Homological Methods in Banach Space Theory

559 Pages·2023·9.314 MB·English

by  Félix Cabello Sánchez, Jesús M. F. Castillo

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 203
EditorialBoard
J. BERTOIN, B. BOLLOBA´S, W. FULTON, B. KRA, I. MOERDIJK,
C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO
HOMOLOGICALMETHODSINBANACH
SPACETHEORY
Manyresearchersingeometricfunctionalanalysisareunawareofalgebraicaspectsof
the subject and the advances they have permitted in the last half century. This book,
writtenbytwoworldexperts onhomological methods inBanach spacetheory, gives
functionalanalystsanewperspectiveontheirfieldandnewtoolstotackleitsproblems.
All techniques and constructions from homological algebra and category theory are
introduced from scratch and illustrated with concrete examples at varying levels of
sophistication.Thesetechniquesarethenusedtopresentbothimportantclassicalresults
andpowerfuladvancesfromrecentyears.Finally,theauthorsapplythemtosolvemany
oldandnewproblemsinthetheoryof(quasi-)Banachspacesandoutlinenewlinesof
research.Containingalotofmaterialunavailableelsewhereintheliterature,thisbook
isthedefinitiveresourceforfunctionalanalystswhowanttoknowwhathomological
algebracandoforthem.
Fe´lixCabelloSa´nchezisProfessorofMathematicsattheUniversidaddeExtremadura.
Heisco-authorofthemonographSeparablyInjectiveBanachSpaces(2016).
Jesu´sM.F.CastilloisProfessorofMathematicsattheUniversidaddeExtremadura.
He is co-author of the monographs Three-Space Problems in Banach Space Theory
(1997)andSeparablyInjectiveBanachSpaces(2016).
Published online by Cambridge University Press
CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS
EditorialBoard
J.Bertoin,B.Bolloba´s,W.Fulton,B.Kra,I.Moerdijk,C.Praeger,P.Sarnak,B.Simon,B.Totaro
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.Fora
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192E.KowalskiAnIntroductiontoProbabilisticNumberTheory
193V.GorinLecturesonRandomLozengeTilings
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196F.Durand&D.PerrinDimensionGroupsandDynamicalSystems
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201J.vanNeervenFunctionalAnalysis
202A.SchmedingAnIntroductiontoInfinite-DimensionalDifferentialGeometry
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204G.P.Paternain,M.Salo&G.UhlmannGeometricInverseProblems
Published online by Cambridge University Press
Homological Methods in Banach
Space Theory
FE´LIX CABELLO SA´ NCHEZ
UniversidaddeExtremadura
JESU´ S M. F. CASTILLO
UniversidaddeExtremadura
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Informationonthistitle:www.cambridge.org/9781108478588
DOI:10.1017/9781108778312
©CambridgeUniversityPress&Assessment2023
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andtotheprovisionsofrelevantcollectivelicensingagreements,
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Firstpublished2023
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Published online by Cambridge University Press
Contents
Preface pageix
Preliminaries 1
1 ComplementedSubspacesofBanachSpaces 9
1.1 BanachandQuasi-BanachSpaces 9
1.2 ComplementedSubspaces 13
1.3 UncomplementedSubspaces 16
1.4 LocalPropertiesandTechniques 19
1.5 The Dunford–Pettis, Grothendieck, Pełczyn´ski
andRosenthalProperties 25
