Encyclopaedia of Mathematical Sciences Volume 125

Operator Algebras and Non-Commutative Geometry

Subseries Editors: Joachim Cuntz Vaughan F. R. Jones

Springer-Verlag Berlin Heidelberg GmbH

M. Takesaki

Theory of Operator Algebras II

, Springer

Author

Masamichi Takesaki University of California

Department of Mathematics

Los Angeles, CA 90095-1555 USA

e-mail: mt@math.ucla.edu

Founding editor of the Encyclopaedia of Mathematical Sciences: R. V. Gamkrelidze

Mathematics Subject Classification (2000): 22D25, 46LXX, 47CXX, 47DXX

Theory of Operator Algebras 1 by M. Takesaki was published as VoI. 124 ofthe Encyclopaedia of Mathematical Sciences, ISBN 3-540-42248-X,

Theory of Operator Algebras III by M. Takesaki was published as VoI. 127 of the Encyclopaedia of Mathematical Sciences, ISBN 3-540-42913-1

ISSN 0938-0396

ISBN 978-3-642-07689-3 ISBN 978-3-662-10451-4 (eBook) DOI 10.1007/978-3-662-10451-4

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Preface to the Encyclopaedia Subseries

on Operator Algebras and Non-Commutative Geometry

The theory of von Neumann algebras was initiated in a series of papers by Murray and von Neumann in the 1930's and 1940's. A von Neumann algebra is a self-adjoint unital subalgebra M of the algebra of bounded operators of a Hilbert space which is closed in the weak operator topology. According to von Neumann's bicommutant theorem, M is closed in the weak operator topology if and only if it is equal to the commutant of its commutant. A factor is a von Neumann algebra with trivial centre and the work of Murray and von Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, IT and III.

C* -algebras are self-adjoint operator algebras on Hilbert space which are closed in the norm topology. Their study was begun in the work of Gelfand and Naimark who showed that such algebras can be characterized abstractly as involutive Banach algebras, satisfying an algebraic relation connecting the norm and the involution. They also obtained the fundamental result that a commutative unital C* -algebra is isomorphic to the algebra of complex valued continuous functions on a compact space - its spectrum.

Since then the subject of operator algebras has evolved into a huge mathematical endeavour interacting with almost every branch of mathematics and several areas of theoretical physics.

Up into the sixties much of the work on C* -algebras was centered around rep­resentation theory and the study of C* -algebras of type I (these algebras are char­acterized by the fact that they have a well behaved representation theory). Finite dimensional C* -algebras are easily seen to be just direct sums of matrix algebras. However, by taking algebras which are closures in norm of finite dimensional al­gebras one obtains already a rich class of C* -algebras - the so-called AF-algebras - which are not of type I. The idea of taking the closure of an inductive limit of finite-dimensional algebras had already appeared in the work of Murray-von Neu­mann who used it to construct a fundamental example of a factor of type II - the "hyperfinite" (nowadays also called approximately finite dimensional) factor.

One key to an understanding of the class of AF-algebras turned out to be K­theory. The techniques of K -theory, along with its dual, Ext-theory, also found im­mediate applications in the study of many new examples of C* -algebras that arose

VI Preface to the Subseries

in the end of the seventies. These examples include for instance "the noncommuta­tive tori" or other crossed products of abelian C* -algebras by groups of homeomor­phisms and abstract C* -algebras generated by isometries with certain relations, now known as the algebras (!In' At the same time, examples of algebras were increasingly studied that codify data from differential geometry or from topological dynamical systems.

On the other hand, a little earlier in the seventies, the theory of von Neumann algebras underwent a vigorous growth after the discovery of a natural infinite family of pairwise nonisomorphic factors of type III and the advent of Tomita-Takesaki theory. This development culminated in Connes' great classification theorems for approximately finite dimensional ("injective") von Neumann algebras.

