文档介绍：该【Completely Compact Herz–Schur Multipliers of Dynamical Systems 2022 Weijiao He 】是由【小舍儿】上传分享，文档一共【25】页，该文档可以免费在线阅读，需要了解更多关于【Completely Compact Herz–Schur Multipliers of Dynamical Systems 2022 Weijiao He 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:.
JournalofFourierAnalysisandApplications(2022)28:47
/s00041-022-09942-6
CompletelyCompactHerzSchurMultipliersofDynamical
Systems
WeijiaoHe1··LyudmilaTurowska3
Received:11August2020/Revised:7February2022/Accepted:12February2022/
Publishedonline:13May2022
©TheAuthor(s)2022
Abstract
WeprovethatifGisadiscretegroupand(A,G,α)isaC*-dynamicalsystemsuch
thatthereducedcrossedproductAr,αGpossessesproperty(SOAP)thenevery
completelycompactHerzSchur(A,G,α)-multipliercanbeapproximatedinthe
completelyboundednormbyHerzSchur(A,G,α)-
consequence,ifGhastheapproximationproperty(AP)thenthecompletelycompact
HerzSchurmultipliersofA(G)coincidewiththeclosureofA(G)inthecompletely
SchurmultipliersofAr,αGandprovideadescriptionofthisclassinthecaseofthe
irrationalrotationalgebra.
KeywordsCrossedproduct·Multiplier·Completelycompactmap
MathematicsSubjectClassiflcationPrimary37A55·Secondary46L07·43A55
.
BLyudmilaTurowska
******@
WeijiaoHe
******@
******@
1DepartmentofMathematics,TaiyuanNormalUniversity,JinzhongCity030619,Shanxi,China
2DepartmentofMathematicalSciences,UniversityofDelaware,501EwingHall,Newark,
DE19716,USA
3DepartmentofMathematicalSciences,ChalmersUniversityofTechnologyandtheUniversityof
Gothenburg,Gothenburg41296,Sweden:.
47Page2of25JournalofFourierAnalysisandApplications(2022)28:47
1Introduction
Fouriermultipliersaretransformationsonfunctionspacesassociatedwithabelian
topologicalgroupsthatactterm-
caseofthegroupZ,suchmappingshaveanaturalsigniflcanceinclassicalAnalysis,
wheretheapproximationofagivenfunctionbytrigonometricpolynomialshasbeen
,
,
compactnessofFouriermultipliershasbeenstudiedfromotherperspectivesaswell,
forexample,inrelationwiththecompactnessofpseudo-differentialoperators(see
.[8]).
Fornon-commutativelocallycompactgroupsG,whereFouriertransformisnot
readilyavailable,thenaturalsettingforthestudyofFouriermultipliersisprovidedby
theFourieralgebraA(G)ofthegroupGacommutativeBanachalgebraconsisting
ofallcoefflcientsoftheleftregularrepresentationofGwhoseGelfandspectrum
[6],withillustrious
subsequenthistoryandsomefar-reachingapplicationstoapproximationtechniquesin
operatoralgebratheory,whereflniterankmultipliershaveplayedacornerstonerole
(see[5]).In[22],LaushowedthattheFourieralgebraA(G)hasanon-zeroweakly
compactleftmultiplierifandonlyifGisdiscreteandthat,fordiscreteamenable
groups,A(G)
thereaderto[7,11,15]forfurtherrelatedresults.
Inthecaseofnon-abeliangroups,thenewpropertyofcompleteboundedness
broughtaboutbynon-commutativitybecomesanaturalrequirement,andthe
associatedmultipliersofA(G)areknownasHerzSchurmultipliers[6].HerzSchur
multipliersarerelatedtoSchurmultiplierstransformationsonthealgebraofall
boundedoperatorsonL2(G)thatextendpoint-wisemultiplicationofintegralkernels
byagivenflxedfunctionviaoperatortransference,pioneeredintheareabyBo˙zejko
andFendler[3].Wereferthereaderto[34,35]forasurveyoftheseresultsandideas.
Anaturalquantisedversionofthenotionofcompactnessinthenon-commutative
settingcompletecompactnesswasflrstintroducedbySaarinhisDiplomarbeit[32]
ssupervisionandfurtherdevelopedinunpublishedpaper[37]by
[16]andcompletelyalmostperiodicfunctionalsoncompletelycontractiveBanach
algebrasin[31].Thenotionhasbeenimportantforthestudyofvariousoperatorspace
analoguesoftheGrothendieckapproximationpropertyand,inparticular,forquestions
thestudyofcompletecompactnessforHerz
Schurmultipliersofcrossedproductswere
introduced,andtheirrelationwiththesurroundingclassofoperator-valuedSchur
multipliersinvestigated,in[25](seealso[1]fordiscretedynamicalsystems).The
completelycompactoperator-valuedSchurmultiplierswerecharacterisedin[18].
