Banach Space Theory Proceedings of a Research Workshop held July 5-25, 1987

AMERICAN MATHEMATICAL SOCIETY VOLUME 85 http://dx.doi.org/10.1090/conm/085 Titles in This Series

Volume 1 Markov random fields and their 18 Fixed points and nonexpansive applications, Ross Kindermann and mappings, Robert C. Sine, Editor J. Laurie Snell 19 Proceedings of the Northwestern 2 Proceedings of the conference homotopy theory conference, on integration, topology, and Haynes R. Miller and Stewart B. geometry in linear spaces, Priddy, Editors William H. Graves. Editor 20 Low dimensional topology, 3 The closed graph and P-closed Samuel J. Lomonaco. Jr .. Editor graph properties in general 21 Topological methods in nonlinear topology, T. R. Hamlett and , S. P. Singh, L. L. Herrington S. Thomeier, and B. Watson, Editors 4 Problems of elastic stability and 22 Factorizations of b" ± 1, b = vibrations, Vadim Komkov. Editor 2, 3, 5, 6, 7,10, 5 Rational constructions of 11,12 up to high powers, modules for simple Lie algebras. John Brillhart, D. H. Lehmer, George B. Seligman J. L. Selfridge, Bryant Tuckerman. and 6 Umbral calculus and Hopf algebras, S. S. Wagstaff, Jr. Robert Morris, Editor 23 Chapter 9 of Ramanujan's second 7 Complex contour integral notebook-Infinite series identities, representation of cardinal spline transformations, and evaluations, functions, Walter Schempp Bruce C. Berndt and Padmini T. Joshi 8 Ordered fields and real algebraic 24 Central extensions, Galois groups, geometry, D. W. Dubois and and ideal class groups of number T. Recio, Editors fields, A. Frohlich 9 Papers in algebra, analysis and 25 Value distribution theory and its statistics, R. Lidl, Editor applications, Chung-Chun Yang, 10 algebras and K-theory, Editor Ronald G. Douglas and Claude 26 Conference in modern analysis Schochet, Editors and probability, Richard Beals, 11 Plane ellipticity and related Anatole Beck. Alexandra Bellow, and problems, Robert P. Gilbert. Editor Arshag Hajian, Editors 12 Symposium on algebraic topology in 27 Microlocal analysis, M. Salah honor of Jos~ Adem, Samuel Gitler, Baouendi, Richard Beals, and Editor Linda Preiss Rothschild, Editors 13 Algebraists' homage: Papers in 28 Fluids and plasmas: geometry and ring theory and related topics, dynamics, Jerrold E. Marsden, Editor S. A. Amitsur. D. J. Saltman. and 29 Automated theorem proving, G. B. Seligman, Editors W. W. Bledsoe and Donald Loveland, 14 Lectures on Nielsen fixed point Editors theory, Boju Jiang 30 Mathematical applications of 15 Advanced analytic number theory. category theory, J. W. Gray, Editor Part 1: Ramification theoretic 31 Axiomatic set theory, James E. methods, Carlos J. Moreno Baumgartner, Donald A. Martin, and 16 Complex representations of Saharon Shelah, Editors GL(2. K) for finite fields K. 32 Proceedings of the conference llya Piatetski-Shapiro on Banach algebras and several 17 Nonlinear partial differential complex variables, F. Greenleaf and equations, Joel A. Smoller, Editor D. Gulick, Editors Titles in This Series

