Tensor products of C-star-algebras and operator spaces

495 Pages·2020·2.51 MB·English

by  Pisier G

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56 Logic,inductionandsets,THOMASFORSTER
57 IntroductiontoBanachalgebras,operatorsandharmonicanalysis,GARTHDALESetal
58 Computationalalgebraicgeometry,HALSCHENCK
59 Frobeniusalgebrasand2-Dtopologicalquantumfieldtheories,JOACHIMKOCK
60 Linearoperatorsandlinearsystems,JONATHANR.PARTINGTON
61 AnintroductiontononcommutativeNoetherianrings(2ndEdition),K.R.GOODEARL&
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62 Topicsfromone-dimensionaldynamics,KARENM.BRUCKS&HENKBRUIN
63 Singularpointsofplanecurves,C.T.C.WALL
64 AshortcourseonBanachspacetheory,N.L.CAROTHERS
65 ElementsoftherepresentationtheoryofassociativealgebrasI,IBRAHIMASSEM,
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66 Anintroductiontosievemethodsandtheirapplications,ALINACARMENCOJOCARU&
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67 Ellipticfunctions,J.V.ARMITAGE&W.F.EBERLEIN
68 Hyperbolicgeometryfromalocalviewpoint,LINDAKEEN&NIKOLALAKIC
69 LecturesonKa¨hlergeometry,ANDREIMOROIANU
70 Dependencelogic,JOUKUVA¨A¨NA¨NEN
71 ElementsoftherepresentationtheoryofassociativealgebrasII,DANIELSIMSON&
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72 ElementsoftherepresentationtheoryofassociativealgebrasIII,DANIELSIMSON&
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73 Groups,graphsandtrees,JOHNMEIER
74 RepresentationtheoremsinHardyspaces,JAVADMASHREGHI
75 Anintroductiontothetheoryofgraphspectra,DRAGOSˇCVETKOVIC´,PETERROWLINSON&
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76 NumbertheoryinthespiritofLiouville,KENNETHS.WILLIAMS
77 Lecturesonprofinitetopicsingrouptheory,BENJAMINKLOPSCH,NIKOLAYNIKOLOV&
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78 Cliffordalgebras:Anintroduction,D.J.H.GARLING
79 IntroductiontocompactRiemannsurfacesanddessinsd’enfants,ERNESTOGIRONDO&
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80 TheRiemannhypothesisforfunctionfields,MACHIELVANFRANKENHUIJSEN
81 Numbertheory,Fourieranalysisandgeometricdiscrepancy,GIANCARLOTRAVAGLINI
82 Finitegeometryandcombinatorialapplications,SIMEONBALL
83 Thegeometryofcelestialmechanics,HANSJO¨RGGEIGES
84 Randomgraphs,geometryandasymptoticstructure,MICHAELKRIVELEVICHetal
85 Fourieranalysis:PartI–Theory,ADRIANCONSTANTIN
86 Dispersivepartialdifferentialequations,M.BURAKERDOG˘AN&NIKOLAOSTZIRAKIS
87 Riemannsurfacesandalgebraiccurves,R.CAVALIERI&E.MILES
88 Groups,languagesandautomata,DEREKF.HOLT,SARAHREES&CLAASE.RO¨VER
89 AnalysisonPolishspacesandanintroductiontooptimaltransportation,D.J.H.GARLING
90 Thehomotopytheoryof(∞,1)-categories,JULIAE.BERGNER
91 TheblocktheoryoffinitegroupalgebrasI,M.LINCKELMANN
92 TheblocktheoryoffinitegroupalgebrasII,M.LINCKELMANN
93 Semigroupsoflinearoperators,D.APPLEBAUM
94 Introductiontoapproximategroups,M.C.H.TOINTON
95 RepresentationsoffinitegroupsofLietype(2ndEdition),F.DIGNE&J.MICHEL

LondonMathematicalSocietyStudentTexts96
Tensor Products of C*-Algebras and
Operator Spaces
The Connes–Kirchberg Problem
GILLES PISIER
TexasA&MUniversity
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Informationonthistitle:www.cambridge.org/9781108479011
DOI:10.1017/9781108782081
©GillesPisier2020
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Firstpublished2020