1.6 C(K)-SpacesandTheirComplementedSubspaces 26
1.7 Sobczyk’sTheoremandItsDerivatives 29
1.8 NotesandRemarks 36
1.8.1 TopologicalStuff 36
1.8.2 Orlicz,Young,FenchelandL Too 38
0
1.8.3 UltrapowersofL When0< p<1 39
p
1.8.4 Sobczyk’sTheoremStrikesBack 42
2 TheLanguageofHomology 46
2.1 ExactSequencesofQuasi-BanachSpaces 48
2.2 BasicExamplesofExactSequences 54
2.3 TopologicallyExactSequences 67
2.4 CategoricalConstructionsforAbsoluteBeginners 70
2.5 PullbackandPushout 72
2.6 PushoutandExactSequences 75
2.7 ProjectivePresentations:theUniversalPropertyof(cid:96) 78
p
2.8 PullbacksandExactSequences 81
2.9 InjectivePresentations:theUniversalPropertyof(cid:96) 82
∞
2.10 AllaboutThatPullback/PushoutDiagram 84
v
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vi Contents
2.11 DiagonalandParallelPrinciples 94
2.12 HomologicalConstructionsAppearinginNature 98
2.13 TheDevice 105
2.14 ExtensionandLiftingofOperators 111
2.15 NotesandRemarks 119
2.15.1 CategoricalLimits 119
2.15.2 HowtoDrawMoreDiagrams 120
2.15.3 AmalgamationofSequences 124
2.15.4 CategoriesofShortExactSequences 125
3 QuasilinearMaps 128
3.1 AnIntroductiontoQuasilinearMaps 129
3.2 QuasilinearMapsinAction 131
3.3 QuasilinearMapsversusExactSequences 138
3.4 LocalConvexityofTwistedSumsandK-Spaces 149
3.5 ThePullbackandPushoutinQuasilinearTerms 154
3.6 SpacesofQuasilinearMaps 155
3.7 HomologicalPropertiesof(cid:96) andL When0< p≤1 161
p p
3.8 ExactSequencesofBanachSpacesandDuality 167
3.9 DifferentVersionsofaQuasilinearMap 176
3.10 LinearisationofQuasilinearMaps 179
3.11 TheTypeofTwistedSums 181
3.12 AGlimpseofCentralizers 185
3.13 NotesandRemarks 190
3.13.1 Doman´ski’sWorkonQuasilinearMaps 190
3.13.2 ACohomologicalApproachtoQuasilinearity 193
3.13.3 TableofCorrespondencesbetweenDiagrams
andQuasilinearMaps 194
4 TheFunctorExtandtheHomologySequences 197
4.1 TheFunctorExt 198
4.2 TheHomologySequences 204
4.3 HomologyinQuasilinearTerms 212
4.4 AlternativeConstructionsofExt 216
4.5 TopologicalAspectsofExt 224
4.6 NotesandRemarks 234
4.6.1 AdjointFunctors 234
4.6.2 DerivedFunctors 237
4.6.3 UnknownKnownsaboutExt2 240
4.6.4 OpenProblemsabouttheTopologyofExt 241
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Contents vii
5 LocalMethodsintheTheoryofTwistedSums 243
5.1 LocalSplitting 244
5.2 UniformBoundednessPrinciplesforExactSequences 258
5.3 TheMysteriousRoleoftheBAP 270
5.4 NotesandRemarks 283
5.4.1 WhichBanachSpacesAreK-Spaces? 283
5.4.2 TwistingaFewExoticBanachSpaces 284
6 Fra¨ısse´ LimitsbythePound 287
6.1 Fra¨ısse´ ClassesandFra¨ısse´ Sequences 288
6.2 AlmostUniversalDisposition 290
6.3 AlmostUniversalComplementedDisposition 299
6.4 AUniversalOperatoronG 316
p
6.5 NotesandRemarks 324
6.5.1 WhatIfε=0? 324
6.5.2 BeforeG SpacesFadeOut 325
p
6.5.3 Fra¨ısse´ ClassesofBanachSpaces 326
7 ExtensionofOperators,IsomorphismsandIsometries 329
7.1 Operators:ExtensibleandUFOSpaces 331
7.2 Isomorphisms:theAutomorphicSpaceProblem 336
7.3 Isometries:UniversalDisposition 348
7.4 PositionsinBanachSpaces 354
7.5 NotesandRemarks 365
7.5.1 IsomorphicbutDifferentTwistedSums 365
7.5.2 HowManyTwistedSumsofTwoSpacesExist? 366
7.5.3 MovingtowardstheAutomorphicSpaceProblem 368
7.5.4 TheProductofSpacesof(Almost)Universal
Disposition 369
8 ExtensionofC(K)-ValuedOperators 372
8.1 ZippinSelectors 374
8.2 TheLindenstrauss–Pełczyn´skiTheorem 378
