Table Of ContentLecture Notes in Mathematics 2132
Antonio Avilés
Félix Cabello Sánchez
Jesús M. F. Castillo
Manuel González
Yolanda Moreno
Separably
Injective
Banach Spaces
Lecture Notes in Mathematics 2132
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis,Zurich
MariodiBernardo,Bristol
AlessioFigalli,Austin
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GaborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,ParisandNY
CatharinaStroppel,Bonn
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Antonio Avilés (cid:129) Félix Cabello Sánchez (cid:129)
Jesús M.F. Castillo (cid:129) Manuel González (cid:129)
Yolanda Moreno
Separably Injective
Banach Spaces
123
AntonioAvilés FélixCabelloSánchez
Dpto.deMatemáticas Dpto.deMatemáticas
UniversidaddeMurcia UniversidaddeExtremadura
Murcia,Spain Badajoz,Spain
JesúsM.F.Castillo ManuelGonzález
Dpto.deMatemáticas Dpto.deMatemáticas
UniversidaddeExtremadura UniversidaddeCantabria
Badajoz,Spain Santander,Spain
YolandaMoreno
Dpto.deMatemáticas
UniversidaddeExtremadura
Cáceres,Spain
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-319-14740-6 ISBN978-3-319-14741-3 (eBook)
DOI10.1007/978-3-319-14741-3
LibraryofCongressControlNumber:2016935425
MathematicsSubjectClassification(2010):46A22,46B03,46B08,46M10,46M18,46B26,54B30
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Preface
ThePlot
Injective Banach spaces are those spaces that allow the extension of any operator
withvaluesinthemtoanysuperspace.Finitedimensionaland` arethesimplest
1
examplesofinjectivespaces.WhenaBanachspaceisinjective,thereautomatically
appears a constant that controls the norms of the extensions. At the other end of
the line, we encounter the Banach spaces allowing the “controlled” extension of
finite-rankoperators,whichformthewell-knownclassofL -spaces.
1
The main topic of this monograph lies between these two extremes: Banach
spacesthatallowacontrolledextensionofoperatorsundercertainrestrictionsonthe
sizeoftheirrangeorthesizeofthespacestowheretheycanbeextended.Thebasic
caseisthatofBanachspacesallowingtheextensionofoperatorsfromsubspacesof
separable spaces, called separablyinjective, for which c0 is the simplest example.
ThesecondmoreimportantcaseisthatofBanachspacesallowingtheextensionof
operators from separable spaces elsewhere, called universally separably injective,
forwhichthespace`c .(cid:2)/istheperfectexample.
1
This monographcontains most of what is currently known about (universally)
separablyinjectivespaces;andcertainlyitcontainsallweknowplusagoodpartof
allwedon’t.Chapters1–5,whosecontentwedescribebelow,givearatherdetailed
account of the current theory of separable injectivity, with its many connections
andapplications.Atthesametime,the“Notesandremarks”sectionsattheendof
each chapter and the entire “Open problems” section in Chap.6 provide a large-
scale map of the land beyond the sea. We have decided not to reproduce here
severalitemsalreadycoveredinbooks.ThebestexampleofthiswouldbeZippin’s
theoremassertingthatc0istheonlyseparableseparablyinjectiveBanachspace,for
whichanexceptionallylimpidexpositioncanbefoundin[253];seealso[33].Other
seemingly relevantpieces of informationcan be foundin the literature, but as we
wereunabletoestablishpreciseconnectionswithourtopic,wechosetoomitthem.
v
vi Preface
Injectivity vs.(Universal)SeparableInjectivity
ThereaderisreferredtoeitherthePreliminariesortheAppendixforallunexplained
notation.
A BanachspaceE issaidtobeinjectiveifforeveryBanachspaceX andevery
subspaceY ofX,eachoperatortWY !EadmitsanextensionTWX !E.Thespace
issaidtobe(cid:3)-injectiveif,besides,T canbechosensothatkTk(cid:2)(cid:3)ktk.
The space ` is the perfect example of 1-injective space, and a key point is
1
to determine to what extent other injective spaces share the properties of ` .
1
On the positive side, one should score the results of Nachbin and Kelley who
characterized the 1-injective spaces as those Banach spaces linearly isometric to
aC.K/space,withKanextremelydisconnectedcompactspace.Ontheotherhand,
the research of Argyros, Haydon, Rosenthal, and other authors provided several
generalstructuretheorems,goodexamplesofexoticinjectivespaces,andthefeeling
that the complete classification of injective Banach spaces is an unmanageable
problem.
Most of the research on injective spaces has revolved around the following
problems:
(1) Iseveryinjectivespaceisomorphictoa1-injectivespace?
(2) IseveryinjectivespaceisomorphictosomeC.K/-space?
(3) Whatisthestructureofaninjectivespace?