Volume 33 Contributions to group theory, 50 Random matrices and their Kenneth I. Appel, John G. Ratcliffe. applications, Joel E. Cohen, Harry and Paul E. Schupp, Editors Kesten, and Charles M. Newman, 34 Combinatorics and algebra, Editors Curtis Greene, Editor 51 Nonlinear problems in geometry, 35 Four-manifold theory, Cameron Dennis M. DeTurck, Editor Gordon and Robion Kirby, Editors 52 Geometry of normed linear 36 Group actions on manifolds, spaces, R. G. Bartle. N. T. Peck. Reinhard Schultz, Editor A. L. Peressini, and J. J. Uhl, Editors 37 Conference on algebraic topology 53 The Selberg trace formula and in honor of Peter Hilton, related topics, Dennis A. Hejhal, Renzo Piccinini and Denis Sjerve, Peter Sarnak, and Audrey Anne Terras, Editors Editors 54 Differential analysis and 38 Topics in complex analysis. infinite dimensional spaces, Dorothy Browne Shaffer, Editor Kondagunta Sundaresan and 39 Errett Bishop: Reflections Srinivasa Swaminathan. Editors on him and his research, 55 Applications of algebraic K-theory Murray Rosenblatt, Editor to algebraic geometry and 40 Integral bases for affine Lie algebras number theory, Spencer J. Bloch, and their universal enveloping R. Keith Dennis, Eric M. Friedlander, algebras, David Mitzman and Michael R. Stein, Editors 41 Particle systems, random 56 Multiparameter bifurcation theory, media and large deviations, Martin Golubitsky and John Richard Durrett, Editor Guckenheimer. Editors 42 Classical real analysis, Daniel 57 Combinatorics and ordered sets, Waterman. Editor Ivan Rival, Editor 43 Group actions on rings, Susan 58.1 The Lefschetz centennial Montgomery, Editor conference. Part 1: Proceedings 44 Combinatorial methods in on algebraic geometry, topology and algebraic geometry, D. Sundararaman. Editor John R. Harper and Richard 58.11 The Lefschetz centennial Mandelbaum. Editors conference. Part II: Proceedings on 45 Finite groups-coming of age, algebraic topology, S. Gitler, Editor John McKay, Editor 58.111 The Lefschetz centennial 46 Structure of the standard modules conference. Part Ill: Proceedings for the affine Lie algebra A11 ). on differential equations, James Lepowsky and Mirko Prime A. Verjovsky, Editor 47 Linear algebra and its role in 59 Function estimates, J. S. Marron, systems theory, Richard A. Brualdi, Editor David H. Carlson, Biswa Nath Datta, 60 Nonstrictly hyperbolic conservation Charles R. Johnson, and Robert J. laws. Barbara Lee Keyfitz and Plemmons. Editors Herbert C. Kranzer, Editors 48 Analytic functions of one complex 61 Residues and traces of differential variable, Chung-chun Yang and forms via Hochschild homology, Chi-tai Chuang, Editors Joseph Lipman 49 Complex differential geometry and 62 Operator algebras and mathematical nonlinear differential equations, physics, Palle E. T. Jorgensen and Yum-Tong Siu, Editor Paul S. Muhly, Editors Titles in This Series

Volume 63 Integral geometry, Robert L. Bryant, 74 Geometry of group representations. Victor Guillemin, Sigurdur Helgason, William M. Goldman and Andy R. and R. 0. Wells. Jr., Editors Magid, Editors 64 The legacy of Sonya Kovalevskaya, 75 The finite calculus associated Linda Keen, Editor with Bessel functions, Frank M. 65 Logic and combinatorics, Cholewinski Stephen G. Simpson. Editor 76 The structure of finite algebras. 66 Free group rings, Narian Gupta David C. Hobby and Ralph Mckenzie 67 Current trends in arithmetical 77 Number theory and its applications algebraic geometry, Kenneth A. in China. Wang Yuan. Yang Ribet, Editor Chung-chun, and Pan Chengbiao, 68 Differential geometry: The Editors interface between pure and applied 78 Braids. Joan S. Birman and Anatoly . Mladen Luksic, Libgober. Editors Clyde Martin, and William Shadwick, 79 Regular differential forms. Editors Ernst Kunz and Rolf Waldi 69 Methods and applications of 80 Statistical inference from stochastic mathematical logic, Walter A. processes. N. U. Prabhu, Editor Carnielli and Luiz Paulo de Alcantara, 81 Hamiltonian dynamical systems, Editors Kenneth R. Meyer and Donald G. 70 Index theory of elliptic operators, Saari, Editors foliations. and operator algebras, 82 Classical groups and related topics. Jerome Kaminker, Kenneth C. Millett, Alexander J. Hahn, Donald G. James. and Claude Schochet, Editors and Zhe-xian Wan. Editors 71 Mathematics and general relativity, 83 Algebraic K-theory and algebraic James A. Isenberg, Editor number theory. Michael R. Stein and 72 Fixed point theory and its R. Keith Dennis, Editors applications, R. F. Brown. Editor 84 Partition problems in topology, 73 Geometry of random motion, Stevo Todorcevic Rick Durrett and Mark A. Pinsky, 85 theory. Bor-Luh Lin, Editors Editor COnTEMPORARY MATHEMATICS Volume85

Banach Space Theory Proceedings of a Research Workshop held July 5-25, 1987 with support from the National Science Foundation

Bor-Luh Lin, Editor

AMERICAn MATHEMATICAL SOCIETY Providence • RhOda Island EDITORIAL BOARD

Irwin Kra, managing editor M. Salah Baouendi Jonathan Goodman Daniel M. Burns Gerald J. Janusz David Eisenbud Jan Mycielski

The Research Workshop on Banach Space Theory was held at the University of Iowa, Iowa City, on July 5-25, 1987 with support from the National Science Foundation, Grant DMS- 8604481. 1980 Mathematics Subject Classification (1985 Revision). Primary 46-06, 46B10. 46B20, 46B22. 46B25.