Perhaps the most significant area in which operator algebras have been used is mathematical physics, especially in quantum statistical mechanics and in the foun­dations of quantum field theory. Von Neumann explicitly mentioned quantum the­ory as one of his motivations for developing the theory of rings of operators and his foresight was confirmed in the algebraic quantum field theory proposed by Haag and Kastler. In this theory a von Neumann algebra is associated with each region of space-time, obeying certain axioms. The inductive limit of these von Neumann algebras is a C* -algebra which contains a lot of information on the quantum field theory in question. This point of view was particularly successful in the analysis of superselection sectors.

In 1980 the subject of operator algebras was entirely covered in a single big three weeks meeting in Kingston Ontario. This meeting served as a review of the classification theorems for von Neumann algebras and the success of K -theory as a tool in C* -algebras. But the meeting also contained a preview of what was to be an explosive growth in the field. The study of the von Neumann algebra of a foliation was being developed in the far more precise C* -framework which would lead to index theorems for foliations incorporating techniques and ideas from many branches of mathematics hitherto unconnected with operator algebras.

Many of the new developments began in the decade following the Kingston meeting. On the C* -side was Kasparov's K K -theory - the bivariant form of K­theory for which operator algebraic methods are absolutely essential. Cyclic coho­mology was discovered through an analysis of the fine structure of extensions of C*-algebras These ideas and many others were integrated into Connes' vast Non­commutative Geometry program. In cyclic theory and in connection with many other aspects of noncommutative geometry, the need for going beyond the class of C*­algebras became apparent. Thanks to recent progress, both on the cyclic homol­ogy side as well as on the K -theory side, there is now a well developed bivariant K -theory and cyclic theory for a natural class of topological algebras as well as a bivariant character taking K-theory to cyclic theory. The 1990's also saw huge progress in the classification theory of nuclear C* -algebras in terms of K -theoretic invariants, based on new insight into the structure of exact C* -algebras.

On the von Neumann algebra side, the study of subfactors began in 1982 with the definition of the index of a subfactor in terms of the Murray-von Neumann the­ory and a result showing that the index was surprisingly restricted in its possible

Preface to the Subseries VII

values. A rich theory was developed refining and clarifying the index. Surprising connections with knot theory, statistical mechanics and quantum field theory have been found. The superselection theory mentioned above turned out to have fasci­nating links to subfactor theory. The subfactors themselves were constructed in the representation theory of loop groups.

Beginning in the early 1980's Voiculescu initiated the theory of free probability and showed how to understand the free group von Neumann algebras in terms of random matrices, leading to the extraordinary result that the von Neumann algebra M of the free group on infinitely many generators has full fundamental group, i.e. pMp is isomorphic to M for every non-zero projection p E M. The subsequent in­troduction of free entropy led to the solution of more old problems in von Neumann algebras such as the lack of a Cartan subalgebra in the free group von Neumann algebras.

Many of the topics mentioned in the (obviously incomplete) list above have become large industries in their own right. So it is clear that a conference like the one in Kingston is no longer possible. Nevertheless the subject does retain a certain unity and sense of identity so we felt it appropriate and useful to create a series of encylopaedia volumes documenting the fundamentals of the theory and defining the current state of the subject.

In particular, our series will include volumes treating the essential technical re­sults of C* -algebra theory and von Neumann algebra theory including sections on noncommutative dynamical systems, entropy and derivations. It will include an ac­count of K -theory and bivariant K -theory with applications and in particular the index theorem for foliations. Another volume will be devoted to cyclic homology and bivariant K -theory for topological algebras with applications to index theorems. On the von Neumann algebra side, we plan volumes on the structure of subfactors and on free probability and free entropy. Another volume shall be dedicated to the connections between operator algebras and quantum field theory.

October 2001 subseries editors: Joachim Cuntz Vaughan Jones

In loving memory of our daughter

Yuki

whose childhood was greatly influenced by events and developments in Operator Algebras.

Contents of Theory of Operator Algebras I, II and III

Theory of Operator Algebras I

Introduction

Chapter I. Chapter II. Chapter III. Chapter IV. ChapterV.

