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ISOMETRIES on BANACH SPACES: Function Spaces CHAPMAN & HALUCRC Monographs and Surveys in Pure and Applied Mathematics 1 29 ISOMETRIES ON BANACH SPACES: function spaces O 2003 by Chapman & HallICRC CHAPMAN & HALLICRC Monographs and Surveys in Pure and Applied Mathematics Main Editors H. Brezis, Universite' de Paris R.G. Douglas, Texas A&M University A. Jeffrey, University of Newcastle upon Tyne (Founding Editor) Editorial Board R. Aris, University of Minnesota G.I. Barenblatt, University of California at Berkeley H. Begehr, Freie Universitat Berlin P. Bullen, University of British Columbia R.J. Elliott, University of Alberta R.P. Gilbert, University of Delaware R. Glowinski, University of Houston D. Jerison, Massachusetts Institute of Technology K. Kirchgassner, Universitat Stuttgart B. Lawson, State University of New York B. Moodie, University of Alberta L.E. Payne, Cornell University D.B. Pearson, University of Hull G.E Roach, University of Strathclyde I. Stakgold, University of Delaware W.A. Strauss, Brown University J. van der Hoek, University of Adelaide O 2003 by Chapman & HallICRC CHAPMAN & HALUCRC Monographs and Surveys in Pure and Applied Mathematics 1 29 ISOMETRIES ON BANACH SPACES: function spoces RICHARD J. FLEMING JAMES E. JAMISON CHAPMAN & HALUCRC A CRC Press Company Boca Raton London NewYork Washington, D.C. O 2003 by Chapman & HallICRC Library of Congress Cataloging-in-PublicationData Fleming, Richard J. Isometries on Banach spaces : function spaces I by Richard J. Fleming and James E. Jamison. p. cm. - (Chapman & HallICRC monographs and surveys in pure and applied mathematics ; 129) Includes bibliographical references and index. ISBN 1-58488-040-6 (alk. paper) 1. Function spaces. 2. Banach spaces. 3. Isometrics (Mathematics) I. Jamison, James E. 11. Title. 111. Series. QA323 .F55 2002 515'.73--dc21 2002041118 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquyies to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com O 2003 by Chapman & HallICRC No claim to original U.S. Government works International Standard Book Number 1-58488-040-6 Library of Congress Card Number 2002041 118 Winted in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper O 2003 by Chapman & HallICRC Contents Preface Chapter 1. Beginnings 1.1. Introduction 1.2. Banach's Characterization of Isometries on C(&) 1.3. The Mazur-Ulam Theorem 1.4. Orthogonality 1.5. The Wold Decomposition 1.6. Notes and Remarks Chapter 2. Continuous Function Spa,ces-The Banach-Stone Theorem 25 2.1. Introduction 25 2.2. Eilenberg's Theorem 26 2.3. The Nonsurjective Case 29 2.4. A Theorem of Vesentini 39 2.5. Notes and Remarks 42 Chapter 3. The LP Spaces 3.1. Introduction 3.2. Lamperti's Results 3.3. Subspaces of LP and the Extension Theorem 3.4. Bochner Kernels 3.5. Notes and Remarks Chapter 4. Isornetries of Spaces of Analytic Functions 4.1. Introduction 4.2. Isometries of the Hardy Spaces of the Disk 4.3. Bergman Spaces 4.4. Bloch Spaces 4.5. 9 Spaces 4.6. Notes and Remarks Chapter 5. Rearrangement Invariant Spaces 5.1. Introduction 5.2. Lumer's Method for Orlicz Spaces 5.3. Zaidenberg's Generalization 5.4. Musielak-Orlicz Spaces @ 2003 by Chapman & HallICRC 5.5. Notes and Remarks Chapter 6. Banach Algebras 6.1. Introduction 6.2. Kadison's Theorem 6.3. Subdifferentiability and Kadison's Theorem 6.4. The Nonsurjective Case of Kadison's Theorem 6.5. The Algebras and AC 6.6. Douglas Algebras 6.7. Notes and Remarks Bibliography O 2003 by Chapman & HallICRC Preface Herman Weyl has said that in order to understand any mathematical structure, one should investigate its group of symmetries. In the class of Ba- nach spaces, such a goal leads naturally to a study of isometries. A principal theme in geometry from the earliest times has been the study of transforma- tions preserving lengths and angles. If the origins of the theory of Banach spaces are assigned to the appearance of Banach's book in 1932, then the study of Banach space isometries must be assigned the same starting date. In his book, Banach included the first characterizations of all isometries on certain classical spaces. The body of literature concerning isometries that has grown up since that time is large, perhaps surprisingly so. An isometry, of course, is a transformation which preserves the distance between elements of a space. When Banach showed in 1932 that every linear isometry on the space