 are proved in Tonev [123]. The inclusion d(k+')A(S2)c Fk(s2) in Example 3.3.9(2) has been proved by Basener [12]. As shown also by Basener [ll]each strictly k-pseudoconvex point z belongs to d(k+1)A(s2n B ( z , r ) ) for some ball B ( z ,r ) and, more precisely, near any strictly k-pseudoconvex point z there exists an analytic variety of codimension k 1 touching 3 exactly at z. The last result generalizes the well known fact for the existence of analytic support function at each strictly pseudoconvex boundary point in S2, due to Andreotti [2]. The inclusions in Example 3.3.9(1) and the equality d(k+l)A(R)= Nk(S2), k 2 1, in Example 3.3.9(3) are due to Sibony [91]. The sets N k ( Q ) , Ic 1, historically are the first examples of non-trivial multi-tuple Shilov boundaries. The case k = 0 for pseudoconvex domains with C2smooth plurisubharmonic defining functions e (and, in particular, for any strictly pseudoconvex domain, when dA(S2) = bS2) the above result has been proved earlier by Bremermann [21]; for intersections of Stein domains the case has been proved by Rossi [85]. Theorem 3.3.20 and Proposition 3.3.21 are from Basener's paper [12]. For Shilov boundaries the proofs of Theorem 3.3.22 and Corollary 3.3.23 can be found e.g. in Stout's book [107].

+

>

3.4. SHILOV BOUNDARIES OF TENSOR PRODUCTS Let A and B be two function spaces and let A 8 B denote the tensor product of A and B. By definition A @ B is the smallest closed function subspace of C ( X x Y )which contains all monomials f(z)g(y), where f E A , g E B , z E X , y E Y .

206

Chavter III. n-Tuple Shilov Boundaries

In this section we prove Basener-Slodkowski theorem for multi-tuple Shilov boundaries of function spaces of type A @ B in terms of the multi-tuple Shilov boundaries of generating spaces A and B. First we show that Shilov boundary of tensor product of two spaces A and B can be expressed in a very simple way by Shilov boundaries of A and B . Namely

3.4.1. PROPOSITION. d ( A @ B ) = d A x dB. PROOF.Let (x,,yo) E d(A8.B)and let U and V be neighborhoods of x, and yo in X and Y respectively. Clearly, (x,,yo) € U x V c X x Y and by Corollary 3.2.10(2) we can find a function in A @ B which does not vanish on X x Y and such that

Let (x1,yl) E

r x v be such that lf(xl,y1)I

= min If(x,y)( and XXY

let f'(x) = f(x:,yl) E A , f"(y) = f(x1,y) E B. Clearly, both functions f' and f" do not vanish on X and Y respectively. By (78) we have

Thus for each neighborhood U of x, and V of yo we could find functions f' E A and f" E B which do not attain the minimum of its modulus outside and 7respectively. By Corollary 3.2.10(2) we have that x o E dA, yo E dB; and therefore, d ( A @ B ) c d A x dB. For the opposite inclusion take points x, E dA, yo E d B and neighborhoods U c X , V c Y of x o and yo respectively. By Corollary 3.2.10(2) we can find functions f E A and g E B which do not vanish on X and Y respectively and such that

3.4. Shilov boundaries of tensor products

207

Now Corollary 3.2.10(2) asserts that (zo,yo) E d ( A @ B )because of the liberty of choice of neighborhood U x V . The proposition is proved. The next theorem generalizes Proposition 3.4.1 for multi-tuple Shilov boundaries of tensor products of function spaces.

3.4.2. THEOREM [Basener-Slodkowski].

PROOF.First we shall show that

Fix a point ( x o ,yo) in a(")(A@ B )and let U and V be neighbor€ hoods of x, and yo in X and Y respectively. Clearly, (z,,y,) U x V C X x Y. By Corollary 3.2.10(2) we can find a regular n-tuple (f1, .. . ,fn) of functions from A @ B for which

Chapter I l l . n-Tuple ShiJov Boundaries

208

for every j = 1,.. . ,n. Clearly, both n-tuples

(fi', . . . ,f:)

(fi, . . . ,f;)

and

are regular. By (80) we have that

Thus for each neighborhood U of x, and V of yo we could find regular n-tuples over A and B respectively, which do not attain the minimum of their norm outside and respectively. By Corollary 3.2.10(2) we conclude that x, E d(")Aand yo E d(")B. Consequently, a(")(A@ B ) c d(")Ax 8 " ) B as claimed.

v

The inclusion d(")(A@ B ) II

6 d ( k ) Ax d("-k+l)Bis ele-

k= 1

mentary. Indeed, let (z,, yo) E a ( k ) Ax d ( n - k + l ) Bfor some fixed k = 1,. . . ,n, and let U x V c X x Y be a fixed neighborhood of (sc,,~,). Because of z, E &'))A and U 3 zo, we have that UnaAV(fl,...,fk-l) f 0 for a (.f1(x)7".7fk-1(x)) over A, according to (50). Similarly, we can find an ( n- k)-tuple (gl(Y), g n - k ( Y ) ) over B , such that v n a B V ( g l ,. . . , g n - k ) # 0. ConWquent1y7 V ( f 1 ,. . . f k - 1 ) dBv(91 ,...,gn - k ) n U x V # 0. According to Proposition 3.4.1, *

3.4. Shilov boundaries of tensor products

209

Hence a(.)( A 8 B ) nU x V # 0 for every neighborhood U x V of (z,,, yo), and thus (z,,, yo) E d (" )(A @ Bbecause ) of the closedness of the multi-tuple Shilov boundaries. The opposite inclusion

u d("A n

we prove in two steps. Denote for a while the set

x

k=l

d(n-k+l)Bby E . STEP 1. Every function of type

where fi(z) E A and gj(y) E B , i = 1,. . . ,k, j = 1,. . . ,n - k, satisfies the inclusion

(83)

bF(X x Y ) c F(E).

Indeed, let (zo,yo) E X x

we have that

Y and let llF(z,y)ll # 0. Since

Chapter III. n-TuDfe Shifov Boundaries

210

If the ( n - k)-tuple (gl(y), ...,gn-k(y))

vanishes on Y , then certainly the k-tuple (fl(z), . . . ,fk(z)) does not vanish on X and X min

11 (fl(4,- - - f k W ) 11 = min 11 (fl(4, ,fk(4) 11 7

a(k)A

* *

by Corollary 3.2.10(3), so that

If the k-tuple (fl(z), . . . , f k ( z ) ) vanishes on X , then certainly the (n - k)-tuple (gl(y), . . . , gn-k(y)) does not vanish on Y ; and we can obtain in a similar way that

If both tuples (fi(z), . . . ,fk(z)) and (gi(y), vanish, then again as above we get that

. . .gn-k(y))

do not

Consequently,

for any regular n-tuple F of type (82). If we assume that the inclusion (83) is violated for some F of type (82), we can find a E X x Y such that F(z,,y,) E bF(X x Y ) \ F ( E ) . point (z,,y,) Consequently, we can find a point z, in C" \ a ( F ) close enough to F(z,,y,) such that the function H = F - z, does not vanish on X x Y and at the same time does not satisfy the inequality (84) in contradiction with the result just proved. We conclude that the inclusion (83) is fulfilled for each n-tuple F of type (82).

3.4. Shilov boundaries of tensor products

211

STEP 2. The inclusion (83) holds for every mapping F ( z ,y) in the space ( A @ PROOF.We prove the case n = 2 of the statement. Let F E A k , G E B‘ and let PI and P2 are polynomials in k and I variables respectively. It is not difficult to check that

for each j and for every F E A k . Indeed, if (PI,. .. , p j ) is a regular j-tuple in P ( F ( X ) ) ,then we have ( p l ( z ) , . . . , p j ( z ) ) =

11

2

min

EEF( a(j ) A )

11

( 1 ( p l ( z ) ,...,pn(z))11 for each z E F ( X ) . Corollary

3.2.10(2) implies that d ( j ) P ( F ( X ) )c F ( d ( j ) A ) ,as claimed. By Slodkowski’s result for spaces of type P ( X ) , X c C ” , we have that

d ( ’ ) ( P ( F ( X ) x G ( Y ) )= a ” ’ ( P ( F ( X ) ) @ P ( G ( Y ) ) ) = ( d P ( F ( X ) )x d ( ” P ( G ( Y ) ) ) U ( d ( 2 ) P ( F ( X )x) d P ( G ( Y ) ) ) ;

and therefore,

212

Chapter III. n-Tuwle Shilov Boundaries

= (Pl,P2) 0 ( F ,G)(E ) = (Pl( F ,G),p2 ( F ,G))( E ) .

Hence (83) is fulfilled for every mapping of type (PI ( F ,G), P2 ( F , G)). Since the mappings of this type are uniformly dense in the space we conclude that (83) holds for every function in ( A@ This proves the case n = 2 of Theorem 3.4.2, because by Corollary 3.2.10( 1) aC2)(A@ B) is the smallest closed subset of X x Y for which (83) holds for any mapping in ( A @ The proof of the general case ( n > 2) of Theorem 3.4.2 follows the same lines as the proof of the case n = 2.

3.4.3. NOTES AND REMARKS. Basener-Slodkowski's Theorem 3.4.2 was conjectured originally and also proved for several particular cases of uniform algebras by Basener (e.g. [lo]) in 1975. It generalizes the well known property of the usual Shilov boundary (Proposition 3.4.1) as well as the corresponding formula for the multi-tuple Shilov boundaries of polydisc algebras. Applications of the theorem are pointed out by Kumagai [68]. Slodkowski [92] (also [93]) gave a complete proof of Theorem 3.4.2 in 1983 for spaces of type P ( K ) , ' A c C", by utilizing the delicate machinery of qplurisubharmonic functions, and also for arbitrary uniform algebras by applying a specific approximation procedure for algebra elements. The proof of Theorem 3.4.2, presented here, which avoids this approximation procedure, is from Tonev [135].

3.5. MULTI-TUPLE HULLS The classical hulls - polynomial, rational, holomorphic, Aconvex etc. are tightly connected with functional approximations and interpolations. Recently, a new generation of hulls related

3.5. Multi-tuple hulls

213

to vector-valued functions have come into appearance. It turns out that these hulls are essential in the investigation of so called g-holomorphic functions, topological stable rank of algebras, analytic perturbations of Taylor spectrum for several commuting operators, multi-tuple Shilov boundaries etc. In this section we create a unified approach to these new families of hulls, investigate their properties and interrelations with the multi-tuple Shilov boundaries of function spaces. Recall that given a uniform algebra A and a closed subset E of S P A ,the A-convex hull of E is the set

Remember, that a subset E of s p A is A-convex if h ( E ) = E , or equivalently, if s p A = ~ E. It is not hard to see that h ( E ) is also the biggest among all closed subsets N of S P A ,such that min If(m)l = min If(m)l for every f E A which does not mEN

mEsp A

vanish on N . The multi-tuple version of this property generates a family of hulls of compact sets K c X , which are tightly connected with n-tuples of functions in function spaces over X and extend naturally the notion of the A-convex hull.

3.5.1. DEFINITION. Let B be a function space on X , let E be a closed subset of X and let 11 . 11 be the Euclidean norm in C". The n-tuple B-convex hull hn(E) of E is the biggest among all subsets N of X which contain E and for which the equality

holds for every non-vanishing on N n-tuple F of functions in B. E is an n-tuple A-convex set if it coincides with its n-tuple A-convex hull h,(E).

214

Chapter 111.n-TuDle Shilov Boundaries

n-tuple B-convex hulls are closed subsets of X , since the equality (85) is true for the closure N of any set N C X for which (85) is true. In fact, h,(E) coincides with the union of all subsets N of X for which (85) is true.

3.5.2. PROPOSITION [Tonev]. Let E be a closed subset of s p A . The n-tuple A-convex hull h n ( E ) of E coincides with the biggest among all subsets N of X which contain E and for which (86)

b F ( N ) c F ( E ) for any F E B".

PROOF.First we show that if N contains E , and if in addition it satisfies (86), then (85) holds for every non-vanishing on N mapping F E B". If c = min llF(z)11,then b F ( N ) C F ( E ) cC"\ xEE

B(c) = {z E C" : llzll 2 c}; and consequently, F ( N ) c C" \ B( c) because of 0 4 F( E ) 3 F ( E ) . Hence

Because of N IIE the opposite inequality is obviously fulfilled and we conclude that (85) holds for every N for which (86) is true. Now let N be a closed subset of X for which (85) is fulfilled for every F E B" which does not vanish on N . Assume that (86) is false, i.e. that b F ( N ) \ F ( E ) # 0. Let z, be a point in BF(N) \ F ( E ) and let z o E F-'(z,). Obviously,

for the n-tuple H ( z ) = F ( s ) - z,. We can find also a point z1 E

C"\F(N) close enough to z,, such that (87)holds for the n-tuple H l ( z ) = F ( z ) - z1. But this contradicts (85) since obviously H , ( z ) does not vanish on N . The proposition is proved.

