Banach Algebras and Several Complex Variables

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1 PROCEEDINGS OF THE CONFERENCE ON Banach Algebras and Several Complex Variables AMERICAn MATHEMATICAL SOCIETY VOLUME 32

2 COnTEMPORARY MATHEMATICS Titles in this Series Volume 1 Markov random fields and their applications, Ross Kindermann and J. Laurie Snell 2 Proceedings of the conference on integration, topology, and geometry in linear spaces, William H. Graves. Editor 3 The closed graph and P-closed graph properties in general topology, T. R. Hamlett and L. L. Herrington 4 Problems of elastic stability and vibrations, Vadim Komkov. Editor 5 Rational constructions of modules for simple Lie algebras, George B. Seligman 6 Umbral calculus and Hopf algebras, Robert Morris. Editor 7 Complex contour integral representation of cardinal spline functions, Walter Schempp 8 Ordered fields and real algebraic geometry, D. W. Dubois and T. Recio. Editors 9 Papers in algebra, analysis and statistics, R. Lidl. Editor 10 Operator algebras and K-theory, Ronald G. Douglas and Claude Schochet. Editors 11 Plane ellipticity and related problems, Robert P. Gilbert. Editor 12 Symposium on algebraic topology in honor of Jose Adem, Samuel Gitler. Editor 13 Algebraists' homage: Papers in ring theory and related topics, S. A. Amitsur. D. J. Saltman and G. B. Seligman. Editors 14 Lectures on Nielsen fixed point theory, Boju Jiang 15 Advanced analytic number theory. Part 1: Ramification theoretic methods, Carlos J. Moreno 16 Complex representations of GL(2, K) for finite fields K, llya Piatetski-Shapiro 17 Nonlinear partial differential equations, Joel A. Smoller. Editor 18 Fixed points and nonexpansive mappings, Robert C. Sine. Editor 19 Proceedings of the Northwestern homotopy theory conference, Haynes R. Miller and Stewart B. Priddy. Editors 20 Low dimensional topology, Samuel J. Lomonaco. Jr.. Editor

3 Titles in this Series Volume 21 Topological methods in nonlinear functional analysis, S. P. Singh. S. Thomeier. and B. Watson. Editors 22 Factorizations of b" ± 1, b == 2, 3, 5, 6, 7,10,11,12 up to high powers, John Brillhart. D. H. Lehmer. J. L. Selfridge. Bryant Tuckerman. and S. S. Wagstaff. Jr. 23 Chapter 9 of Ramanujan's second notebook-infinite series identities, transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 24 Central extensions, Galois groups, and ideal class groups of number fields, A. Frohlich 25 Value distribution theory and its applications, Chung-Chun Yang, Editor 26 Conference in modern analysis and probability, Richard Beals. Anatole Beck. Alexandra Bellow and Arshag Hajian. Editors 27 Microlocal analysis, M. Salah Baouendi. Richard Beals and Linda Preiss Rothschild. Editors 28 Fluids and plasmas: geometry and dynamics, Jerrold E. Marsden. Editor 29 Automated theorem proving, W. W. Bledsoe and Donald Loveland. Editors 30 Mathematical applications of category theory, J. W. Gray. Editor 31 Axiomatic set theory, James E. Baumgartner. Donald A. Martin and Saharon Shelah. Editors 32 Proceedings of the conference on Banach algebras and several complex variables, F. Greenleaf and D. Gulick. Editors 33 Contributions to group theory, Kenneth I. Appel. John G. Ratcliffe and Paul E. Schupp, Editors

4 Proceedings of the Conference on Banach Algebras and Several Complex Variables

5 EDITORIAL BOARD R. 0. Wells, Jr., managing editor Jeff Cheeger Adriano M. Garsia Kenneth Kunen James I. Lepowsky Johannes C. C. Nitsche Irving Reiner PROCEEDINGS OF THE CONFERENCE ON BANACH ALGEBRAS AND SEVERAL COMPLEX VARIABLES HELD AT YALE UNIVERSITY NEW HAVEN, CONNECTICUT JUNE 21-24, 1983 These proceedings were prepared by the American Mathematical Society with partial support from the National Science Foundation Grant MCS Mathematics Subject Classification. Primary 32Axx, 32Bxx, 32Exx, 32Fxx, 46Hxx, 46Jxx. Library of Congress Cataloging in Publication Data Conference on Banach Algebras and Several Complex Variables ( 1983: Yale University) Proceedings of the Conference on Banach Algebras and Several Complex Variables. (Contemporary mathematics; v. 32) Held to honor Prof. Charles E. Rlckart. Bibliography: p. 1. Banach algebras-congresses. 2. Functions of several complex variables-congresses. 3. Rickart, C. E. (Charles Earl), 1913) I. Greenleaf, Frederick P. 11. Gulick, Denny. Ill. Rickart, C. E. (Charles Earl), (American Mathematical Society); v IV. Title. V. Series: Contemporary mathematics QA326.C ' ISBN (alk. paper) Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication!including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island The appearance of the code on the first page of an article in this volume indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to Copyright Clearance Center, Inc., 21 Congress Street, Salem, Massachusetts This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotion purposes, for creating new collective works or for resale. Copyright 1984 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government This volume was printed directly from author prepared copy. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