HereweprovideacharacterisationofcompletelycompactHerzSchurmultipliersof
thereducedcrossedproductof(unital)C*-algebrasAbytheactionofadiscretegroup
G,:.
JournalofFourierAnalysisandApplications(2022)28:47Page3of2547
,werecallsomeresultsaboutoperator-
valuedSchurmultipliers,HerzSchurmultipliersofC*-dynamicalsystems(A,G,α)
andtheirinterrelation[26],weassociate
witheverycompletelyboundedmapwithdomainAα,rGaHerzSchurmultiplier
F,andidentifysomepropertiesofthemap→F.Thisisthemaintechnical
tool,,since
thedynamicalsystemsofnon-discretegroupshavenonon-trivialcompactHerz
Schurmultipliers().
completelycompactHerzSchur(A,G,α)-multipliersinthecasewhereAα,rG
possessesthestrongoperatorapproximationproperty(SOAP)[9].Asaconsequence,
onecanchoosethemaps,approximatingtheidentity,tobeinthiscaseHerzSchur
,weobtainthatforgroupswith
theapproximationproperty(AP)[17],aHerzSchurmultiplieru:G→Cdeflnes
acompletelycompactmapSu:C∗(G)→C∗(G)ifandonlyifubelongstothe
rr
closureAcb(G)ofA(G)inthespaceMcbA(G)ofHerz-
resultofBo˙zejko[2],weprovethattheclassesofcompactandcompletelycompact
HerzSchurmultipliersu:G→Careingeneraldifferent.
Therestofthepaperisdevotedtospecialclassesofcompletelycompact(A,G,α)-
,
C*-dynamicalsystem(A,G,α),obtainedbyliftingcompletelyboundedmapson
theC*-algebraAthatsatisfyanaturalcovarianceproperty,andexhibitacanonical
,,wecon-
sidercompletelycompactmultipliersofthedynamicalsystem(c0(G),G,α),where
,viatheStone-vonNeumann
Theorem,mappingsonthespaceKofallcompactoperatorson2(G)thatarechar-
acterisedintermsoftheHaageruptensorproductK⊗,
weshowthatanycompactSchurmultiplieronKgivesrisetoanaturalcompletely
compact(c0(G),G,α)-multiplier.
Weflnishthisintroductionwithageneralcommentaboutaseparabilityassumption.
ManyofourresultsrelyonthedevelopmentofthetheoryofHerzSchurmultipliers
ofC*-dynamicalsystems(A,G,α),undertakenin[25].In[25],theC*-algebraA
,ifthegroupGofthe
dynamicalsystemisdiscrete,aspointedoutbefore[26,],aninspection
oftheproofsfrom[25]revealsthattheseparabilityassumptiononAcanbelifted.
Inthepresentpaper,allC*-dynamicalsystems(A,G,α)willbeassumedtobeover
discretegroupsandarbitraryC*-algebrasA.
2SchurandHerz–SchurMultipliers
WedenotebyB(H)thealgebraofallboundedlinearoperatorsacingonaHilbert
spaceH,andbyIH(orIwhenHisclearfromthecontext)theidentityoperatoron
⊆B(H),weletMn(X)bethespaceofallnbynmatrices
withentriesinX,andidentifyitwithasubspaceofB(Hn)(whereHnisthedirectsum
ofncopiesofH).IfXandYareoperatorspaces,actingonHilbertspacesHandK,:.
47Page4of25JournalofFourierAnalysisandApplications(2022)28:47
respectively,andϕ:X→Yisalinearmap,weletasusualϕ(n):Mn(X)→Mn(Y)
(n)
bethemapgivenbyϕ(xi,j)i,j=ϕ(xi,j)i,
(n)
boundedifϕcb:=supn∈Nϕ<∞.WewriteCB(X,Y)forthespaceofall
⊗minYbetheminimaltensor
productofXandY,thatis,theclosureofthealgebraictensorproductX⊗Ywhen
consideredasasubspaceofB(H⊗K).Wewillusethroughoutthepaperbasicresults
fromoperatorspacetheory,andwereferthereadertothemonographs[10,29]forthe
necessarybackground.
ForalocallycompactgroupG,weletλ0beitsleftregularrepresentationonL2(G);
thus,
λ0g(s)=g(t−1s),g∈L2(G),s,t∈G.
t
Weusethesamesymbol,λ0,todenotetheleftregularrepresentationofL1(G)on
L2(G).Let
C∗(G)={λ0(f):f∈L1(G)}⊆B(L2(G))
r
w∗
bethereducedC*-algebraofG,VN(G):=Cr∗(G)thevonNeumannalgebraofG
(herew∗denotestheweak*topologyofB(L2(G))),andA(G)betheFourieralgebra
ofG,thatis,thecollectionofthefunctionsonGoftheforms→(λ0ξ,η),whereξ,η∈
s
L2(G).ThealgebraA(G)willbeequippedwiththeoperatorspacestructurearising
fromitsidentiflcationwiththepredualofVN(G);itsnormwillbedenotedby·A,and
by·A(G)(G)(resp.
