http://dx.doi.org/10.1090/pspum/038.1
PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 38, Part 1
OPERATOR ALGEBRAS AND APPLICATIONS
AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1982 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY
HELD AT QUEENS UNIVERSITY KINGSTON, ONTARIO JULY 14-AUGUST 2, 1980
EDITED BY RICHARD V. KADISON
Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 79-27061
Library of Congress Cataloging in Publication Data Symposium in Pure Mathematics (1980: Queens University, Kingston, Ont.) Operator algebras and applications. (Proceedings of symposia in pure mathematics; v. 38) Includes bibliographies and index. 1. Operator algebras-Congresses. I. Kadison, Richard V., 1925— II. Title. III. Series. QA326.S95 1982 512'.55 82-11561 ISBN 0-8218-1441-9 (v. 1) ISBN 0-8218-1444-3 (v. 2) ISBN 0-8218-1445-1 (set)
1980 Mathematics Subject Classification. Primary 46L05, 46L10; Secondary 43A80, 81E05, 82A15. Copyright © 1982 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. TABLE OF CONTENTS PARTI
Author index xi Preface RICHARD V. KADISON xvn* Operator algebras—the first forty years 1 RICHARD V. KADISON On the structure theory of C*-algebras: some old and new problems 19 EDWARD G. EFFROS Homological invariants of extensions of C*-algebras 35 JONATHAN ROSENBERG Geometric resolutions and the Kunneth formula for C*-algebras 77 CLAUDE SCHOCHET The J£-groups for free products of C*-algebras 81 JOACHIM CUNTZ The internal structure of simple C*-algebras 85 JOACHIM CUNTZ K homology and index theory 117 PAUL BAUM AND RONALD G. DOUGLAS Extensions of AF algebras: the projection lifting problem 175 LAWRENCE G. BROWN On the A'-theory of the C*-algebra generated by a projective representation ofZ" 177 GEORGE A. ELLIOTT Extensions of Cuntz-Krieger algebras 181 MASATOSHI FUJII EXT-groups as A'-groups 185 WILLIAM L. PASCHKE On a conjecture of Effros-Shen 187 NORBERT RlEDEL Positive matrices and dimension groups affiliated to topological Markov chains 191 DAVID HANDELMAN A graph theory for C*-algebras 195 YASUO WATATANI The harmonic analysis of automorphism groups 199 WILLIAM ARVESON
in IV TABLE OF CONTENTS
Dynamical systems and crossed products 271 GERT K. PEDERSEN Morita equivalence for operator algebras 285 MARC A. RIEFFEL Applications of strong Morita equivalence to transformation group C*-algebras 299 MARC A. RIEFFEL On certain properties of crossed products 311 ELLIOT C. GOOTMAN Minimal abelian group actions on type I C*-algebras 323 ELLIOT C. GOOTMAN AND DORTE OLESEN Twisted crossed products, the "Mackey Machine," and the Effros-Hahn con• jecture 327 PHILIP GREEN Equivariant ^-theory and crossed product C*-algebras 337 PHILIP GREEN C*-algebras of groupoids and foliations 339 JEAN N. RENAULT Period two automorphisms 351 RICHARD H. HERMAN Fixedpoint algebras versus crossed products 357 OLA BRATTELI The dual topology of a crossed product and applications to group representa• tions 361 JONATHAN ROSENBERG Ideals of C*-crossed products by locally compact abelian groups 365 A. KISHIMOTO Duality for C*-crossed products and its applications 369 HIROSHI TAKAI A correspondence between subgroups and subalgebras in a C*-crossed product 375 HISASHI CHODA Tensor products and nuclear C*-algebras 379 E. CHRISTOPHER LANCE Stone-Weierstrass theorems for separable C*-algebras 401 JOHN W. BUNCE On maximal abelian subalgebras of C*-algebras 409 R. J. ARCHBOLD Pure states as a dual object for C*-algebras 413 FREDERIC W. SHULTZ L2 -index theory on homogeneous spaces and discrete series representations 419 ALAIN CONNES AND HENRI MOSCOVICI Unitary representation of Lie groups and generalized symplectic geometry 435 L. PUKANSZKY TABLE OF CONTENTS V