Appendix. Bibliography Notation Index Subject Index

Fundamentals of Banach Algebras and C* -Algebras Topologies and Density Theorems in Operator Algebras Conjugate Spaces Tensor Products of Operator Algebras and Direct Integrals Types of von Neumann Algebras and Traces

Polish Spaces and Standard Borel Spaces

Theory of Operator Algebras II

Preface

Chapter VI. Chapter VII. Chapter VIII. Chapter IX. ChapterX. Chapter XI. Chapter XII.

Appendix Bibliography Notation Index Subject Index

Left Hilbert Algebras Weights Modular Automorphism Groups Non-Commutative Integration Crossed Products and Duality Abelian Automorphism Group Structure of a von Neumann Algebra of Type III

XII Contents of Volumes I, II and III

Theory of Operator Algebras III

Preface

Chapter XIII. Ergodic Transformation Groups and the Associated von Neumann Algebras

Chapter XN. Approximately Finite Dimensional von Neumann Algebras Chapter XV. Nuclear C* -Algebras Chapter XVI. Injective von Neumann Algebras Chapter XVII. Non-Commutative Ergodic Theory Chapter XVIII. Structure of Approximately Finite Dimensional Factors Chapter XIX. Subfactors of an Approximately Finite Dimensional

Appendix Bibliography Notation Index Subject Index

Factor of Type III

Contents Theory of Operator Algebras II

Preface XIX

Chapter VI

Left Hilbert Algebras 1

§ 0 Introduction.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 § 1 Left Hilbert Algebras and Right Hilbert Algebras . . . . . . . . . . . . . . . . . . . . 2 § 2 Tomita Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22 § 3 Direct Integral of Left Hilbert Algebras and Tomita Algebras . . . . . . . . .. 28

ChapterVll

Weights 40

§ 0 Introduction... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 § 1 Weights and Semi-Cyclic Representations. . . . . . . . . . . . . . . . . . . . . . . . .. 40 § 2 Left Hilbert Algebras and Weights ................................ 58 § 3 The Plancherel Weight and the Fourier Algebra. . . . . . . . . . . . . . . . . . . . .. 65 §4 Weights on a C*-Algebra ........................................ 88

ChapterVllI

Modular Automorphism Groups 91

§ 0 Introduction................................................... 91 § 1 Modular Automorphism Group of a Weight. . . . . . . . . . . . . . . . . . . . . . . .. 92 § 2 The Centralizer of a Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 § 3 The Connes Cocycle Derivative .................................. 106 § 4 Tensor Product and Direct Integrals of Weights . . . . . . . . . . . . . . . . . . . . . . 133

Chapter IX

Non-Commutative Integration 141

§ 0 Introduction....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 § 1 Standard Form of a von Neumann Algebra .......................... 142 § 2 Measurable Operators and Integral for a Trace ....................... 167

XIV Contents

§ 3 Bimodules, Spatial Derivatives and Relative Tensor Products .......... 186 § 4 Conditional Expectations and Operator Valued Weights ............... 210

Chapter X

Crossed Products and Duality 237

§ 0 Introduction ................................................... 237 § 1 Crossed Products and Dual Weights ............................... 238 § 2 Duality for Crossed Products by Abelian Groups .................... 257 § 3 Equivariant Disintegration ....................................... 279 § 4 Induced Covariant System and Crossed Product ..................... 290

Chapter XI

Abelian Automorphism Group 311

§ 0 Introduction ................................................... 311 § 1 Spectral Analysis ............................................... 312 § 2 Connes Spectrum r(a) ................. ......................... 332 § 3 Derivations and Inner Automorphisms ............................. 352

Chapter XII

Structure of a von Neumann Algebra of Type III 363

§ 0 Introduction ................................................... 363 § 1 Structure of a von Neumann Algebra of Type III, Part I ............... 364 § 2 Structure of Factors of Type Ill ... , 0 < A < 1 ........................ 380 § 3 Structure of Factors of Type lIIo .................................. 384 § 4 The Flow of Weights ............................................ 403 § 5 Action of Int(M) on 2I10 and 6* .................................. 421 § 6 Structure of a von Neumann algebra of Type III,