215

3.5. Multi-tuple hulls

3.5.3. EXAMPLES. 1.) Let A be a uniform algebra. The I-tuple A-convex hull h l ( E ) of any closed subset E of spA coincides with its usual A-convex hull h(E). Indeed, since b f ( h l ( E ) )= b f ( s p A ~ )E f ( i 3 A ~ )c f ( E ) ,we have that max If(m)l 5 max If(m)l for every function f E mEhl(E)

A^;

mEE

and consequently, h l ( E ) c h ( E ) by the definition of A-convex hull h(E). Assume that h ( E ) contains properly hl(E). Then by the Proposition 3.5.2 there exists a function f in A such that b f ( h ( E ) ) @ f ( E ) . Let mo be a point in h(E), such that f ( r n , ) E b f ( h ( E ) )\ f ( E ) . By choosing a point zo in C \ f ( h ( E ) ) and close enough to f ( m , ) , we can get 1g(rn,)l < min Ig(m)l mEE

for g(z) = f ( z ) - zo E A-l h ( E ) ’ Hence Vls(m0)l > l/ls(m)l* Since l / g E &(E) and s p A h ( ~ = ) h ( E ) C SPA, there exists a function g1 in A such that Igl(m,)l > maxIgl(m)l, i.e. m, @ mEE h ( E ) in contradiction with the choice of m,.

2.) The n-tuple polynomial convex hulls 7rrn(E). Let A be an arbitrary set and let C” be the Cartesian product of A copies of the complex plane, equipped by the natural topology. Given a compact subset E in C”, denote by P ( E ) the closure in C ( E ) of the set of all polynomials in C”. Since the polynomials are dense in P(E),the n-tuple P(E)-convex hull of E is the biggest among all closed subsets N of C” such that Ql? - - - ,Pn)( N ) c (Ply * * ,Pn)( E ) for every n-tuple (P1 ,* * * ,P n ) of polynomials in C”. We call this hull the n-tuple polynomial conzlez hull of E and denote it by .rrn(E). Being the n-tuple P(E)-convex hull of E , 7rn(E)is the biggest among all sets N in C” which contain E and such that the inequality

-

holds for every non-vanishing on N n-tuple ( p l , . . . , p n ) of polynomials in C”.

Chapter 1x1. n-Tuple Shilov Boundaries

216

A subset E

c

C" is n-tuple polynomially convex if it coincides with its n-tuple polynomial hull r n ( E ) . By applying the inclusion

to the identity mapping in C n , we get that b ( r n ( E ) )c E for

every compact subset E in C". This implies that the n-tuple polynomial convex hull r n ( E ) of E is contained in the union of E and the union of all bounded components of its complement in C n . Because of its maximality property, r n ( E ) actually coincides with this union. Therefore, E is n-tuple polynomidly convex i f and only i f its complement C n \ E does not possess bounded components. n-tuple polynomial convex hulls rn(E ) of subsets E of C" are !,. = { z E C" : natural generalizations of usual polynomial hulls ? Ip(z)l 5 max Ip(z)l for every polynomial p in C " } . Namely, as SEE

h

Example 3.5.3(1) shows, a l ( E ) = E. In general, the hulls r n ( E ) are distinct from the usual polynoFor instance the 2-tuple polynomial hull a2(E) of mial hulls the set E = ( ( ~ 1 ~ 2 E 2 )C2 : 1 5 lzll 5 2, 1221 = 0} c C2 is the set E itself, which does not coincide with the usual polynomial hull E^ = ((z1,zZ) E C 2 : lzll 5 2, 1221 = O}. Indeed, for any z , $ E we can choose an E > 0 small enough, so that the regular pair of polynomials (21 - zo,z2 E ) attains the minimum of its norm near z, and outside E at the same time. This means that zo 4 7r2(E) for any z , $ E , i.e. that r 2 ( E ) c E , and hence 7rz(E)= E since r2(E) 3 E by Definition 3.5.1.

E.

+

Observe that because of bF( h,(E)) c F ( E ) ,for every F E Bn we have that F ( h n ( E ) )c T n ( F ( E ) )and hence

217

3.5. Multi-tuple hulls

In fact both sets are equal. Indeed, denote the latter set by K and take a F E B" with llF(x)11 # 0 on K . Clearly, llell 2 min llF(x)11for every z E ?rn(F(E)) because F ( x ) does not vanish zEE

on h,(E)

c K . Thus llF(xo)ll 2 min llF(x)11 for every point x, mEE

in F-'(7rn(F(E))); and therefore, for every point x o in K as well. Hence K C h"(E), i.e.

n F-l

3.5.4. COROLLARY [Tonev]. h,(E) =

for any compact set E in C".

(7rn(F(E)))

FEBn

3.5.5. PROPOSITION. Let E be a closed subset of X . Then E C . * . C hn+l(E) c h n ( E ) c . * * c h l ( E ) = h ( E ) c X and

n ~,(E)=E.

n2l

PROOF.The first part of the statement is obvious because if (85) is fulfilled for every (n + 1)-tuple F of functions in B which does not vanish on K , then it is fulfilled also for every non-vanishing on K n-tuple over B. The inclusion h n ( E ) 3 E is clear from the first part. If

n

rill

x, 4 E , then for every x in E we can find a function f, E l3-l such that If,(x)l > 1 and If,(x,)l < 1. Let U, be an open neigh> 1 on U,. By the compactness borhood of m such that If,(y)l argument, there exist finitely many points 2 1 , . .. , x k in E such that E C UzlU - . U Uzk. By replacing each f Z j by some ot its powers we can assume from the beginning that lfZj(xo)1 < l/k and IfZj(y)I > 1 on UZj. Hence for F = (fz1,.. . , f Z k ) E Bk we have that llF(x)11 # 0 on X , llF(x)11 > 1 on E and llF(xo)11 < 1. Thus x, 4 hk(E); and moreover, x, E n h , ( E ) . We conclude

-

that

n h,(E) c E , as required. n2l

n

Chapter III. n-Tude Shilov Boundaries

218

The next theorem establishes a relation between the n-tuple A-convex hulls and the A-convex hulls of certain subsets of s p A of a uniform algebra A. 3.5.6. THEOREM [Tonev]. The n-tuple A-convex hull h n ( E ) of a closed set E in s p A coincides with the set of these points m E s p A which belong to the Av(s)-convexhulls h(E n V ( S ) ) of the sets E n V(S)for any set S in A whose cardinality does not exceed n - 1 and such that V ( S )contains m, i.e.

h n ( E ) = {m E spA : m E h ( E n V ( S ) )for every (88)

k-tuple S E Ak, k

5 n, such that V ( S )contains m}.

PROOF.Denote for a while the set (88), by K and suppose that K \ h,(E) # 0. By Definition 3.5.1 we can find an n-tuple F = ( f i , . . . ,f n ) c A" which does not vanish on I?' and such that IIF(m,)ll < min IIF(rn)ll for an m, E K \ hn(E). Without loss mEE of generality (applying, if necessary, an orthogonal transformation in C")we can assume from the beginning that F(m,) = (fl(rno),O,...,O). Hence m, E V ( S ) for S = (f2 ,..., fn). Since f1 does not vanish on K n V ( S ) , we have Ifl(rn,)l = IX-lin

Ifl(rn)l, i-e.

mEh(EnV(S))

max m € h ( EnV(S))

ls(m>l < 1 9 ( ~ 0 ) lfor g =

l/fi E A v ( s ) . Consequently, m, $ h ( E n V ( S ) )in contradiction with m, E K . We conclude that K c hn(E). Suppose conversely that h,(E) \ K # 0 and let m, E r n ( E )\ I?'. Let S be an (n-1)-tuple over A such that m, E V(S)\h(En V ( S ) .) Consequently, there exists a function f E A v ( s ), such that If(m,)l > max If(m)l. Without loss of generality we can EnV(S) assume that f does not vanish on spA. For g = l/f E A-l we have Ig(m,)l < r = min Ig(rn>l. For any positive E < r EnV( S)

3.5. Multi-tuule hulls

219

we can find a neighborhood V, of the set E n V ( S ) on which Ig(m)l > r - e. Hence for every m E E we have n-1

j=1

for some large enough positive constant Cc. The inequality (89) holds also on h,(E) because the n-tuple (C,fi, ..., C c f n - l , g ) is regular. In particular, at m, we have Ig(m,)l > r - e; and henceforth, Ilg(rno)ll 2 r because of the liberty of the choice for E. Since this contradicts the initial inequality Ig(m,,)l < r , we conclude that h,(E) c K . The theorem is proved. Denote by A@), S E A, the uniform algebra on s p A which is generated by A and the set 3 = (J : f E S } .

3.5.7. THEOREM [Basener]. Let E be a closed subset of s p A . The n-tuple A-convex hull h,(E) of E coincides with the intersection of its A@)-convex hulls, where S c A and #S n - 1,


ll= min ~ ~ ~ ofor~every ( n-tuple m ) ~of type ~ h

(x(

mEK

(x(

mEsp A

P O XE A" which does not vanish on the set z-'(h,(;i(spA)))

OX(

=

(X(

s p A. Since the set of functions p j m) is dense in A , r n K)) = rn(X(SpA)) if and only if min llF(m)11 = min IIF(m)ll for mEK

mE8p A

every regular n-tuple F E An . Since the n-tuple Shilov boundary d(")A is the smallest closed subset of s p A with the last property, we conclude that X ( 8 n ) A )is the smallest among all compact subsets of C n whose n-tuple polynomial hulls are equal to r n ( ~ ( S PA ) ) , claimed. The following proposition, wich we give without a proof, establishes an interesting relation between the n-tuple A-convex hulls and n-tuple Shilov boundary of a uniform algebra A. 3.5.25. PROPOSITION [Basener]. h,(E) consists of all points n

m E s p A which belong to n

U Ej

j= 1

U h(Ej) for any decomposition E =

j=l

of E into n compact subsets Ej; d(")A is the smallest

compact subset E of s p A such that

n

U h(Ej) = s p A for every

j=l

n

decomposition E =

U Ej

j=l

of E into n compact subsets E j .

230

ChaDter HI. n-Tude Shilov Boundaries

3.5.26. NOTESAND REMARKS. Multi-t uple hulls were introduced originally for uniform algebras. The sets described in Theorem 3.5.6 are precisely the generalized hulls introduced by Basener in [12] and used by him in his study of the q-holomorphic functions. Propositions 3.5.14, 3.5.15, Theorem 3.5.24 and the uniform algebra versions of Propositions 3.5.2, 3.5.10, 3.5.23 and Theorems 3.5.5, 3.5.18, 3.5.24 are from Tonev 11371; Corollaries 3.5.12 and 3.5.20 are from Corach-Subez 1281; Theorem 3.5.7 and Proposition 3.5.25 are from Basener 1121. The uniform algebra version of the class of sets described in Corollary 3.5.16 is precisely the class of n-th rational hulls considered by Slodkowski [94] and used by him in his recent investigations on analytic perturbation of Taylor spectrums s p T of commuting n-tuples T = (TI,. . . ,Tn)of operators. The sets on the right hand side of (91) are precisely the generalized rational hulls introduced by Corach and Sukez [28] and utilized by them in their recent investigation on the algebra's topological stable rank, the completeness of an algebra in any of its superalgebras and the dimension of minimal generating subset of an algebra. Lupacciolu [75] has found interesting characterizations for n-tuple polynomial and n-tuple rational hulls of compact subsets of Stein spaces in the course of his study of removable singularities of CR-functions. The 1-tuple case of Propositions 3.5.14 and 3.5.15 can be found e.g. in Gamelin's book [32]. The proofs of Proposition 3.5.24 and Theorem 3.5.25 for classical Shilov boundaries and classical polynomial hulls can be found e.g. in Stout's book (1071. The case n = 1 of Proposition 3.5.15 is the well known equality r ( E ) = h'(E)(E) (e.g. [32]) between the rational hull and the P(E)-convex hull of a set E C s p A , which justifies the names "n-tuple rational hull" and "n-tuple rational B-convex hull". The description of multituple polynomial hulls 7rn(E),E c Cn (Example 3.5.3(2)), and Propositions 3.5.14 and 3.5.15 indicate that n-tuple rational and n-tuple B-convex hulls are the "precise" analogues of usual rational and polynomial hulls of planar sets.