Admittedly, (3) is rather vague and (2) is a particular case of the more general
problemoffindingoutwhetherornotacomplementedsubspaceofaC.K/-spaceis
againisomorphicto a C.K/-space(the compactspacemay vary).These problems
haveremainedopenfor50years,(1)and(3)eveninthecaseinwhichtheinjective
space is a C.K/-space. We refer the reader to Chap.1 for a summary of what is
knownaboutinjectivespaces.
In this monograph, we deal with several weak forms of injectivity, mainly
separableinjectivityanduniversalseparableinjectivity.ABanachspaceE issaid
to be separably injective if it satisfies the extension property in the definition
of injective spaces under the restriction that X is separable; it is said to be a
universally separably injective if it satisfies the extension property when Y is
separable.Obviously,injectivespacesareuniversallyseparablyinjective,andthese,
inturn,areseparablyinjectives;the converseimplicationsfail. Thecorresponding
definitionsof(cid:3)-separablyinjectiveanduniversally(cid:3)-separablyinjectiveshouldbe
clear.
The study of separably injective spaces was initiated by Phillips [216] and
Sobczyk[235],whoshowedthatc(resp.c0)is2-separablyinjective.LaterZippin
[252] proved that c0 is the only (infinite dimensional) separable space that is
separablyinjective.Moreover,Ostrovskii[208]provedthata(cid:3)-separablyinjective
space with (cid:3) < 2 cannot be separable, Baker [25] and Seever [228] studied
separablyinjectiveC.K/spaces,andseveralresultsintheliteratureontheextension
ofoperatorscanalsobeformulatedintermsofseparablyinjectivespaces.Separably
Preface vii
injective spaces have been studied more recently by several authors such as
Rosenthal[225],Zippin[253],JohnsonandOikhberg[146],andalsobythepresent
authors in [20], where the notion of universal separable injectivity was formally
introduced.
The theory of separably injective and universally separably injective spaces is
quitedifferentfromthatofinjectivespaces,ismuchricherinexamples,andcontains
interestingstructureresultsandhomologicalcharacterizations.Forinstance,prob-
lems (1) and (2) abovehave a negativeanswer for separablyinjective spaces, and
quiteinterestinginformationaboutthestructureof(universally)separablyinjective
spaces can be offered. Moreover this theory is far from being complete: many
openproblemscanbeformulated(seeChap.6)thatweexpectcanbeattractivefor
Banach spacers and could foster the interestin studyinginjectivity-like properties
ofBanachspaces.
ABrief Description ofthe Contents ofThisMonograph
After this Preface, “Preliminaries” section contains all due preliminaries about
notation and basic definitions. An Appendix at the end of the book describes
the basic facts about L1 and L1-spaces, homologicaltechniques, and transfinite
chains that will appear and be used throughoutthe monograph.Other definitions,
properties,orrequiredconstructionswillbegivenwhenneeded.
In Chap.1, we gather together properties of injective Banach spaces, basic
examples and counterexamples, and criteria that are useful to prove that certain
spacesarenotinjective.Thesefactswillbeusedlaterandwillallowthereaderto
comparethestabilitypropertiesandthevarietyofexamplesofinjectivespaceswith
thoseof(universally)separablyinjectivespaces.
In Chap.2, we introduce the separably injective and the universally separably
injectiveBanachspaces,aswellastheirquantitativeversions,andobtaintheirbasic
properties and characterizations. We establish that infinite-dimensional separably
injective spaces are L1-spaces, contain c0, and have Pełczyn´ski’s property .V/.
Universallyseparablyinjectivespaces,moreover,areGrothendieckspaces,contain
` ,andenjoyRosenthal’sproperty.V/.Wealsoproveanumberofstabilityresults
1
thatallowustopresentmanynaturalexamplesofseparablyinjectivespaces,such
asC.K/-spacewhenKiseitheranF-spaceorhasfiniteheight,twistedsumsandc0-
vectorsumsofseparablyinjectivespaces,etc.,includinganexampleofaseparably
injective space that is not isomorphic to any complemented subspace of a C.K/-
space, which solves problem (2) above for separable injectivity. In passing, these
factsprovideamajorstructuraldifferencebetween(cid:3)-separablyinjectivespacesfor
various values of (cid:3), something that currently does not exist for injective spaces:
1-separablyinjectivespacesareGrothendieckandLindenstraussspaces;hencethey
must be nonseparable when they are infinite dimensional. However, 2-separably
injectivespacescanbeevenseparable.
viii Preface
Thefundamentalstructuretheoremforuniversallyseparablyinjectivespacesis
thataBanachspaceEisuniversallyseparablyinjectiveifandonlyifeveryseparable
subspace is contained in a copy of ` inside E. This establishes a bridge toward
1
thestudyofthepartiallyautomorphiccharacterof(universally)separablyinjective
spaces, namely, toward determining in which cases an isomorphism between two
subspacesextendstoanautomorphismofthewholespace.