Part II - Functoriality and the Characteristic Square .................. 437

Appendix 463

Bibliography 491

Notation Index 509

Subject Index 513

Contents Theory of Operator Algebras I

Introduction

Chapter I

Fundamentals of Banach Algebras and C* -Algebras

§ 0 Introduction § 1 Banach Algebras § 2 Spectrum and Functional Calculus § 3 Gelfand Representation of Abelian Banach Algebras § 4 Spectrum and Functional Calculus in C* -Algebras § 5 Continuity of Homomorphisms § 6 Positive Cones of C* -Algebras § 7 Approximate Identities in C* -Algebras § 8 Quotient Algebras of C*-Algebras § 9 Representations and Positive Linear Functionals § 10 Extreme Points of the Unit Ball of a C* -Algebra § 11 Finite Dimensional C* -Algebras

Chapter II

Topologies and Density Theorems in Operator Algebras

§ 0 Introduction § 1 Banach Spaces of Operators on a Hilbert Space § 2 Locally Convex Topologies in £(S) § 3 The Double Commutation Theorem of J. von Neumann § 4 Density Theorems

Chapterffi

Conjugate Spaces

§ 0 Introduction § 1 Abelian Operator Algebras § 2 The Universal Enveloping von Neumann Algebra of a C* -Algebra § 3 W* -Algebras

XVI Contents of Volume I

§ 4 The Polar Decomposition and the Absolute Value of Functionals § 5 Topological Properties of the Conjugate Space § 6 Semicontinuity in the Universal Enveloping von Neumann Algebra*

Chapter IV

Tensor Products of Operator Algebras and Direct Integrals

§ 0 Introduction § 1 Tensor Products of Hilbert Spaces and Operators § 2 Tensor Products of Banach Spaces § 3 Completely Positive Maps § 4 Tensor Products of C* -Algebras § 5 Tensor Products of W*-Algebras § 6 Integral Representations of States § 7 Representation of L2(r, JL) ®.fj, L 1(r, JL) ®y .M*, and L(r, JL) ®.M § 8 Direct Integral of Hilbert Spaces, Representations,

and von Neumann Algebras

Chapter V

Types of von Neumann Algebras and Traces

§ 0 Introduction § 1 Projections and Types of von Neumann Algebras § 2 Traces on a von Neumann Algebra § 3 Multiplicity of a von Neumann Algebra on a Hilbert Space § 4 Ergodic Type Theorem for von Neumann Algebras* § 5 Normality of Separable Representations* § 6 The Borel Space of von Neumann Algebras § 7 Construction of Factors of Type IT and Type m

Appendix. Polish Spaces and Standard Borel Spaces

Bibliography

Notation Index

Subject Index

Contents Theory of Operator Algebras III

Preface

Chapter XIII

Ergodic Transformation Groups and the Associated von Neumann Algebras

§ 0 Introduction § I Factors Associated with Ergodic Transformation Groups § 2 Krieger's Construction and Orbit Structure § 3 Approximately Finite Measured Groupoids § 4 Amenable Groups and Groupoids

Chapter XIV

Approximately Finite Dimensional von Neumann Algebras

§ 0 Introduction § I Inductive Limit and Infinite Tensor Products § 2 Uniqueness of Approximately Finite Dimensional Factors of Type III § 3 The Group von Neumann Algebras of Free Groups § 4 Strongly Stable Factors § 5 Maximal Abelian Subalgebras

Chapter XV

Nuclear C* -Algebras

§ 0 Introduction § I Completely Positive Approximation and Nuclear C* -Algebras § 2 Completely Positive Lifting § 3 Nuclear C* -Algebras and Injective von Neumann Algebras § 4 Grothendieck-Haagerup-Pisier Inequality