CHAPTER IV

ANALYTIC STRUCTURES IN UNIFORM ALGEBRA SPECTRA

Recognizing various analytic structures which live in algebra spectrum is one of the central themes in uniform algebra theory. Recall that a subset of algebra spectrum is said to carry an analytic structure if it can be given a structure, according to which the restrictions of algebra functions are holomorphic. It was thought one time that there must be analytic structure somewhere in s p A whenever s p A is larger than dA. However, these hopes were shattered first by an example due to Stolzenberg, who has found a set X in C" with polynomial hull properly containing X but with no analytic structures in If analytic structures can not be expected always in s p A , one can ask when it could happen. One of the general results connected with the problem for global existence of one-dimensional analytic structures in uniform algebra spectrum is Bear-Hile's theorem which says: Let U be an open subset of the spectrum of a uniform algebra A which does not meet its Shilov boundary. If there exists a function in A which is isolated-to-one on U , then there is an open dense subset R of U which can be given the structure of a onedimensional complex analytic manifold such that all functions in A are analytic on R. In this chapter we extend Bear-Hile's result and find sufficient conditions for existence of n-dimensional analytic manifolds, big-manifolds, analytic big-sets, and related analytic Talmost-periodic structures in uniform algebra spectrum.

2 2.

232

Chapter IV. Analvtic Structures in Algebra Spectra

4.1. n-DIMENSIONAL MANIFOLDS IN SPECTRA One of the most significant results concerning n-dimensional analytic structures in algebra spectrum is a theorem due to Basener [9], which turned out to be the very f i s t indication on the necessity and importance of multi-tuple Shilov boundaries. In this section we state Basener’s theorem, extend Bear-Hile’s theorem for n-dimensions and discuss also other possibilities for existence of multi-dimensional analytic structures in algebra spectra. We begin with a multi-dimensional version of Wermer’s maximality theorem (Theorem 1.4.5).

4.1.1. PROPOSITION [Sibony]. Let X be a compact polynomially convex set in Cn with connected interior. Let A be a closed subalgebra of C ( b X ) , containing the polynomials and whose n-tuple Shilov boundary d(”)Ais identified with bX. Then s p A can be identified either with bX or with X and in the last case every f in A possesses an analytic continuation on X . For the proof we need the following two lemmas. 4.1.2. LEMMA.Let n 2 1 and let F E A”. If W is a component of C” \ F ( 8 ” ) A ) such that W n a ( F ) # 0, then c a(F).

w

PROOF.Clearly, the set

W1

=W

\ a ( F ) is open in W . We

claim that W1 is also closed in W . Indeed, if z, E w1 n a ( F ) , then z, E b o ( F ) C F(a(”)A)by (51); and consequently, bW1 c F(@”)A) C C” \ W . Since W is connected, its complement W \ W , is clopen in W and W \ W, = W n a ( F ) # 0 , by

4.1. n-dimensional manifolds an svectra

233

the hypothesis. We conclude that W \ W1 = W ;and therefore, Wl = W \ a ( F ) = 0, i.e. W c a ( F ) as claimed. 4.1.3. LEMMA.Let X be a polynomially convex set in C, let A be a uniform algebra on bX containing the function z md let 7r : s p A + C 1: y H y ( z ) denote the spectral mapping of z. If n ( s p A ) = bX or 7r(spA) = X , then 7r is one-to-one and in the second case A = P ( X ) .

PROOF.By definition ~ ( m=)m ( z ) for any point m in X = 2. A contains P ( b X ) = P ( X ) and s p P ( X ) = 2. Therefore, 7r( s p A ) c c.(X) = X . The algebra generated by the polynomials is a Dirichlet algebra on b X , so ~ - ' ( w ) reduces to the evaluation at w for every w E bX. Therefore, 7r is one-to-one on ?r-l(bX). This completes the proof of the lemma when 7r(spA)= bX. Suppose that ..(SPA) = X # bX and that w is a point in the interior of X . Let cp1,cp2 E n-l(w) and let dpl,dp2 be their representing measures. Since for every polynomial p we have Jp(l)dpj

= p ( w ) , j = 1,2, and since P ( b X ) is a Dirichlet

bX

algebra, it follows that dp1 = dp2 and so 9 1 = 9 2 . Thus the mapping 7r is injective; and since ~ ( sAp) = X by the hypothesis, s p A can be identified with X . Let U = { z E C1 : Iz--zol < E } be a neighborhood in X \ bX and consider the algebra A p According to the local maximum modulus principle dA- c bU. Hence u!. (Al& z l P ( q u , wherefrom (Al& P(U>laub ecause of the maximdity of P(r)lau(clearly (Aler)lau # C(bU)). We obtain that all functions in A are holomorphic in X \ bX. By Mergelyan's theorem A = P ( X ) .

PROOFOF PROPOSITION 4.1.1. Since X \ bX is connected, Lemma 4.1.2 implies that either 7r(spA) = bX or 7r(spA) = X . Let w = ( q..., , wn) E 7r(spA) c Cn. Denote by M i the

234

Chapter IV. Analytic Structure in Algebra Spectra

compact set

M i = { m E spA : Zi(m) = wi, i # j } . Since &")A b X , we see that the Shilov boundary of A M i is contained in the set

S i = { z E bX : z, = w ; , i

#j}.

-. Denote by S& the compact polynomially convex set

3;

= { Z E ..(SPA)

: zi

= w;, i

# j).

The algebra As& of restriction of A on S& contains z j and 2' -. projects spAsi onto S&. According to Lemma 4.1.3 M A is homeomorphic with g&via 2j. Consequently, s p A is homeomorphic with ~ ( s p A ) If. ~ ( s p A= ) X , then, clearly, # S& 3 aAM&and AM&c A(3;) by Lemma 4.1.3.By the theorem on separate analyticity A c A ( K ) . Let M be a Hausdorff space. Recall that M is a complex analytic manifold of dimension n if there exists an open covering {U,},E~of M (so that U, = M ) and a family { ~ , } , E I

u

,€I

of homeomorphisms of U , onto open subsets cp(U,) of C", such that y p ( y ; ' ( z ) ) is an analytic map of p,(U,nUp) onto yp(Uan Up) for each pair a , p E I . A complex valued function f defined on an n-dimensional complex analytic manifold 1M is holomorphic if for each set U,, a E I, the function f o cp;'(z), which is well defined on p,(U,) c C" is analytic.

4.1.4. THEOREM. Let A be a uniform algebra. I f the mapping F = (fi,. . . ,fn) E A" is locally one-to-one on an open subset U of spA\d(")A, then U can begiven the structure of an n-dimensional complex analytic manifold such that the functions in A are holomorphic on 27.

4.1. n-dimensional manifolds an spectra

235

PROOF.By the hypothesis of the theorem, every point m, E

U has an open neighborhood Vm, c Vmoc U of type: vm,

= { m E S P A : Ig3(m) -g3(mo)1 gj € A ,

S>O, j = 1 ,

< 6,

...,k },

such that F maps Vmohomeomorphically onto a closed subset K c C";and moreover, F(bVm,)= bK. Hence F(rn,)= (fi(mo),...,fn(mo)) E K\bK. Let An(F(mo),c) = {z E C " : Izi - fi(m0)l < E , i = 1,... ,n} be a polydisc in K \ bK and consider the set

fi(mo>l

< E , Igj(m) - gj(mo)l < 6, i = 1,... ,n,j

lc).

= I,. . . ,

Clearly, N is an A-convex neighborhood of rn, which lies in V and F( N ) = Zn (F(m,),E ) . By the minimum norm principle (Theorem 3.3.24) the function IIG(m)ll assumes its minimum within bN for every regular n-tuple G = (gl, . . . ,g n ) in .A : Hence the n-tuple Shilov boundary of the algebra AF = 'OF-' I z " ( F ( m , ) , e ) of functions on the polydisc (F(m,),E ) C C" is contained in bA" (F(m,),E ) (note that because of N c

zn

V m o ) ,F ( b N ) = bZn(F(m,),~)). S'ince all coordinate functions zj = fi o F - l ( z ) are in AF, so are all polynomials. By Proposition 4.2.1 either S P A F = bA"((F(rn,),~) or SPAF = -n A (F(rn,), E ) and AF C A ( A n ( F ( m , ) , ~ )Because ). the first possibility fails to be true, it turns out that every function in AF is analytic in A" (F(m,),e ) , Let Urn, denote the open set F-l (A"(F(mo))Emo))nVmo and v m , = urn,. v m , is a homeomorphism between Um, and An(F(mo),&mo), since by the hypothesis F is homeomorphic in Vmo. Clearly, U U, = U.Now

~1

mEU

let A" ( F ( m l ) ,E ~ n An ~ (F(m2), ) E ~ # ~0 for ) some ml # m2. Obviously, F ( F - ' ( z ) ) = z for every point z E An(F(ml),~rnl)n

236

Chapter

N.Analytic Structures in Algebra Spectra

An(F(mz),&m2).Therefore, v m , ( ~ ~ ~ ( = z ) z) for every point E A" ( F ( ~ IErn,) ) , n A" ( F ( m z ) ,Emz). Hence v m l ocpi; = - id on A"(F(ml),s,,) n A n ( F ( m 2 ) , ~ m 2 We ) . conclude that U is a complex analytic manifold of dimension m. If f E A , then foqm =f 0 la"(F(m),c,) E AF is an analytic function for each m E U as shown above. Consequently, any function f E A is holomorphic on U . The theorem is proved. Note that locally any function in A depends analytically on the variable F ( m ) = ( f I ( m ) ,. . . ,fn(m)) in U,which turns out to be a locally uniformizing variable in U for all functions in A. In particular an open subset U of s p A can be given the structure of an n-dimensional complex analytic manifold if there exists an F E A" which is one-to-one on U . The purpose of what follows is to strenghten this assertion.

4.1.5. DEFINITION. Let F be a function which maps a topological space Y t o the topological space 2. We say that F is: (1) k-to-one; ( 2 ) finite - t o - one; (3) countable-to-one; (4) light (or isolated-to-one) at a point y in Y , i f the set F-' (~(9)): (1) consists of k points; (2) is at most finite; (3) is discrete and countable or finite; (4) is discrete respectively. F is k-to-one, finite-to-one, countable-to-one or light on Y if it possesses the corresponding property at each point in Y . 4.1.6. PROPOSITION. Let F E A". I f f F is light on an open subset U of sp A \ $")A, then F is locally one-to-one on an open dense subset of U.

4.1. n-dimensional manifolds in spectra

237

For the proof we need several topological facts. 4.1.7. LEMMA [Bear, Hile]. If m, E spA \ dA, then every neighborhood of m, contains uncountable many compact A-convex neighborhoods of m, with nonempty and pairwise disjoint boundaxies.

PROOF.Let m, E spA \ d A and let U be an open set in sp A \ dA which contains m,. Take functions g l ,. . . ,gn E A such that Qe = { m E spA : Igj(m)-gj(mo)l 5 c , j = 1 , . . . , Y Z }c U for some E > 0 small enough. Qc is a compact neighbourhood of m, and bQc # 0 since bQc 2 ~ A G If ~ 0. < 77 < c , &,, c QE so bQ, n bQc = 0. If we let hj = g j - gj(rn,), then Qd = { m E spA : Ihj(m)l 5 c , j = 1,.. . , n } and QE is obviously A-convex. 4.1.8. COROLLARY [Bear, Hile]. Every nonempty open subset of sp A \ dA is uncountable. 4.1.9. PROPOSITION [Tonev]. Let A be a uniform algebra. I f the mapping F = (fi, . . . ,fn) E A" is light on an open subset U of spA \ d(")A, then F is open on U .

PROOF.Assume first that F is finite-to-one but not open on U. There is an open set V c U and a point m, E V such that F(m,) E bF(V). It is clear that for every closed neighborhood Q C V of m, we have F(m,) E b F ( g ) . If g is A-convex, then sp A q = and d(n)Agc bQ by Corollary 4.1.8. Hence by the Theorem 3.2.1 applied to the algebra Ag, F(d(")Ag) c F(bQ) and F(m,) E bF@) c F(d(")Aq) c F(bQ). Thus for every closed A-convex neighborhood c V of m, there exists an TCQ E bQ with F(mQ)= F(m,). Since d(")A 3 dA, g C V C U C sp A\dA. By Lemma 4.1.7 there are uncountable many A-convex neighborhoods Q c U of m, with pairwise disjoint boundaries.

238

Chapter N.Analytic Structures in Algebra SDectra

Thus there are uncountable many points mQ E bQ C U with F ( ~ Q=)F(m,) in contradiction with the hypothesis on F . Let now F be a light mapping on U. Any open set V in U is a union of open sets Q with c U. By the compactness argument, F is finite-to-one on every such &. Hence F is open on & according to the first part of the statement. Consequently, F ( Q ) is m open set in each Q with & E U. Of course F ( V ) = U F ( & ) is also an open set. The proposition is proved. Q

4.1.10. COROLLARY. Let F = ( f 1 , . . . ,fn) E A" be a regular n-tuple. If F is countable-to-one on an open subset U of s p A \ acn)A, then the function IIF(rn)ll does not assume its minimum within U . 4.1.11. LEMMA.Let F be a continuous open map of a Hausdorffspace Y to the topological space 2, Then (1) The set Yi+)where F is at least k-to-one is open in Y ; (2) If F is at most n-to-one on Y for some n, then F is locally one-to-one on a dense open subset of Y .

