Operator Algebras and Mathematical Physics l'roceedings of a Summer Conference held June 17-211 1985

,,MERICAft MATHEMATICAL SOCIETY VOLUME 62

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 functional analysis, 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. 11. 12 up to high

5 Rational constructions of powers. John Brillhart. D. H. Lehmer. modules for simple Lie algebras. J. L. Selfridge, Bryant Tuckerman. and George B. Seligman S. S. Wagstaff. Jr.

6 Umbral calculus and Hopf algebras. 23 Chapter 9 of Ramanujan's second Robert Morris. Editor notebook-Infinite series Identities, 7 Complex contour Integral transformations. and evaluations.

representation of cardinal spline Bruce C. Berndt and Padmini T. Joshi functions, Walter Schempp

24 Central extensions, Galois groups, 8 Ordered fields and real algebraic and Ideal class groups of number

geometry, D. W. Dubois and fields, A. Frohlich T. Recio. Editors

25 Value distribution theory and Its 9 Papers In algebra. analysis and applications. Chung-Chun Yang, statistics, R. Lidl. Editor Editor 10 Operator algebras and K-theory,

26 Conference In modern analysis Ronald G. Douglas and Claude and probability, Richard Beals. Schochet. Editors Anatole Beck. Alexandra Bellow. and

11 Plane ellipticity and related Arshag Hajian. Editors problems. Robert P. Gilbert. Editor

27 Mlcrolocal analysis, M. Salah 12 Symposium on algebraic topology In Baouendi. Richard Beals. and honor of Jose Adem, Samuel Gitler. Linda Preiss Rothschild. Editors Editor Fluids and plasmas: geometry and 13 Algebraists' homage: Papers In 28

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 Saharan 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. Smaller. Editor D. Gulick. Editors

http://dx.doi.org/10.1090/conm/062

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 Comblnatorlcs 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

38 Topics In complex analysis, 54 Differential analysis and

Dorothy Browne Shaffer. Editor Infinite dimensional spaces. 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 Multlparameter bifurcation theory, media and large deviations, Martin Golubitsky and John Richard Durrett. Editor Guckenheimer. Editors

42 Classical real analysis, Daniel 57 Comblnatorlcs and ordered sets. Waterman. Editor Ivan Rival. Editor

43 Group actions on rings, Susan 58 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 59 Function estimates. J. S. Marron. Mandelbaum. Editors Editor

45 Finite groups-coming of age, 60 Nonstrlctly hyperbolic conservation John McKay, Editor laws. Barbara Lee Keyfitz and

46 Structure of the standard modules Herbert C. Kranzer. Editors for the affine Lie algebra Ai1), 61 Residues and traces of differential James Lepowsky and Mirko Prime forms via Hochschlld homology,

47 Linear algebra and Its role In Joseph Lipman

systems theory, Richard A. Brualdi, 62 Operator algebras and mathematical David H. Carlson. Biswa Nath Datta. physics, Palle E. T. Jorgensen and Charles R. Johnson. and Robert J. Paul S. Muhly, Editors Plemmons. Editors

48 Analytic functions of one complex variable, Chung-chun Yang and Chi-tai Chuang. Editors

49 Complex differential geometry and nonlinear differential equations. Yum-Tong Siu. Editor

Operator Algebras and Mathematical Physics

COnTEMPORARY MATHEMATICS

Volume62

Operator Algebras and Mathematical Physics Proceedings of a Summer Conference held June 17-211 19851 with support from the National Science Foundation and the University of Iowa

Palla E. T. Jorgensen and Paul S. Muhly1 Editors

AMERICAn MATHEMATICAL SOCIETY Prouldence • RhOde Island

EDITORIAL BOARD Irwin Kra.

managing editor Thomas F. Banchoff M. Salah Baouendi

W. H. Jaco Gerald J. Janusz Jan Mycielski Johannes C. C. Nitsche

Alan D. Weinstein

The Proceedings of the Summer Conference on Algebras and Mathematical Physics was held at the University of Iowa. Iowa City. Iowa. June 17-21. 1985. Work was supported by the National Science Foundation. Grant OMS 85-00879 and the University of Iowa.

1980 Mathematics Subject Classification. {1985 Revision). Primary 46LXX. 22025. 46L60. 81E05. 82A15.

Library of Congress Cataloging-In-Publication Data Operator algebras and mathematical physics.

(Contemporary mathematics. ISSN 0271-4132; v. 62) Proceedings of the Summer Conference on Operator Algebras and Mathematical Physics

in honor of the 60th birthdays of Profs. H-J. Borchers. N. M. Hugenholtz. R. V. Kadison. and D. Kastler.

Bibliography: p. 1. c• -algebras-Congresses. 2. Von Neumann algebras-Congresses. 3. Mathematical

physics-Congresses. I. Jorgensen, Palle E. T .. 1947- . II. Muhiy. Paul S. Ill. Summer Conference on Operator Algebras and Mathematical Physics {1985: University of Iowa) IV. Series: Contemporary mathematics (American Mathematical Society); v. 62. QA326.0645 1987 5121.55 86-32070 ISBN 0-8218-5066-0 (alk. paper)

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CONTENTS

Preface xi

I.

II.

