Characterisations of crossed products by partial a

arXiv:funct-an/9604001v13Apr1996CHARACTERISATIONSOFCROSSEDPRODUCTSBYPARTIALACTIONSJOHNQUIGGANDIAINRAEBURNAbstract.PartialactionsofdiscretegroupsonC∗-algebrasandtheassociatedcrossedproductshavebeenstudiedbyExelandMc-Clanahan.Wecharacterisethesecrossedproductsintermsofthespectralsubspacesofthedualcoaction,generalisingandsimplify-ingatheoremofExelforsinglepartialautomorphisms.WethenusethischaracterisationtoidentifytheCuntzalgebrasandtheToeplitzalgebrasofNicaascrossedproductsbypartialactions.IntroductionExelhasrecentlyintroducedandstudiedpartialautomorphismsofaC∗-algebraA:isomorphismsofoneidealinAontoanother[5].HehasshownthatmanyinterestingC∗-algebrascanbeviewedascrossedproductsbypartialautomorphisms,andthatthesecrossedproductshavemuchincommonwithordinarycrossedproductsbyactionsofZ.McClanahansubsequentlyextendedExel’sideastocoverpartialactionsofmoregeneralgroupsbypartialautomorphisms,andshowedthat,rathersurprisingly,manyimportantresultsoncrossedproductsbyfreegroupscarryovertocrossedproductsbypartialactions[9].Herewegiveacharacterisationof(reduced)crossedproductsbypartialactionsofdiscretegroups,whichissimilarinspirittothatofLandstadforordinarycrossedproducts(see[7]or[12,7.8.8]),andwhichbothgeneralisesandsimplifiesExel’scharacterisationofcrossedproductsbysinglepartialautomorphisms[5,Theorem4.21].OurmainresultsaysthataC∗-algebraBisacrossedproductbyapartialactionofGifandonlyifitcarriesacoactionδofGandthereisapartialrepresentationofGbypartialisometriesinthedoubledualB∗∗whichinducessuitableisomorphismsamongthespectralsubspacesofδ;thisresulttakesaparticularlyelegantformwhenGisthefreegroupFn.WethenuseourclassificationtoidentifytheCuntzalgebrasOn,theDate:February13,1996.1991MathematicsSubjectClassification.Primary46L55.ThisresearchwaspartiallysupportedbytheAustralianResearchCouncilandbytheNationalScienceFoundation(underGrantNo.DMS9401253).12JOHNQUIGGANDIAINRAEBURNCuntz-KriegeralgebrasOA,andtheToeplitzorWiener-HopfalgebrasofNica[10]ascrossedproductsbypartialactionsofFn.WehavenotpreviouslyseenthecanonicalcoactionofFnonOnusedinaseriousway,sotheseideasmayhaveinterestingimplicationsforthestudyofcoactionsofdiscretegroups.Webeginwithadiscussionofpartialactionsandcovariantrepre-sentations.ApartialactionαofGonAisacollection{Ds:s∈G}ofidealsinA,andisomorphismsαsofDs−1ontoDs,suchthatαstextendsαs◦αtfromitsnaturaldomainDt−1∩α−1t(Ds−1).Calculationsinvolvingthedomainscanbetricky,sowehavetakencaretomakethevariousrelationshipsexplicit.ApartialrepresentationofGisamapuofGintothesetofpartialisometriesonaHilbertspace(orinaC∗-algebra)suchthatustextendsusut,andacovariantrepresenta-tion(π,u)of(A,G,α)consistsofarepresentationπofAandapartialrepresentationuofGsuchthatπ(αs(a))=usπ(a)u∗sfora∈Ds−1.Mc-Clanahandidnotdiscusspartialrepresentationsintheirownright,sowehaveincludedadetaileddiscussionofthemandtheirrelationshiptocovariantrepresentations.AkeytechnicalinnovationinourtreatmentistheimplementationofHilbert-moduleisomorphismsofspectralsubspacesbymultipliersofimprimitivitybimodules,asintroducedin[4];theparticularmul-tipliersinvolvedherewillformthepartialrepresentationofGinthedoubledualofthecrossedproduct.Wethereforerecallin§2somefactsaboutmultipliersofbimodules,relatethemtoHilbert-moduleisomor-phisms,anddiscusshowincertainsituationsthewholestructurecanbeembeddedinthedoubledualofaC∗-algebra.In§3,weconstructthecrossedproductA×αGofapartialactionα,astheC∗-algebrageneratedbyauniversalcovariantrepresentationof(A,G,α)in(A×αG)∗∗.AssociatedtoanyfaithfulrepresentationπofAisaregularrepresentationofA×αG;uptoisomorphism,theimageisindependentofthechoiceofπ,andiscalledthereducedcrossedproductA×α,rG.Ofcourse,bothcrossedproductsturnouttobetheonesstudiedin[9],butouremphasisonuniversalpropertiesallowsustoseequicklythattheycarryadualcoactionofG.OurcharacterisationofthereducedcrossedproductintermsofthisdualcoactionisTheorem4.1.AnordinaryactionαofthefreegroupFnisdeterminedcompletelybythenautomorphismsαgicorrespondingtogenerators{gi}ofFn.ItisquiteeasytoconstructpartialactionsofFnfromnpartialautomor-phisms[9,Example2.3],butingeneralevenpartialactionsofF1=Zneednotarisethisway.Soweconcentratein§5onafamilyofpartialCHARACTERISATIONSOFCROSSEDPRODUCTSBYPARTIALACTIONS3actionsαofFnwhicharedeterminedby{αgi};crossedproductsbysuchmultiplicativepartialactionscanbecharacterisedintermsofthespectralsubspacesofthedualcoactioncorrespondingtothegenera-torsofFn.ThemainresulthereisTheorem5.6,anditsapplicationstoCuntzalgebras,Cuntz-KriegeralgebrasandNica’sToeplitzalgebrasarethecontentofourlastsection.Acknowledgements.Asthispaperwasbeingwrittenup,theauthorsreceivedacopyof[6],inwhichExelprovesaresultrelatedtoourTheorem4.1.However,hisresultconcernsC∗-algebraicbundles,andusestechniquessubstantiallydifferentfromours.ThebulkofthisresearchwascarriedoutwhilethefirstauthorwasvisitingtheUniversityofNewcastle.HewishestothankIainRaeburnforhishospitality.TheauthorsthankMarceloLacaforhelpfulcon-versations,andforintroducingthemtotheworkofNicathroughacollaborationwiththesecondauthor[8].1.Partialactionsandcovariant