PROOF.(1)Let y1 E Yk(+)= { y E Y : # F - l ( F ( y ) ) 2 k) and let 91,. . . ,y k be k distinct points in Y such that F(y1) = * = F ( y k ) . Let VI, . . . V k be disjoint neighborhoods of y 1 , . . . ,yk n F(Vk). Then U is a respectively, and let U = F ( V l ) fl neighborhood of z, = F ( y j ) and F - ' ( V ) is a neighborhood of every y j . Clearly, F is at least k-to-one at every point in the set F - l ( U ) n VI which contains y1. (2) Denote by Yk the set {y E Y : # F - ' ( F ( y ) ) = k). If F is at most n-to-one on Y , then the set Y, is open by the first part of the statement. Let F(y1) = = F(y,) = z, for n distinct points y l ) . . . ,yn. Let V1,.. . ,V, be disjoint open neighborhoods of y 1 , . . . yn respectively, and let as before U = F(V1) n - . n F(V,). For each z E V the set F - l ( z ) contains * -

)

)

4.1. n-dimensional manifolds an spectra

239

exactly one point of each Ui. Hence F is one-to-one on each of the sets v , n F - l ( V ) . Therefore, F is a homeomorphism on some neighborhood of each point in Yn. Clearly, Y,-l c Y \ Y,. Let Xn-1 = Y,-l \ Pn. By the above argument Xn-1 is an open subset of Y \ Pn; and therefore, an open subset of Y , and F is a local homeomorphism Now remove X,-1 from Y \ Yn and at each point of X,-1. repeat the argument. Eventually we obtain disjoint open sets Yn,X,-l,Xn-2,. . . ,- X2, X I SO that F is locally - one-to-one except possibly on Yn \ YnYXn-1 \ Xn-1,. . . ,X2 \ X2. We claim that the interior of the union of these sets in Y is empty. Assume on the contrary that W is a nonempty open set and W C bYn U bXn-1 U -..U bX2. TO be specific, assume that W contains a point of by, and therefore a point y of Yn. Clearly, y is not a boundary point of Yn since Y, is open. If y E bXj, then the open neighborhood Yn of y must contain points of X j , which is impossible. Therefore, the set Yn U X,-1 U . U X I , where F is locally one-to-one, is dense in the set Y .

--

4.1.12. LEMMA[Bear, Hile]. Let F be a continuous open map on a locally compact Hausdo# space Y to a topological space 2. I f F is light on Y , then F is locally one-to-one on a dense open set.

PROOF.Consider first the case when F is finite-to-one. By Lemma 4.1.11(1) the set Y,(+)of all points in Y where F is at least n-to-one is open for each n. We have Y = Yj+)2 Yz(+)3

..., and by the hypothesis

n Y,(+)= 0. 0

The set on which

n= 1

F is locally one-to-one is open by definition, so we need only to show that it is dense. Suppose on the contrary that there is a non-empty open subset U of Y such that F is not a local homeomorphism at each point in U. Hence Y,(+)n V # 0 for every n and for every nonempty open set V in U , by the Lemma 4.1.11(2) (if assume the opposite, F will be at most n-to-one on

240

Chapter IV.Analvtic Structures in Algebra Spectra

Yi+)

V ) . Therefore, each is dense in U and also open, by the Lemma 4.1.11(1). Since Y is locally compact and Hausdorff, it is regular (see e.g. [59]); and hence any countable intersection

n Yn(+)is 00

of open dense subset in Y is dense ([59]). Therefore,

n=l

dense in U . This proves the statement for finite-to-one mappings

n Y;+)= 0. 00

F since for such mappings

n= 1

Suppose now that F is light. Let y be a fixed point of Y and let Q denotes an open neighborhood of y with compact closure. The restriction FIQ is finite-to-one so F is a local homeomorphism on a dense open subset of Q, by the first part of the statement. Therefore, F is a local homeomorphism on a dense open subset of Y . Now the proof of Proposition 4.1.6 follows easily. Indeed, if F is light on U , then according to Proposition 4.1.9, F is open on U ; and therefore F is locally one-to-one on a dense open set of U , by Lemma 4.1.12. Theorem 4.1.4 and Proposition 4.1.6 imply the following

4.1.13. THEOREM [Tonev]. Let A be a uniform algebra. If F = ( f i , . . . , f n ) E A" is light on an open subset U of s p A which does not meet a(")A,then there exists an open dense subset R of U which can be given the structure of an n-dimensional complex analytic manifold such that the functions in A are holomorphic on R. In fact F ( m ) is a locally uniformizing variable on R and every F in A is locally holomorphic function in F ( m ) , m E s p A. Observe that we are interested in analytic structures which might exist within the set a(n)A c spA \ U only. It is quite possible some subsets of n-tuple Shilov boundary to admit analytic structure as well. Example of an algebra, whose Shilov boundary contains a (nonopen) subset with an (one-dimensional) analytic

241

4.1. n-dimensional manifolds i n spectra

structure is the so called "tomato can" algebra, defined as follows: Let X = A x [ O , l ] and let A = {f E C ( X ) : f( . , 0 ) E A ( A ) } . It is not hard to see that s p A = dA = X and that A x ( 0 ) is an analytic disc lying in dA. Of course the set A x (0) is not open in X . As we know, ba(F) c F(B(")A)for every F E A". If W is a component of the set o ( F )\ F(a(")A),then F - l ( W ) is open in spA and F - ' ( W ) does not meet d(")A. Theorem 4.1.13 implies now the following

4.1.14. COROLLARY [Basener]. I f F E A" and W is a component of a ( F )\ F(a(")A)such that F is at most countable-to-one on F - ' ( W ) , then there exists an open dense subset R o f F - l ( W ) which can be given the structure of an n-dimensional complex analytic manifold and all functions in A are holomorphic on R. Consider, for example the algebra A = A(O), where R = A x D,D = { z E A : Re z 5 Im z + l}. As it not hard to see, spA and d(2)A= bR = bA x D U E x bD = S1 x B U z x 37T ( { z = eie,O 5 6' 5 ,>UK), where K is the segment [-i, 11. For

=n

-2

the 2-tuple F ( z ~ , z=~ 0 such that B(F(rn,),S) = { z E C" : 112~ ( r n , ) J< l S> c F ( u ) . Clearly, F-1(B(F(mo),6)) nint # 0 and hence there exists an integer k, 1 5 k I: 1 such that F ( B ( F ( ~ , )S), n int uk n int (u/-))# 0.Similarly, F(int u,>n B(F(rn,),b) is a nonempty open set in C". Fix a point z, E B(F(m,),S). It is not hard to show that there exists a w E Cn\O such that if L = {zo [w : [ E Cl}, then the set L n F(int Uz) n B(F(rn,), S) is open. For simplicity let zo = 0 and w = ( l , O , . . . ,O). Let G = (f2,. . . ,fn) and let Q be the set { z E C : ( z , 0,. . . ,o) E fl(int uk)n B ( O ,s)]. Clearly, f l is ICto-one on the set fT1 n U . Let X = [f;l(bB(O,S)) n U ] and consider the algebra A x . We have spAx = [j;'(B(O,S)) n U ] , f 1 ( , E A x , #fc'(z) = k on fF1(Q) n U and Q is an open set in the connected component of the set C1 \ bB(0,S) = C1 \ f l ( X ) . According to the corresponding result in the oneIC for every dimensional case (see [48, Theorem 11.2]), #j;'



+


3/4}, so that U is a neighbourhood of F-'(z0). Choose an E > 0 such that { m E spA : IIF(m) - z,lI 5 E } c U. The set N = { m E spA : IIF(m) - z,ll 5 e,Ih(rn)l 5 1/4} is a compact A-convex neighborhood of J and N c Q. If m E bN, then either IIF(m) - z,ll = E or Ih(m)l = 1/4 so that in both cases z , $! F ( b N ) (because of Ih(m)l # 1/4 on { m E spA : IIF(m) zoII 5 E } ) . Finally, N c spA \ a("-l)A, so by Corollary 4.18 AN C bN. Since z , E F ( N ) \ F ( b N ) , Lemma 4.1.2 implies that F ( N ) is a neighborhood of z,. The lemma is proved.

4.1.21. L E M M A .Let F E A" and let W be a component of C" \ F(a(")A). If T is a relatively closed subset of W and if F is finite-to-one on F - l ( W ) , then for every 1 2 1 the set F((F-~(w)):-)n ) T is relatively closed and consequently the set F ( (F-l ( W ) )[) n T is measurable.

PROOF.Suppose z , E T, z , E F ( ( F - ' ( W ) ) ; - ) ) n T and let z n tend to zo as n -+ 00. Let k = #F-'(z,) < 00. By Lemma 4.1.20 there exist k disjoint compact sets N1,. . . ,Nk such that F ( N j ) is a neighborhood of z , for j = 1,.. . ,Ic. Thus Ic 5 #F-'(z,) 5 E for n large enough; therefore, z , E ( F - ' ( W ) ) ; - ) and we are done.

Chapter IV. Analytic Structures in Algebra Spectra

248

Let U be an open set in s p A \ 8 " ) A and such that F ( U ) is an open and connected subset of C" which does not meet F(a(")A>. Denote 21 = { m E U : for any S > 0 there exists a measurable set Kg c B ( F ( m ) , S )f7F(U,!-)) c C" with 7 7 2 2 k ( K a ) > O}. It is clear that 21 is a closed subset of U ; and therefore, F(Z1) is closed in F ( U ) . The next lemma shows that only two possibilities exist for 21 - either to be empty or to coincide with U . 4.1.22. LEMMA.F(Z1) is an open subset of F ( U ) which is contained in F(u:-'). PROOF.Let m, E 21 and let z, = F(m,). Choose a 6 > 0 such that B(z,,S) c F ( U ) and let Kb be a closed subset of B(z,,S) n F(U,'-') such that m z n ( K b ) > 0. Denote X = [ U n F-'(bB(z,,S))] and consider the algebra A x . We have: s p A x = [F-'(B(z,,6)) nu] and FIX E A F . By Lemma 4.1.21 applied to A x , the set K6 n F(Uj(-)) n B ( z , , 6) is measurable for each j. Therefore, there exists some integer k, 1 5 k 5 1, such that r n z n ( F ( U k )n K6) > 0. Fix a z, E B ( z , , 6 ) . It is not hard to show that there exists a w E Cn \ ( 0 ) such that if L = { z cw : E C ' ) , then m z n ( L n F ( U k ) n K ) > 0. For simplicity let z, = 0 and w = (1,0,. . . ,O). Let G = ( f 2 , . . . ,f n ) , let % be the component of C' \ f 1 (bA(0,S) n V ( G ) )which confl K b } . tains 0 and let I< = { C E C' : ( < , O , . . . , O ) E f~(Uk) Then f l X E A x is Ic-to-one on fF'(K) n U . According to the one-dimensional result due to Wermer (see [64, Theorem 11.21) #fl(C) 5 Ic for any C E @. Thus #fyl(0) n U = # ( F - I ( o ) n U ) 5 k. So # ( F - ' < z ) n U ) 5 Ic for each z E B ( z , , S) and hence F(Z2) 3 B(z,,S). We conclude that F(Z2) is open in

+


0.

Denote the

u K6 f'l F(Ul), Lemma 00

set K fl B ( z , , S ) by

Kb.

Since Kg =

I='

4.1.21 implies that each Kb n F ( U / - ) ) n B ( z o ,6) is measurable, so there is a positive integer Ic such that m2n(K6 n F(Uk)) > 0. Then F ( Z k ) # 0 so the connectedness of U and L e m m a 4.1.22 k

imply that F ( Z k ) = F ( U ) . Thus

U c Ul-) = U Ul. Note I='

also that if I < k the same argument implies that we must have rnzn(Ui) = 0. We conclude that Uk is dense in U ,as desired.

As we know, ba(F) c F(O(")A)for a given F E A"; and therefore, every component W of a(F)\ F( a(")A)is an open set in C " with F - ' ( W ) open and contained in a ( F )\ F(a(")A).Theorem 4.1.17 and Proposition 4.1.19 together imply the following 4.1.23. THEOREM [Basener]. Let F E A" and let W be a component of Cn \ F ( 8 " ) A ) which meets a ( F ) . Suppose that there exists a set K c W such that rnzn(K) > 0 and #F-'(z) < 00 for every z E K . Then there exists a s h e a f 0 of germs offunctions on F - ' ( W ) such that (F-'(w), 0 ) is an analytic space of pure dimension n and all functions in A are holomorphic on F - ' ( W ) . Actually, there exists a positive integer k such that (F-l(W),FIF-l(w), W ) is an n-dimensional k-sheeted branched analytic cover with negligible set ( F - l ( W ) ) k - l ,and 0 is the sheafof germs of a.U functions in Alu. In fact F ( m ) is a locally uniformizing variable on F - ' ( W ) , and each function in A is locally holomorphic function in F ( m ) , rn E F - ' ( W ) . The only fact which needs a proof is that the analytic cover

(F-l ( W ) , FlF - l ( W ) 7 W ) is an analytic space, which actually follows from a theorem of Grauert and Remmert (see e.g. in [42, Theorem 321).

