EncyclopaediaofMathematicalSciences Volume122 OperatorAlgebrasandNon-Commutative GeometryIII SubseriesEditors: JoachimCuntz VaughanF.R.Jones B. Blackadar Operator Algebras ∗ Theory of C -Algebras and von Neumann Algebras ABC BruceBlackadar DepartmentofMathematics UniversityofNevada,Reno Reno,NV89557 USA e-mail:[email protected] FoundingeditoroftheEncyclopediaofMathematicalSciences: R.V.Gamkrelidze LibraryofCongressControlNumber:2005934456 MathematicsSubjectClassification(2000): 46L05,46L06,46L07,46L08,46L09,46L10,46L30,46L35,46L40,46L45,46L51, 46L55,46L80,46L85,46L87,46L89,19K14,19K33 ISSN0938-0396 ISBN-10 3-540-28486-9SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-28486-4SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinGermany Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorandTechBooksusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11543510 46/TechBooks 543210 to my parents and in memory of Gert K. Pedersen 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 topologyifandonlyifitisequaltothecommutantofitscommutant.Afactor isavonNeumannalgebrawithtrivialcentreandtheworkofMurrayandvon Neumann contained a reduction of all von Neumann algebras to factors and a classification of factors into types I, II 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 operatoralgebrashas evolvedinto a huge math- ematicalendeavourinteracting withalmosteverybranchofmathematics 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 dimensionalC∗-algebrasare easilyseento be just directsums ofmatrixalgebras.However,bytakingalgebraswhichareclosuresinnormof finitedimensionalalgebrasoneobtainsalreadyarichclassofC∗-algebras–the so-calledAF-algebras–whicharenotoftypeI.Theideaoftakingtheclosure ofaninductivelimitoffinite-dimensionalalgebrashadalreadyappearedinthe workofMurray-vonNeumannwhousedittoconstructafundamentalexample of a factor oftype II – the ”hyperfinite” (nowadaysalsocalledapproximately finite dimensional) factor. VIII Preface to the Subseries One key to an understanding of the class of AF-algebras turned out to be K-theory. 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 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 gener- ated by isometries with certain relations, now known as the algebras On. At thesametime,examplesofalgebraswereincreasinglystudiedthatcodifydata from differential geometry or from topological dynamical systems. Onthe other hand, a little earlier in the seventies,the theory of von Neu- mann 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 usedismathematicalphysics,especiallyinquantumstatisticalmechanicsand inthefoundationsofquantumfieldtheory.VonNeumannexplicitlymentioned 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 proposedby Haag andKastler.In this theory a vonNeumann algebra is associated with each region of space-time, obeying certain axioms. The inductivelimitofthesevonNeumannalgebrasisaC∗-algebrawhichcontains 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 re- view of the classification theorems for von Neumann algebras and the suc- cess 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 preciseC∗-frameworkwhichwouldleadtoindextheoremsforfoliationsincor- porating techniques and ideas from many branches of mathematics hitherto unconnected with operator algebras. ManyofthenewdevelopmentsbeganinthedecadefollowingtheKingston meeting. On the C∗-side was Kasparov’s KK-theory – the bivariant form of K-theoryforwhichoperatoralgebraicmethodsareabsolutelyessential.Cyclic cohomologywasdiscoveredthroughananalysisofthe finestructureofexten- sionsofC∗-algebrasTheseideasandmanyotherswereintegratedintoConnes’ vast Noncommutative Geometry program. In cyclic theory and in connection withmanyotheraspects ofnoncommutativegeometry,the needforgoingbe- yond 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 nowa welldevelopedbivariantK-theoryandcyclic theoryfor a naturalclass of topological algebras as well as a bivariant character taking K-theory to Preface to the Subseries IX cyclictheory.The 1990’salsosawhugeprogressinthe classificationtheoryof 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 withthedefinitionoftheindex ofasubfactorintermsoftheMurray-vonNeu- mann theory and a result showing that the index was surprisingly restricted in its possible 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 aboveturnedouttohavefascinatinglinkstosubfactortheory.Thesubfactors themselves were constructed in the representation theory of loop groups. Beginning in the early 1980’sVoiculescu initiated the theory of free prob- ability and showed how to understand the free group von Neumann algebras interms ofrandommatrices,leading to the extraordinaryresultthat the von Neumann algebra M of the free group on infinitelymany generators has full fundamental group, i.e. pMp is isomorphic to M for every non-zero projec- tion p ∈ 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. Manyofthetopicsmentionedinthe(obviouslyincomplete)listabovehave become largeindustriesintheir ownright.Soit isclearthataconferencelike theoneinKingstonisnolongerpossible.Neverthelessthesubjectdoesretain 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 techni- cal results of C∗-algebra theory and von Neumann algebra theory including sections on noncommutative dynamical systems, entropy and derivations. It willinclude anaccountofK-theoryandbivariantK-theorywithapplications 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 shallbe dedicatedto the connections between oper- ator algebras and quantum field theory. October 2001 subseries editors: Joachim