Operator Algebras and Quantum Statistical Mechanics 1: C*- and W*-Algebras Symmetry Groups Decomposition of States

509 Pages·1987·22.868 MB·English

by  Professor Ola Bratteli, Professor Derek W. Robinson (auth.)

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
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© 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
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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