Texts and Monographs in Physics Series Editors: R. Balian, Gif-sur-Yvette, France W. Beiglböck, Heidelberg, Germany H. Grosse, Wien, Austria E. H. Lieb, Princeton, NJ, USA N. Reshetikhin, Berkeley, CA, USA H. Spohn, München, Germany W. Thirring, Wien, Austria Springer-Verlag Berlin Heidelberg GmbH ONLINE LlBRARY Physics and Astronomy http://www.springer.de/phys/ DIa Bratteli Derek W. Robinson Operator Aigebras and Quantum Statistical Mechanics 1 C* - and W* -Algebras Symmetry Groups Decomposition of States Second Edition t Springer Professor DIa Bratteli Universitetet i Os10 Matematisk Institutt Moltke Moes vei 31 0316 Os10, Norway e-mail: [email protected] Horne page: http://www.math.uio.no/~bratteli/ Professor Derek W. Robinson Australian National University School of Mathematical Sciences ACT 0200 Canberra, Australia e-mail: [email protected] Horne page: http://wwwmaths.anu.edu.au/. .. derek/ Library of Congress Cataloging-in·Publication Data BraUeli, Ola. Operator algebras and quantum statistical mechanics. (Texts and monographs in physics) Bibliography; v. I, p. lncludes index. Contents: v. I. C· -and W· -algebras, symmetry groups, decomposition of states. I. Operator a1gebras. 2. Statistical mechanics. 3. Quantum statistics. I. Robinson, Derek W. Il. Title. IIl. Series. QA 326.B74 1987 512'.55 86-27877 Second Edition 1987. Second Printing 2002 ISSN 0172-5998 ISBN 978-3-642-05736-6 ISBN 978-3-662-02520-8 (eBook) DOI 10.1007/978-3-662-02520-8 This work is subject to copyright. All rights are reserved. whether the whole or part of the material is concemed, speeifically the rights of translation, reprinting, reuse of illustrations, reeitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of Ibis publication or parts thereof is permiued only under the provisions ofthe German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are Iiable for prosecution under the German Copyright Law. http://www.springer.de © Springer-Verlag Berlin Heidelberg 1979, 1987 Originally published by Springer-Verlag Berlin Heidelberg New York in 1987. Softcover reprint ofthe hardcover 2nd edition 1987 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speeific statement, that such names are exempt from the relevant probreak tective laws and regulations and therefore free for general use. Cover design: desigll & pmdllction GmbH, Heidelberg Printed on acid-free paper 55/311l/ba 5432 I Preface to the Second Printing of the Second Edition In this second printing of the second edition several minor and one major mathe matical mistake have been corrected. We are indebted to Roberto Conti, Sindre Duedahl and Reinhard Schaflitzel for pointing these out. Canberra and Trondheim, 2002 Oia Bratteli Derek W. Robinson Preface to the Second Edition The second edition of this book differs from theoriginal in three respects. First, we have eliminated a large number of typographical errors. Second, we have corrected a small number of mathematical oversights. Third, we have rewritten several subsections in order to incorporate new or improved results. The principal changes occur in Chapters 3 and 4. In Chapter 3, Section 3.1.2 now contains a more comprehensive discussion of dissipative operators and analytic elements. Additions and changes have also been made in Sections 3.1.3, 3.1.4, and 3.1.5. Further improvements occur in Section 3.2.4. In Chapter 4 the only substantial changes are to Seetions 4.2.1 and 4.2.2. At the time of writing the first edition it was an open question whether maximal orthogonal probability measures on the state space of a C*-algebra were automatically maximal among all the probability measures on the space. This question was resolved positively in 1979 and the rewritten seetions now incorporate the result. All these changes are nevertheless revisionary in nature and do not change the scope of the original edition. In particular, we have resisted the temptation to describe the developments of the last seven years in the theory of derivations, and dissipations, associated with C*-dynarnical systems. The current state of this theory is summarized in [[Bra 1]] published in Springer-Verlag's Lecture Notes in Mathematics series. Canberra and Trondheim, 1986 Ola Bratteli Derek W. Robinson v Preface to the First Edition In this book we describe the elementary theory of operator algebras and parts of the advanced theory which are of relevance, or potentially of relevance, to mathematical physics. Subsequently we describe various applications to quantum statistical mechanics. At the outset of this project we intended to cover this material in one volume but in the course of develop ment it was realized that this would entail the omission ofvarious interesting topics or details. Consequently the book was split into two volumes, the first devoted to the general theory of operator algebras and the second to the applications. This splitting into theory and applications is conventional but somewhat arbitrary. In the last 15-20 years mathematical physicists have realized the importance of operator algebras and their states and automorphisms for problems of field theory and statistical mechanics. But the theory of 20 years aga was largely developed for the analysis of group representations and it was inadequate for many physical applications. Thus after a short honey moon period in which the new found tools of the extant theory were applied to the most amenable problems a longer and more interesting period ensued in which mathematical physicists were forced to redevelop the theory in relevant directions. New concepts were introduced, e.g. asymptotic abelian ness and KMS states, new techniques applied, e.g. the Choquet theory of barycentric decomposition for states, and new structural results obtained, e.g. the existence of a continuum of nonisomorphic type-three factors. The results