250

Chapter IV.Analytic Structures in Algebra Spectra

Note that the finiteness-to-one requirement on F is not too artificial, since, as shown by Bishop, given a G E A " ( 0 ) , the set G-'(z) is finite for almost every z E 0. of As shown by Sibony [91], if the envelope of holomorphy W is a finitely sheeted Riemann domain contained in a ( F ) ,then W ) in Theothe branched analytic cover ( F - '( W), FIF-l(w), rem 4.1.23 extends to a Stein space S in s p A and the functions in A are holomorphic on it. Let 0 be a domain in C" and let F = (f1,. . . ,fP) be a p-tuple of functions analytic in 52 which separate the points of 0. Let h' be a compact set in f2 and consider the uniform subalgebra P ( F ( K ) )of C ( K ) which is generated by the constants and the functions f i , . . . , fP. Observe that s p P ( F ( K ) )is homeomorphic to h ( F ( K ) ) ,the polynomially convex hull of F ( K ) in CP.

4.1.24. THEOREM [Sibony]. Let K be a compact set in C". I f h ( F ( K ) ) c CP, then there exists an analytic structure of dimension n in an open dense set u in ~ ( F ( K \) B ) (")P(F(K)). More precisely, any point m in U has a neighborhood in h( F ( K ) ) = s p P ( F ( K ) ) which can be equipped by the structure of an ndimensional branched analytic cover.

4.1.25. NOTESAND REMARKS. Stolzenberg's example for nonexistence of analytic structure in sp A \ d A was published first in [106]. The search for analytic structures in algebra spectra was initiated by the classical result of Bishop [18]from 1963, which is precisely the one-dimensional case of Theorem 4.1.23. The result presents in some sense an opposite assertion to the following well known fact from the classical function theory: Given an analytic function f on some domain D in C1 and a component W of C1 \ f(D), then the set f - l ( w ) is finite for every w E W . In a similar sense, Theorem 4.1.23 is an opposite statement to the following abovementioned result of Bishop [18], utilized by him for proving the O h ' s the-

4.2. Big-manifolds an algebra spectra

251

convex domain in C ” , the sets G-’(z) are finte for almost every point z in 52 (see also [143]). Aupetit and Wermer [6] have strenghtened Bishop’s theorem by removing the Lebesgue measure from the context and replacing it by the logarithmic capacity. Basener [8] has removed the finitenes requirement from the area and has replaced it by the countability. Basener [9] passed first beyond the framework of single functions in A and onedimensional analytic structures respectively. Soon after, Sibony [91] has extended Basener’s results to various directions. Corollary 4.1.14 had been proved originally by Basener [9] in a different way (see also [143]). The proof of Theorem 4.1.23 follows the same line as Wermer’s proof of Bishop’s theorem from [143]. It has been extended afterwards by Aupetit [6] for sets K C W with nonzero outher capacity. The notion of branched analytic covering (Definition 4.1.15) is due to Gunning and Rossi [48]. Theorem 4.1.1, Lemma 4.1.3 and Theorem 4.1.24 are proved in [91]. Lemmas 4.1.7, 4.1.11, 4.1.12 and the one-dimensional cases of Propositions 4.1.6, 4.1.9 and Corollary 4.1.10 are proved in Bear and Hile’s paper [151. The one-dimensional case of Theorem 4.1.13, due to Bear and Hile [15], presents a uniform-algebraic generalization of a classical result of Stoilov [lo51 about functions on one-dimensional complex analytic manifolds. Kramm [64] has utilized the multi-tuple analytic structures and multituple Shilov boundaries to help obtain a characterization of Stein algebras. Other conditions for locating n-dimensional analytic structures have been found by Rusek [89] and, in a more general setting, by Kumagai [68], [69].

4.2. BIG-MANIFOLDS I N ALGEBRA

SPECTRA

In this section we give r-analytic versions of most of the results in the previous section. In particular, we extend Bear Hile’s theorem for T-analytic big-manifolds. Though most of

Chapter IV. Analytic Structures in Algebra Spectra

252

the proofs are just the same as the corresponding proofs for the n-dimensional case from Section 4.1, we give them completely to show what changes they require in order to work in the analytic situation as well. We begin with the following version of a theorem due to Rudin which we have made use of in Section 4.1.

r-

Let the group F possesses property 4.2.1. PROPOSITION. (*) and let A be an algebra of continuous functions on a basis say Q = neighborhood Q in the big-plane C G , where G = Q ( Z ~ , E , U )= { z E C G : (za- z z l < E } such that:

pr,

(i) Ail functions xa, a E TO= [O,w]n belong to A; (ii) A satisfies a maximum modulus principle with respect to the topoiogicd boundary bQ of Q, i.e. the inequality

r

(93) holds for each z E and for every F E A. Then all functions in A axe r-analytic in Q. The statement follows immediately from Theorem 2.7.5 because the uniform closure 2 of A also satisfies (i) and (ii) from above and the hypotheses of Theorem 2.7.5 are satisfied for the algebra A = XIbQ with a(f2) = &.

r

Let A4 be a Hausdorf€ space, let the group possesses property (*) and let G = PI'. Below we introduce the notion of big-manifolds. The definition is quite similar to the definition of complex manifolds; but instead of Euclidean-space-valued charts we use big-plane-valued ones. We say that M is a big-manifoZd (or manifold over the bigplane C , ) if there exists an open covering {U,},,l of M (so that U , = M ) and a family {va}aEl of homeomorphisms

u

aEI

4.2. Biu-manifolds i n aluebra spectra

253

of U, onto open subsets cp(U,) of C G , such that cpp(cp,'(z)) is a continuous map of cp,(U, n Up) onto pp(U, n Up) for each pair a , p E I. If the functions x0 (pp(cp;'(z))) are r-analytic on (p,(U, n Up) for every a E To and for each a , p E I, we say that M is an r - a n d y t i c big-manifold (or analytic manifold over C G ) . A complex valued function f defined on a r-analytic bigmanifold M is r-holomorphic if for each set U,, a E I, the function f o cp,'(z), which is well defined on cp,(U,) c C G , is r-analytic.

4.2.2. THEOREM. Let A be a uniform algebra. Let thegroup satisfies condition (*) and let R = {fa}a,=r, be a multiplicative subsemigroup o f functions in A which is algebraically isomorphic to the semigroup To. If the spectral mapping r, : s p A + C G of R is locally one-to-one on an open subset U of s p A \ dA, then the set U can be given the structure of a I'-analytic bigmanifold such that the Gelfand extensions o f all functions in A are r-holomorphic on U .

r

For the proof we shall need the following two auxilliary lemmas. 4.2.3. LEMMA.I f the group r possesses property (*) and if 0 = { f a } a E r , , then the inclusion

holds for every uniform algebra A which contains 52.

PROOF.Suppose that boL? \ r , ( d A ) # 0 and let m, be a point in s p A such that ~ , ( m , ) $ b a ( 0 ) \ r,(dA). Then one can find a number a E To and a neighborhood U = { z E C G : - x " ( ~ , ( m ) )< l E } which does not meet .r,(dA>. Since U = ( x " ) - ' ( V ) ,where V = { z E C : 12 - xa(.r,(m,))l
)= E a ( f u ) .If we assume that a whole neighborhood of the above type, say V 3 x a ( T , ( m o ) ) is contained in a ( f a ) = A ( s p A ) , then (x")-'(V) c ( x " ) - l ( L ( s p A )C ) a(52)in contradiction with T,(m,) E ba(52). Consequently, fa(m,>E b a ( f a ) , i.e. L(m,> = x a ( T n ( m o > ) E bo(f,)\ X ( a A ) which is absurd. The lemma is proved. E},

L(m,)

.E@A)

h

4.2.4. LEMMA.Let E be a closed boundary of A, let W be a component of the set CG \ rn( E ) and let 52 be as in Theorem 4.2.2. Then W C a(52) whenever W n a ( 0 ) = 0.

PROOF.Clearly, the set W1 = W \ ~ ( 5 2 is) open in W . We claim that W1 is also closed in W . Indeed, if z E w1 na(52) then z E bo( 52) c T, ( a A ) c r, (E) by Lemma 4.2.3 and consequently bVV1 C T, ( E ) C CG \ W . Since W is a connected set, W \ W1 is a clopen subset of W and W \ W1 = W n a ( Q ) # 0 by the hypothesis. We conclude that W \ Wl = W ; and therefore, W I = W \ a ( 0 ) = 0, i.e. W c a ( Q ) ,as desired. PROOFOF THEOREM 4.2.2. By the hypothesis each point m, E U has an open neighborhood Vm, c Vm, c U of type

vmo

homeomorphically onto a closed subset such that T, maps Ir' of C G and for which T ~ ( ~=VbK. ) Hence T, (m,) E Ii' \ b K , and there exists a neighborhood Qm, = Q ( T , ( ~ ~a ,)E,) = { z E C G : 12" - ( ~ , ( m , ) ) " l< E } lying in K \ bK, where a E I'. Consider the set N = T;'(Q) n Vm, = { m E s p A : If"(rn) f u ( r n o ) l < E , lcj(m) - cj(m,)l < 6, j = I , . . . , m } . Clearly, h

h

4.2. Bio-manifolds an alaebra spectra

255

N is an A-convex neighborhood of m, which lies in Vm and r , ( N ) = Qm,. By Rossi's local maximum modulus principle (1.5.14) i 3 A p c bN; and consequently, each f in AT attains the maximum of its modulus within bN. Hence Am, = 20 -1

70

Ism,

is an algebra of functions on GrnO c CG whose Shilov

fl, r.

boundary is contained in bQm, = r, ( b N ) . By (28) (ri'(z>) = X=(T, o r;'(z)) = ~ " ( 2 )= za for every a E Therefore, A,, is an algebra on which satisfies a maximum modulus principle with respect to bQm, In addition Am, contains all the functions xa, a E To and dAmo c bQm,. By Proposition 4.2.1 Am, c A(Gma), i.e. every f in A is a r-analytic function on Qm,. Let Um, be the set r i l ( Q m a ) n Vm, and by Qm, the mapping r, Evidently, U Urn, = U. Clearly, ymois a

gm,

Iuma.

m,EU

homeomorphic correspondence between Urn, and Q m , , because by the hypothesis r, is a homeomorphism on the set Vm,. Let ~ = z for now Qml nQm, # 0 for some ml # m2. Since r, ( T (z)) each z E Q m l f I Q m z , we have that v m l (pz:(z)) = z for every z E Qml nQmz, Hence p m l o pi: = id on Q m l n Qm,. Consequently, U is a big-manifold, and moreover, a r-analytic big-manifold. If f E A, then cp;' = (r, =fo E Am, for each m, E U as we have seen above. Consequently, every function E A^ is r-holomorphic on U. The theorem is proved.

70

70 Ivma)

h

~

;

~

1

7

A^

Note that locally each function in depends F-analytically on the variable T, (2) on U , which actually is a locally uniformizing variable on U for all functions in

A^.

In particular, if the spectral mapping r, is one-to-one on an open subset U of SPA, then U can be given a structure of a F-analytic big-manifold. Below we extend this observation for more general situations.

~

~

256

Chapter IV. Analytic Structures in AlPebra Spectra

4.2.5. PROPOSITION. Let S2 be as in Theorem 4.2.2. I f the spectral mapping r, is light on an open subset U of s p A \ aA, then it is locally one-to-one on an open dense subset o f U .

For the proof we need several auxiliary topological results. 4.2.6. PROPOSITION[Tonev]. Let A be a uniform algebra and let 0 be as in Theorem 4.2.2. If the spectral mapping r, of f2 is light on an open subset U of s p A \ aA, then r, is open on

U.

PROOF.Assume first that

is finite-to-one but not open on a point m, E V such that r, (m,) E br, ( V ) . It is clear that for every closed neighborhood Q c V of m, we have r,(rn,) E brn(Q). If G is A-convex, then s p A= ~ and ~ A Q c @ by Rossi’s local maximum modulus principle. Hence by Lemma 4.2.3 applied to the algebra A ~ Jwe have rn(m,) E br,(Q) C r,(dAq) C rn(@). Thus for every closed A-convex neighborhood Q c V of m, there exists an mQ E bQ with r , ( m ~ )= ~,(m,). By Lemma 4.1.7 there are uncountable many A-convex neighborhoods Q c U of m, with pairwise disjoint boundaries. Thus there are uncountable many points rn, E bQ c U with r , ( m g ) = rn(m,) in contradiction with our hypothesis on 7., Let now r, be a light mapping on U . Every open set V C U is a union of open sets Q such that c U . By the compactness argument the mapping r, is finite-to-one on each set of type Q. Hence TQ is open on U according to the first part of the statement. Consequently, the set T,(Q) is open for every Q with Q C U . Naturally, r,(V) = U r n ( & )is also an open set. The T,

U. There is an open set V c U and

v

proposition is proved.

