Several Complex Variables and Banach Algebras

264 Pages·1998·3.159 MB·English

by  Herbert Alexander, John Wermer (auth.)

Graduate Texts in Mathematics 35
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and Its Applications.
Herbert Alexander
John Wermer
Several Complex
Variables and
Banach Algebras
Third Edition
Springer
Herbert Alexander John Wermer
University of Illinois at Chicago Brown University
Department of Mathematics Department of Mathematics
Chicago, IL 60607-7045 Providence, RI02912
Editorial Board
F.W. Gehring S. Axler K. Ribet
Mathematics Department Mathematics Department Department of Mathematics
East Hall San Francisco State University University of Califomia
University of Michigan San Francisco, CA 94132 at Berkeley
Ann Arbor, MI 48104 Berkeley, CA 94720-3840
AMS Subject Classifications
MSC 1991: 32DXX, 32EXX, 46JXX
Libraiy of Congress Cataloging-in-Publication Data
Alexander, Herbert.
Several complex variables and Banach algebras / Herbert Alexander,
John Wermer. — 3rd ed.
p. cm. — (Graduate texts in mathematics; 3 5)
Rev. ed. of: Banach algebras and several complex variables / John
Weimer. 2nd ed. 1976.
Includes bibliographical references (p. - ) and index.
ISBN 0-387-98253-1 (alk. paper)
1. Banach Algebras. 2. Functions of several con^lex variables.
1. Wermer, John. II. Wermer, John. Banach algebras and several
conq)lex variables. III. Title. FV. Series.
QA326.W47 1997
512'.5S-dc21 97-16661
© 1998 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
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USA), except for brief excerpts in coimection with reviews or scholarly analysis. Use in connection with
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is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks
Act, may accordingly be used freely by anyone.
ISBN 0-387-98253-1 Springer-Veriag New York Beriin Heidelbeig SPIN 10524438
to Susan and to the memory of Kerstin
Table of Contents
PrefacetotheSecondEdition ix
PrefacetotheRevisedEdition xi
Chapter1 PreliminariesandNotation 1
Chapter2 ClassicalApproximationTheorems 5
Chapter3 OperationalCalculusinOneVariable 17
Chapter4 DifferentialForms 23
Chapter5 The∂-Operator 27
Chapter6 TheEquation∂ u (cid:1) f 31
Chapter7 TheOka-WeilTheorem 36
Chapter8 OperationalCalculusinSeveralVariables 43
Chapter9 TheSˇilovBoundary 50
Chapter10 MaximalityandRado´’sTheorem 57
Chapter11 MaximumModulusAlgebras 64
Chapter12 HullsofCurvesandArcs 84
Chapter13 IntegralKernels 92
Chapter14 PerturbationsoftheStone–WeierstrassTheorem 102
Chapter15 TheFirstCohomologyGroupofaMaximalIdealSpace 112
Chapter16 The∂-OperatorinSmoothlyBoundedDomains 120
vii
viii TableofContents
Chapter17 ManifoldsWithoutComplexTangents 134
Chapter18 SubmanifoldsofHighDimension 146
Chapter19 BoundariesofAnalyticVarieties 155
Chapter20 PolynomialHullsofSetsOvertheCircle 170
Chapter21 Areas 180
Chapter22 TopologyofHulls 187
Chapter23 PseudoconvexsetsinCn 194
Chapter24 Examples 206
Chapter25 HistoricalCommentsandRecentDevelopments 224
Chapter26 Appendix 231
Chapter27 SolutionstoSomeExercises 237
Bibliography 241
Index 251
Preface to the Second Edition
During the past twenty years many connections have been found between the
theory of analytic functions of one or more complex variables and the study of
commutativeBanachalgebras.Ontheonehand,functiontheoryhasbeenusedto
answeralgebraicquestionssuchasthequestionoftheexistenceofidempotentsin
aBanachalgebra.Ontheotherhand,conceptsarisingfromthestudyofBanach
algebrassuchasthemaximalidealspace,theSˇilovboundary,Gleasonparts,etc.
haveledtonewquestionsandtonewmethodsofproofinfunctiontheory.
Roughlyonethirdofthisbookisconcernedwithdevelopingsomeoftheprinci-
palapplicationsoffunctiontheoryinseveralcomplexvariablestoBanachalgebras.
Wepresupposenoknowledgeofseveralcomplexvariablesonthepartofthereader
butdevelopthenecessarymaterialfromscratch.Theremainderofthebookdeals
with problems of uniform approximation on compact subsets of the space of n
complex variables. For n > 1 no complete theory exists but many important
particularproblemshavebeensolved.
Throughout, our aim has been to make the exposition elementary and self-
contained. We have cheerfully sacrificed generality and completeness all along
thewayinordertomakeiteasiertounderstandthemainideas.
RelationshipsbetweenfunctiontheoryinthecomplexplaneandBanachalge-
bras are only touched on in this book. This subject matter is thoroughly treated
inA.Browder’sIntroductiontoFunctionAlgebras,(W.A.Benjamin,NewYork,
1969)andT.W.Gamelin’sUniformAlgebras,(Prentice-Hall,EnglewoodCliffs,
N.J.,1969).Asystematicexpositionofthesubjectofuniformalgebrasincluding
manyexamplesisgivenbyE.L.Stout,TheTheoryofUniformAlgebras,(Bogden
andQuigley,Inc.,1971).
The first edition of this book was published in 1971 by Markham Publishing
Company.ThepresenteditioncontainsthefollowingnewSections:18.Subman-
ifoldsofHighDimension,19.Generators,20.TheFibersOveraPlaneDomain,
21.ExamplesofHulls.Also,Section11hasbeenrevised.
Exercisesofvaryingdegreesofdifficultyareincludedinthetextandthereader
shouldtrytosolveasmanyoftheseashecan.Solutionstostarredexercisesare
giveninSection22.
ix