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Contents
Introduction page1
1 Completelyboundedandcompletelypositivemaps:Basics 11
1.1 Completelyboundedmapsonoperatorspaces 11
1.2 ExtensionpropertyofB(H) 18
1.3 Completelypositivemaps 23
1.4 Normalc.p.mapsonvonNeumannalgebras 30
1.5 Injectiveoperatoralgebras 31
1.6 Factorizationofcompletelybounded(c.b.)maps 33
1.7 Normalc.b.mapsonvonNeumannalgebras 37
1.8 Notesandremarks 39
2 Completelyboundedandcompletelypositivemaps:Atoolkit 41
2.1 Rowsandcolumns:operatorCauchy–Schwarzinequality 41
2.2 Automaticcompleteboundedness 43
2.3 Complexconjugation 44
2.4 Operatorspacedual 48
2.5 Bi-infinitematriceswithoperatorentries 50
∗
2.6 FreeproductsofC -algebras 53
∗
2.7 UniversalC -algebraofanoperatorspace 57
2.8 Completelypositiveperturbationsofcompletelyboundedmaps 58
2.9 Notesandremarks 61
∗
3 C -algebrasofdiscretegroups 63
∗
3.1 Full(=Maximal)groupC -algebras 63
∗
3.2 FullC -algebrasforfreegroups 66
∗
3.3 ReducedgroupC -algebras:Fell’sabsorptionprinciple 71
3.4 Multipliers 73
3.5 GroupvonNeumannAlgebra 77
v
vi Contents
3.6 Amenablegroups 78
3.7 OperatorspacespannedbythefreegeneratorsinC∗(F ) 83
λ n
3.8 Freeproductsofgroups 84
3.9 Notesandremarks 85
∗
4 C -tensorproducts 87
∗
4.1 C -normsontensorproducts 87
∗
4.2 NuclearC -algebras(abriefpreliminaryintroduction) 91
∗
4.3 TensorproductsofgroupC -algebras 92
∗
4.4 AbriefrepertoireofexamplesfromgroupC -algebras 95
4.5 Statesonthemaximaltensorproduct 96
4.6 Statesontheminimaltensorproduct 99
∗
4.7 TensorproductwithaquotientC -algebra 103
4.8 Notesandremarks 104
5 Multiplicativedomainsofc.p.maps 106
5.1 Multiplicativedomains 106
5.2 Jordanmultiplicativedomains 108
5.3 Notesandremarks 112
6 Decomposablemaps 113
6.1 Thedec-norm 113
6.2 Theδ-norm 121
6.3 Decomposableextensionproperty 125
6.4 Examplesofdecomposablemaps 129
6.5 Notesandremarks 135
7 Tensorizingmapsandfunctorialproperties 136
7.1 (α →β)-tensorizinglinearmaps 136
7.2 (cid:5)(cid:5) isprojective(i.e.exact)butnotinjective 141
max
7.3 max-injectiveinclusions 144
7.4 (cid:5)(cid:5) isinjectivebutnotprojective(i.e.notexact) 150
min
7.5 min-projectivesurjections 153
∗
7.6 GeneratingnewC -normsfromoldones 157
7.7 Notesandremarks 160
∗
8 Biduals,injectivevonNeumannalgebras,andC -norms 161
∗
8.1 BidualsofC -algebras 161
8.2 Thenor-normandthebin-norm 162
8.3 NuclearityandinjectivevonNeumannalgebras 163
8.4 Localreflexivityofthemaximaltensorproduct 170
8.5 Localreflexivity 174
8.6 Notesandremarks 179
Contents vii
9 Nuclearpairs,WEP,LLP,QWEP 180
9.1 Thefundamentalnuclearpair(C∗(F∞),B((cid:6)2)) 181
9.2 C∗(F)isresiduallyfinitedimensional 186
9.3 WEP(WeakExpectationProperty) 188
9.4 LLP(LocalLiftingProperty) 193
9.5 Toliftornottolift(globallifting) 198
9.6 LinearmapswithWEPorLLP 202
9.7 QWEP 204
9.8 Notesandremarks 208
10 Exactnessandnuclearity 210
10.1 Theimportanceofbeingexact 210
10.2 Nuclearity,exactness,approximationproperties 216
10.3 Moreonnuclearityandapproximationproperties 222
10.4 Notesandremarks 224
11 Tracesandultraproducts 225
11.1 Traces 225
11.2 TracialprobabilityspacesandthespaceL (τ) 228
1
11.3 ThespaceL (τ) 230
2
11.4 Anexamplefromfreeprobability:semicircularand
circularsystems 235
11.5 Ultraproducts 238
11.6 FactorizationthroughB(H)andultraproducts 246
11.7 Hypertracesandinjectivity 256
11.8 Thefactorizationpropertyfordiscretegroups 259