Q

Now the proof of Proposition 4.2.5 follows easily. Indeed, if r, is light on U , then according to Proposition 4.2.6 the mapping

4.2. Baa-manifolds an alaebra spectra

257

r, is open on U and by Lemma 4.1.12 it is locally one-to-one on an open dense set of U . Together with Theorem 4.2.2 Proposition 4.2.5 implies the following [Tonev]. Let A be a uniform algebra, let 4.2.7. THEOREM the group r possesses property (*) and let 0 = { f a } a E r , be a multiplicative subsemigroup of functions in A which is algebraically isomorphic to the semigroup To. If the spectrd mapping rn : s p A + CG is light on an open subset U of s p A \ dA, then an open dense subset R of U can be given the structure of a I-'-analytic big-manifold such that the Gelfand extensions of all functions in A axe r-holomorphic in R. In fact r,(rn) is a locally uniformizing variable on R and the Gelfand extension of each function in A is a locally r-holomorphic function in the variable r,(rn), rn E spA.

By (94) T, ( d A ) 3 ba(f2)and if W is a component of a( 52) \ r, ( d A ) then U = T;'( W ) is an open set in s p A and U does not meet dA. Theorem 4.2.7 now implies the following

4.2.8. COROLLARY. If L?is as in Theorem 4.2.7 and if W is a component of a(f2)\r, ( d A ) such that T, is at most countableto-one on r;'(W), then there exists an open dense subset U of T;'(W) which can be provided by the structure of a r-analytic big-manifold such that Gelfand extensions of all functions in A are r-holomorphic on U . Actually Theorem 4.2.7 is stronger than Corollary 4.2.8. It helps to recognize r-analytic big-manifolds not only within the , on set r i 1 ( W ) , where W is a component of o(52)\ T , ( ~ A )but the whole set s p A \ d A as well.

258

Chapter IV.Analytic Structures in Aljzebra Spectra

4.2.9. DEFINITION. k-sheeted branched r-analytic big-coveT we call any triple ( U ,T , V ) , where:

(i) U is a locally compact HausdorfT space; (ii) V is a domain in C G ; (iii) T is a proper continuous mapping of U onto V (i.e. .-'(IT) is compact in U for every compact K c V ) which is light on U ; (iv) There exists a negligible set A c V and an integer k such that T is a k-sheeted covering mapping of U\T-'(A) onto \ A; (v) The set U \ ..-'(A) is dense in U . If A = 0, then the r-analytic big-cover ( U ,T , V ) is called nonbranched and if all conditions except (iii) in Definition 4.2.9 hold, then ( U ,T , V ) is called improper.

v

Similarly to the n-dimensional case, by a negligible set here we understand a nowhere dense in D C C G set A such that for every subdomain D1 c D any function f which is r-holomorphic on D1 \ A and locally bounded in D1 admits a r-holomorphic extension on D1. 4.2.10. DEFINITION. A continuous complex valued function f defined on an open subset W of a branched r-analytic bigcover ( U ,T , V ) is r-holomorphic on W if for any open subset W1 c W\T-'(A) on which T is one-to-one the function f ~ w , o ~ - l is T-analytic on the set T ( W ~c)C G . If a spectral mapping T, is E-to-one on U c s p A \ d A , then by Theorem 4.2.7 one can get that the triple (U, T, T,(U)) is a (possibly improper) k-sheeted non-branched r-analytic bigcover; and all functions in A^ are F-holomorphic on U . In fact, r n ( z )is a locally uniformizing variable on U and every Gelfand extension ?E is T-holomorphic function in the variable T , ( z ) . The structure of U will be more delicate if r, is only E-to-one

Iu,

A^

4.2. Baa-manifolds an alaebra spectra

259

at most on U . To show this we need some preparations. First of all we shall fix some notations. Let U be a subset of s p A and l2 c A. Denote as before Eu

:#T;'(T,(~))

=k},

Ui+) = { m E U

:#T;'(T,(~))

2 k},

Ui-) = { m E U : # T ; ' ( T , ( ~ ) )

5 k}.

u k

=

(772

4.2.11. PROPOSITION. Let T satisfies condition (*) and let 0 be an isometrically isomorphic to To multiplicative subsemigroup of functions in a uniform algebra A. If U is an open subset of s p A \ d A such that u k is dense in U for some k , then the triple (U, T, T,(U)) is a (possibly improper) k-sheeted

Iu,

branched r-analytic big-cover with U i I i as a negligible set, such that Gelfand extensions o f all functions in A are r-holomorphic on U .

PROOF.By Lemma 4.2.6 the spectral mapping T, is open on U . According to Lemma 4.1.11(i) U k is open in U and Lemma 4.1.11(ii) asserts that r, is locally one-to-one on u k . By Lemma 4.1.11(i) every UI(+) with 1 > k is open in U and thus empty because Uk is open, dense and disjoint from any Ul with 1 > k. Consequently U.1; = U \ U k is nowhere dense in U . By Theorem 4.2.2 u k can be given the structure of a T-analytic bigmanifold such that the Gelfand extensions of all functions in A are T-holomorphic in u k . It follows that T, + @Uk) is a k-sheeted covering mapping. Let zo E T , ( u k ) and let T ; ' ( Z ~ ) f l u = { p l ( z , , ) , .. ., p k ( z , ) } . For each function g E A which separates the points p l ( z o ) ,. . . , p k ( z 0 ) the function

Iv,

ChaDter IV. Analytic Structures in Algebra Spectra

260

is continuous on r,(Uk). Since T, is locally homeomorphic in Uk,then the proof of Theorem 4.2.2 demonstrates that H , is I'analytic on the open set T, ( U E )which is dense in T, ( U ) because Uk itself is dense in U . Clearly, # ( T ; ~( z ' ) n U ) < k whenever z' E T,(U) \ T ~ ( U , )and to be specific let T ; ~ ( z ' ) n U = { P l ( z ' ) >... , p l ( ~ ' ) > I, < k . Let Zm E T , ( U ~ )and let zm tend to z' E T,(U). The cluster points in U of all sets { p j ( ~ ~ j) = } ~ ~ ~ , 1 , . . . , 1, are among the points p l ( z ' ) , . . . ,pl(z'). Indeed, if m, = lim p a r ( z r l ) ,m o E u, then ~ , ( m , )= lim 7, ( p , , ( z m , ) ) = 1'00 l+OO lim Zm, = z' i.e. m, E T;'(z') n U and hence m, is one of the f '00

points P ~ ( z ' >.,- - , ~ t ( z ' ) ~ . h u {s1S(Pi(Zrn))-g(pj(zrn))l},#j as m -+ 00 and hence Hg(zm) =

n

(XPi(zrn>)- ii(pj(zrn)))

+

-+

0

0

i#j

-+ 00. If we define now H S ( z ' )as 0 for z' E T,(U)\T,(U~), we obtain a continuous function on T,(U) which is T-analytic on the set { z E T,(U) : H , ( z ) # 0 } c T,(U~).Now we apply Rado's theorem to conclude that H g is r-analytic on T,(U). Since the functions in A separate the points of s p A , we have

as m

7-,

(?$',)

= .,(U)

\ T,(U,)

=

u

{ z E T,(U) : H , ( z ) = 0).

SEA

Thus 7, (@;) is a proper I'-analytic big-subvariety of T, ( U ) and hence it is a negligible subset of T, ( V ) . Since 7;' (T, ( U ) ) \ T ~ ( U ~n I )U~=) U \ UiI', = U k is dense in U = T;~(T,(U)), we see that the triple (U, T, ,T, ( U ) ) possesses all features of a k-sheeted branched r-analytic big-covering with the exception of condition (iii) in Definition 4.2.9, which actually might be violated if T,(U) n T , ( ~ A#) 0. The theorem is proved.

4.2. Bag-manifolds i n alaebra spectra

261

4.2.12. PROPOSITION. Let 52 be as in Proposition 4.2.11, let U be an open subset o f s p A \ d A such that is a connected set in C G and let # r i 1 ( z ) 5 k for some k, 1 5 k < 00, and for each zET , ( U ) . Then there exists a positive integer k l 5 k such that U k l is dense in U .

PROOF. Without loss of generality we can assume that Uk # 0. As in the proof of Proposition 4.2.11 we observe that r, is an open mapping in U , that Ur, is open set in U and that the function H , in (95) vanishes identically on 7, (Uj:',). Hence the set r, ( U j I ; ) is nowhere dense in T, ( U ) ;and therefore, UiI; is nowhere dense in U because of the openness of T, on U . By Propositions 4.2.11 and 4.2.12 we get the following

r

[Tonev]. Let satisfies condition (*) and 4.2.13. THEOREM let 52 be a multiplicative subsemigroup of functions in a uniform algebra A which is isometrically isomorphic to I f U is an open subset of s p A \ d A such that r n ( U ) is connected in C G , and if there exists a k , 1 5 k < 00, such that r i l (T, ( m ) )5 k for each m E U , then there exists a positive integer kl 5 k such that:

r,,.

(i) The triple ( V , T , ~ ~ , T , ( U )is) a branched kl-sheeted I"-

analytic big-cover (probably improper), with &', as a negligible set; (ii) T h e Gelfand extensions of all functions in A are rholomorphic on U ; (iii) T h e covering is proper i f and only i f r, ( U ) does not meet 7 ,

(84.

Clearly, Theorem 4.2.13 holds for each set U of type r i l (W ) , where W is a component of g( 52) \ r, ( d A ) .

262

Chapter IV. Analvtic Structures in Algebra Spectra

4 . 2 . 1 4 . NOTESAND

REMARKS.

The 2-case of Theorem 4.2.7 coincides with the Bear and Hile's theorem [15]. The Z-case of Proposition 4.2.1 is due to Wermer (see e.g. [143]). Proposition 4.2.1 and Lemma 4.2.3 are from Tonev [127]. The 2-cases of Propositions 4.2.5 and 4.2.6 are due: to Bear and Hile [15]. The 2-case of Corollary 4.2.8 is due to Basener [8]. The case r = Q of Theorem 4.2.13 and Proposition 4.2.11 were originally proved in Tonev [118]. See also [127].

4.3. ALMOSTPERIODIC

AND

F-ANALYTIC STRUCTURES

In this section we establish explicit conditions which guarantke local existence of structures of F-analytic big-covers in uniform algebra spectrum. Throughout this section we shall consider, without mentioning especially, that the group possesses property (*) and that L? = {fa}aEro is an algebraically isomorphic to Tomultiplicative subsemigroup of functions in A . As before 7;, will stand for the spectral mapping of 0.

r

[Tonev]. Let A be a uniform algebra on 4 . 3 . 1 . THEOREM X and let rn maps an open subset U of s p A homeomorphically to some open connected set in the big-plane C G . If 9 E U is a linear and multiplicative functional of A which possesses the property

Iu

then the restriction rn of spectral mapping of fl is a homeomorphic mapping between U and T,(U) c C G and h o r;' is a T-analytic function on r n ( U ) c CG for every function h in A. A

4.3. Almost periodic and r-analvtic structures

263

PROOF.Note first that (97) is equivalent to the identity

where, as before, T' = (0,1] n F . Because of (98) one can easily observe that the functional cp is uniquely determined by its values on the functions in 0. Clearly, by X " ( T , ( ( P ) ) = which holds for all a E the point z, = T,((P,) is also uniquely determined by the given data, wherefrom we conclude that the full preimage T;'(z,) of z, consists of the element cp only. We : U + T,(U) is a one-to-one claim that the restriction T, mapping. Note that the full preimage of each point z E T,(V) consists of all maximal ideals in A which contain the ideal J ( z ) = [ (J (fa - z a ) A ] . We shall show that codim J ( z ) = dim A / J ( z )

A(cp),

r,,

Iu

aEP

+

= 1 for each z E T, (U). Under the ordering " b a if and only if bk = a for some integer k" the set F' is a system directed to the right, i.e. for every a, b E there exists a c E which follows both a and b. Clearly, [ ( f a -A(cp))A] C [(fb-&(cp))A]

r'

whenever b

+ a.

r'

Therefore, there arises an injective spectrum

{ [(fa - & ) A + C ];ig; a , b E P} of algebras A a ( z ) =

+

A(cp))A C ], where

it

(b

[(fa

+ a ) , is the natural inclusion.

