Operator Algebras and Quantum Statistical Mechanics: C*- and W*-Algebras Symmetry Groups Decomposition of States

502 Pages·1979·9.92 MB·English

by  Ola Bratteli

Texts and
Monographs
in Physics
w. Beiglbock
M. Goldhaber
E. H. Lieb
W. Thirring
Series Editors
ala Bratteli
Derek W. Robinson
Operator Algebras and
Quantum Statistical
1
Mechanics
C*- and W*-Algebras
Symmetry Groups
Decomposition of States
[I]
Springer Science+Business Media, LLC
Ola Bratteli Derek W. Robinson
Institute of Mathematics School of Mathematics
University of Oslo University of New South Wales
Blindem, Oslo 3 P.O. Box 1
Norway Kensington, NSW
Australia 2033
Editors:
Wolf Beiglbock Maurice Goldhaber
Institut fUr Angewandte Mathematik Department of Physics
Universităt Heidelberg Brookhaven National Laboratory
Im Neuenheimer Fe1d 5 Associated Universities, Inc.
D-69oo Heidelberg 1 Upton, NY 11973
Federal Republic of Germany USA
Elliott H. Lieb Walter Thirring
Department of Physics Institut fUr Theoretische Physik
Joseph Henry Laboratories der Universităt Wien
Princeton University Boltzmanngasse 5
P.O. Box 708 A-I090 Wien
Princeton, NJ 08540 Austria
USA
ISBN 978-3-662-02315-0 ISBN 978-3-662-02313-6 (eBook)
DOI 10.1007/978-3-662-02313-6
Library of Congress Cataloging in PubIication Data
Bratteli, Ola.
Operator algebras and quantum statistical mechanics.
(Texts and monographs in physics)
Includes bibliographical references and index.
1. Operator algebras. 2. Quantum statistics.
1. Robinson, Derek W., joint author.
II. Title.
QA326.B74 512'.55 78-27159
AlI rights reserved.
No part of this book may be translated or reproduced in
any form without written permission from Springer-Verlag.
© 1979 by Springer Science+Business Media New York
Originally published by Springer-Verlag New York Inc. in 1979
Softcover reprint of the hardcover 1s t edition 1979
9 8 7 6 5 4 3 2 1
Preface
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 of various 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 offield theory and statistical mechanics. But the theory of 20 years
ago 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 of this period had a substantial impact on the subsequent development
of the theory of operator algebras and led to a continuing period of fruitful
collaboration between mathematicians and physicists. They also led to an
v
vi Preface
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 of the 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 a brief historical introduction and it is the five subsequent
chapters that form the main body of material. 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 lJJ [[Eva IJ].
This book was written between September 1976 and July 1979
Most of Chapters 1-5 were written whilst the authors were in Marseille at
the Universite d'Aix-Marseille II, 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, John 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 errors.
Oslo and Sydney, 1979. Ola Bratteli
Derek W. Robinson
Contents Volume 1
Introduction
Notes and Remarks 16
C*-Algebras and von Newmann Algebras 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 Algebras 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
vii
viii Contents
2.4. von Neumann Algebras 65
2.4.1. Topologies on 2'(fl) 65
2.4.2. Definition and Elementary Properties of von Neumann Algebras 71
2.4.3. Normal States and the Predual 75
2.4.4. Quasi-Equivalence of Representations 79
2.S. Tomita-Takesaki Modular Theory and Standard Forms of von Neumann
Algebras 83
2.5.1. a-Finite von Neumann Algebras 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. Miscellaneous Results and Structure 136
2.7.1. Dynamical Systems and Crossed Products 136
2.7.2. Tensor Products of Operator Algebras 142
2.7.3. Weights on Operator Algebras; Self-Dual Cones of General von
Neumann Algebras; 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 183
3.1.4. Perturbation Theory 189
3.1.5. Approximation Theory 198
3.2. Algebraic Theory 205
3.2.1. Positive Linear Maps and Jordan Morphisms 205
3.2.2. General Properties of Derivations 228
Contents IX
3.2.3. Spectral Theory and Bounded Derivations 244
3.2.4. Derivations and Automorphism Groups 259
3.2.5. Spatial Derivations and Invariant States 263
3.2.6. Approximation Theory for Automorphism Groups 285
Notes and Remarks 298
Decomposition Theory 309
4.1. General Theory 311
4.1.1. Introduction 311
4.1.2. Barycentric Decompositions 315
4.1.3. Orthogonal Measures 333
4.1.4. Borel Structure of States 344
4.2. Extremal, Central, and Subcentral Decompositions 353
4.2.1. Extremal Decompositions 353
4.2.2. Central and Subcentral Decompositions 362
4.3. Invariant States 367
4.3.1. Ergodic Decompositions 367
4.3.2. Ergodic States 386
4.3.3. Locally Compact Abelian Groups 400
4.3.4. Broken Symmetry 416
4.4. Spatial Decomposition 432
4.4.1. General Theory 433
4.4.2. Spatial Decomposition and Decomposition of States 442
Notes and Remarks 451
References 459
Books and Monographs 461
Articles 464
List of Symbols 481
Subject Index 487
Contents Volume 2
States in Quantum Statistical Mechanics
5.1. Introduction
5.2. Continuous Quantum Systems I
5.2.1. The CAR and CCR Relations
5.2.2. The CAR and CCR Algebras
5.2.3. States and Representations
5.2.4. The Ideal Fermi Gas
5.2.5. The Ideal Bose Gas
5.3. KMS States
5.3.1. The KMS Condition
5.3.2. The Set of KMS States
5.4. Stability and Equilibrium
5.4.1. Stability of KMS States
5.4.2. Stability and the KMS Condition
5.4.3. Gauge Groups and the Chemical Potential
5.4.4. Passive Systems
Notes and Remarks
XI
xii Contents
Models of Quantum Statistical Mechanics
6.1. Introduction
6.2. Quantum Spin Systems
6.2.1. Kinematical and Dynamical Descriptions
6.2.2. Equilibrium States
6.2.2.A The Gibbs Condition
6.2.2.B The Maximum Entropy Principle
6.2.2.C Translationally Invariant States
6.2.2.D Uniqueness of KMS States
6.2.2.F Non-uniqueness of KMS States
6.2.2.F Ground States
6.3. Continuous Quantum Systems
6.3.1. Functional Integration Techniques
6.3.2. Correlation Functions and Equilibrium States
Notes and Remarks
References
Books and Monographs
Articles
List of Symbols