Br(G))fortheFourier-Stieltjes(-Stieltjes)
spaceB(G)((G))isgeneratedbycontinuouspositive-deflnitefunctionsonG
(-deflnitefunctionsweaklyassociatedtotheleftregularrepresentation
λ0ofG),andonehastheinclusionsA(G)⊆Br(G)⊆B(G).By[12,],
B(G)andB(G)canbeidentifledwiththedualspaceofthefullC*-algebraC∗(G)
r
andreducedC*-algebraC∗(G)ofG,,whenB(G)isequipped
r∗∗
withthenormarisingfromtheidentiflcationB(G)=C(G),itbecomesaBanach
algebrawithrespecttothepointwisemultiplication,andA(G)andBr(G)areclosed
idealsofB(G).ThenormsonA(G)andBr(G)inheritedfromB(G)coincidewith
thenormsarisingfromtheidentiflcationsA(G)∗=VN(G)andBr(G)=C∗(G)∗.
r
Wereferthereadertothemonograph[21]fornecessaryfurtherbackgroundfrom
AbstractHarmonicAnalysis.
Afunctionu:G→CiscalledamultiplierofA(G)ifuv∈A(G)forevery
v∈A(G).WedenotebyMA(G)thealgebraofallmultipliersofA(G).Anelement
u∈MA(G)iscalledaHerzSchurmultiplierofA(G)[6]ifthemapv→uvon
A(G)iscompletelybounded(here,andinthesequel,weequipA(G)andB(G)with
theoperatorspacestructures,arisingfromtheidentiflcationsA(G)∗=VN(G)and
B(G)=C∗(G)∗).WeletMcbA(G)bethealgebraofallHerzSchurmultipliersof
A(G).Wenotethatu∈McbA(G)ifandonlyifthemapSu:C∗(G)→C∗(G),
rr
λ0(f) →λ0(uf),f∈L1(G),iscompletelybounded,whichisprovedusingsimilar
argumentstotheonesin[6,].:.
JournalofFourierAnalysisandApplications(2022)28:47Page5of2547
WehenceforthflxaHilbertspaceHandanon-degenerateC*-algebraA⊆B(H).
LetGbeadiscretegroupandα:G→Aut(A)beapoint-normcontinuoushomo-
morphism;thus,(A,G,α)isaC*-{δs:s∈G}forthe
canonicalorthonormalbasisof2(G).Welet1(G,A)betheconvolution*-algebra
ofallsummablefunctionsf:G→A,setH:=2(G)⊗Handidentifyitwith
theHilbertspace2(G,H)ofallsquaresummableH-
λ:G→B(H),t→λt,betheunitaryrepresentationofGgivenby
λtξ(s)=ξ(t−1s),s,t∈G,ξ∈H;
notethatλt=λ0⊗:A→B(H)bethe*-representationgivenby
t
π(a)ξ(s)=αs−1(a)(ξ(s)),a∈A,ξ∈H,s∈G.
Wenotethecovariancerelation
π(αt(a))=λtπ(a)λ∗,a∈A,t∈G.(1)
t
Thepair(π,λ)givesrisetoa*-representationπ˜:1(G,A)→B(H),givenby
1
π(˜f)=π(f(s))λs,f∈(G,A).(2)
s∈G
(Notethattheseriesontherighthandsideof(2)convergesinnormforeveryf∈
1(G,A)).ThereducedcrossedproductAα,rGisdeflnedbyletting
Aα,rG=π(˜1(G,A)),
wheretheclosureistakenintheoperatornormofB(H).Notethat,afteridentifyingA
withπ(A),wemayconsiderAasaC*-subalgebraofAα,-knownthat
ifρ:A→B(K)isafaithfulnon-degenerate*-representationandα
isthecanonical
actionofGonρ(A),arisingfromα,thenAα,rG∼=∼=ρ(A)α
,rGcanonically(see
.[30,]).
IdentifyingHwith⊕s∈GH,weassociatetoeveryoperatorx∈B(H)amatrix
(xp,q)p,q,wherexp,q∈B(H),p,q∈G;thus,
xp,qξ,η=
x(δq⊗ξ),δp⊗η,ξ,η∈H,p,q∈G.
Inparticular,ifa∈Aandt∈Gthen
α−1(a)ifpq−1=t
(π(a)λ)=p(3)
tp,q−1
0ifpq=t.
Equation(3)impliesthat
(π(a)λt)e,qp−1=δt,pq−1aEvaluationWar