-

We

have

=

[U

aEP

[(fa-E(V))A+c]]= =

[5

aEP

[(fa-z:)A+C]

1

[ 1 4Aa(~o)]. aEr'

Since the spectrum of an injective limit of algebras is the limit of the naturally arising projective spectrum of their spectra, sp A is

264

Chapter IV. Analvtic Structures in Algebra Spectra

homeomorphic to lim c { sp Aa(z,);(ii)*; a , b E

I"} where all in-

aEr'

jective and projective spectra are considered with respect to the partial ordering '' " introduced above. The homeomorphic corc spAa(z,) can be expressed respondence between s p A and lim

+

aEr'

explicitly in the following way: To each maximal ideal M = [ Ma] = [ l@ Ma] of algebra A it corresponds the element

u

aEP

l${Ma;z:} aEr'

Let now z ideal

aEr'

E lim c spAa(Zo), where Ma E spAa(z,), a E

# z,

aEr'

E rn ( U ) . Fix a number a in

Ja(z) =

[(fa

- za)A] n Aa(zo) C

r'.

r' and consider the Aa(zo).

Denote by Sa(z) the unit sphere of Ja(z). We apply below Kato theorem for perturbation of semi-Fredholm pairs of closed linear subspaces, which says: Given two closed linear subspaces M and N of a normed space 2 such that codim M = dim Z / M = k < 00 and for which S ( M ,N ) = sup g(u, N ) < 1, where S ( M ) is the uES(M)

unit sphere of M and e ( a , b ) = Ila - bll is the natural distance in 2 , then codimN I c o d i m M . Take Aa(z0) for 2 , J a ( Z 1 ) for M and Ja(z2) for N , where z1, z2 E T, ( U ) . By Kato theorem codim Ja(z2) 5 codim J a ( z 1 ) whenever S(Ja(zl),J a ( ~ 2 ) )< 1. Let h be an arbitrary function in A such that the functions tij = (fa - zg)h, j = 1,2, belong to Ja(zj). We have:

Suppose that the point z1 is contained in T,(V) together with a basis neighborhood Q(z1, E , a ) c C G ,where a E F' and 0 < E < 1. Since

265

4.3. A l m o s t periodic and r - a n a l v t i c structures

we have

wherefrom we conclude that

(99)

sup UESla(Z1)

1

e(U, Ja(z2) < 5 whenever

&

- z,"l < -2 '

Consequently, by Kato theorem we have that

whenever 2 2 E Q(z1,&/2,a). If dimA,(z,)/J,(zl) 5 1 for a z1 E T,(U) and any a E P , then dimA,(z,)/J,(z) 5 1 for every e in Q(z1, e/2, a ) c T, ( U ) . Consequently the closed set (a E T,(U) :dimA,(z,)/ Ja(zl) 5 l} is also open in T,(U). It is nonempty because it contains at least the point z,. By the connectedness of the set T,(U), we see that dimA,(z,)/J,(z) 5 1 on T,(U). Actually, dimAa(z,)/Ja(z,) = 1 for all z E T,(U) because Ja(z) is a proper ideal in the algebra A,(e,). Recall that we can choose the number a close enough to 0. As an ideal of codimension one, Ja(z) is a maximal ideal in A a ( z o ) .We can consider that the number a in r' from above is far enough with respect to the ordering >. Consequently, the ideal l@ Ja(z) aEr'

belongs to the projective limit lim c spA,(z) and by the above aEr'

maximal ideal of A for each z E T, ( U ) . Since, according to the initial remark no other maximal ideal contains J(z), we obtain that the full preimage of each point z in T,(U) consists of J ( z ) only. Consequently, the restriction T, : U -+ T, ( U ) of the

lu

266

ChaDter IV. Analytic Structures in Alnebra Spectra

spectral mapping T, on U is one-to-one. Now the case k = 1 of Theorem 4.2.13 completes the proof of the theorem.

-

In the case when a(R) = AG and cp E 7 i 1 ( 0 ) Theorem 4.3.1 yields the following result for embedding big-discs in uniform algebra spectrum.

4.3.2. COROLLARY. Let A be a uniform algebra on X and let in addition X = { m E s p A : If,(rn)l = l} for some (and hence for every) fa E R. If Kercp = [ U f, A] for some linear QEP

multiplicative functional cp of A then A is isomorphic to a subalgebra of the algebra H w ( A c ) of bounded r-analytic functions in the big-disc A G .

c rn ( a A ) c T, ( X ) c G and since a(R) = &. Moreover, U = T;'(AG)=

Indeed, because of bo( R) 0 E a(R), we see that

I

s p A\aA. By Theorem 4.3.1 T is a homeomorphic corresponn,u dence between U and AG and f o T;' are r-analytic functions in the open big-disc AG for each function f in A .

4.3.3. COROLLARY. Under the hypothesis of Theorem 4.3.1 there exists a continuous embedding j of some open subset W of the complex plane C into s p A whose range is dense in U , such that j is a bounded analytic r-almost-periodic function in W for each function f in A . Namely, for j we can take as before the composition T;' o j , of T ; ~ with any standard embedding j,, g E G, of C into CG; for W we take the set j;' o T,(U). Note that for the r-almostperiodic version of the result of Corollary 4.3.2 the set W should be choosen to be the upper half-plane H. The next theorem is a generalized version of Theorem 4.3.1.

4.3. Almost periodic and r-analvtic structures

267

4.3.4.THEOREM. Let A be a uniform algebra on X and let U be an open subset of s p A \ d A such that r,(U) is an open and connected subset of the big-plane CG. If the ideal

is of codimension k < 00 for some linear and multiplicative functional cp0 which belongs to U , then U can be given the structure of a kl-sheeted branched T-analytic big-cover over r,(U) for some k1 5 k, and h o r;' is a r-analytic function on this bigcover for each function h in A . The cover is proper if and only if r,(U) does not meet the set T , ( ~ A ) . h

PROOF.We claim that the number of elements in the preimages T ; ' ( z ) of each z E T,(U) is bounded from above by k. Proceeding in a similar way as in the proof of Theorem 4.3.1, we show first that dimA/J(z) 5 k for every z E ~ , ( y ) ,cp E U , where J(Z)

=

[

U

(fa

- .L(v))A] *

aEP

Fix a linear basis of the space A / J ( z , ) , say ( u 1 , .. . ,uk}, where uj = u j J ( z o ) , u j E A, and denote by B a ( z o )the Banach space Cul Cuk [ ( f a - x a ( z o ) ) A ]where , z0 = 7,(cp0). We have

+

+ + -

-

a

+

268

Chapter IV. Analytic Structures in Algebra Spectra

where the inductive limit is taken with respect to the partial ordering introduced in Theorem 4.3.1. For a z E r n(U)consider the space

By making use of Kato perturbation theorem for semi-Fredholm pairs of closed 1inea.r subspaces, in a similar way as in Theorem 4.3.1 we get that d imB , (z , )/E , (z ) 5 dim Ba(zo)/Ea(zo)for Because of E,( z ) c Ba(zo) we every z E T~ ( U ) and each a E have that E,(z) = Ba(zo)and hence A = C u l + . . . + C u k + J ( z ) for each z E T~ ( U ) ,wherefrom dim A/J(z) 5 k for every z E W . Let z be a fixed point in T,(U). Each element g in the factorspace A/J(z) can be presented in a unique way as

r'.

for some kl 5 k, where o,,, u = 1,.. . , kl, are functions in A whose cosets v, = o, J(z) form a basis in the factor-space A/J(z) and a, = a, J ( z ) , a , E A, are Icl linearly independent elements in ( A / J ( z ) ) *which form a basis in the dual space (A/J(z))* C k l .Consequently, each element g E A possesses a representation of type

+

+

v= 1

where h E J(z) and a,, v = 1,, . . ,Icl , are elements in A* such that a,(f) = a,(f) for each f E A. In particular = 0. If q is a linear multiplicative functional of A such that T~ ( q ) = z,, then fa = ~ " ( z )= za for every a E To.Since the Gelfand extension of each function in the ideal J(z) = [ U (f" - zz)A] h

aEP

4.3. Almost periodic and I'-analutic structures

269

h

vanishes on every such q, we have that h(q) = 0 €or the function h from above, and consequently

for every function g in A and for each linear multiplicative funcThis means that considered as an element tional Q in r;'(z). of A*, any functional q in r;'(z) coincides with the linear funct ional

v= 1

on A . As linear multiplicative functionals, the points in T;'(Z) are linearly independent elements in the dual space A*. Since a, are linear functionals over A*, identity (102)shows that the set r;l(z) can not contain more than k1 5 k elements, as claimed. Now we apply Theorem 4.2.13 to complete the proof. We use Theorem 4.3.4 to strengthen some of the results from Section 4.2. Note that both Theorems 4.3.1and 4.3.4hold for the set U = r;'(W), where W is a component of the set a(Q)\ r, ( X ) .

4.3.5.PROPOSITION. Let W be a component Of CG \r,(X). Suppose that the interior of the set r, ((r;'(W))k) is nonempty. Then #r;'(z) 2 Ic for every z E W . For the proof we need several auxiliary results. In what follows r;'(W)will be denoted by U. Observe that under this notation the hypothesis in Proposition 4.3.5now says the set int (r, (Uk)) to be nonempty.

ChaDter IV.Analytic Structures in Algebra Spectra

270

4.3.6. LEMMA.Let the spectrd mapping rn of 0 be light on U and let Q be a basis neighborhood of CG which is centered at z, and for which Q c & c W . If J is a component of T;'(&) such that T,(J) n Q # 0, then T,( J) = &.

PROOF.The set T;'(&) is A-convex. Indeed, if Q = { m E s p A : I&(m) - zZI < a} and if If^(m)l I max- f ( m ) l m € 7; ( 9 )

for some m E s p A and for every f E A, then in particular _Consequently, T;'@) is an A-convex subset of S P A ;and there) T;'(&). According to Rossi's local maxifore, s p A T i l t ~ =

mum modulus principle,

I$((m)l 5 br;

max

I

Is^(m>for

(G)u(aAnr; (G) all g E A and for each m E T;'@). Now dA n T;'(&) C X n T;'(&) = 0 because of & c W c C G \ T , ( X ) , and hence iF(m)l 5 max Ic(m)l for each g E A. By the continuity b

(GI

argument T;'(int&)

c

int

(Q)); and therefore, 3 A r n l f ~ )

(T; 1 -

(&) c T;' (bQ). Let now J be a connected component of T;'(&). We claim that S ~ A =J J . Let cp belong to S ~ A and J ~ ) is let @ be the linear multiplicative functional of A T n 1 (which ) T;'(&) C S P A , generated by 9. Because of @ E s p A , - , ( ~ = n we can consider (Z as a point in sp A . Suppose that $! J. Since J is a component, there exists a clopen subset J' c T;'@) for which J' c J and cp 4 J'. We can find a function e E X r i l t ~ ) with e ( 9 ) = 1 and e 0 on J'. Hence one can find a function

c

E A^ such that e ' ( 3 is close enough to 1 but whose values on J' are close to 0. Considered as a function in A J , the restriction

el

"1

Jf

satisfies the inequality

4.3. Almost periodic and r-analvtic structures

271

which is an absurd since IIqII = 1. Hence E J ; and consequently, S ~ A =J J . As shown above b ( ~ ; l ( g )c) ~ ; l ( b Q ) , which implies that bJ c r i l ( b Q )n J . Denote by X1 the set bJ. We have .,(XI) = r,(bJ) c r,(~;l(bQ)) nT,(J) c bQ; and therefore, .,(XI) n Q = 0. Hence Q is contained in some component of the set CG \ r,(X,). By Proposition 4.2.6 r, is open on U . Therefore, .,(XI) = r,(BJ) 3 br,(J) = br,(spAJ). By Lemma 4.2.4 applied to the set Q,the algebra AJ and the boundary r i 1 ( b Q )n J 3 X1 of A J we obtain that Q c r n ( J ) . Thus C r, ( J ) as desired. Denote by z1,. . . ,zk the points which are in the preimage ri1(z0) of a fixed point z, in T , ( W ~ )If. Q is a basis neighborC W and if z, € Q, then by J, hood of W such that Q C will be denoted in the sequel the component of contains the point z,, u = 1,.. . ,k.

T;'(&)

which

4.3.7. LEMMA. Under hypotheses of Lemma 4.3.6 on r,, Q and J , the following two assertions hold: (9 If Q is centered at a point z, in the set r,(Uk), then ..

u= 1

(ii) If & c r,(Uk) for some integer k, then: (a) r, maps injectively the set J , n ril(g) onto for every v = 1,2,. . . , k, and (b) For each function f in A and for every u = 1,.. . ,k there exists a r-analytic function F,, on Q such that f = F, o r, on the set J , n.7';

g

h

k

PROOF.Suppose that r;'(&) k

ril(g) \ u J,. u=l

\ U J, # 0

and let m E

u=l

Denote by K the component of r;'(&) which

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Chapter IV. Analvtic Structures in Algebra Spectra

contains m. Clearly, K does not meet any J,. According to Lemma 4.3.6 Q c T,(K); and therefore, zo = rn(rn) E C T h i s r,(K). Hence the point z, has more than k preimages. contradicts the hypothesis on the point z, and completes the c T,(U~), then by Lemma 4.3.6 and by the proof of (i). If first part of the statement it follows that T;'(Q)

(J,) = g for

k

=

U J,.