8.3 Kalton’sApproachtotheC-ExtensionProperty 383
8.4 SequenceSpaceswiththeC-ExtensionProperty 394
8.5 C-ExtensibleSpaces 400
8.6 TheDarkSideoftheJohnson–ZippinTheorem 411
8.7 TheAstoundingStorybehindtheCCKYProblem 424
8.8 NotesandRemarks 434
8.8.1 HomogeneousZippinSelectors 434
8.8.2 Lindenstrauss-ValuedExtensionResults 436
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viii Contents
8.8.3 TheLastStrokeontheExtensionofC-Valued
LipschitzMaps 437
8.8.4 Property(M)andM-Ideals 440
8.8.5 SetTheoreticAxiomsandTwistedSumAffairs 440
9 SingularExactSequences 444
9.1 BasicPropertiesandTechniques 445
9.2 SingularQuasilinearMaps 451
9.3 AmalgamationTechniques 452
9.4 NotesandRemarks 462
9.4.1 SuperSingularity 462
9.4.2 DisjointSingularity 463
9.4.3 Cosingularity 465
9.4.4 TheBasicSequenceProblem 465
10 BacktoBanachSpaceTheory 468
10.1 Vector-ValuedVersionsofSobczyk’sTheorem 468
10.2 PolyhedralL -Spaces 471
∞
10.3 LipschitzandUniformlyHomeomorphicL -Spaces 473
∞
10.4 PropertiesofKernelsofQuotientMapsonL Spaces 476
1
10.5 3-SpaceProblems 484
10.6 ExtensionofL -ValuedOperators 495
∞
10.7 KadecSpaces 502
10.8 TheKalton–PeckSpaces 505
10.9 ThePropertiesofZ ExplainedbyItself 516
2
Bibliography 521
Index 543
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Preface
This is a book about homological techniques applied to Banach and quasi-
Banach space theory. It has been our aim to show how much this modern
branch of functional analysis has to offer analysts. To do so, we will present
the basic elements and techniques of homological algebra in the concrete
categories of (quasi-) Banach spaces and will show how to use them and
how they work to approach and solve classical problems in analysis. Years
ofworkingcollaborationwithmanymathematiciansconvincedusofasimple
truth:peopledonotunderstanddiagrams.Thisisthecauseoftheseasickness
so customary when sailing the homological sea. There are good reasons for
that;mainly,thatthefoundationalideas,basicprinciplesandvisualtechniques
thatmakethetheorypowerfularecouchedincategoricaljargonandnotinthe
languageoffunctionalanalysis.Thusconsiderableeffortisrequiredsimplyto
decodetheinformation.
Accordingly,ithasbeenoneofourmainconcernstopresentallhomological
tools without losing contact with classical (quasi-) Banach space theory: all
categorical constructions are introduced to explain some concrete (quasi-)
Banach space constructions appearing in nature. Our guide in this line was
settledbyOlekPełczyn´ski:examplesalwaysgofirst.Infact,probablythefirst
paper in Banach space theory with clear homological content is Pełczyn´ski’s
monograph [377]. Semadeni’s book [430], which followed, was already
writteninanentirelycategoricalmood.Todiscoverwhathappenedafterthat,
just turn back the page. The role of 3-space problems for Hilbert spaces
and local convexity cannot go unmentioned. From these came the seminal
works of Enflo, Lindenstrauss and Pisier [167], Ribe [401] and Kalton and
Peck[280],whereexactsequencesandquasilinearmapscametolight.After
that,categoricalmethodsin(quasi-)Banachspacetheoryemergedatasteady
pace: pullback/pushout constructions, splitting, extension/lifting of operator
connectionsand,finally,homologysequences.
ix
https://doi.org/10.1017/9781108778312.001 Published online by Cambridge University Press