C-Crossed Products by Twisted Inverse Semigroup Ac

arXiv:funct-an/9712011v125Dec1997C∗-CROSSEDPRODUCTSBYTWISTEDINVERSESEMIGROUPACTIONSN´andorSiebenAbstractThenotionsofBusby-SmithandGreentypetwistedactionsareextendedtodis-creteunitalinversesemigroups.Theconnectionbetweenthetwotypes,andtheconnectionwithtwistedpartialactions,areinvestigated.Decompositiontheoremsforthetwistedcrossedproductsaregiven.1991MathematicsSubjectClassification.Primary46L55.1.IntroductionTheworkofRenault[Re1]connectingalocallycompactgroupoidtoitsam-pleinversesemigroupmakesthestudyofinversesemigroupsinteresting.SomeoftheresultsoninversesemigroupsarePaterson’s[Pa1]andDuncanandPaterson’swork[DP1],[DP2]ontheC∗-algebrasofinversesemigroups,Kumjian’slocaliza-tionC∗-algebras[Kum]andNica’s˜F-inversesemigroups[Nic].Wehaveseenin[Sie]thatthetheoryofcrossedproductscanbegeneralizedtoinversesemigroups.ThestrongconnectionbetweentheC∗-algebrasoflocallycompactgroupoidsandinversesemigroupsfoundbyPaterson[Pa2]promisesasimilarconnectionbetweenthegroupoidcrossedproductsof[Re2]andinversesemigroupcrossedproducts.Green[Gre]andPackerandRaeburn[PR]showedhowtousetwistedcrossedproductstodecomposecrossedproductC∗-algebras.Inthispaperwepartiallygeneralizetheirresultstodiscreteinversesemigroups.WeprovedecompositiontheoremsforbothGreenandBusby-Smithstyletwistedactions.Weshowthatunlikeinthegroupcase,GreentwistedactionsseemslightlymoregeneralthanBusby-Smithtwistedactions.Weshowthatthecloseconnectionbetweenpartialactions[Ex1],[McC]andinversesemigroupactionsseenin[Sie]and[Ex3]stillholdsforthetwistedpartialactionsof[Ex2]andBusby-Smithtwistedinversesemigroupactions.Itisanaturalquestiontoask,whetherthereisasimilarconnectionbetweenGreentwistedinversesemigroupactionsandsomesortofGreentwistedpartialactions.TheresearchforthispaperwascarriedoutwhiletheauthorwasastudentatArizonaStateUniversityunderthesupervisionofJohnQuigg.PartoftheresearchwasdoneduringashortvisitattheUniversityofNewcastle.TheauthorisgratefultoProfessorIainRaeburnforhishospitality.2N´andorSieben2.TwistedinversesemigroupactionsTwistedactionsoflocallycompactgroupswereintroducedin[BS].TheinversesemigroupversioncloselyfollowsExel’sdefinitionoftwistedpartialactionsin[Ex2].RecallthatasemigroupSisaninversesemigroupifforeverys∈Sthereexistsauniqueelements∗ofSsothatss∗s=sands∗ss∗=s∗.Themaps7→s∗isaninvolution.Anelementf∈Ssatisfyingf2=fiscalledanidempotentofS.Thesetofidempotentsofaninversesemigroupisasemilattice.ThereisanaturalpartialorderonSdefinedbys≤tifandonlyifs=ts∗s.Definition2.1.LetAbeaC∗-algebra.ApartialautomorphismofAisanisomorphismbetweentwoclosedidealsofA.Definition2.2.LetAbeaC∗-algebra,andletSbeaunitalinversesemigroupwithidempotentsemilatticeE,andunite.ABusby-SmithtwistedactionofSonAisapair(β,w),whereforalls∈S,βs:Es∗→EsisapartialautomorphismofA,andforalls,t∈S,ws,tisaunitarymultiplierofEst,suchthatforallr,s,t∈Swehave(a)Ee=A;(b)βsβt=Adws,t◦βst;(c)ws,t=1M(Est)ifsortisanidempotent;(d)βr(aws,t)wr,st=βr(a)wr,swrs,tforalla∈Er∗Est.Wealsorefertothequadruple(A,S,β,w)asaBusby-Smithtwistedaction.Notethatβr(aws,t)makessensesincea=a1a2forsomea1,a2∈Er∗Estandsoaws,t=a1a2ws,t∈a1Est⊂Er∗.Ourdefinitionisageneralizationofinversesemigroupactionsdefinedin[Sie].EveryinversesemigroupactionisatriviallytwistedBusby-Smithinversesemigroupactionbytakingwr,s=1M(Ers).Conversely,everytriviallytwistedBusby-Smithinversesemigroupactionisaninversesemigroupaction.ThedefinitionisalsoageneralizationofdiscretetwistedgroupactionsincaseourinversesemigroupSisactuallyagroup.ThebasicpropertiesofBusby-Smithtwistedinversesemigroupactionsarecollectedinthefollowing.Lemma2.3.If(A,S,β,w)isaBusby-Smithtwistedactionandr,s,t∈S,then(a)Es=Ess∗;(b)βss∗=idEs;(c)βe=idA;(d)βs∗=Adws∗,s◦β−1s;(e)βr(Er∗Es)=Ers;(f)βr(aw∗s,t)=βr(a)wr,stw∗rs,tw∗r,sforalla∈Er∗Est;(g)βr(ws,ta)=wr,swrs,tw∗r,stβr(a)foralla∈Er∗Est;twistedsemigroupactions3(h)βr(w∗s,ta)=wr,stw∗rs,tw∗r,sβr(a)foralla∈Er∗Est;(i)ws∗r∗r,s=ws∗,r∗rs;(j)ws∗,s1M(Es∗r∗)=ws∗,r∗rs;(k)βs(ws∗,s)=ws,s∗.Proof.(a)and(b)followfromthecalculationsEs=dom(βsβs∗)=dom(Adws,s∗◦βss∗)=Ess∗βf=βfβfβ−1f=Ad1M(Ef)◦βffβ−1f=βfβ−1f=idEf,wheref=ss∗.(c)isaspecialcaseof(b).Using(b)wehave(d)sinceβs∗=βs∗βsβ−1s=Adws∗,s◦βs∗sβ−1s=Adws∗,s◦β−1s.Wehaveβr(Er∗Es)=im(βrβs)=im(Adwr,s◦βrs)=Ers,whichproves(e).Replacingabyaw∗s,tinDefinition2.2(d)gives(f).Applying(f)wehave(g)becauseβr(ws,ta)=βr(a∗w∗s,t)∗=(βr(a∗)wr,stw∗rs,tw∗r,s)∗=wr,swrs,tw∗r,stβr(a).Replacingabyw∗s,tain(g)wehave(h).Toshow(i)letb∈Es∗r∗.Thenby(a),(b)and(e)thereisana∈EsEr∗=βr∗r(Er∗rEs)=Er∗rssuchthatb=βs∗(a).Henceb=βs∗(awr∗r,s)(2.2(c))=βs∗(a)ws∗,r∗rws∗r∗r,sw∗s∗,r∗rs(2.2(d))=bws∗r∗r,sw∗s∗,r∗rs((a)and2.2(c)),whichmeansws∗r∗r,sw∗s∗,r∗rsistheidentityofEs∗r∗.Thisimpliesourstatementsinceby(a)bothws∗r∗r,sandws∗,r∗rsareunitarymultipliersofEs∗r∗.Toshow(j)letaandbasintheproofofpart(i).Thenwehaveb=βs∗(aws,s∗r∗rs)(2.2(c))=βs∗(a)ws∗,sws∗s,s∗r∗rsw∗s∗,ss∗r∗rs(2.2(d))=bws∗,s1M(Es∗r∗)w∗s∗,r∗rsandthestatementfollowsasinpart(i).Finally,(k)followsfromDefinition2.2(c,d)ifweextendβstothemultipliersofEs∗.4N´andorSiebenRecallfrom[Pet],[CP],[How]thatacongruencerelationonaninversesemi-groupSisanequ