Operator algebras: theory of C*-algebras and von Neumann algebras

529 Pages·2006·4.03 MB·English

by  Bruce Blackadar

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
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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