M-Ideals in Banach Spaces and Banach Algebras
This book provides a comprehensive exposition of M-ideal theory, a branch ofgeometric functional analysis which deals with certain subspaces of Banach spaces arising naturally in many contexts. Starting from the basic definitions the authors discuss a num
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1547
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen
1547
P. Harmand D. Werner W. Werner
M-Ideals in Banach Spaces and Banach Algebras
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Authors Peter Hannand Fachbereich Mathematik Universitat Oldenburg Postfach 2503 D-26111 Oldenburg, Germany Dirk Werner 1. Mathematisches Institut Freie Universitat Berlin Arnimallee 3 D-14195 Berlin, Germany Wend Werner Universitat-GH-Paderbron Fachbereich 17 Postfach 1621 D-33095 Paderborn, Germany
Mathematics Subject Classification (1991): 46BXX, 41A50, 41A65, 43A46, 46A55, 46E15, 46HIO, 46J10, 46L05, 46M05, 47A12, 47D15, 52A07 ISBN 3-540-56814-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56814-X Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data Harniand, P. (Peter), 1953M-idea1s in Banach spaces and Banach algebras / P. Harmand, D. Werner, W. Werner. p. em. - (Lecture notes in mathematics; 1547) Includes bibliographical references and index. ISBN 3-540-56814-X (Berlin: pbk.: acid-free) - ISBN 0-387-56814-X (New York: pbk.: acid-free) 1. Banach spaces. 2. Ideals (Algebra) 3. Approximation theory. 1. Werner, D. (Dirk), 1955- . II. Werner, W. (Wend), 1958- . III. Title. IV. Series: Lecture notes in mathemtics (Springer-Verlag); 1547. QA3.L28 no. 1547 (QA326) 510 s-dc20 (515'.732) 93-16064 CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 46/3140-543210 - Printed on acid-free paper
Preface
The present notes centre around the notion of an M-ideal in a Banach space, introduced by E. M. Alfsen and E. G. Effros in their fundamental article "Structure in real Banach spaces" from 1972. The key idea of their paper was to study a Banach space by means of a collection of distinguished subspaces, namely its M-ideals. (For the definition of an M-ideal see Definition 1.1 of Chapter 1.) Their approach was designed to encompass structure theories for C'-algebras, ordered Banach spaces, L1-preduals and spaces of affine functions on compact convex sets involving ideals of various sorts. But Alfsen and Effros defined the concepts of their M -structure theory solely in terms of the norm of the Banach space, deliberately neglecting any algebraic or order theoretic structure. Of course, they thus provided both a unified treatment of previous ideal theories by means of purely geometric notions and a wider range of appli
