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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 www.cambridge.org Information on this title: www.cambridge.org/9780521816694 © 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 isbn--10 0-521-81669-6 hardback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. 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