Completely Bounded Maps and Operator Algebras

314 Pages·2002·1.518 MB·English

by  Vern Paulsen

This page intentionally left blank
CAMBRIDGE STUDIES IN
ADVANCED MATHEMATICS 78
EDITORIAL BOARD
B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK
CompletelyBoundedMapsandOperatorAlgebras
In this book the reader is provided with a tour of the principal results and
ideasinthetheoriesofcompletelypositivemaps,completelyboundedmaps,
dilationtheory,operatorspaces,andoperatoralgebras,togetherwithsomeof
theirmainapplications.
Theauthorassumesonlythatthereaderhasabasicbackgroundinfunctional
analysis and C∗-algebras, and the presentation is self-contained and paced
appropriately for graduate students new to the subject. The book could be
usedasatextforacourseorforindependentreading;withthisinmind,many
exercisesareincluded.Expertswillalsowantthisbookfortheirlibrary,since
theauthorpresentsnewandsimplerproofsofsomeofthemajorresultsinthe
area,andmanyapplicationsarealsoincluded.
Thiswillbeanindispensableintroductiontothetheoryofoperatorspaces
forallwhowanttoknowmore.
Alreadypublished
2 K.PetersenErgodictheory
3 P.T.JohnstoneStonespaces
5 J.-P.KahaneSomerandomseriesoffunctions,2ndedition
7 J.Lambek&P.J.ScottIntroductiontohigher-ordercategoricallogic
8 H.MatsumuraCommutativeringtheory
9 C.B.ThomasCharacteristicclassesandthecohomologyoffinitegroups
10 M.AschbacherFinitegrouptheory
11 J.L.AlperinLocalrepresentationtheory
12 P.KoosisThelogarithmicintegralI
14 S.J.PattersonAnintroductiontothetheoryoftheRiemannzeta-function
15 H.J.BauesAlgebraichomotopy
16 V.S.VaradarajanIntroductiontoharmonicanalysisonsemisimpleLiegroups
17 W.Dicks&M.DunwoodyGroupsactingongraphs
18 L.J.Corwin&F.P.GreenleafRepresentationsofnilpotentLiegroupsandtheirapplications
19 R.Fritsch&R.PiccininiCellularstructuresintopology
20 H.KlingenIntroductorylecturesonSiegelmodularforms
21 P.KoosisThelogarithmicintegralII
22 M.J.CollinsRepresentationsandcharactersoffinitegroups
24 H.KunitaStochasticflowsandstochasticdifferentialequations
25 P.WojtaszczykBanachspacesforanalysts
26 J.E.Gilbert&M.A.M.MurrayCliffordalgebrasandDiracoperatorsinharmonicanalysis
27 A.Frohlich&M.J.TaylorAlgebraicnumbertheory
28 K.Goebel&W.A.KirkTopicsinmetricfixedpointtheory
29 J.F.HumphreysReflectiongroupsandCoxetergroups
30 D.J.BensonRepresentationsandcohomologyI
31 D.J.BensonRepresentationsandcohomologyII
32 C.Allday&V.PuppeCohomologicalmethodsintransformationgroups
33 C.Souleetal.LecturesonArakelovgeometry
34 A.Ambrosetti&G.ProdiAprimerofnonlinearanalysis
35 J.Palis&F.TakensHyperbolicity,stabilityandchaosathomoclinicbifurcations
37 Y.MeyerWaveletsandoperators1
38 C.WeibelAnintroductiontohomologicalalgebra
39 W.Bruns&J.HerzogCohen-Macaulayrings
40 V.SnaithExplicitBrauerinduction
41 G.LaumonCohomologyofDrinfeldmodularvarietiesI
42 E.B.DaviesSpectraltheoryanddifferentialoperators
43 J.Diestel,H.Jarchow,&A.TongeAbsolutelysummingoperators
44 P.MattilaGeometryofsetsandmeasuresinEuclideanspaces
45 R.PinskyPositiveharmonicfunctionsanddiffusion
46 G.TenenbaumIntroductiontoanalyticandprobabilisticnumbertheory
47 C.PeskineAnalgebraicintroductiontocomplexprojectivegeometry
48 Y.Meyer&R.CoifmanWavelets
49 R.StanleyEnumerativecombinatoricsI
50 I.PorteousCliffordalgebrasandtheclassicalgroups
51 M.AudinSpinningtops
52 V.JurdjevicGeometriccontroltheory
53 H.VolkleinGroupsasGaloisgroups
54 J.LePotierLecturesonvectorbundles
55 D.BumpAutomorphicformsandrepresentations
56 G.LaumonCohomologyofDrinfeldmodularvarietiesII
57 D.M.Clark&B.A.DaveyNaturaldualitiesfortheworkingalgebraist
58 J.McClearyAuser’sguidetospectralsequencesII
59 P.TaylorPracticalfoundationsofmathematics
60 M.P.Brodmann&R.Y.SharpLocalcohomology
61 J.D.Dixonetal.Analyticpro-Pgroups
62 R.StanleyEnumerativecombinatoricsII
63 R.M.DudleyUniformcentrallimittheorems
64 J.Jost&X.Li-JostCalculusofvariations
65 A.J.Berrick&M.E.KeatingAnintroductiontoringsandmodules
66 S.MorosawaHolomorphicdynamics
67 A.J.Berrick&M.E.KeatingCategoriesandmoduleswithK-theoryinview
68 K.SatoLevyprocessesandinfinitelydivisibledistributions
69 H.HidaModularformsandGaloiscohomology
70 R.Iorio&V.IorioFourieranalysisandpartialdifferentialequations
71 R.BleiAnalysisinintegerandfractionaldimensions
72 F.Borceaux&G.JanelidzeGaloistheories
73 B.BollobasRandomgraphs
COMPLETELY BOUNDED MAPS
AND OPERATOR ALGEBRAS
VERN PAULSEN
UniversityofHouston
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge  , United Kingdom
Published in the United States of America by Cambridge University Press, New York
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© Vern Paulsen 2002
This book is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2003
isbn--13 978-0-511-06103-5 eBook (NetLibrary)
isbn--10 0-511-06103-X eBook (NetLibrary)
isbn--13 978-0-521-81669-4 hardback
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ToJohn,Ival,Effie,Susan,StephenandLisa.
Mypast,presentandfuture.

Contents
Preface pageix
1 Introduction 1
2 PositiveMaps 9
3 CompletelyPositiveMaps 26
4 DilationTheorems 43
5 CommutingContractionsonHilbertSpace 58
6 CompletelyPositiveMapsinto M 73
n
7 Arveson’sExtensionTheorems 84
8 CompletelyBoundedMaps 97
9 CompletelyBoundedHomomorphisms 120
10 PolynomiallyBoundedandPower-BoundedOperators 135
11 Applicationsto K-SpectralSets 150
12 TensorProductsandJointSpectralSets 159
13 AbstractCharacterizationsofOperatorSystems
andOperatorSpaces 175
14 AnOperatorSpaceBestiary 186
15 InjectiveEnvelopes 206
16 AbstractOperatorAlgebras 225
17 CompletelyBoundedMultilinearMapsandtheHaagerup
TensorNorm 239
18 UniversalOperatorAlgebrasandFactorization 260
19 SimilarityandFactorization 273
Bibliography 285
Index 297
vii