XVIII Contents of Volume III

Chapter XVI

Injective von Neumann Algebras

§ 0 Introduction § 1 Equivalence of Injectivity and Approximately Finite Dimensionality § 2 Finite Injective von Neumann Algebras (Second Approach)

Chapter XVII

Non-Commutative Ergodic Theory

§ 0 Introduction § 1 Non-Commutative Rokhlin Type Theorem § 2 Stability of Outer Conjugacy Classes § 3 Outer Conjugacy of Approximately Inner Automorphisms

of Strongly Stable Factors

Chapter XVIII

Structure of Approximately Finite Dimensional Factors

§ 0 Introduction § 1 AFD Factors of Type III}.. 0 < A < 1 § 2 The Flow of Weights and AFD Factors of Type IIIo § 3 Asymptotic Centralizer § 4 AFD Factors of Type III 1

Chapter XIX

Subfactors of an Approximately Finite Dimensional Factor of Type III

§ 0 Introduction § 1 AF-Algebras § 2 Index of Subfactors § 3 Construction of Subfactors § 4 Classification of Subfactors of Approximately Finite Dimensional

Factors of Type III with Finite Index and Depth

Appendix

Bibliography

Notation Index

Subject Index

Preface

The author believes that the theory of operator algebras should be viewed as a num­ber theory in analysis. Number theory has been attracting the interest of humans ever since civilization began. Every culture in the world throughout history has given special meanings to certain numbers.

For example, a number may represent a position, quantity and/or qUality. To­day's civilization would be just impossible without numbers. People have been at­tracted to the mysteries of numbers throughout history. Accordingly, number theory is the oldest and most developed area of mathematics. Throughout the mathemat­ical path to the present day, people have gradually learned properties of numbers. It is surprising to find that the number zero was not recognized until Hindus found it about one thousand years ago (although it is recognized that Mayans found it as well). Compared to this old field of mathematics, the theory of operator algebras is very new; its foundation was given by the pioneering work of J. von Neumann and his collaborator F. J. Murray in the early part of the twentieth century, i.e. in the thirties. Subsequent major development occurred only a decade later in the late forties and the early fifties. But since then it has marked steady progress reaching new heights today. The theory handles self-adjoint algebras of bounded operators on a Hilbert space. The advent of quantum physics at the turn of century forced one to consider non-commutative variables. One needed to broaden the concept of numbers. Integers, rational numbers, real numbers and complex numbers are all commutative. Among the few noncommutative mathematical systems available at the beginning of quantum mechanics were matrix algebras, which did not accom­modate the needs of quantum physics because the Heisenberg uncertainty principle and/or Heisenberg commutation relation do not allow one to stay in the realm of finite matrices. One needs to consider algebras of operators on a Hilbert space of infinite dimension. Some of these operators correspond to important physical quan­tities. One has to include operators in the list of "numbers". Number theory tells us to put numbers in a field to study them more efficiently. Similarly, the theory of oper­ator algebras puts operators of interest in an algebra and we study the algebra and its structure first. The infinite dimensionality of the underlying Hilbert space poses big challenges and also presents interesting new phenomenon which do not occur in the classical frame work. We have already seen some of them in the first volume. For ex­ample, the continuity of dimensions in a factor of type III is one of them. The infinite dimensionality of our objects forces us to create sophisticated methods to handle ap­proximations. Simple minded counting does not lead to the heart of the matter. For

xx Preface

ex~ple, it is impossible to introduce a simple minded coordinate system in an in­finite dimensional operator algebra, thus mathematical induction based on a basis does not fly. The early part of the theory, in the period of the forties through the early sixties were spent on this issue. Luckily there is a remarkable similarity be­tween the theory of measures on a locally compact space and the theory of operator algebras. The first volume was devoted to the pursuit of this similarity.

