Operator Algebras and Quantum Statistical Mechanics 1 : C*- and W*-Algebras. Symmetry Groups. Decomposition of States (Texts and Monographs in Physics)

520 Pages·1987·38.84 MB·English

by  Ola Bratteli

Texts and Monographs in Physics
Series Editors:
R. Balian, Gif-sur-Yvette, France
W. Bei9lb6ck, Heidelberg, Germany
t,
H. Grosse,Wien,Austria
E.H. Lieb, Princeton, NJ, USA
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01a Bratteli
Derek W. Robinson
Operator Algebras
d Q
uantum
an
StatisticaIM hanic I
ec s
C*- and W*-Algebras
Symmetry Groups
Decomposition of States
Second Edition
4
Springer
ProfessorOlaBratteli
UniversitetetiOslo
MatematiskInstitutt
MoltkeMoesvei31
0316Oslo,Norway
e-mail:[email protected]
Homepage: http:Hwww.math.uio.no/-bratteli/
ProfessorDerekW. Robinson
AustralianNationalUniversity
SchoolofMathematicalSciences
ACT0200Canberra,Australia
e-mail:[email protected]
Homepage: http:Hwwwmaths.anu.edu.au/-derek/
LibraryofCongressCataloging-in-PublicationData
Bratteli,Ola. Operatoralgebrasandquantumstatisticalmechanics.(Textsandmonographsinphysics) Bibliography;v.
1, p. Includesindex. Contents:v.1.C*-andW*-algebras,symmetrygroups,decompositionofstates. 1.Operator
algebras. 2.Statisticalmechanics. 3.Quantumstatistics. 1.Robinson,DerekW.H.Title. III.Series.
QA326.B74 1987 512'.55 86-27877
SecondEdition 1987. SecondPrinting2002
ISSN0172-5998
ISBN3-540-17093-6 2ndEdition Springer-VerlagBerlinHeidelbergNewYork
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Preface to the Second Printing of the Second Edition
Inthis secondprinting ofthe secondedition severalminorandonemajormathe-
matical mistake have been corrected. We are indebted to Roberto Conti, Sindre
Duedahl andReinhardSchaflitzelforpointingtheseout.
CanberraandTrondheim, 2002 OlaBratteli
DerekW. Robinson
Preface to the Second Edition
The second edition ofthis book differs from the original in threerespects. First,
we have eliminated a large number of typographical errors. Second, we have
corrected a small number ofmathematical oversights. Third, we have rewritten
severalsubsectionsinordertoincorporateneworimprovedresults. Theprincipal
changes occurinChapters 3 and4.
In Chapter 3, Section 3.1.2 now contains a more comprehensive discussion
ofdissipative 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
Section3.2.4. InChapter4the only substantialchanges areto Sections4.2.1 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 sections now
incorporatetheresult.
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 ofthe last seven years in the theory ofderivations,
anddissipations, associatedwithC*-dynamical systems. Thecurrentstateofthis
theoryissummarizedin [[Bra 1]] publishedinSpringer-Verlag'sLectureNotesin
Mathematics series.
CanberraandTrondheim, 1986 OlaBratteli
DerekW. 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 ofthis project
weintendedtocoverthismaterialinonevolumebutinthecourseofdevelop-
mentitwasrealizedthatthiswouldentailtheomissionofvariousinteresting
topics or details. Consequently the book was split into two volumes, the
firstdevoted tothegeneraltheoryofoperatoralgebrasandthesecondto 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 offieldtheoryand statisticalmechanics. Butthetheoryof20years
ago was largely developed for the analysis ofgroup representations and it
was inadequate for many physical applications. Thus after a short honey-
moonperiodinwhichthenewfoundtools ofthe extant theorywereapplied
to themostamenableproblemsalongerandmore interestingperiodensued
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 ofa continuum ofnonisomorphic type-three factors. The
resultsofthisperiodhadasubstantialimpactonthesubsequentdevelopment
ofthe theory ofoperator algebras and led to a continuing period offruitful
Vil
viii Preface to the First Edition
collaboration between mathematicians and physicists. They also led to an
