CONTEMPORARY 321

Trends in Banach Spaces and Theory

A Conference on Trends in Banach Spaces and October 5-9, 2001 University of Memphis Anna Kaminska Editor http://dx.doi.org/10.1090/conm/321 CoNTEMPORARY MATHEMATICS

321

Trends in Banach Spaces and Operator Theory

A Conference on Trends in Banach Spaces and Operator Theory October 5-9, 2001 University of Memphis Anna Kaminska Editor

American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius

This volume contains the proceedings of a conference on Trends in Banach Spaces and Operator Theory which was held at the University of Memphis, October 5-9, 2001.

2000 Mathematics Subject Classification. Primary 22A22, 46Axx, 46Bxx, 46E30, 46Lxx, 47Axx, 47Bxx, 47Hxx, 47Lxx, 51F15.

Library of Congress Cataloging-in-Publication Data Trends in Banach spaces and operator theory : a conference on trends in Banach spaces and operator theory, October 5-9, 2001, University of Memphis/ Anna Kaminska, editor. p. em. ,.--(Contemporary mathematics, ISSN 0271-4132; 321) Includes bibliographical references. ISBN 0-8218-3234-4 (softcover : acid-free paper) 1. Banach spaces-Congresses. 2. Operator theory-Congresses. I. Kaminska, Anna, 195Q- II. Contemporary mathematics (American Mathematical Society) ; v. 321.

QA322.2.T69 2003 515'. 732-dc21 2003041485

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissionlllams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 2003 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. !§ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www. ams. org/ 10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03 Contents

Preface v

Characterizations of the reflexive spaces in the spirit of James' Theorem MARIA D. AcosTA, JuLIO BECERRA GuERRERO, AND MANUEL Rmz GMAN 1

Uniqueness of unconditional bases in quasi-Banach spaces FERNANDO ALBIAC, NIGEL J. KALTON, AND CAMINO LERANOZ 15

A note on the method of minimal vectors GEORGE ANDROULAKIS 29

The projective tensor product I JOE DIESTEL, JAN FOURIE, AND JOHAN SWART 37

Spectrum of a weakly hypercyclic operator meets the unit circle S. J. DILWORTH AND VLADIMIR G. TROITSKY 67

The dynamics of cohyponormal operators NATHAN S. FELDMAN 71

Hilbert-Schmidt composition operators on Dirichlet spaces EvA A. GALLARDo-GuTIERREZ AND MARIA J. GoNzALEZ 87

A remark on sectorial operators with an H 00-calculus N. J. KALTON 91

Borel injective tensor product and convolution of vector measures and their weak convergence JUN KAWABE 101

On linear operator pencils and inclusions of images of balls V. A. KHATSKEVICH AND V. S. SHULMAN 115

Ordinal indices and £1-spreading models DENNY H. LEUNG AND WEE-KEE TANG 125 Characterizing the existence of local Smith forms for c= families of matrix operators JULIAN LOPEZ-GOMEZ AND CARLOS MORA-CORRAL 139

iii iv CONTENTS

Convex geometry of Coxeter-invariant polyhedra NICHOLAS MCCARTHY, DAVID OGILVIE, NAHUM ZOBIN, AND VERONICA ZOBIN 153 Commutators on bounded symmetric domains in en JIE MIAO 181 Growth conditions and decomposable extensions T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN 197 Differences of composition operators JENNIFER MOORHOUSE AND CARL TOEWS 207 Complex vs. real variables for real 3-homogeneous polynomials on .er A counterexample GUSTAVO A. MUNOZ 215 The Fourier-Stieltjes and Fourier algebras for locally compact groupoids ALAN L. T. PATERSON 223 Preserving the commutant under functional calculus GABRIEL T. PRAJITURA 239 Lp-spaces associated with a von Neumann algebra without trace: a gentle introduction via complex interpolation YVES RAYNAUD 245 Banach and structure of C* -algebras HASKELL P. ROSENTHAL 275 How many operators exist on a ? TH. SCHLUMPRECHT 295 Maximal algebra norms GEOFFREY V. WoOD 335 On Banach spaces with small spaces of operators ANDRAS ZSAK 347 A remark on p-summing norms of operators ARTEM ZVAVITCH 371 Preface