BIOGRAPHICAL SKETCHES

Introduction

A biographical sketch of Hans-Jiirgen Borchers by KLAUS FREDENHAGEN

A biographical sketch of Nico M Hugenholtz by MARINUS WINNINK

A biographical sketch of Richard V. Kadison by ERLING ST~RMER

A biographical sketch of Daniel Kastler by RUDOLF HAAG

THE ALGEBRAIC APPROACH TO QUANTUM THEORY OF FIELDS AND STATISTICAL MECHANICS

Bogoliubov automorphisms and Fock representations of canonical anticommutation relations

3

5

9

13

17

by HUZIHIRO ARAKI 23

On the interplay between spectrum condition and locality in quantum field theory by H. J. BORCHERS 143

The structure of local algebras of observables byKLAUSFREDENHAGEN 153

How the C*-algebraic formulation of statistical mechanics helps understanding the approach to equilibrium by N. M. HUGENHOL TZ 167

vii

viii CONTENTS

KMS structures in geometric quantization .. by GERARD G. EMCH 175

Perturbations and KMS states in C*-dynamical systems (Generalization of the absence theorem of phase transition in continuous quantum systems)

by SHOICHIRO SAKAI 187

III. METHODS FROM DIFFERENTIAL GEOMETRY AND K-THEORY

Z/2-graded cyclic cohomology within the differential envelope

by DANIEL KASTLER 221

Yang-Mills for non-commutative two-tori by ALAIN CONNES and MARC A. RIEFFEL 237

Non-stable K-theory and non-commutative tori by MARC A. RIEFFEL 267

IV. POSITIVITY IN OPERA TOR ALGEBRAS

Nonlinear states on C*-algebras by WILLIAM AR VESON 283

Cones of positive maps by ERLING ST{l)RMER 345

On the geometry of positive maps in matrix algebras by JUN TOMIY AMA 357

V. DYNAMICAL SYSTEMS AND DERIVATIONS

Differential operators on C*-algebras by DEREK W. ROBINSON 367

Derivations and free group actions by OLA BRA TTELI 385

Automatic continuity of derivations on eigenspaces by OLA BRA TTELI and AKIT AKA KISHIMOTO 403

Dynamics in continuous quantum systems 1\ 1\

by SHOICHIRO SAKAI 413

CONTENTS ix

VI. OPERA TORS IN HILBERT SPACES: SOME ALGEBRAIC METHODS

Kuiper's theorem for Hilbert modules by JOACHIM CUNTZ and NIGEL HIGSON 429

Feynman's operational calculus, generalized Dyson series and the Feynman integral

by GERALD W. JOHNSON and MICHEL L. LAPIDUS 437

An index theory for continuous semigroups of *-endomorphisms of D( S2 )and II1 factors

by ROBERT T. POWERS 447

Multivariable Toeplitz operators, Jordan algebras and index theory

by HARALD UPMEIER 461

VII. C*-ALGEBRAS: SOME RECENT RESULTS

On multilinear completely bounded module maps by EDWARD G. EFFROS 479

On the classification of noncommutative tori, III by BERNDT BRENKEN, JOACHIM CUNTZ,

GEORGE A. ELLIOTT and R YSZARD NEST 503

VIII. VON NEUMANN ALGEBRAS

The joint modular structure of an inclusion of von Neumann algebras

by ROBER TO LONGO 529

Central sequences in crossed products by RICHARD H. HERMAN and

VAUGHAN F. R. JONES 539

PREFACE

This volume contains papers presented at the summer conference in honor of the sixtieth birthdays of Professors H.-J. Borchers, N. M. Hugenholtz, R. V. Kadison and D. Kastler held at the University of Iowa from June 17 through 21, 1985. The conference did not attempt to cover all subspecialties within operator algebras, let alone mathematical physics, rather it focused on some operator algebraic developments which are of interest to students and research workers in pure as well as in applied areas of mathematics. Special attention is given to an overview of the algebraic approach to quantum field theory, and, in particular, to quantum statistical mechanics. The individual papers are to a large extent self-contained. More than half of the papers culminate with presentations of new results which have not appeared previously in journals. With a few exceptions, these new results are presented with complete proofs. The other half of the papers are devoted to a review of developments in the subject since the 1980 Summer Institute in Kingston.

The material should be accessible to a relatively wide audience, including graduate students and research workers in a broad spectrum of mathematics and mathematical physics: Functional analysis, operator algebras, operator theory, differential geometry, cyclic cohomology, K-theory, and index theory are applied to questions in the quantum theory of fields and statistical mechanics. While the reader should have some familiarity with basic concepts from functional analysis and opera tor theory (at the level of the first volume of the book series by Kadison and Ringrose, the first two chapters of Bratteli-Robinson, or volume one of Reed-Simon), no physics background is assumed.

Xl

Xll PREFACE

The past two decades have witnessed a fruitful interaction between operator algebras and mathematical physics. This development is given a systematic and up-to-date treatment.

While the conference emphasized connections between different branches of mathematics and mathematical physics as they connect to operator algebras, we have found it convenient to divide the papers into eight subject groups. It is in the nature of the material that the division line between the different groups (making up separate chapters) is not a sharp one, and, in some cases, a given paper might be considered as belonging to two or more subject categories.

We should like to stress that these proceedings contain material covered in one form or another at the conference. A number of researchers were invited and could not come, and their results unfortunately are not included. Someone could have been overlooked when invitations were mailed out, and the organizers want to apologize in advance for such oversights.

The editors and conference organizers wish to thank the National Science Foundation and the University of Iowa for generous financial support, and for help and encouragement in the planning phase. We also wish to thank the Departments of Mathematics and Physics at the University of Iowa for facilities and support. The American Mathematical Society, and Ms. M. C. Almeida in the production department, have been very helpful in the production of this book. We are pleased to record our gratitude. Last, but not least, thanks are due to the participants and speakers for making the conference possible.

"This standard construction will GNS-construction after I. M. Gel'fand, M. I. E. Segal who share the credit for the many of the mathematical concepts used."

be called the A. Naimark and development of

R. Haag, N. M. Hugenholtz, and M Winnink (Commun. Math. Phys. 5(1967), 216)