Characters and semicharacters on connected Lie groups 467 NIELS VIGAND PEDERSEN Classical groups and classical Lie algebras of operators 477 PIERRE DE LA HARPE Some representations of U(°°) 515 SERBAN STRATILA A surve» y of foliations and operator algebras 52 1 ALAIN CONNES Some properties of the C*-algebra associated with a foliation 629 THIERRY FACK AND GEORGES SKANDALIS This page intentionally left blank TABLE OF CONTENTS PART 2
Author index Celebration of Tomita's theorem ALFONS VAN DAELE Positive cones for von Neumann algebras HUZIHIRO ARAKI Geometric aspects of the Tomita-Takesaki theory CHRISTIAN F. SKAU Left Hilbert systems F. COMBES ITPFI factors—a survey E. J. WOODS Classification des facteurs ALAIN CONNES Automorphisms and von Neumann algebras of type III MASAMICHI TAKESAKI A survey of H/*-categories P. GHEZ Actions of non-abelian groups and invariant T R. LIMA Cohomological invariants for groups of outer automorphisms algebras COLIN E. SUTHERLAND Automorphism groups and invariant states ERLING ST0RMER Compact ergodic groups of automorphisms MAGNUS B. LANDSTAD Actions of discrete groups on factors V. F. R. JONES Ergodic theory and von Neumann algebras CALVIN C. MOORE Topologies on measured groupoids ARLAN RAMSAY
vn vm TABLE OF CONTENTS
Cohomology theory for operator algebras 229 J. R. RlNGROSE Low dimensional cohomology of Banach algebras 253 B. E. JOHNSON Derivations and their relation to perturbations of operator algebras 261 ERIK CHRISTENSEN Invariants of C*-algebras stable under perturbations 275 JOHN PHILLIPS The map Ad: U(B) —> Inn B 281 JOHN PHILLIPS Automorphisms of continuous trace C*-algebras 285 IAIN RAEBURN On lifting automorphisms 289 DENNIS SULLIVAN AND J. D. MAITLAND WRIGHT Derivations and automorphisms of Jordan C*-algebras 291 HARALD UPMEIER Order structure and Jordan Banach algebras 297 J. BELLISSARD AND B. IOCHUM A Tomita-Takesaki theory for JBW-algebras 301 HARALD HANCHE-OLSEN Faithful normal states on JBW-algebras 305 G. G. EMCH AND W. P. C. KING Developments in the theory of unbounded derivations in C*-algebras 309 SHOICHIRO SAKAI Derivations of abelian C*-algebras 333 C. J. K. BATTY Closed *-derivations commuting with an automorphism group 339 F. GOODMAN Differentiable structure on non-abelian C*-algebras 343 HIROSHI TAKAI Unbounded *-derivations and infinitesimal generators on operator algebras 353 OLA BRATTELI AND PALLE E. T. J^RGENSEN Semiderivations on group Cf-algebras 367 MARTIN E. WALTER Perturbations 371 RICHARD H. HERMAN Generators of dynamical semigroups 377 DAVID E. EVANS The characterization of the analytic generator of *-automorphism groups 381 LASZLO ZSIDO TABLE OF CONTENTS IX
Almost uniformly continuous dynamical systems 385 LASZLO ZSIDO (report on joint work with GEORGE A. ELLIOTT) Algebras of unbounded operators 389 ROBERT T. POWERS C*-algebras and statistical mechanics 407 N. M. HUGENHOLTZ Does ergodicity plus locality imply the Gibbs structure? 467 DANIEL KASTLER Stability in statistical mechanics 491 GEOFFREY L. SEWELL Phase transitions 499 OLA BRATTELI Nature of Bose condensation 503 A. VERBEURE Perturbations of free evolutions and Poisson measures 509 PH. COMBE, R. HOEGH-KROHN, R. RODRIGUEZ, M. SIRUGUE AND M. SlRUGUE-COLLIN Structural questions in quantum field theory 513 RUDOLF HAAG New light on the mathematical structure of algebraic field theory 523 JOHN E. ROBERTS Algebraic and modular structure of von Neumann algebras of physics 551 ROBERTO LONGO Local nets of C*-algebras and the structure of elementary particles 567 DETLEV BUCHHOLZ AND KLAUS FREDENHAGEN Quasi-periodic Hamiltonians. A mathematical approach 579 J. BELLISSARD AND D. TESTARD Positive linear maps 583 MAN-DUEN CHOI Representations of inseparable C*-algebras 591 SIMON WASSERMANN C*-algebras with a countable dual 593 HORST BEHNCKE On the extensions of C*-algebras relative to factors of type 11^ 597 HIDEO TAKEMOTO A structure theory in the regular monotone completion of C*-algebras 601 KAZUYUKI SAITO Commuting pairs of C*-algebras in the Calkin algebra 605 KENNETH R. DAVIDSON X TABLE OF CONTENTS