ofthis period had a substantial impact on the subsequent development of the theory of operator algebras and led to a continuing period of fruitful VII viii Preface to the First Edition collaboration between mathematicians and physicists. They also led to an intertwining of the theory and applications in which the applications often forced the formation of the theory. Thus in this context the division of this book has a certain arbitrariness. The two volumes ofthe book contain six chapters, four in this first volume and two in the second. The chapters of the second volume are numbered consecutively with those of the first and the references are cumulative. Chapter 1 is abrief historical introduction and it is the five subsequent chapters that form the main body ofmaterial. We have encountered various difficulties in our attempts to synthesize this material into one coherent book. Firstly there are broad variations in the nature and difficulty of the different chapters. This is partly because the subject matter lies between the main streams of pure mathematics and theoretical physics and partly because it is a mixture of standard theory and research work which has not previously appeared in book form. We have tried to introduce a uniformity and structure and we hope the reader will find our attempts are successful. Secondly the range of topics relevant to quantum statistical mechanics is certainly more extensive than our coverage. For example we have completely omitted discussion of open systems, irreversibility, and semi-groups of completely positive maps because these topics have been treated in other recent mono graphs [[Dav 1]] [[Eva 1]]. This book was written between September 1976 and luly 1979. Most of Chapters 1-5 were written whilst the authors were in Marseille at the Universite d'Aix-Marseille 11, Luminy, and the Centre de Physique Theorique CNRS. During a substantial part of this period O. Bratteli was supported by the Norwegian Research Council for Science and Humanities and during the complementary period by a post of "Professeur Associe" at Luminy. Chapter 6 was partially written at the University of New South Wales and partially in Marseille and at the University of Oslo. Chapters 2, 3,4 and half of Chapter 5 were typed at the Centre de Physique Theorique, CNRS, Marseille. Most of the remainder was typed at the Department of Pure Mathematics, University of New South Wales. It is a pleasure to thank Mlle. Maryse Cohen-Solal, Mme. Dolly Roche, and Mrs. Mayda Shahinian for their work. We have profited from discussions with many colleagues throughout the preparation of the manuscript. We are grateful to Gavin Brown, Ed Effros, George Elliott, Uffe Haagerup, Richard Herman, Daniel Kastler, Akitaka Kishimoto, lohn Roberts, Ray Streater and Andre Verbeure for helpful comments and corrections to earlier versions. We are particularly indebted to Adam Majewski for reading the final manuscript and locating numerous errots. Oslo and Sydney, 1979 Ola Bratteli Derek W. Robinson Contents (Volume 1) Introduction Notes and Remarks 16 C*-Algebras and von Neumann Aigebras 17 2.1. C*-Algebras 19 2.1.1. Basic Definitions and Structure 19 2.2. Functional and Spectral Analysis 25 2.2.1. Resolvents, Spectra, and Spectral Radius 25 2.2.2. Positive Elements 32 2.2.3. Approximate Identities and Quotient Aigebras 39 2.3. Representations and States 42 2.3.1. Representations 42 2.3.2. States 48 2.3.3. Construction of Representations 54 2.3.4. Existence of Representations 58 2.3.5. Commutative C*-Algebras 61 ix x Contents (Volume 1) 2.4. von Neumann Algebras 65 2.4.1. Topologies on ~(f,) 65 2.4.2. Definition and Elementary Properties of von Neumann Aigebras 71 2.4.3. Normal States and the Predual 75 2.4.4. Quasi-Equivalence of Representations 79 2.5. Tomita-Takesaki Modular Theory and Standard Forms of von Neumann Algebras 83 2.5.1. u-Finite von Neumann Aigebras 84 2.5.2. The Modular Group 86 2.5.3. Integration and Analytic Elements for One-Parameter Groups of Isometries on Banach Spaces 97 2.5.4. Self-Dual Cones and Standard Forms 102 2.6. Quasi-Local Algebras 118 2.6.1. Cluster Properties 118 2.6.2. Topological Properties 129 2.6.3. Algebraic Properties 133 2.7. MisceUaneous Results and Structure 136 2.7.1. Dynamical Systems and Crossed Products 136 2.7.2. Tensor Produc'ts ofOperator Algebras 142 2.7.3. Weights on Operator Algebras; Self-Dual Cones of General von Neumann Aigebras; Duality and Classification of Factors; Classification of C*-Algebras 145 Notes and Remarks 152 Groups, Semigroups, and Generators 157 3.1. Banach Space Theory 159 3.1.1. Uniform Continuity 161 3.1.2. Strong, Weak, and Weak* Continuity 163 3.1.3. Convergence Properties 184 3.1.4. Perturbation Theory 193 3.1.5. Approximation Theory 202 3.2. Aigebraic Theory 209 209 3.2.1. Positive Linear Maps and Jordan Morphisms 233 3.2.2. General Properties of Derivations Contents (Volume 1) Xl 3.2.3. Spectrai Theory and Bounded Derivations 249 3.2.4. Derivations and Automorphism Groups 264 3.2.5. Spatial Derivations and Invariant States 269 3.2.6. Approximation Theory for Automorphism Groups 290 Notes and Remarks 303 Decomposi~ion Theory 315 4.1. General Theory 317 4.1.1. Introduction 317 4.1.2. Barycentric Decompositions 321 4.1.3. Orthogonal Measures 339 4.1.4. Borel Structure of States 350 4.2. Extremal, Central, and Subcentral Decompositions 359 4.2.1. Extremal Decompositions 359 4.2.2. Central and Subcentral Decompositions 370 4.3. Invariant States 374 4.3.1. Ergodie Decompositions 374 4.3.2. Ergodie States 393 4.3.3. Locally Compact Abelian Groups 407 4.3.4. Broken Symmetry 423 4.4. Spatial Decomposition 439 4.4.1. General Theory 440 4.4.2. Spatial Decomposition and Decomposition of States 449 Notes and Remarks 458 467 References Books and Monographs 469 Articles 473 List of Symbols 489 Subject Index 495