Library of Congress Cataloging-In-Publication Data Research Workshop on Banach Space Theory (1987: University of Iowa) Banach space theory: proceedings of a research workshop held July 5-25. 1987, with support from the National Science FoundationfBor-Luh Lin, editor. p. em. -(Contemporary mathematics, ISSN 0271-4132; v. 85) "The Research Workshop on Banach Space Theory was held at the University of Iowa. Iowa City"-T.p. verso. Includes bibliographies. ISBN 0-8218-5092-x (alk. paper) 1. Banach spaces-Congresses. I. Lin, Bor-Luh. II. American Mathematical Society. Ill. Title. VI. Series: Contemporary mathematics (American Mathematical Society); v. 85. QA322.2.R47 1987 88-38106 515.7'32-dc 19 CIP

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 acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this pub- lication (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.O. Box 6248, Providence, Rhode Island 02940. The appearance of the code on the first page of an article in this book 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 the Copyright Clearance Center, Inc., 21 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale.

Copyright @1989 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. 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. €9 CONTENTS

Preface ix Workshop Participants xi Titles of seminars given during the Workshop xiii Isometries of James space STEVEN F. BELLENOT 1 Quasi-reflexive and tree spaces constructed in the spirit of R. C. James STEVEN BELLENOT, RICHARD HAYDON and EDWARD ODELL 19 Projections onto block bases PETER G. CASAZZA 45 The Schroeder-Bernstein property for Banach spaces PETER G. CASAZZA 61 Rotund reflexive Orlicz spaces are fully convex SHUTAO CHEN, BOR-LUH LIN and XINTAI YU 79 Biorthogonal systems and big quotient spaces CATHERINE FINET and GILLES GODEFROY 87 Analytic martingales and plurisubharmonic barriers in complex Banach spaces N. GHOUSSOUB, J. LINDENSTRAUSS and B. MAUREY 111 Existence and uniqueness of isometric preduals: A survey GILLES GODEFROY 131 The ball topology and its applications G. GODEFROY and N.J. KALTON 195 On the closedness of the sum of closed operators on a UMD space SYLVIE GUERRE 239 Ultrapowers of rearrangement-invariant function spaces I RICHARD HAYDON and PEI-KEE LIN 253 KMP, RNP, and PCP for Banach spaces R. C. JAMES 281

vii viii CONTENTS

Independence in separable Banach spaces N.J. KALTON 319 On the weak* -fixed point property M.A. KHAMSI 325 Some geometrical and topological properties of the unit sphere in a normed linear space BOR-LUH LIN, PEI-KEE LIN and S. L. TROYANSKI 339 The >.-function in Banach spaces ROBERT H. LOHMAN 345 Smooth functions in Orlicz spaces R. P. MALEEV and S. L. TROYANSKI 355 Fefferman spaces and C* -algebras PAUL S. MUHLY 371 J H* has the PCP E. ODELL and C. S. SCHUMACHER 387 An analogue of the F. and M. Riesz theorem for spaces of differential functions A. PELCZYNSKI 405 Almost isometric methods in some isomorphic embedding problems YVES RAYNAUD 427 Sub-simplexes of convex sets and some characterizations of simplexes with the RNP HASKELL P. ROSENTHAL 447 Some more remarkable properties of the James-tree space WALTER SCHACHERMAYER 465 A note on the Lavrientiev index for the quasi-reflexive Banach spaces A. SERSOURI 497 w-independence in non separable Banach spaces ABDERRAZZAK SERSOURI 509

The canonical injection for C{[O, 1]) into L2,1 is not of cotype 2 MICHEL TALAGRAND 513 PREFACE

These Proceedings are from a Research Workshop on Banach Space Theory held at the University of Iowa in Iowa City, Iowa from July 5th to July 25th, 1987. The intent of the Workshop was to provide participants with a close, working atmosphere, one in which the exchange of ideas could be promoted at an informal level. This included two hours daily seminars on current problems. In addition, participants were provided with adjacent offices to facilitate collaboration and consultation. At the conclusion of the Workshop, these Proceedings were compiled. This volume includes several papers initiated during the Workshop, which are presented here in their final form. The volume also includes some papers whose contributors would like to participate in the Workshop but were unable to come. No papers in this volume will be published elsewhere. We would like to express our thanks to Professor A. Pelczynski for attending the Workshop at our invitation as an Ida Beam Distinguished Visiting Professor. We are deeply indebted to Professor J. Lindenstrauss for his tremendous help in organizing the Workshop and to Professors W.B. Johnson, G. Pisier and H.P. Rosenthal for their valuable suggestions and help. We would also like to thank those colleagues and participants, in many cases coming a great distance, whose encouragement and contributions led to the success of the project. The Workshop was supported by a grant from the National Science Foundation, the Department of Mathematics, and the Ida Beam Foundation at the University of Iowa. We thank them for their generous help in making the project possible. Finally, we would like to thank Profes59r Irwin Kra for including the Proceedings in the series of Contemporary Mathematics. Bor-Luh Lin