The second volume of "Theory of Operator Algebras" is devoted to the study of the structure of von Neumann algebras of type ill and their automorphism groups, cf. Chapter VI through Chapter XII; and the third volume is devoted to the study of the fine structure analysis of approximately finite dimensional factors and their automorphism groups, cf. Chapter XIII through Chapter XVill. The last chapter, Chapter XIX, is an introduction to the theory of subfactors and their symmetries. One should note that the class of von Neumann algebras of type ill is given by exclusion, i.e., by the absence of a non-trivial trace or a non-zero finite projection. This situation presented the major obstruction for the study of von Neumann alge­bras from the beginning of the subject until the advent of Tomita-Takesaki theory in the late sixties whilst many examples had been found to be of type ill: the infinity of non-isomorphic factors were first established for factors of type ill by Powers in 1967, [670], before the discovery of infinitely many non-isomorphic factors of type III or 1100 , [635,686], and most examples from quantum physics were shown to be of type ill, [430]. It was the Tomita-Takesaki theory which broke the ice. It is still amazing that the subject defined by exclusion admits such a fine structural anal­ysis since usually exclusion does not allow one to find any alternative and is viewed as pathological. Of course, a von Neumann algebra of type ill had been pathological until we discover their fine structure. We will explore this in full detail through the second volume.

Each chapter has its own introduction which describes the content of that chapter and the basic strategy so that the reader can get a quick overview of the chapter.

In the second and third volume, we present two major items in the theory of von Neumann algebras: one is the analogy with integration theory on an abstract mea­sure space and the other is the emphatic importance of automorphisms of algebras, i.e. we emphasize the symmetries of our objects following the modem point of view of E. Galois.

In general, the theory of von Neumann algebras is considered to be non­commutative integration. In Volume I, the similarity between von Neumann algebras and measure spaces are examined from the point of view of Banach space duality. In the second and third volume, non-commutative integration goes far beyond the analogy with ordinary integration, Since it is not our main interest to examine how ordinary integration should be formulated based on commutative von Neumann al­gebras, it is not discussed here in detail beyond a few comments. Still it is possible to develop a theory which covers the ordinary integration theory based on the oper­ator algebra approach. In fact; such a theory has been explored by G. K. Pedersen, [653, Chapter 6], and it does eliminate pathological uninteresting measure spaces easily. The main difference between the operator algebra approach and the con­ventional approach to integration theory relies on the fact that in operator algebras

Preface XXI

one considers functions first, or equivalently variables, and then one views the un­derlying points as the spectrum of the variables; whilst in the ordinary approach one considers points first and views variables as functions on the set of points. We would like to point out here, however, that in practice we never observe points di­rectly only approximately by successive evaluations of coordinates. Besides this philosophical difference, there is another major difference between the ordinary in­tegration theory and the non-commutative integration theory which rests on the fact that a weight, a non-commutative counterpart of a a-finite measure, gives rise to a one-parameter automorphism group, called the modular automorphism group, of the von Neumann algebra in question. This modular automorphism group can be considered as the time evolution of the system, i.e., in the non-commutative world a state determines the associated dynamics. The appearance of the modular automor­phism group distinguishes our theory sharply from the classical theory. The modular automorphism group gives us abundant non-trivial information precisely when there is no trace on the algebra in question. Since the ordinary integration is a trace, the modular automorphism group is trivial in that case and cannot be appreciated. Fur­thermore, thanks to the Connes cocycle derivative theorem, Theorem VllI.3.3, the modular automorphism group is unique up to perturbation by a one unitary cocy­cle, which allows us to relate the structure of a von Neumann algebra of type ill to that of the associated von Neumann algebra of type IToo equipped with a trace scaling one parameter automorphism group, cf. Chapter XII. As a byproduct of our non-commutative integration theory, a duality theorem attributed to Pontrjagin, van Kampen, Tannaka, Stinespring, Eymard, Saito and Tatsuuma, is presented in §3, Chapter VII. With this exception, no discussion of examples is presented in the sec­ond volume, Chapter VI through Chapter XIT. Extensive discussions of examples and constructions of factors occupy the third volume starting in Chapter Xill and through Chapter XVITI.