C*-algebras of multivariable Wiener-Hopf operators 607 PAUL S. MUHLY AND JEAN N. RENAULT The concept of bibounded operators on H/*-algebras 611 BURKHARD KUMMERER Contractive projections on C*-algebras 615 Y. FRIEDMAN AND B. RUSSO Problems on joint quasitriangularity for TV-tuples of essentially commuting, essentially normal operators 619 NORBERTO SALINAS Spectral permanence for joint spectra 623 RAUL E. CURTO Unbounded negative definite functions and property T for locally compact groups 625 CHARLES A. AKEMANN Qoint work with MARTIN E. WALTER) AUTHOR INDEX
Part Page AKEMANN, CHARLES A. 0oint work with WALTER, MARTIN E.). Un• bounded negative definite functions and property T for locally 2 625 compact groups 2 5 ARAKI, HUZIHIRO. Positive cones for von Neumann algebras 1 ARCHBOLD, R. J. On maximal abelian subalgebras of C*-algebras 409 ARVESON , WILLIAM . The harmonic analysis of automorphism groups 1 199 BATTY, C. J. K. Derivations of abelian C*-algebras 2 333 BAUM, PAUL AND DOUGLAS, RONALD G. K homology and index theory 1 117 BEHNCKE, HORST. C*-algebras with a countable dual 2 593 BELLISSARD, J. AND IOCHUM, B. Order structure and Jordan Banach algebras 2 297 BELLISSARD, J. AND TESTARD, D. Quasi-periodic Hamiltonians. A math• ematical approach 2 579 BRATTELI, OLA. Fixedpoint algebras versus crossed products 1 357 . Phase transitions 2 499 BRATTELI, OLA AND J^RGENSEN, PALLE E. T. Unbounded ^deriva• tions and infinitesimal generators on operator algebras 2 353 BROWN, LAWRENCE G. Extensions of AF algebras: the projection Hfting problem 1 175 BUCHHOLZ, DETLEV AND FREDENHAGEN, KLAUS. Local nets of C*- algebras and the structure of elementary particles 2 567 BUNCE, JOHN W. Stone-Weierstrass theorems for separable C*-algebras 1 401 CHODA, HISASHI. A correspondence between subgroups and subalgebras in a C*-crossed product 1 375 CHOI, MAN-DUEN. Positive linear maps 2 583 CHRISTENSEN, ERIK. Derivations and their relation to perturbations of operator algebras 2 261 COMBE, PH., HOEGH-KROHN, R., RODRIGUEZ, R., SIRUGUE, M. AND SIRUGUE-COLLIN, M. Perturbations of free evolutions and Poisson measures 2 509 COMBES, F. Left Hilbert systems 2 21 CONNES, ALAIN. A survey of foliations and operator algebras 1 521 . Classification des facteurs 2 43 CONNES, ALAIN AND MOSCOVICI, HENRI. Z,2-index theory on homo• geneous spaces and discrete series representations 1 419 CUNTZ, JOACHIM. The ^-groups for free products of C*-algebras 1 81
XI Xll AUTHOR INDEX
Part Page CUNTZ, JOACHIM. The internal structure of simple C*-algebras 1 85 CURTO, RAUL E. Spectral permanence for joint spectra 2 623 DAVIDSON, KENNETH R. Commuting pairs of C* algebras in the Calkin algebra 2 605 DOUGLAS, RONALD G. See BAUM, PAUL AND DOUGLAS, RONALD G. EFFROS, EDWARD G. On the structure theory of C*-algebras: some old and new problems 1 19 ELLIOTT, GEORGE A. On the ^-theory of the C*-algebra generated by a projective representation of Z" 1 177 . See ZSIDO, LASZLO (report on joint work with ELLIOTT, GEORGE A.) EMCH, G. G. AND KING, W. P. C. Faithful normal states on JBW-alge- bras 2 305 EVANS, DAVID E. Generators of dynamical semigroups 2 377 FACK, THIERRY AND SKANDALIS, GEORGES. Some properties of the C*-algebra associated with a foliation 1 629 FREDENHAGEN, KLAUS. See BUCHHOLZ, DETLEV AND FREDENHAGEN, KLAUS. 