Random Operators and Crossed Products

arXiv:math-ph/9902030v126Feb1999RANDOMOPERATORSANDCROSSEDPRODUCTSDanielH.LenzJanuary1999Abstract.Thisarticleisconcernedwithcrossedproductsandtheirapplicationstorandomoperators.WestudythevonNeumannalgebraofadynamicalsystemusingtheunderlyingHilbertalgebrastructure.ThisgivesaparticularlyeasywaytointroduceatraceonthisvonNeumannalgebra.Wereviewseveralformulasforthistrace,showhowitcomesasanapplicationofConnes’noncommutativeintegrationtheoryanddiscussShubin’straceformula.WethenrestrictourselvestothecaseofanactionofagrouponagroupandincludenewproofsforsometheoremsofBellissardandTestardonananalogueoftheclassicalPlancherelTheorem.Weshowthattheintegrateddensityofstatesisaspectralmeasureintheperiodiccase,therbygeneralizingaresultofKaminkerandXia.Finally,wediscussdualityresultsandapplyamethodofGordonetal.toestablishadualityresultforcrossedproductsbyZ.0.IntroductionFamiliesofrandomoperatorsariseinthestudyofdisorderedmedia.Moreprecisely,oneisgivenatopologicalspaceXandafamilyofoperators(Hx)x∈XonL2(G).Here,Xrepresentsthesetof”allmanifestations”ofafixedkindofdisorderonthelocallycompactabeliangroupG[3,4].Thesimplestexampleofadisorderedmediumisgivenbytheperiodicstructureofacrystal.InthiscaseXisthequotientofGbythesubgroupofperiods.InthegeneralcaseXwillnotbeaquotientofG,buttherewillstillbeanactionαofGonX.ThefactthatallpointsofXstemfromthesamekindofdisorderstructureistakenaccountofbyrequiringtheactiontobeergodic.Whereasforafixedx∈XtheoperatorHxmaynothavealargesymmetrygroup,thewholefamilyofoperatorswillbeG-invariant.Thisleadstothestudyofthisfamilyasanewobjectofinterest.ThisstudyisbestperfomedinthecontextofC∗-algebras.Infact,itturnsoutthatthecrossedproductsG×αC(X)provideanaturalframworkfortheseobjects[3,6,7,10,36].Asitis,oneisevenledtoamoregeneralalgebraicstructure,vizC∗-algebrasofgroupoidswhenstudyingcertainquasicrystalsmodelledbytilings[21,26,27].Butthisisnotconsiderdhere.InaremarkableseriesofpapersBellissardandhisCo-workersintroducedaK-theorybasedmethodcalled”gaplabelling”forthestudyofrandomoperators[3,4,5,19].1UsingresultsofPimsnerandVoiculescu[32],theywereabletogetadescriptionofthepossiblegapsinmanyimportantcases.AsK-theoryisbestknowninthecases,whereeitherGisdiscreteorXstemsfromanalmostperiodicfunction,muchoftheirworkwasdevotedtothesecases.However,therearemanyimportantexamplesofmoregeneralrandomoperators[11,12,28,29].Thisisoneofthestartingpointsofthisarticle.Infact,themainpurposeofSections1and2istostudytheframeworkofgeneralrandomoperators.ThisisdonebymeansofHilbertalgebras.Sections3and4arethendevotedtospecialresultsinthefieldofrandomoperators.Moreprecisely,thisarticleisorganizedasfollows.InSection1weintroducecrossedproducts,studytwoimportantrepresentationsandrevisetheirbasictheory.Specialattentionispaidtotherelationshipbetweensymmetrypropertiesofrandomoperatorsanddirectintegraldecompositions.InSection2weuseHilbertalgebrastothestudyofthevonNeumannalgebrasandtheC∗-algebrasofthedynamicalsystemsofSection1.WeusethemtointroduceatraceonthesevonNeumannalgebras.WeshowthatthistracecoincideswiththetraceintroducedbyShubinforalmostperiodicoperators[36]andwiththetracestudiedbyBellissardandothersfordiscreteG[3,4,5].Moreover,wediscusshowitisconnectedwithConnes’noncommutativeintegrationtheory[13].InSection3westudythecasethatXisagroupitself.WestudytherelationbetweenthetworepresentationsintroducedinSection1.Weprovideproofsforsometheoremsfirstannouncedin[6]and[7](cf.[2]aswell),whoseproofsneverseemtohaveappearedinprint.Moreover,werevisetheBlochtheoryforperiodicoperatorsfromthealgebaricpointofview.Thispointofviewhastheadvantagethattheoperatorsinquestionsareneitherrequiredtohavepurepointspectrumnortohaveakernel.Weshowthattheintegrateddensityofstatesisaspectralmeasureinthiscaseforpurelyalgebraicreasons.ThisgeneralizesaresultofKaminkerandXia[25]andsimplifiestheirproof.Finally,inSection4,weadaptamethoddeveloppedbyGordonetal.[22]forthestudyofthealmostMathieuoperatortogeneralcrossedproductsbyZ.1.TheC∗-algebraG×αC0(X)Toeverydynamicalsystem(G,α,X)aC∗-algebracanbeconstructedcalledthecrossedproductofGandC0(X)anddenotedbyG×αC0(X).IfXconsistsofonlyonepoint,thenG×αC0(X)isnothingbutthegroupC∗-algebraC∗(G).WewillbeconcernedwithtwospecialrepresentationsofG×αC0(X).Forfurtherdetailsongeneralcrossedproductswereferto[31,38],fordetailsontopologicaldynamicsandcrossedproductssee[39,40,41].1.1BasicDefinitions.Adynamicalsystemisatriple(G,α,X)consistingof-aseparable,metrizable,locallycompact,abeliangroupG,whoseHaarmea-surewillbedenotedbyds,-aseparable,metrizablespaceX,-acontinuousactionαofGonX,Moreoverwewillneed-anα-invariantmeasureonXwithsuppm=Xtodefinetherepresentationsconsideredbelow.Weemphasizethatthismeasureisnotneededtodefinethecrossedproduct.2ThegroupGisactingonL2(G):=L2(G,ds)byTt:L2(G)→L2(G),Ttξ(s):=ξ(s−t),s,t∈G,ξ∈L2(G)andonL2(X):=L2(X,dm)bySt:L2(X)→L2(X),Stξ(x):=ξ(α(−t)(x)),t∈G,x∈X,ξ∈L2(X).GivenatopologicalspaceY,wedenotebyCc(Y)(C0(Y),Cb(Y)resp.)thealgebraofcontinuousfunctionswithcompactsupport(vanishingatinfinity,beingboundedresp).Letk·k∞denotethesupremumnormoneitherofthese