ix Workshop Participants*

Dale E. Alspach, Oklahoma State University, Stillwater, Oklahoma S. Argyros, University of Crete, Iraklion, Crete, Greece Peter G. Casazza, University of Missouri, Columbia, Missouri William J. Davis, Ohio State University, Columbus, Ohio Guanggui Ding, Nankai University, Tianjin, China Catherine Finet, Universite de 1' Etat a Mons, Mons, Belgium Ji Gao, University of Pittsburgh, Pittsburgh, Pennsylvania Nassif Ghoussoub, University of British Columbia, Vancouver, British Columbia, Canada Sylvie Guerre, Universite Paris VI, Paris, France Richard Haydon, Brasenose College, Oxford, England William B. Johnson, Texas A & M University, College Station, Texas Nigel J. Kalton, University of Missouri, Columbia, Missouri M. Amine Khamsi, Universite Paris VI, Paris, France Pei-Kee Lin, Memphis State University, Memphis, Tennessee , The Hebrew University, Jerusalem, Israel Gun-Marie Lovblem, University of Stockholm, Stockholm, Sweden Aleksander Pelczynski, Polish Academy of Sciences, Warszawa, Poland Gilles Pisier, Universite Paris VI, Paris, France Edward Odell, University of Texas, Austin, Texas Yves Raynaud, Universite Paris VI, Paris, France Haskell P. Rosenthal, University of Texas, Austin, Texas

*other than those from the University of Iowa

XI xii WORKSHOP PARTICIPANTS

Gideon Schechtman, Weismann Institute of Sciences, Rehovot, Israel Abderrazak Sersouri, Universite Paris VI, Paris, France Stanimir L. Troyanski, University of Sofia, Sofia, Bulgaria J. Barry Turett, Oakland University, Rochester, Michigan Xintai Yu, East China Normal University, Shanghai, China Junfeng Zhao, Wuhan University, Wuhan, China TITLES OF THE SEMINARS GIVEN BY THE PARTICIPANTS

Seminars were held daily at 2:30-3:30 p.m. and 3:45--4:45 p.m. during the Workshop. The following is the list of titles in the seminars arranged in chronological order.

SOME CHARACTERIZATIONS OF SIMPLICES WITH RADON-NIKODYM PROPERTY H.P. Rosenthal

WEAK HILBERT SPACES P.G. Casazza APPROXIMATING ZONOIDS BY ZONOTOPES I J. Lindenstrauss ALMOST ISOMETRIC METHODS IN SOME ISOMORPHIC EMBEDDING PROBLEMS Yves Raynaud SUMS OF INDEPENDENT RANDOM VARIABLES IN REARRANGEMENT INVARIANT FUNCTION SPACES W .B. Johnson ON THE GEOMETRY OF SPACES WHERE ANALYTIC MARTINGALES CONVERGE N. Ghoussoub APPROXIMATING ZONOIDS BY ZONOTYPES II J. Lindenstrauss

ON MATHEMATICAL OLYMPIADS A. Pelczynski

COMPLEXITY OF WEAKLY NULL SEQUENCES D. Alspach

NON RNP SETS WITH PCP S. Agyros

OPERATORS INDUCED BY H1 MULTIPLIERS AND c*-ALGEBRAS I A. Pelczynski

OPERATORS INDUCED BY H1 MULTIPLIERS AND c*-ALGEBRAS II A. Pelczynski WEAK HILBERT SPACES I G. Pisier

xiii xiv TITLES OF THE SEMINARS

WEAK HILBERT SPACES II G. Pisier SHELAH'S SPACE WITH FEW OPERATORS R.G. Haydon EMBEDDING FINITE METRIC SPACES IN NORMED SPACES G. Schechtman CONTRACTIVE PROJECTIONS ON lp W.J. Davis

ON BAIRE 1/4-FUNCTIONS E. Odell ON FRECHET DIFFERENTIABILITY IN ORLICZ SPACES S. Troyanski

ISOMETRICS ON Lp,l J.B. Turett

ON THE CLOSEDNESS OF THE SUMS OF CLOSED OPERATORS S. Guerre THE APPROXIMATION PROBLEM OF ALMOST ISOMETRIC OPERATORS BY ISOMETRIC OPERATORS Guanggui Ding ON QUASI-REFLEXIVE BANACH SPACES C. Finet ON BANACH SPACES WHICH ARE ISOMETRIC TO THEIR BIDUALS A. Sersouri AN EXAMPLE OF AN IRREGULAR RANDOM MEASURE S. Guerre

ON THE WEAK* FIXED POINT PROPERTY A. Khamsi ON KUR SPACES Xintai Yu THE ORDERING STRUCTURES ON BANACH SPACES Junfeng Zhao