Hecke algebras of semidirect products and the fini

arXiv:math/0609316v2[math.OA]18Oct2007HECKEALGEBRASOFSEMIDIRECTPRODUCTSANDTHEFINITEPARTOFTHECONNES-MARCOLLIC∗-ALGEBRAMARCELOLACA1,NADIAS.LARSEN2,ANDSERGEYNESHVEYEV2Abstract.WestudyaC∗-dynamicalsystemarisingfromtheringinclusionofthe2×2integermatricesintherationalones.TheorientationpreservingaffinegroupsoftheseringsformaHeckepairthatiscloselyrelatedtoarecentconstructionofConnesandMarcolli;ourdynamicalsystemconsistsoftheassociatedreducedHeckeC∗-algebraendowedwithacanonicaldynamicsdefinedintermsofthedeterminantfunction.WeshowthattheSchlichtingcompletionalsoconsistsofaffinegroupsofmatrices,overthefiniteadeles,andweobtainresultsaboutthestructureandinducedrepresentationsoftheHeckeC∗-algebra.InasomewhatunexpectedparallelwiththeonedimensionalcasestudiedbyBostandConnes,thereisagroupofsymmetriesgivenbyanactionofthefiniteintegralideles,andthecorrespondingfixedpointalgebradecomposesasatensorproductovertheprimes.ThisdecompositionallowsustoobtainacompletedescriptionofanaturalclassofequilibriumstateswhichconjecturallyincludesallKMSβ-statesforβ6=0,1.IntroductionLetHdenotetheupperhalfplaneandletAfbetheringoffiniteadeles.ThegroupGL+2(Q)of2×2rationalmatriceswithpositivedeterminantactsonH×Mat2(Af),byM¨obiustransformationsonHandbyleftmultiplicationonMat2(Af).Roughlyspeaking,theC∗-algebraunderlyingtheGL2-systemofConnesandMarcolli[3]canbeconstructedbyeffectingtwomodificationsonthecorrespondingtransformationgroupoidGL+2(Q)×(H×Mat2(Af));thefirstoneistocutdownfromAftothecompactopensubringoffiniteintegraladelesˆZ=QpZpyieldingakindofsemigroupcrossedproductbyMat+2(Z),andthesecondoneistoeliminatethedegeneracyduetotheΓ=SL2(Z)symmetriesbyfactoringouttheactionofΓ×Γgivenby(γ1,γ2)(g,x)=(γ1gγ−12,γ2x).Thesemodificationsdestroytheinitialgroupoidandsemigroupcrossedproductstructures,buttheresultingC∗-algebraC∗r(Γ\GL+2(Q)⊠Γ(H×Mat2(ˆZ))),forwhichweusethenotationof[12],retainsenoughoftheoriginaltransformationgroupoidflavorthatitispossibletouseslightlymodifiedcrossedproducttechniquesinitsstudy.TheconvolutionformulathatgivestheproductontheConnes-MarcolliC∗-algebraisbasedontheconvolutionformulafortheclassicalHeckealgebra.ThisconnectionwaspursuedearlyonbyTzanev[23],whopointedoutthattheConnes-MarcolliC∗-algebracouldalsobedescribedasC∗r(P0\P×P0H),where(P,P0)=1Mat2(Q)0GL+2(Q),1Mat2(Z)0SL2(Z)isaHeckepairandtheactionofPonHisdefinedbyM¨obiustransformationsthroughtheobvioushomomorphismP→GL+2(Q)(technically,theactionofP0isnotproper,buttheconstructionofC∗r(P0\P×P0H)stillmakessenseby[12,Remark1.4]).Thus,theConnes-MarcolliC∗-algebracanbeDate:October16,2006;minorcorrectionsJune20,2007.PartofthisworkwascarriedoutthroughseveralvisitsofthefirstauthortotheDepartmentofMathematicsattheUniversityofOslo.Hewouldliketothankthedepartmentfortheirhospitality,andtheSUPREMAprojectforthesupport.1)SupportedbytheNaturalSciencesandEngineeringResearchCouncilofCanada.2)SupportedbytheResearchCouncilofNorway.12M.LACA,N.S.LARSEN,ANDS.NESHVEYEVthoughtofasanewtypeofcrossedproduct:ofthealgebraC0(Γ\H)bytheHeckepair(P,P0),and,inparticular,thereducedHeckeC∗-algebraC∗r(P,P0)iscontainedinthemultiplieralgebraoftheConnes-MarcolliC∗-algebra,see[12,Lemma1.3].Becauseofthis,abusingslightlytheterminology,wewillrefertoC∗r(P,P0)asthefinitepartoftheConnes-MarcolliC∗-algebra,andwepointoutthatthisfinitepartcorrespondstothequotientofthedeterminantpartoftheGL2-system,cf.[3,Section1.7],bytheaboveactionofΓ×Γ.ThegoalofthepresentworkistostudythestructureofC∗r(P,P0)andthephasetransitionofthecorrespondingC∗-dynamicalsystem.WewereinitiallymotivatedbyourbeliefthatitshouldbepossibletoexploitthecrossedproductstructureobservedbyTzanevinordertostudythephasetransitionoftheConnes-Marcollisystem,andthatinordertodothisonewouldhavetounderstandfirstthestructureandthephasetransitionofthefinitepart.WewerealsomotivatedbytheobservationthattheHeckepair(P,P0)consistsoftheorientationpreservingaffinetransformationsoftheringsof2×2matricesovertherationalsandovertheintegers,andhencetheassociatedC∗-dynamicalsystemisaverynatural(albeitsomewhatna¨ıve)higherdimensionalversionoftheonestudiedbyBostandConnes[1],whichcertainlydeservesconsideration.Inaddition(P,P0)isaveryinterestingexampleofaHeckeinclusionofsemidirectproductgroups,aclassthathasreceivedconsiderableattentioninrecentyears,seee.g.[2,11,15,6].Asitturnedout,wewereabletostudytheConnes-MarcolliphasetransitionandtoprovetheuniquenessoftheKMSβ-statesforβinthecriticalintervalbyamoredirectmethodthatdoesnotrequireconsiderationofC∗r(P,P0),althoughitdoesrelyonitforinsight,see[12].Interestinglyenough,thephasetransitionofthefinitepartoftheConnes-MarcollisystemseemstobeamoreresilientproblemthanfortheirfullGL2-system.Themainreasonforthisisthatthefreenessresultingfromthe‘infinitepart’,thatistosay,freenessofGL+2(Q)actingonH×(Mat2(Af)\{0}),isacrucialingredientinreducingKMS-statesoftheConnes-MarcolliGL2-systemtomeasuresonH×Mat2(Af).BecauseofthisourclassificationoftheKMS-statesofthefinitepartreliesonanextrahypothesisofregularitywhichallowsustousetechniquessimilartothoseof[1,10,16,3,12].Thisregularitypropertyseemsnaturalandwebel