2 615 FRIEDMAN, Y. AND RUSSO, B. Contractive projections on C*-algebras 1 181 Fum, MASATOSHI. Extensions of Cuntz-Krieger algebras GHEZ,P. A survey of W*-categories 2 137 GOODMAN, F. Closed *-derivations commuting with an automorphism 2 339 group 1 311 GOOTMAN, ELLIOT C. On certain properties of crossed products GOOTMAN, ELLIOT C. AND OLESEN, DORTE. Minimal abelian group actions on type I C*-algebras 1 323 GREEN, PHILIP. Twisted crossed products, the "Mackey Machine", and the Effros-Hahn conjecture 1 327 . Equivariant A'-theory and crossed product C*-algebras 1 337 HAAG, RUDOLF. Structural questions in quantum field theory 2 513 HANCHE-OLSEN, HARALD. A Tomita-Takesaki theory for JBW-algebras 2 301 HANDELMAN, DAVID. Positive matrices and dimension groups affiliated to topological Markov chains 1 191 DE LA HARPE, PIERRE. Classical groups and classical Lie algebras of operators 1 477 HERMAN, RICHARD H. Period two automorphisms 1 351 . Perturbations 2 371 HOEGH-KROHN, R. See COMBE, PH., HOEGH-KROHN, R., RODRIGUEZ, R., SlRUGUE, M. AND SlRIGUE-COLLIN, M. HUGENHOLTZ, N. M. C*-algebras and statistical mechanics 2 407 IOCHUM, B. See BELLISSARD, J. AND IOCHUM, B. JOHNSON, B. E. LOW dimensional cohomology of Banach algebras 2 253 JONES, V. F. R. Actions of discrete groups on factors 2 167 AUTHOR INDEX Xlll
Part Page J0RGENSEN, PALLE E. T. See BRATTELI, OLA AND JJZ>RGENSEN, PALLE E. T. KADISON, RICHARD V. Preface 1 xvii . Operator algebras—the first forty years 1 1 KASTLER, DANIEL. Does ergodicity plus locality imply the Gibbs struc• ture? 2 467 KING, W. P. C. See EMCH, G. G. AND KING, W. P. C. KISHIMOTO, A. Ideals of C*-crossed products by locally compact abelian groups 1 365 KUMMERER, BURKHARD. The concept of bibounded operators on W*~ algebras 2 611 LANCE, E. CHRISTOPHER. Tensor products and nuclear C*-algebras 1 379 LANDSTAD, MAGNUS B. Compact ergodic groups of automorphisms 2 163 LIMA, R. Actions of non-abelian groups and invariant T 2 141 LONGO, ROBERTO. Algebraic and modular structure of von Neumann algebras of physics 2 551 MOORE, CALVIN C. Ergodic theory and von Neumann algebras 2 179 MOSCOVICI, HENRY. See CONNES, ALAIN AND MOSCOVICI, HENRY MUHLY, PAUL S. AND RENAULT, JEAN N. C*-algebras of multivariable Wiener-Hopf operators 2 607 OLESEN, DORTE. See GOOTMAN, ELLIOT C. AND OLESEN, DORTE. PASCHKE, WILLIAM L. EXT-groups as ^-groups 1 185 PEDERSEN, GERT K. Dynamical systems and crossed products 1 271 PEDERSEN, NIELS VIGAND. Characters and semicharacters on connected 1 467 Lie groups 2 275 PHILLIPS, JOHN. Invariants of C*-algebras stable under perturbations 2 281 . The map Ad: U{B) —* Inn B 2 389 POWERS, ROBERT T. Algebras of unbounded operators PUKANSZKY, L. Unitary representations of Lie groups and generalized 1 435 symplectic geometry 2 285 RAEBURN, IAIN. Automorphisms of continuous trace C*-algebras 2 227 RAMSAY, ARLAN. Topologies on measured groupoids 1 339 RENAULT, JEAN N. C*-algebras of groupoids and foliations . See MUHLY, PAUL S. AND RENAULT, JEAN N. 1 187 RIEDEL, NORBERT. On a conjecture of Effros-Shen 1 285 RIEFFEL, MARC A. Morita equivalence for operator algebras . Applications of strong Morita equivalence to transformation 1 299 group C*-algebras 2 229 RINGROSE, J. R. Cohomology theory for operator algebras ROBERTS, JOHN E. New light on the mathematical structure of algebraic field theory 2 523 RODRIGUEZ, R. See COMBE, PH., HOEGH-KROHN, R., RODRIGUEZ, R., SlRUGUE, M. AND SlRUGUE-COLLIN, M. XIV AUTHOR INDEX