6 COITEIYIPORARY MATHEMATICS Valume32 PROCEEDINGS OF THE CONFERENCE ON Banach Algebras and Several Complex Variables F. Greenleaf and D. Gulick, Editors AMERICAn MATHEMATICAL SOCIETY Providence RhOde Island

8 A conference in honor of CHARLES E. RICKART upon his retirement from Yale University.

9 CONTENTS Introduction Signatures of Participants H. Alexander G. Allan B. Aupetit J. Bachar w. Bade w. Bade & P. Curtis F. Bonsall H. Dales J. Esterle I. Glicksberg B. Kramm D. Kumagai K. Laursen J. McClure A. O'Farrell, K. Preskenis & D. Walsh Capacities in (n Holomorphic Left-Inverse Functions Geometry of Pseudoconvex Open Sets and Distribution of Values of Analytic Multivalued Functions Some Results on Range Transformations Between Function Spaces Recent Results in the Ideal Theory of Radical Convolution Algebras Module Derivations from Commutative Banach Algebras Criteria for Boundedness and Compactness of Hankel Operators Algebra and Topology in Banach Algebras Mittag-Leffler Methods in the Theory of Banach Algebras and a New Approach to Michael's Problem Orthogonal and Representing Measures Nuclearity (resp. Schwartzity) Helps to Embed Holomorphic Structure into Spectra Maximum Modulus Alge~s Analytic Structure and Multi-Dimensional Central Factorization in C*-Algebras and its Use in Automatic Continuity Nonstandard Ideals and Approximations in Primary Weighted t 1-Algebras Holomorphic Approximation in Lipschitz Norms xi xiii ix

10 X CONTENTS M. Rajagopalan & P. Ramakrishnan R. Rochberg W. Rudin S. Sakai D. Sarason Z. Slodkowski E. Stout J. Wada J. Wermer B. Yood W. Zame Uses of ~S in Invariant Means and Extremely Left Amenable Semigroups Deformation Theory for Uniform Algebras: Introduction Nevanlinna's Interpolation Theorem Revisited Unbounded Derivations in C*-Algebras and Statistical Mechanics Remotely Almost Periodic Functions Analytic Multifunctions, q-plurisubharmonic Functions and Uniform Algebras Algebraic Domains in Stein Manifolds Sets of Best Approximations to Elements in Certain Function Spaces Green's Functions and Polynomial Hulls Continuous Homomorphisms and Derivations on Banach Algebras The Maximal Ideal Space of a Commutative Banach Algebra An

11 INTRODUCTION A Conference on Banach Algebras and Several Complex Variables was held on June 21-24, 1983, to honor Professor Charles E. Rickart as he retired after 40 years at Yale University. * These Proceedings contain articles submitted for the Conference, The topics, both at the Conference and in these Proceedings, represent recent advances in a wide spectrum of topics related to Banach algebras, function algebras and infinite dimensional holomorphy. Professor Rickart has profoundly affected these areas, and the many participants who have been either associates or graduate students of Professor Rickart would like to join in thanking him for his inspiration. The preparatory work for the conference rested primarily with R. R, Coifman and Phil Curtis, as well as Phyllis Stevens of the Department of Mathematics at Yale University, who lent an expert hand that was invaluable to the conference. The painstaking job of retyping each manuscript to appear in these Proceedings fell onto the shoulders of Donna Belli, Caroline Curtis, Mary Ellen Del Vecchio and Bernadette Highsmith ef the Department of Mathematics, under the careful and patient supervision of Regina Hoffman. all those who helped make the conference successful and memorable, as well as to the National Science Foundation ** for its generous support, we wish to express our gratitude. The editors would also like to thank the following conference speakers whose results are appearing elsewhere: T. Lyons, M. Sibony, B. Cole, E. Effros, T. Gamelin, With this volume we extend our thanks to Professor Rickart, who has for such a long time served as teacher, researcher, administrator and friend. Fred Greenleaf Denny Gulick To * After completing his undergraduate study at the University of Kansas, Professor Rickart began his move east, earning his Ph.D. at the University of Michigan in 1941 and serving as Benjamin Peirce Instructor at Harvard University from 1941 to He joined the Yale faculty in 1943, and became one of the pioneers in the study of Banach algebras. In 1960, Professor Rickart published his classical treatise General Theory of Banach Algebra, More recently his interests have turned to the study of infinite dimensional holomorphy, culminating in his 1979 book Natural Function Algebras, an important contribution to this subject. ** NSF Grant: MCS xi

12 SIGNATURES OF PARTICIP.AN'l'S xiii

13 ISBN a o B z 1 Is s o CONi\ 1/32

Axiomatic Set Theory. AMERICAII mathematical SOCIETY. u m 0 LU E 31

Axiomatic Set Theory. AMERICAII mathematical SOCIETY. u m 0 LU E 31 Axiomatic Set Theory AMERICAII mathematical SOCIETY u m 0 LU E 31 http://dx.doi.org/10.1090/conm/031 COITEIYIPORARY MATHEMATICS Titles in this Series Volume 1 Markov random fields and their applications,