Cuntz Vaughan Jones Preface This volume attempts to give a comprehensive discussion of the theory of operator algebras (C*-algebras and von Neumann algebras.) The volume is intendedtoservetwopurposes:torecordthestandardtheoryintheEncyclo- pediaofMathematics,andtoserveasanintroductionandstandardreference for the specialized volumes in the series on current research topics in the subject. Since there are already numerous excellent treatises on various aspects of thesubject,howdoesthisvolumemakeasignificantadditiontotheliterature, and how does it differ from the other books in the subject? In short, why another book on operator algebras? The answer lies partly in the first paragraph above. More importantly, no other single reference covers all or even almost all of the material in this volume. I have tried to cover all of the main aspects of “standard” or “classi- cal” operator algebra theory; the goal has been to be, well, encyclopedic. Of course, in a subject as vast as this one, authors must make highly subjective judgments as to what to include and what to omit, as well as what level of detail to include, and I have been guided as much by my own interests and prejudices as by the needs of the authors of the more specialized volumes. A treatment of such a large body of material cannot be done at the detail level of a textbook in a reasonably-sized work, and this volume would not be suitableasatextandcertainlydoesnotreplacethemoredetailedtreatments of the subject. But neither is this volume simply a survey of the subject (a finesurvey-levelbookisalreadyavailable[Fil96].)Myphilosophyhasbeento not only state what is true, but explain why: while many proofs are merely outlined or even omitted, I have attempted to include enough detail and ex- planation to at least make all results plausible and to give the reader a sense ofwhatmaterialandlevelofdifficultyisinvolvedineachresult.Whereanar- gument can be given or summarized in just a few lines, it is usually included; longer arguments are usually omitted or only outlined. More detail has been included where results are particularly important or frequently used in the sequel, where the results or proofs are not found in standard references, and XII Preface inthefewcaseswherenewargumentshavebeenfound.Nonetheless,through- outthe volume the reader should expectto have tofill out compactly written arguments, or consult references giving expanded expositions. Ihaveconcentratedontryingtogiveacleanandefficientexpositionofthe details of the theory, and have for the most part avoided general discussions of the nature of the subject, its importance, and its connections and applica- tions in other parts of mathematics (and physics); these matters have been amply treated in the introductory article to this series. See the introduction to [Con94] for another excellent overview of the subject of operator algebras. Thereisverylittleinthisvolumethatistrulynew,mainlysomesimplified proofs. I have tried to combine the best features of existing expositions and arguments, with a few modifications of my own here and there. In preparing thisvolume,Ihavehadthepleasureofrepeatedlyreflectingontheoutstanding talents of the many mathematicians who have brought this subject to its presentadvancedstate,andthetheorypresentedhereisamonumenttotheir collective skills and efforts. Besides the unwitting assistance of the numerous authors who originally developed the theory and gave previous expositions, I have benefited from comments, suggestions, and technical assistance from a number of other spe- cialists, notably G. Pedersen, N. C. Phillips, and S. Echterhoff, my colleagues B.-J. Kahng, V. Deaconu, and A. Kumjian, and many others who set me straight on various points either in person or by email. I am especially grate- ful to Marc Rieffel, who, in addition to giving me detailed comments on the manuscript, wrote an entire draft of Section II.10 on group C*-algebras and crossed products. Although I heavily modified his draft to bring it in line with the rest of the manuscript stylistically, elements of Marc’s vision and refreshing writing style still show through. Of course, any errors or misstate- ments in the final version of this (and every other) section are entirely my responsibility. Speaking of errors and misstatements, as far-fetched as the possibility mayseemthatanystillremainafteralltheonesIhavealreadyfixed,Iwould appreciate hearing about them from readers, and I plan to post whatever I find out about on my website. No book can start from scratch, and this book presupposes a level of knowledge roughly equivalent to a standard graduate course in functional analysis (plus its usual prerequisites in analysis, topology, and algebra.) In particular,thereaderisassumedtoknowsuchstandardtheoremsastheHahn- Banach Theorem, the Krein-Milman Theorem, and the Riesz Representation Theorem (the Open Mapping Theorem, the Closed Graph Theorem, and the Uniform Boundedness Principle also fall into this category but are explicitly statedinthetextsincedtheyaremoredirectlyconnectedwithoperatortheory and operator algebras.) Most of the likely readers will have this background, orfarmore,andindeeditwouldbedifficulttounderstandandappreciatethe material without this much knowledge. Beginning with a quick treatment of the basics of Hilbert space and operator theory would seem to be the proper Preface XIII pointofdepartureforabookofthissort,andtheearlysectionswillbeuseful even to specialists to set the stage for the work and establish notation and terminology. September 2005 Bruce Blackadar