On a localization of the UKK property and the Fixe

ONALOCALIZATIONOFTHEUKKPROPERTYANDTHEFIXEDPOINTPROPERTYINLw;1N.L.Carothers,S.J.DilworthandC.J.LennardAbstract.WeintroducealocalizationoftheuniformKadec-Kleepropertytoweaklycom-pactconvexsetswhichimpliesweaknormalstructure.WecharacterizethispropertyfortheLorentzspacesLw;1(0;1)andthusestablishthexedpointpropertyforweaklycompactconvexsetsinthesespaceswhenevertheweightisstrictlydecreasing.Finally,weshowthatallnon-reflexivesubspacesofLw;1(0;1)failthexedpointpropertyforclosedboundedconvexsets,whichistheanalogueforLw;1ofarecentresultofDowlingandLennardfornon-reflexivesubspacesofL1.1.IntroductionLetw:(0;1)![0;1)beanon-increasingweightfunctionsatisfyingthenormalizationconditionR10w(t)dt=1.TheLorentzspaceLw;1(0;1)consistsofallreal-valuedmeasur-ablefunctionsfdenedon(0;1)forwhichjfjpossessesanon-increasingrearrangementfwhichsatiseskfk=Z10f(t)w(t)dt1:Spacesofthistypewereintroducedin[17].Inadditiontothenormalizationconditionabove,weshallmakeonlytwostandingassumptionsaboutw:(i)w(t)!0ast!1,and(ii)R10w(t)dt=1.Inparticular,wedonotneedtoassumethatw(t)!1ast!0.ForourresultsinSection4weshallrequirewtobestrictlydecreasing.LetXbeaBanachspaceandletbeavectorspacetopologyonXthatisweakerthanthenormtopology(inthispaperonlytheweakandweak-startopologieswillappear).RecallthatXhastheKadec-Kleepropertyw.r.t.,denotedKK(),ifkxn−xk!0wheneverkxnk!kxkandxn!xw.r.t..AsequencehxniintheclosedunitballofX(denotedBa(X))issaidtobe-separated,denotedsep(hxni),ifinfn6=mkxn−xmk.WesaythatXhastheuniformKadec-Kleepropertyw.r.t.,denotedUKK(),if,given0,thereexists0suchthatwheneverhxniisa-convergent-separatedsequenceinBa(X)with-limitx,thenkxk1−.ThispropertywasintroducedbyHu[12]fortheweaktopologyandbyLennard[15]inthegeneralsetting.ResearchofS.J.DilworthwasdonewhileonsabbaticalleaveatBowlingGreenStateUniversityResearchofC.J.LennardwaspartiallysupportedbyaUniversityofPittsburghFASGrantTypesetbyAMS-TEX12N.L.CAROTHERS,S.J.DILWORTHANDC.J.LENNARDMuchoftherecentinterestinUKKpropertiesstemsfromthediscoveriesofvanDulstandSims[9]andLennard[15]thatUKK()impliesnormalstructurefor-compactconvexsets,whichinturnimpliesthexedpointpropertyfornon-expansivemappingsbyatheoremofKirk[14].Bygeneralizingthepaper[3],inwhichtheimportantspecialcaseofLp;1(0;1)(cor-respondingtotheweightw(t)=(1=p)t1p−1)isconsidered,DilworthandHsu[5]recentlycharacterizedtheweightsw(t)forwhichLw;1(0;1)hasUKKforitsnaturalweak-startopology.Subsequentlyitwasshown[7]thatthesearepreciselytheweightsforwhich(t)=Rt0w(s)dsisuniformlyconcave:thatis,given2(0;1),thereexists()0suchthatforall0st1,wehaves+t2(s)+(t)2+t−st(t):ThestartingpointofthepresentpaperistheobservationthattheUKKcanbelocalizedtoweaklycompactsets,andthatthislocalizationisstillsucienttodeducethexedpointpropertyforweaklycompactconvexsets(Theorem1).WecallthisnewpropertytheweakuniformKadec-Kleeproperty,denotedWUKK,andweobservethatitliesstrictlybetweentheKKandUKKproperties.InSection4weprove(Theorem8)thatinLw;1(0;1)theWUKKpropertyisequivalenttothemildrequirementthatwbestrictlydecreasing(equivalently,thatbestrictlyconcave).ThiscoincideswithSedaev'searliercharacterizationoftheKKproperty[18],andincombinationwithTheorem1itestablishesthexedpointpropertyforamuchlargerclassofLw;1spacesthantheclassofspaceswiththeUKKpropertydescribedabove.Inparticular,theLorentzspacesdiscoveredrecentlyin[6],whichareisomorphictoL1(0;1)andisometrictocertainsubspacesofL1(0;1),allhavethexedpointpropertyforweaklycompactconvexsets.TheproofofTheorem8exploitsthespecialformofweaklycompactsetsinLw;1(0;1)aswellasanintegralrepresentationforelementsofunitnorm;theseauxiliaryresultsarerecordedinSection3.InthenalsectionwegeneralizetherecenttheoremofDowlingandLennard[8]thateverynon-reflexivesubspaceofL1failsthexedpointpropertyforclosedboundedconvexsets.Byestablishingtheexistenceofasymptotic`1sequencesasin[8],weprovetheanalogousresultforLw;1(0;1).2.ALocalizationoftheuniformKadec-KleepropertyHenceforthwedealexclusivelywiththeweaktopology,whichallowsustodropthe`'fromKK(),etc.ThefollowingdenitionisausefullocalizationoftheUKKproperty.Denition.WesaythatXisweaklyuniformlyKadec-Klee(WUKK)ifforeveryweaklycompactsetKBa(X)andforeach0thereexists0(dependingonandonK)ONALOCALIZATIONOFTHEUKKPROPERTY3suchthatwheneverhxniisan-separatedsequencewhosetermsbelongtoKandwhichconvergesweaklytoxthenkxk1−.Remark.NotethatforreflexivespacestheWUKKandUKKpropertiescoincide,andthatingeneralUKK)WUKK)KK.SincetherearereflexivespaceswhichareKKbutnotUKK[12],itfollowsthattheWUKKandKKaredistinctproperties.Theorem8belowprovidesexamplesofLorentzspaceswhichareWUKKbutnotUKK,butwedonotknowofanysimplerexamples.LetCbeaclosedboundedconvexsubsetofX.TheradiusofC,denotedrad(C),isdenedthus:rad(C)=inffsupfkx−yk:y2Cg:x2CgAsusual,thediameterofC,denoteddiam(C),isdenedtobesupfkx−yk:x;y2Cg.RecallthatXhasweaknormalstructure(WNS)ifrad(C)diam(C)foreveryweaklycompactconvexsubsetCwhichcontainsmorethanonepoint.ThenotionofaweaklyuniformlyKadec-Kleespaceseemsanaturaloneinviewofthefollowingtheorem.Theproofisareneme