Homologicalcharacterizationsarepossible:recallthatExt.Z;Y/D0meansthat
wheneveraBanachspaceisomorphictoY iscontainedinaBanachspaceXinsuch
a way that X=Y is isomorphic to Z, it must be complemented.In this language, a
spaceEisseparablyinjectiveifandonlyifExt.S;E/D0foreveryseparablespace
S. For universally separably injective spaces, we have less: if Ext.` =S;U/ D 0
1
for every separable space S, then U is universally separably injective. A problem
thatiscryingouttobesolvediswhetheralsotheconverseholds:Doestheidentity
Ext.` ;U/ D 0 characterize universally separably injective spaces U? A rather
1
detaileddiscussioncanbefoundinSect.6.2.
Section2.4isspecificallydevotedto1-separablyinjectivespaces.Atthispoint,
settheoryaxiomsenterthegame.Indeed,Lindenstraussobtainedinthemid-1960s
what can be understood as a proof that, under the continuum hypothesis CH,
1-separablyinjective spaces are 1-universallyseparably injective;he left open the
question in general. We show how to construct (in a way consistent with ZFC)
an example of a Banach space of type C.K/ that is 1-separably injective but not
1-universally separably injective. The chapter closes with a detailed study of the
separableinjectivityandrelatedhomologicalpropertiesofC.N(cid:2)/.
Chapter 3 focuses on the study of spaces of universaldisposition because they
will providenew (and, in the case of p-Banach spaces, the only currentlyknown)
examplesof (universally)separablyinjective spaces. We presenta basic device to
generatesuchspaces. Thedeviceis ratherflexibleandthus,whenperformedwith
theappropriateinputdata,isabletoproduceagreatvarietyofexamples:theGurariy
spaceG [118],thep-Gurariyspaces[58],theKubis´ space[169],newspacessuch
asF!1 whichisofuniversaldispositionforallfinite-dimensionalspacesbutnotfor
separable spaces, or the L -envelopes obtained in [69]. Remarkable outputs are
1
the examplesof a 1-separablyinjective spaces S!1 and a 1-universallyseparably
injectivespaceU!1 thatarenotisomorphictocomplementedsubspacesofanyC-
space (or an M-space), which solves problem (2) for the classes of (universally)
1-separablyinjectivespaceinasomewhatsurprisingway.Thesespacesturnoutto
beofuniversaldispositionforseparablespaces,whichconfirmsaconjecturemade
by Gurariy in the 1960s. Moreover,under CH, the space S!1 coincides with the
FraïssélimitinthecategoryofseparableBanachspacesconsideredbyKubis´[169]
andwiththecountableultrapowersoftheGurariyspace.
In Chap.4, we study injectivity properties of ultraproducts. We show that
ultraproductsbuiltovercountablyincompleteultrafilters are universallyseparably
injective as long as they are L -spaces, in spite of the fact that they are never
1
injective. Then, we focus our attention on ultraproducts of Lindenstrauss spaces,
with specialemphasisin ultrapowersof C.K/-spacesandthe Gurariyspace. With
theaidofM-idealtheory,theresultscanbeappliedtotheliftingofoperatorsandto
Preface ix
thestudyofthebehaviorofthefunctorExtregardingduality.Onesectionisdevoted
totheHenson-Mooreclassificationproblemofisomorphictypesofultrapowersof
L -spaces.Aparticularconcernoftheauthorshasbeentomakeclearwhichresults
1
canbeprovedwithoutinvoking“modeltheory”andwhichonesstillbelongtothat
domain.
In Chap.5, we consider a natural generalization of separable injectivity to
highercardinals.Namely,givenaninfinitecardinal@,wesaythatE is@-injective
(respectively, universally @-injective) if it satisfies the extension property stated
at the beginning for spaces X (respectively, for subspaces Y) having density
character strictly smaller than @. Stopping at the first uncountable cardinal @1,
onereencounterstheclassesof(universally)separablyinjectivespaces.Somefacts
generalizetothehighercardinalcontext(say,thereexist.1;@/-injectivespacesthat
arecomplementedinnoM-space),butsomeothersofferdifficulties.Forexample,
weneedtorestricttowhatwecallc0.@/-supplementedsubspacesinordertoextend
the results on universallyseparably injective spaces; or, concerningultraproducts,
wehavetodealwith@-goodultrafilters.ThechapterincludesastudyofspacesC.K/
whichare.1;@/-injectiveandastudyoftheinterplaywithtopologicalpropertiesof
thecompactspaceK intheformofarathersatisfactorydualitybetweenextension
propertiesofoperatorsintoC.K/andliftingpropertiesofcontinuousmapsfromK.
At the end of each chapter, we have included a “Notes and Remarks” section
containingadditionalinformation,mainlyconsideredfromthepositiveside.Afinal
Chap.6describesproblemsrelatedtothetopicsofthismonographthatremainopen.
We have chosen to give the available information about them—including partial
solutions,connectionswithotherresults,etc.—inorderthatitcouldbehelpfulfor
furtherresearch.
Murcia,Spain AntonioAvilés
Badajoz,Spain FélixCabelloSánchez
Badajoz,Spain JesúsM.F.Castillo
Santander,Spain ManuelGonzález
Cáceres,Spain YolandaMoreno
