CHU-SPACES, A GROUP ALGEBRA AND INDUCED REPRESENTA

TheoryandApplicationsofCategories,Vol.5,No.8,1999,pp.176{201.CHU-SPACES,AGROUPALGEBRAANDINDUCEDREPRESENTATIONSEVASCHLAPFERTransmittedbyMichaelBarrABSTRACT.UsingtheChu-construction,wedeneagroupalgebrafortopologicalHausdorgroups.Furthermore,forisometric,weaklycontinuousrepresentationsofasubgroupHofaHausdorgroupGinducedrepresentationsareconstructed.1.IntroductionThegroupalgebrakGofanitegroupGisusuallydenedtobeavectorspaceovertheeldkwhosebasisisthegroupelements.ThegroupmultiplicationgivesamultiplicationonthisbasiswhichcanbeextendedbylinearitytokG.Thereisacanonicalbijectionbetweenrepresentationsofthegroupandmodulesoverthegroupalgebra,morepreciselyanisomorphismofcategories.Thegroupalgebraisusedtodeneinducedrepresentationsasthetensorproductoverthesubgroupalgebraofarightandaleftmodule(seeforexample[13]).InthispaperwewilldeneagroupalgebraLGforaHausdortopologicalgroupG.Furthermore,wewillintroduceanotionofweaklycontinuousrepresentationsandforagivenisometricweaklycontinuoussubgrouprepresentationdeneaninducedisometricweaklycontinuousrepresentationofthegroupusingthegroupalgebra.ThealgebraLGwillbedenedinthecategoryofseparatedextensionalChu-spacesovertheautonomouscategoryofBanachspacesandcontractinglinearmaps.TheChu-algebraLGisagroupalgebrainthesensethatthereisabijectionbetweenweaklycontinuousisometricrepresentationsandLG-Chu-modules,morepreciselyacanonicalequivalenceofcategories.Thisisdoneinthethirdsection.LetHbeasubgroupofG,thereisanaturalmapfromLHtoLGthatrestrictsanLG-Chu-moduletoanLH-Chu-module.Inthefourthandlastsection,wewilldeneaninducedLG-Chu-moduleforanygivenleftLH-Chu-module.NotethattheconstructionofthegroupalgebraLGisdoneongeneralHausdortopologicalgroups.InparticularwedonotsupposetheexistenceofaHaarmeasureasintheclassiccaseandtherearenomeasure-theoreticarguments.ThispaperwaswrittenwhentheauthorwasapostdocatMcGillUniversitywithagrantoftheSwissNationalScienceFoundation.TheauthoralsoacknowledgessupportoftheSwissNationalScienceFoundationproject2000-050579.97Receivedbytheeditors1999January1and,inrevisedform,1999May5.Publishedon1999May28.1991MathematicsSubjectClassication:Primary22A25,Secondary18D15.Keywordsandphrases:chu-spaces,groupalgebra,inducedrepresentation.cEvaSchlapfer1999.Permissiontocopyforprivateusegranted.176TheoryandApplicationsofCategories,Vol.5,No.8177InthenextsectionwewillbrieyexplaintheChu-constructionforthereadersun-familiarwiththisnotion.Fortheproofs,whicharerathertechnicalwewillreferto[5].2.Preliminaries2.1.TheChu-construction.TheconstructionthatisnownamedafterhimwasrstdescribedbyP.-H.ChuinhisM.Sc.thesis[5],anothermorerecentreferenceis[1].Itassociatestoanautonomous(symmetricmonoidalclosed)categoryVwithpullbacksandaxedobjectKinVa-autonomouscategorycalledChu(V;K).Thisconstructiongivesusmanyexamplesof-autonomouscategories,whicharediculttoobtaindirectly(see[2]).ForamoredetaileddiscussionoftheadvantagesofusingtheChuconstructiontodeneandstudythe-autonomouscategoriesof[2]see[3].2.2.ThecategoryChu(V;K).LetVbeanautonomouscategorythathaspullbacksandletKbeanobjectinV.ThendeneacategoryChu(V;K)asfollows.Theobjectsareoftheform(V;V0;h;i)whereVandV0areobjectsofVandh;i:VV0!KisamorphismofVcalledapairing.Normally,wewillnotcitethepairingexplicitlyandoftenusethenotationV=(V1;V2)orV=(V;V0)foranobjectinChu(V;K).Amorphism(f;f0):(V;V0)!(W;W0)inChu(V;K)isapairf:V!Wandf0:W0!V0ofarrowsofV,suchthatthefollowingdiagramcommutes.VW0idf0-VV0WW0fid?h;i-K?h;iThiscanbesymbolizedbytheequationhfv;w0i=hv;f0w0i.LetV{WbetheinternalhomonVbetweenVandW,deneaninternalhominChu(V;K)asfollows.Thecanonicalmorphismh;i:VV0!Kdeterminestwomorphisms,thetransposesofthismap,inVV!(V0{K)andV0!(V{K):UsingthesemapsandtheisomorphismsV{(W0{K)=(VW0){K=W0{(V{K)TheoryandApplicationsofCategories,Vol.5,No.8178wedeneV((V;V0);(W;W0))tobetheobjectoccurringinthefollowingpullbackdiagramV((V;V0);(W;W0))-V{WW0{V0?-W0{(V{K)=(VW0){K=V{(W0{K)?ThisdiagramdenesnotonlyanobjectbutalsoanarrowV((V;V0);(W;W0))!(VW0){K,hencedene(V;V0){(W;W0)tobe(V((V;V0);(W;W0));VW0).LetbetheunitforthetensorproductinVandK!Kthecanonicalisomor-phism.ThedualityonChu(V;K)isdenedbythemapthattakes(V;V0)to(V0;V)andinthesamewayonmorphisms.Itisnotdiculttoprovethat(K;)isthedualizingobject,thatis(V0;V)=(V;V0){(K;).ThisdenitionrendersChu(V;K)self-dual.Oncetheinternalhomandthereexivedualityareknown,thetensorproductonChu(V;K)isexplicitlygivenby(V;V0)(W;W0)=((V;V0){(W;W0))=(VW;V((V;V0);(W0;W))):andChu(V;K)isclosed.Theunitforthetensorproductis(;K).ThecategoryChu(V;K)withtheduality,internalhomandtensorproductasex-plainedaboveisa-autonomouscategory.See[5]fordetails.AnobjectinthecategoryChu(V;K)willbecalledaChu-space.Thisismotivatedbythefollowingexample,duetoVaughanPratt.AnobjectofChu(Set;2)isasetStogetherwithasetS0equippedwithamapS0!2S.Ifthismapisinjective(seediscussionbelow),thenS0canbeidentiedwithasetofsubsetsofS,thatisthebeginningsofatopology.2.3.Thecategoryofseparated-extensionalChu-spaces.Supposeafactorisa-tionsyste