Part Page ROSENBERG, JONATHAN. Homological invariants of extensions of C*- algebras 1 35 . The dual topology of a crossed product and applications to group representations 1 361 Russo, B. See FRIEDMAN, Y. AND RUSSO, B. SAITO, KAZUYUKI. A structure theory in the regular monotone comple• tion of C*-algebras 2 601 SAKAI, SHOICHIRO. Developments in the theory of unbounded deriva• tions in C*-algebras 2 309 SALINAS, NORBERTO. Problems on joint quasitriangularity for N-tuples of essentially commuting, essentially normal operators 2 619 SCHOCHET, CLAUDE. Geometric resolutions and the Kunneth formula for C*-algebras 1 77 SEWELL, GEOFFREY L. Stability in statistical mechanics 2 491 SHULTZ, FREDERIC W. Pure states as a dual object for C*-algebras 1 413 SIRUGUE, M. See COMBE, PH., HOEGH-KROHN, R., RODRIGUEZ, R., SlRUGUE, M. AND SlRUGUE-COLLIN, M. SIRUGUE-COLLIN, M. See COMBE, PH., HOEGH-KROHN, R., RODRIGUEZ, R., SIRUGUE, M. AND SIRUGUE-COLLIN, M. SKANDALIS, GEORGES. See FACK, THIERRY AND SKANDALIS, GEORGES. SKAU, CHRISTIAN F. Geometric aspects of the Tomita-Takesaki theory 2 17 ST0RMER, ERLING. Automorphism groups and invariant states 2 149 STRATILA, SERBAN . Some representations of U(°°) 1 515 SULLIVAN, DENNIS AND WRIGHT, J. D. MAITLAND. On lifting auto• morphisms 2 289 SUTHERLAND, COLIN E. Cohomological invariants for groups of outer automorphisms of von Neumann algebras 2 145 TAKAI, HIROSHI. Duality for C*-crossed products and its applications 1 369 . Differentiable structure on the non-abelian C*-algebras 2 343 TAKEMOTO, HIDEO. On the extensions of C*-algebras relative to factors of type 11^ 2 597 TAKESAKI, MASAMICHI. Automorphisms and von Neumann algebras of type HI 2 111 TESTARD, D. See BELLISSARD, J. AND TESTARD, D. UPMEIER, HARALD. Derivations and automorphisms of Jordan (^-alge• bras 2 291 VAN DAELE, ALFONS. Celebration of Tomita's theorem 1 2 VERBEURE, A. Nature of Bose condensation 2 503 WALTER, MARTIN E. Semiderivations on group Cf-algebras 2 367 . See AKEMANN, CHARLES A. (j°int work with WALTER, MARTIN E.). WASSERMANN, SIMON. Representations of inseparable C*-algebras 2 591 AUTHOR INDEX XV
Part Page WATATANI, YASUO. A graph theory for C*-algebras 1 195 WOODS, E. J. ITPFI factors-a survey 2 25 WRIGHT, J. D. MAITLAND. See SULLIVAN, DENNIS AND WRIGHT, J. D. MAITLAND . 2 367 ZSIDO, LASZLO. The characterization of the analytic generator of •-automorphism groups 2 381 . (report on joint work with ELLIOTT, GEORGE A.) Almost uniformly continuous dynamical systems 2 385 This page intentionally left blank PREFACE
These volumes contain an account of the topics dealt with at the Twenty Eighth Summer Institute of the American Mathematical Society held at Queens University in Kingston, Ontario, Canada from July 14 through August 2, 1980. That meeting was the first such to treat the entire subject of Operator Algebras and their applications in over thirteen years. As a consequence, a large part of the program was devoted to a review of developments in the subject over that period. The daily programs consisted of a 90 minute lecture covering a major development in the subject, two forty-five minute lectures focusing on important special aspects of some major topic, and an hour devoted to four fifteen-minute announcements of recent results. The longer lectures were advanced exposition detailing problems as well as results. There are expanded versions of many of the longer lectures in these volumeso In some cases, several articles on a single subject are coordinated as sections of a single account. The first article (by this editor) is a brief chronology of the subject. Taken as a whole, these reports constitute a "state-of- the-art" account of the subject.
Richard V. Kadison
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Conference Participants
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