11.9 Notesandremarks 261
12 TheConnesembeddingproblem 262
12.1 Connes’squestion 262
12.2 Theapproximatelyfinitedimensional(i.e.“hyperfinite”)
II -factor 269
1
12.3 Hyperlineargroups 271
12.4 ResiduallyfinitegroupsandSoficgroups 273
12.5 Randommatrixmodels 276
12.6 CharacterizationofnuclearvonNeumannalgebras 277
12.7 Notesandremarks 279
13 Kirchberg’sconjecture 280
13.1 LLP⇒WEP? 280
13.2 ConnectionwithGrothendieck’stheorem 283
13.3 Notesandremarks 290
viii Contents
14 Equivalenceofthetwomainquestions 291
14.1 FromConnes’squestiontoKirchberg’sconjecture 291
14.2 FromKirchberg’sconjecturetoConnes’squestion 292
14.3 Notesandremarks 296
15 Equivalencewithfiniterepresentabilityconjecture 297
15.1 Finiterepresentabilityconjecture 297
15.2 Notesandremarks 299
16 EquivalencewithTsirelson’sproblem 300
16.1 Unitarycorrelationmatrices 300
16.2 Correlationmatriceswithprojectionvaluedmeasures 303
16.3 StrongKirchbergconjecture 309
16.4 Notesandremarks 310
17 Property(T)andresiduallyfinitegroups:Thom’sexample 311
17.1 Notesandremarks 316
18 TheWEPdoesnotimplytheLLP 317
18.1 TheconstantC(n):W√EP(cid:7)⇒LLP 319
18.2 ProofthatC(n)=2 n−1usingrandomunitarymatrices 323
18.3 Exactnessisnotpreservedbyextensions 327
18.4 AcontinuumofC∗-normsonB⊗B 329
18.5 Notesandremarks 332
19 OtherproofsthatC(n)<n:quantumexpanders 333
19.1 Quantumcodingsequences.Expanders.Spectralgap 333
19.2 Quantumexpanders 336
19.3 Property(T) 338
19.4 Quantumsphericalcodes 341
19.5 Notesandremarks 343
20 LocalembeddabilityintoC andnonseparabilityof(OS ,d ) 344
n cb
20.1 Perturbationsofoperatorspaces 345
20.2 Finite-dimensionalsubspacesofC 346
20.3 Nonseparability of the metric space OS of n-dimensional
n
operatorspaces 351
20.4 Notesandremarks 357
21 WEPasanextensionproperty 358
21.1 WEPasalocalextensionproperty 358
21.2 WEPversusapproximateinjectivity 362
21.3 The(global)liftingpropertyLP 364
21.4 Notesandremarks 365
Contents ix
22 Complexinterpolationandmaximaltensorproduct 366
22.1 Complexinterpolation 366
22.2 Complexinterpolation,WEPandmaximaltensorproduct 371
22.3 Notesandremarks 382
23 Haagerup’scharacterizationsoftheWEP 384
23.1 Reductiontotheσ-finitecase 384
23.2 Anewcharacterizationofgeneralizedweakexpectationsand
theWEP 385
23.3 AsecondcharacterizationoftheWEPanditsconsequences 388
23.4 Preliminariesonself-polarforms 390
+
23.5 max -injectiveinclusionsandtheWEP 395
23.6 Complement 403
23.7 Notesandremarks 408
24 FullcrossedproductsandfailureofWEPforB⊗ B 410
min
24.1 Fullcrossedproducts 410
24.2 Fullcrossedproductswithinneractions 414
24.3 B⊗ BfailsWEP 418
min
24.4 ProofthatC (3)<3(Selberg’sspectralbound) 427
0
24.5 OtherproofsthatC (n)<n 429
0
24.6 Randompermutations 431
24.7 Notesandremarks 432
25 Openproblems 434
Appendix:Miscellaneousbackground 438
A.1 Banachspacetensorproducts 438
A.2 Acriterionforanextensionproperty 439
A.3 UniformconvexityofHilbertspace 441
A.4 Ultrafilters 441
A.5 UltraproductsofBanachspaces 443
A.6 Finiterepresentability 443
A.7 Weakandweak*topologies:bidualsofBanachspaces 444
A.8 Thelocalreflexivityprinciple 446
A.9 AvariantofHahn–Banachtheorem 447
A.10 Thetraceclass 448
∗
A.11 C -algebras:basicfacts 448
∗
A.12 CommutativeC -algebras 450
A.13 StatesandtheGNSconstruction 451
A.14 On∗-homomorphisms 452
∗
A.15 Approximateunits,ideals,andquotientC -algebras 454
A.16 vonNeumannalgebrasandtheirpreduals 456