intertwining ofthe theory and applications in which the applications often
forced the formation ofthe theory. Thus in this context the division ofthis
book has a certain arbitrariness.
Thetwovolumesofthebookcontainsixchapters, fourinthisfirstvolume
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 I is a brief historical introduction and it is the five subsequent
chaptersthatformthemain bodyofmaterial. We have encountered various
difficultiesinourattemptstosynthesizethismaterialintoonecoherentbook.
Firstlythereare broad variations in the natureand difficulty ofthe different
chapters. This is partly because the subject matter lies between the main-
streamsofpuremathematicsandtheoreticalphysicsandpartlybecauseitisa
mixture of standard theory and research work which has not previously
appearedinbookform.Wehavetriedtointroduceauniformityandstructure
and we hope the reader will find our attempts are successful. Secondly the
range oftopics 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 July t979.
Most ofChapters 1-5 were written whilst the authors were in Marseille at
the Universit6 d'Aix-Marseille 11, Luminy, and the Centre de Physique
Th6orique CNRS. During a substantial part ofthis period 0. Bratteli was
supported by the Norwegian Research Council for Science and Humanities
and during the complementary period by a post of"ProfesseurAssoci6" at
Luminy. Chapter 6 was partially written at the University of New South
Wales and partially in Marseille and at the University ofOslo.
Chapters2, 3,4andhalfofChapter5weretypedattheCentredePhysique
Th6orique, CNRS, Marseille. Most of the remainder was typed at the
Department ofPure Mathematics, University ofNew 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 ofthe manuscript. We are grateful to Gavin Brown, Ed Effros,
George Elliott, Uffe Haagerup, Richard Herman, Daniel Kastler, Akitaka
Kishimoto, John Roberts, Ray Streater and Andr6 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 andSydney, 1979 Ola Bratteli
Derek W. Robinson
Contents (Volume 1)
Introduction
Notesand Remarks 16
C*-Algebras and von Neumann Algebras 17
2.1. C* -Algebras 19
2.1.1. BasicDefinitionsand Structure 19
2.2. FunctionalandSpectralAnalysis 25
2.2.1. Resolvents, Spectra, and Spectral Radius 25
2.2.2. Positive Elements 32
2.2.3. Approximate Identitiesand QuotientAlgebras 39
2.3. RepresentationsandStates 42
2.3.1. Representations 42
2.3.2. States 48
2.3.3. Construction ofRepresentations 54
2.3.4. ExistenceofRepresentations 58
2.3.5. Commutative C*-Algebras 61
ix
x Contents (Volume t)
2.4. vonNeumannAlgebras 65
2.4.1. Topologies on Y(.5) 65
2.4.2. Definitionand ElementaryPropertiesofvonNeumannAlgebras 71
2.4.3. Normal StatesandthePredual 75
2.4.4. Quasi-EquivalenceofRepresentations 79
2.5. Tomita-TakesakiModularTheoryandStandardFormsofvonNeumann
Algebras 83
2.5.1. a-FinitevonNeumannAlgebras 84
2.5.2. The ModularGroup 86
2.5.3. IntegrationandAnalytic ElementsforOne-ParameterGroupsof
Isometrieson Banach Spaces 97
2.5.4. Self-Dual Conesand Standard Forms 102
2.6. Quasi-LocalAlgebras 118
2.6.1. ClusterProperties 118
2.6.2. Topological Properties 129
2.6.3. AlgebraicProperties 133
2.7. MiscellaneousResultsandStructure 136
2.7.1. Dynamical SystemsandCrossedProducts 136
2.7.2. TensorProductsofOperatorAlgebras 142
2.7.3. Weights on Operator Algebras; Self-Dual Cones of General
von Neumann Algebras; Duality and Classification ofFactors;
Classification ofC*-Algebras 145
NotesandRemarks 152
Groups, Sernigroups, and Generators 157
3.1. BanachSpaceTheory 159
3.1.1. Uniform Continuity 161
3.1.2. Strong, Weak, andWeak* Continuity 163
3.1.3. ConvergenceProperties t84
3.1.4. PerturbationTheory t93
3.1.5. ApproximationTheory 202
3.2. AlgebraicTheory 209
209
3.2.1. PositiveLinearMapsandJordan Morphisms
233
3.2.2. GeneralPropertiesofDerivations