This volume contains the proceedings of the conference Trends in Banach Spaces and Operator Theory, held at the University of Memphis during the week of October 5-9, 2001. The conference was devoted to the recent advances in theories of Banach spaces and linear operators. These theories are intimately related, and they lie at the core of such diverse fields as harmonic analysis, partial differen- tial equations, approximation theory, dynamical systems, probability, and financial mathematics. Stefan Banach, the inventor of the spaces that were later known by his name, wrote a monograph in 1932 under the very meaningful title Theorie des operations lineaires. The monograph presented the theory of normed and locally convex spaces and showed how the existing examples of function spaces and clas- sical operators motivated the foundations of the new theory. It also described how this in turn influenced the development of the theory of linear operators. The con- stant interplay between the structure of spaces and properties of operators acting on them is crucial for both theories. The conference brought together more than 130 participants from nineteen dif- ferent countries. It gathered mathematicians who represented the entire spectrum of researchers and included both some of the most experienced specialists in the field as well as young mathematicians at the early stages of their research programs. During the meeting the following thirteen principal speakers delivered one-hour plenary lectures.

Yuri Abramovich, Indiana University-Purdue University Indianapolis Sheldon Axler, San Francisco State University John B. Conway, University of Tennessee in Knoxville Carl C. Cowen, Purdue University Joe Diestel, Kent State University , University of Missouri in Columbia Barbara MacCluer, University of Virginia in Charlottesville Edward W. Odell, University of Texas at Austin Aleksander Pelczynski, Polish Academy of Sciences, Warsaw, Poland Gilles Pisier, Texas A&M University in College Station and Universite de Paris VI Haskell P. Rosenthal, University of Texas at Austin Thomas B. Schlumprecht, Texas A&M University in College Station Nicole Tomczak-Jaegermann, University of Alberta in Edmonton

In addition to plenary lectures, there were also eighty 20-minute talks organized in four parallel sessions. A panel discussion held during the meeting provided for a v vi PREFACE discussion of opinions, conclusions, and reflections on the current state of the field. The panel also identified multiple possibilities for future development of the theory and its applications. The discussion evolved into a lively and interesting event thanks to all participants, but especially because of the efforts of the principal speakers who volunteered to be members of the panel. Special recognition should also go out to Professor Sheldon Axler, who very skillfully moderated this rather unusual session. Sadly, one of our principal speakers recently passed away. Yuri Abramovich died on February 5, 2003, after a four-year illness. In his memory, the Department of Mathematical Sciences at Indiana University-Purdue University Indianapolis has established the Yuri Abramovich Memorial Scholarship. The proceedings presented here contain 25 papers; some of them are expository articles, while others are presentations of new results. Most of these results were presented during the conference. Some of the topics include Banach and operator space structure of C* -algebras, Banach spaces with small spaces of operators or with abundance of non-trivial operators, non-commutative £P-spaces, sectorial op- erators, composition operators, tensor products, isometric properties of the norm, bases in quasi-Banach spaces, Fourier algebras for locally compact groupoids, and geometry of Coxeter-invariant polyhedra. The conference was sponsored by grants from the National Science Foundation, the University of Memphis Foundation, the Office of Academic Affairs and the Office of Research of the University of Memphis, and the University of Mississippi. The members of the Mathematical Analysis seminar group from Memphis served as the organizational team for the conference. The group met over the past several years and included Jim Jamison, Anna Kaminska and Pei-Kee Lin from the University of Memphis, as well as Przemo Kranz from the University of Mississippi. Ray Clapsadle from the University of Memphis was also an invaluable addition to the organizational team. He created the web page for the conference, handled registration, and designed the logo and T-shirts for this event. Special acknowledgement should go to Jim Jamison, whose deep involvement and energy were crucial for the successful preparation and running of the meeting. On behalf of the organizing committee I would like to take this opportunity to thank all our colleagues, secretarial staff members, and graduate students from the math department of the University of Memphis. Thanks to all for your efforts in assisting in running the meetings, chairing the sessions, helping with registration and doing several other necessary jobs.

Anna Kaminska This volume contains proceedings of the conference on Trends in Banach Spaces and Operator Theory, which was devoted to recent advances in theories of Banach spaces and linear operators. Included in the volume are 25 papers, some of which are expository, while others present new results. The articles address the following topics: history of the famous James' theorem on reflexivity, projective tensor products, construction of noncommutative Lr-spaces via interpolation, Banach spaces with abundance of nontrivial operators, Banach spaces with small spaces of operators, convex geometry of Coxeter-invariant poly- hedra, uniqueness of unconditional bases in quasi-Banach spaces, dynamics of cohyponormal operators, and Fourier algebras for locally compact groupoids. The book is suitable for graduate students and research mathematicians interested in Banach spaces and operator theory and their applications.

ISBN 0-8218-3234-4

9 780821 832349 CONM/321