of continuous real valued functions on a compact metric space must transform a continuous function x(t) into a continuous function y(t) satisfying where Ih(t)l - 1 and p is a homeomorphism, he was establishing a canon- ical form characterization which fits in an astonishing number of cases. In this volume we are interested primarily in just such explicit descriptions of isometries. Our approach, then, will differ from that of many authors whose interest in isometries lies in showing which spaces are isometric to each other, or to those whose interest is in discovering properties of topological spaces on which the functions in the Banach spaces are defined. Such interests as these have served as excellent motivations for the types of characterizations that interest us. There have been several excellent surveys concerning isometries, including those by Behrends, Loomis, Jarosz, and Jarosz and Pathak, each of which concentrates on some subset of the whole. Our intent was to pro- vide a survey of the entire subject, and our survey article (1993) serves as an inspiration and guide for the current work. Our goal has been to produce a useful resource for experts in the field as well as beginners, and also for those who simply want to acquaint themselves with this portion of Banach space theory. We have tried to provide some history of the subject, some of the important results, some flavor of the wide variety of methods used in attacking the characterization problem for various vii O 2003 by Chapman & HallICRC . vlll PREFACE types of spaces, and an exhaustive bibliography. We have probably underes- timated the enormity of such a project, and perhaps what we have produced is more of a sampler than a full-blown survey. We have chosen to organize the material according to the different classes of spaces under study and this is reflected in the chapter headings. The current volume is the first of two intended volumes, and as can be seen from the table of contents, it is primarily concerned with isometries on function spaces. The first chapter treats some general topics such as linearity, orthogonality, and Wold decompositions, while Chapter 6 contains material on noncommutative C*-algebras, but the rest of the chapters treat the classical function spaces. The second volume will include chapters with the following titles: Chapter 7: The Banach-Stone Property Chapter 8: Other Vector-Valued Function Spaces Chapter 9: Orthogonal Decompositions Chapter 10: Matrix Spaces Chapter 11: Norm Ideals of Operators Chapter 12: Spaces with Trivial Isometries Chapter 13: Epilogue In each chapter we try to include an early result of some historical impor- tance. Other selections are made in order to expose some one of the principal methods that have been used, or perhaps to give an account of work that has not received much attention. The chapters, and even the sections within the chapters, are mostly independent of each other, and the reader can begin at any point of interest. In making our selections we have, no doubt, left out many others just as deserving and quite possibly more important. Hope- fully, most of these omissions in the text are mentioned in the notes and the bibliography. The exposition relies mostly on the original papers, and we have tried to report faithfully on those results, with additional clarifications when possible. Probably we have included more detail than necessary in some instances, but we have chosen to err on that side. There are a few places where we have given only sketchy arguments. In each chapter we include a section on notes and remarks which give related results and other approaches that were not included in the main text. We hope these sections will help to soothe those who disagree sharply with our choice of material. For the most part, all references are given in the notes section. The exceptions are in cases where a reference is needed to justify a statement being made in the text. In the bibliography we give a representative selection of the works on isometries, and a serious investigation of all such works available should prob- ably begin with Math Sci Net using the phrases "isometries on," "isometries in," or "isometries for." Certainly there are many papers on vector-valued function spaces and matrix spaces which one might be expected to be men- tioned here, but we are saving these for the second volume.
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