Since

u= 1

g

u = 1,.. . ,k, then for each v E there exists at least one point in every J,,, u = 1,.. Ic which

z, = T, (2,) E

T,

.,

maps at v. The sets J,, n 7 i 1 ( v ) can not contain more than one point since ~ i ' ( gc)~ k .Consequently, r, is injective on J,,n ril(Q);and this proves the case (;;)(a). For a fixed v , u = 1 , . . . ,k, the mapping T,, = r Jv presents the spectral mapping of semigroup of restrictions of functions onto &. in fi on J , and maps injectively the set J, n Consider the algebra AJ, as a uniform algebra on the set X = r,- l ( b Q ) = bJ,. Because of T , , ( T ; ' ( ~ & ) ) = bQ we see that 6'A J , c bJ, = ril( b Q ) by Rossi's local maximum modulus principle. Consequently, r, maps injectively the set J,,nT,-'(&) onto the component Q of the set C G \ .,(bJ,) = C G \ bQ. The case k = 1 of Theorem 4.2.13 applied to the algebra AJ, now completes the proof of the case (ii)(b) of the lemma. T,

I

~;'(g)

PROOFOF PROPOSITION 4.3.5.Fix a basis neighborhood Q = Q(z,,E,a) c Q c T , ( U ~ ) c int(T,(Uk)). We will use the same notations as in Lemma 4.3.7. Denote by H , the set Ti'(&) n J,, for u = 1,. . . ,k. Lemma 4.3.7(ii)(a) implies that

H , is an open subset of S P A . The set M = s p A

\

k

u H,

is

u=l

compact, and r, does not take the value z, on it. Consequently, for each point z E M there exists a number a(z1) E F' such that f^aa(zl,(z)# z$" = ~ ~ ( " ' ) ( z ,on ) some open subset Vzl of Ad. Choose a finite covering of M consisting of sets of type lcl = 1ra(8,1, j = 1,. . . ,n , and a number a E r' which follows all numbers a ( z j ) with respect to the partial ordering S . Now the

4.3. Almost periodic and r-analutic structures

2 73

function does not take the value zf on M . The algebra B = { f E C(spA) : fix,, E A q v , u = 1,.. . ,k} is a closed subalgebra ,= [ U ( f a - z a ) B ]for each z E W ,where the of C ( s p A ) . Let B aEP

closure is taken with respect to the sup-norm topology on s p A . Let { p l , . . . , p k } = T;’(Z,) be the full preimage of the point h

E T,(Hk).Obviously, BEO C J, = { F E B : Flr-l~Eo)o}. We claim that Bs0 = J,. Let F E J,, i.e. let F E B and let F ( p , ) = 0 for each v = 1,.. . ,k. Because of F E B for every u = 1,. . . ,k there exists a sequence { fn} of functions in A which tends uniformly to F on H,. By Lemma 4.3.7(ii)(b) there exist F-analytic functions on Q for which F 2 ) o T, = fn on Q. By shrinking Q if necessary, we can consider that F?) are continuous on & and that F?) o T, = f n on &. Since { f n } is a uniformly convergent sequence of functions on H , (and hence on If,), { F p ) }tends uniformly on & to a function, say F,(”),which is F-analytic in Q and continuous on &. Because F = FiY)o r, holds on H , and since Fi”)(z,) = limfnoT;l(zo) = F o ~ ; l ( z , ) , Z,

“

n

n

h

h

8’2)

h

n

h

n

each function F;”) can be approximated on & by I‘-polynomials &’) with p ? ) ( z , ) = 0, i.e. by restrictions of functions ( ( x a n ) ( ” ) ( ~ , “ ~ ) ( ” ) ) kon ? ) &. This means that all functions Flpv can be approximated on p, by functions in B. Denote by hn continuous functions on s p A for which h, H, = p p ) o 7, and also h

I

Let the numbers an E F’ follow a and all numbers { a(,Y ) } vk= l with respect to the ordering

hn h

)

+.

Consider the function

whose restrictions on H, belong to

?+bn =

P@,) for ev-

-z?) ery u = 1,.. . ,k. ?+bnis a continuous function on M . Indeed, because of a , >- a the denominator of t+bn does not vanish on M . (fa,

ChaDter IV. Analytic Structures in Algebra Spectra

274

h

Consequently, h , E (fa, - z z n ) B . Since y, tends uniformly to F on s p A , we obtain that every function F in B with F ( p y ) = 0 for each v = 1,. . . , k belongs to the algebra Be,, i.e. J , c B,,, as claimed. Now kl = dim B / B z , = dim B / J o 5 k and for a basis of the factor-space B / B z , we can take the cosets gj = g j +B,, for some g j E B , j = 1,.. . ,k, gj(pj) = S j j . Using the same arguments as in the proof of Theorem 4.3.4 and in a similar way we obtain 5 kl 5 k, for each z E W , as desired. that #r;'(z,) If for some k 2 1 there exists a mea4.3.8. PROPOSITION. surable subset N of r, (U,)such that the Lebesgue measure of the set x " ( N ) c R2 is Ron-zero for some (and thus for each) a E I',, then r,(Uk) has a nonempty interior. For the proof we need the following

4.3.9. LEMMA.Let A be a uniform algebra on X a n d let Q = Q( z , , E , a ) be a basis neighborhood in C G . If

ti) z o E a(Q); (ii) r, (X) = bQ; (iii) There exists a closed subset N C Q such that rn is one-to-one OR r ; ' ( N ) and for some (and thus for each) a E I', the Lebesgue measure of the set > i " ( N )C R2 is Ron-zero. Then for a possibly smaller neighborhood Q1 = &1(z,, E I , a l ) in Q the spectral mapping 7, is a one-to-one mapping from r i ' ( Q 1 ) ORtO & I .

PROOF.By shrinking the neighborhood, if necessary, we can assume that c = 1 and that dO(x"(N1))> 0 for N 1 = N n bQ. For a given Borel measure dp on X we denote by dp' the measure on C = bQ which is induced by the spectral mapping T,, i.e. dpLn(S)= dp(r;'(S)) for every Borel subset S c C. By the

275

4.3. Almost periodic and r-analutic structures

hypothesis a ( 0 ) 3 zo and T , ( X ) 3 bQ. Therefore, g ( 0 ) 3 by Lemma 4.2.4and hence X C 7;r'(bQ) = T ; ~ o (x")-'(S'). Let 'p1,'pz be such linear multiplicative functionals in s p A that ~ , ( ' p l )= r n ( ' p 2 ) = 21 E Q. We claim that 'p1 = 'p2. Suppose on the contrary that 'p f 'p2 and take a function g E A with i j ( ' p 1 ) = 1, 3 9 2 2 ) = 0. Let dpl and d p 2 be representing measures on X for 91 and 'p2 respectively. BY L(7,('pl))= E ( G ( ' p 2 ) ) = L ( Z l ) , which holds for any a E To,we see that dpf' and d p f are representing measures on bQ of the point z 1 considered as an element of spP(Q). Because P ( g ) is a Dirichlet algebra, all functionals in s p P(&) possess unique representing measures. Therefore, dpy = d p f . Since 7, is injective on T;'(N), the restrictions of dpl and d p 2 on T ; ' ( N ) coincide as well. Consequently, if dvj = d ( g p j ) " , j = 1,2, where d ( g p j ) are measures on X defined by

J d ( g p j ) ( x ) = J i j ( x ) d p j ( x ) for any Borel subset K of K

K

X , then

= d v 2 1 N . For our particular choice of g we get

s p ( z ) (du1 - d v 2 ) ( 4 = Jp(.,(.))ij(x) (103)

(4-4

X

C

- d P 2 ) ( 4 = P(G('p1)) = P ( Z 1 )

for every polynomial p on CG. Consider the measure dm on bQ defined as

where K runs within the class of all Borel subsets of bQ. According to (103) dm is orthogonal to all polynomials in CG. For a given measure du on bQ denote by dvS the measure on the unit circle S1 = x " ( C ) - 2: induced by dv, i.e. defined by

S '

c

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Chapter IV. Analytic Structures in Algebra Spectra

+

for any function f f C(S'). If dms = hdB dm:, where dm: is singular with respect to the Lebesgue measure d6 on S' and h E H ( A ) ,is the Lebesgue decomposition of dms, then (103) implies that d m s is orthogonal to the identity id: z --+ z considered as a function on A. The F. and M. Riesz theorem asserts that dms = 0, i.e. that dms = hd6. Because of dullNt = dv2lN1, on x"(N1) C x " ( C ) we have that dmsIxa(N1) = 0; and therefore, the measure h(6)dB = dms is identically equal to zero on x"(N1). As a function in H' which vanishes on the set x"(N1) c S1 of non-zero Lesbegue measure, h(6) is identically equal to zero on S' and therefore dms = h(6)dO = 0. Since z - 2: # 0 on S', then ~ " ( z-)z: # 0 on C ;and we conclude ) zz)(dv: - d u f ) . that du? = duf because of dms = ( ~ " ( z Therefore, the measure dul - du2 vanishes at each polynomial of type p ( z) = i;( xu (2)- z z ) in CG, where i; is some polynomial in C, in contradiction with (103). Consequently, 91 = 9 2 whenever T, (91) = 7, (p2), i.e. 7, is a one-to-one correspondence between T:'(&) and Q. The lemma is proved.

PROOFOF PROPOSITION 4.3.8. By the diagonal principle (e.g. [43]) one can find a z , E N , such that for all u E To the points 2: belong to the closures of sets x " ( N ) . Let p l , . . . ,pk denote the preimages of the point z, with respect to 7, and let Q = &( z,, E , u ) be a basis neighborhood in C G such that the set T;'(G) splits into k disjoint sets, every one of which contains exactly one preimage of z,. If L = N n b Q , , where Q1 = &1(z,,El,u) with e l 5 e small enough, we have d 6 ( x Q ( L ) )> 0, x " ( L ) c { l z l = ~ 1 ) .If J , is the component of T;'(Q) which contains p,, where u = 1,.. . , k, respectively, then dzdy ( x " ( J , ) ) > 0 for all u E To. By Lemma 4.3.7(i) we obtain that T;'(Q) c k

U J u . We claim that the spectral mapping T, u=l

maps the the set

J , n T i ' ( Q 1 ) in a one-to-one way onto Q1 for every u = 1,.. . , k. Let z1 be an arbitrary point in L. By Lemma 4.3.6 we have T ~ ( J , ,3) g1since z , = ~ , ( p , )E T,(J,) for each u = 1,.. . , k.

4.3. Almost periodic and r-analvtic structures

277

Hence in every set J , there exists at least one point which r, maps at zo. Because of z1 E N each set J , contains exactly , z1 E one point in T ; ' ( Z ~ ) , say q,. Note that qv E ~ A J since bQ1 C T , ( ~ A Q , )We . can consider AJ, as a uniform algebra of functions on X u = ~ A Jfor , each v = 1,.. . , k. By Lemma 4.3.9 we see that for some set Q2 = Q ~ ( Ez ,,a ) c Q 1 the mapping T* is a one-to-one correspondence between the sets J , n r;'(Q2) and Q 2 , i.e. that Q2 c W k . Consequently, int Wk # 0 as desired. The proposition is proved.

4.3.10. THEOREM [Tonev]. Let A be a uniform algebra on a compact Hausdorff space X , let be an additive subgroup of R which satisfies property (*), let 52 = {fa}aEr, be a multiplicative subsemigroup of functions in A and let W be a component of the set C G \ r , ( X ) , where r, : s p A --+ C G is the spectral mapping of 52. If there exists a measurable subset N of W such that the Lebesgue measure of the set x a ( N ) is non-zero for some (and thus for each) a E To and if T, is finite-to-one on r;'(N), then:

r

(i) T;'(W) can be given the structure of a proper finitesheeted branched r-analytic big-manifold over W , and (ii) The Gelfand extensions of all functions in A are rholomorphic on T;' (W ) .

PROOF.The sets f ( N ) and Nj = N n ~ , ( ( r ; ' ( W ) ) k )are measurable with respect to the measures d s d y and r d r d g respectively, as one can verify in a similar way as in the classical = Z (see e.g. [103, Sect. 111). Since r i l ( N ) = case when

r

m

j=l

( T ; ~ ( N )and ) ~ since x " ( N ) =

U x " ( r n ( ~ i 1 ( N ) )for k ) each

j=1

a E To, there exists a k 2 1 such that x " ( ~ , ( ~ - l ( N ) ) > k )0 *. for some a E To. By Proposition 4.3.8 the interior of the set T~ ((T;'( W ) ) k )is nonempty and by Proposition 4.3.5 we conclude that T;'(z) k for each z E W , which completes the proof by applying Theorem 4.2.13.