The so-called Murray-von Neumann measure space construction of factors is closely investigated first in Chapter Xill yielding the Krieger construction of fac­tors and the theory of measured groupoids. Systematic study of approximately finite dimensional factors occupies most of the third volume, cf. Chapter XIV through Chapter XIX. The theory is highlighted by the celebrated classification theorem of Alain Connes in the form of Theorems XVI.1.9, XVill.l.1, XVill.2.1 to which W. Krieger made a substantial contribution also, and XVill.4.16 which requires one full section of preparation given by U. Haagerup, [550]. The last chapter, Chap­ter XIX, is devoted to an introduction to the theory of subfactors of an AFD factor created by V. F. R. Jones, and concludes with a classification theorem of Popa, The­orem XIX.4.16, for subfactors of an AFD factor of type II 1 with small indices.

The three volume book, "Theory of Operator Algebras", is a product of the au­thor's research and teaching activities at the Department of Mathematics at Univer­sity of California, Los Angeles, spanning the years from 1969 through the present time. It is important to mention the following: the author's visit to the University of Pennsylvania from 1968 through 1969 where the foundation of Tomita-Takesaki theory was established; the author's participation in various research activities which include several short and long visits to the University of Marseille-Aix-Luminy;

XXII Preface

several short visits to RIMS of Kyoto University; one year participation in the Math­ematical Physics Project of 1975-1976 at ZiP, University of Bielefeld; a full year participation in the operator algebra project of MSRI for 1984-1985; a one year visit to IHES, 1988-1989; two one month long participations in the one year project (1988-1989) on operator algebras at the Mittag-Leffler Institute; several visits to the University of New South Wales; and several month long visits to the Mathematics Institute of University of Warwick. The author would like to express here his sincere gratitude to these institutions and to the mathematicians who hosted him warmly and worked with him. Special thanks are due to Professor Richard V. Kadison with whom the author discussed the philosophy of the subject at length so many times, and to Professor Daniel Kastler who encouraged him in many ways and provided the opportunity to work with him and others including Alain Connes. Throughout the period of the preparation of the book, the author has been continuously supported by the National Science Foundation. Here he would like to record his appreciation of that support. The Guggenheim Foundation also gave the author support at a critical period of his career, for which the author is very grateful. The author also would like to express his gratitude to Professor Masahlro Nakamura who has constantly given his moral support to the author, to Professor Takashi Turumaru whose beau­tifullectures inspired the author to be a functional analyst and to the late Professor Yoshinao Misonou under whose leadership the author started his career as a func­tional analyst. At the final stage of the preparation of the manuscript, Dr. Un Kit Hui and Dr. Toshihiko Masuda took pains to help the author to edit the manuscript. Al­though any misprints and mistakes are the author's responsibility, the author would like to thank them here.

Guidance to the Reader

Each chapter has its own introduction so that one can quickly get an overview of the content of the chapter. Theorems, Propositions, Lemmas and Definitions are numbered in one sequence, whilst formulas and equations are numbered in each section separately without reference to the section. Formulas (respectively, equa­tions) are referred to by the formula number (respectively, equation number) alone if it is quoted in the same section, and by the section number followed by the for­mula number if it is quoted in a different section but in the same chapter, and finally by the chapter number, the section number and the formula number (respectively, equation number) if it is quoted in a different chapter. Some exercises are selected to help the reader to get information and techniques not covered in the main text, so they can be viewed as a supplement to the text. Those exercises taken directly literatures are marked by a t -sign, and the references are cited there.

To keep the book within a reasonable size, this three volume book does not in­clude the materials related to the following important areas of operator algebras: K-theory for C* -algebras, geometric theory of operator algebras such as cyclic co­homology, the classification theory of nuclear C* -algebras, free probability theory and the advanced theory of subfactors. The interested readers are referred to the forthcoming books in this operator algebra series of encyclopedia.