Operator algebras and quantum statistical mechanics 1 2nd edition

Texts and Monographs in Physics Series Editors: R Balian, Gif-sur-Yvette, France W Bei lb6ck, Heidelberg, t, Germany H Grosse, Wien, Austria E H Lieb, Princeton, NJ, USA H Reshetikhin, Berkeley, CA, USA Spohn, Minchen, Germany W Thirring, Wien, N Austria www.pdfgrip.com Springer Berlin Heidelberg New York Hong Kong F London Milan Paris ONLINE LIBRARY - Physics and Astronomy H , Tokyo http://www.springerde/phys/ www.pdfgrip.com 01a Bratteli Derek W Robinson Operator Algebras an d Q uantum S tatisticaIM ec h anic s I C*- and W*-Algebras Symmetry Groups Decomposition of States Second Edition Springer www.pdfgrip.com Professor Ola Bratteli Universitetet i Oslo Matematisk Institutt Moltke Moes vei 31 0316 Oslo, Norway e-mail: bratteli@math.uio.no Home page: http:Hwww math uio no/-bratteli/ Professor Derek W Robinson Australian National University School of Mathematical Sciences ACT 0200 Canberra, Australia e-mail: Derek Robinson@ anu edu au Home page: Library of http:Hwwwmaths.anu.edu.au/-derek/ Congress Cataloging-in-Publication Data Bratteli,Ola Operator algebras and quantum statistical mechanics (Texts and monographs in physics) Bibliography;v and W -algebras, symmetry groups, decomposition of states Operator 1, p Includes index Contents: v C algebras Statistical mechanics Quantum statistics Robinson, Derek W H Title III Series * * - QA 326.B74 1987 512'.55 86-27877 Second Edition 1987 Second Printing 2002 ISSN 0172-5998 ISBN 3-540-17093-6 2nd Edition Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.spfinger.de Springer-Verlag Berlin Heidelberg 1979, Germany 1987 Printed in The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the specific statement, that such names are exempt from the relevant probreak tective laws and regulations and free for general use absence of therefore Cover a design: design Printed on & production GmbH, Heidelberg acid-free paper SPIN 10885981 55/3141/ba www.pdfgrip.com Preface to the Second In this second printing Printing of the Second of the second edition several minor and matical mistake have been corrected We Duedahl and Reinhard Schaflitzel for are pointing one Edition major mathe- indebted to Roberto Conti, Sindre these out Canberra and Trondheim, 2002 Ola Bratteli Derek W Robinson Preface to the Second Edition The second edition of this book differs from the original in three respects First, 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 and 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 the only substantial changes are to Sections 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 sections now incorporate the result All these changes are nevertheless revisionary in nature and 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*-dynamical systems The current state of this theory is summarized in [[Bra 1]] published in Springer-Verlag's Lecture Notes in we have eliminated a large number of Mathematics series Canberra and Trondheim, 1986 Ola Bratteli Derek W Robinson v www.pdfgrip.com www.pdfgrip.com Preface to the First Edition In this book describe the elementary theory of operator algebras and 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 development 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 of field 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 honeymoon 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 abelianness 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 we parts of the advanced Vil www.pdfgrip.com viii Preface to the First Edition collaboration between mathematicians and intertwining of the theory forced the formation of the book has a physicists They also led to an applications applications often in Thus this the division of this context theory and in which the certain arbitrariness The two volumes of the book contain six and two in the second The chapters, four in this first volume of the second volume are numbered chapters consecutively with those of the first and the references are cumulative Chapter I 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 mainstreams 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 monographs [[Dav 1]] [[Eva 1]] This book was written between September 1976 and July t979 Most of Chapters 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 of this period Bratteli was supported by the Norwegian Research Council for Science and Humanities and during the complementary period by a post of Professeur Associ6 at Luminy Chapter was partially written at the University of New South Wales and partially in Marseille and at the University of Oslo Chapters 2, 3, and half of Chapter were typed at the Centre de Physique Th6orique, 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 Andr6 Verbeure for helpful " " comments and corrections to earlier versions We are particularly indebted to Adam Majewski manuscript and locating numerous errors Oslo and Sydney, 1979 for reading the final Ola Bratteli Derek W Robinson www.pdfgrip.com Contents (Volume 1) Introduction Notes and Remarks C*-Algebras and 16 von Neumann Algebras C* -Algebras 19 1 Basic Definitions and Structure 19 2.2 Functional and Spectral Analysis 2.2 Resolvents, Spectra, and 25 Radius 25 Quotient Algebras 39 Spectral 2.2.2 Positive Elements 2.2.3 2.3 17 Approximate Representations 2.3 32 Identities and and States 42 Representations 42 2.3.2 States 48 2.3.3 Construction of 54 2.3.4 2.3.5 Representations Existence of Representations Commutative C*-Algebras 58 61 ix www.pdfgrip.com ... A < 11 A 11 1 and hence :!! B :!! 11 A 11 T But this implies that JIB 11 :! 11 All by a second application of the same formula 11 A 11 1/2) g [ 11 A 11 /2, 11 A 11 /2] and hence u((A 11 A 11 1/2)2)... 12 11 (J) 11 A) + CO (0 `: identity, 11 611 11 (0 11 ) + + + = (o (,)(A*A) > 6(T) (A) + 11 0 )11 by Pro- = 0) (A), and 0 )11 1 which yields + 11 C0 211 the last statement of the = 11 COl + (0 211 , corollary... that o) >_ A(o, and hence Aw, yaj for some < y < I by purity But I 11 0 )11 4( 011 1 + (1 *01W2 11 and one must have 11 wj 11 11 (0 211 Therefore A y and (o w, Similarly, (.o= (02 and hence (,o is