A BOST-CONNES SYSTEM FOR Qp

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ABOST-CONNESSYSTEMFOR Q p BY CODYGUNTON AThesis SubmittedtotheDivisionofNaturalSciences NewCollegeofFloridainpartialfulllmentoftherequirementsforthedegree BachelorofArts UnderthesponsorshipofPatrickMcDonald,ProfessorofMathematics Sarasota,Florida May,2012

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Preface Theyoungeldofarithmeticnoncommutativegeometryexploresideasthatare,in mymind,ofunsurpassedbeauty.Incastingarithmeticquestionsintermsofphysics viathelanguageofnoncommutativegeometry,theeldprovidesanexcitingand promisingapproachtofundamentalproblemsofnumbertheory.Thoughtheeldis steadilygrowing,veryfewpeoplehavemorethanacursoryunderstandingofevenits basicprinciples,makingitdicultforanovicetoenter.Therefore,Ifeelextremely luckytohavehadtheopportunitytowritethisthesis,andforthisowethanksto myadvisor,ProfessorPatrickMcDonaldforallowingmethefreedomtotakeona dicultandsomewhatriskythesisproject.Ithankhimalsoforhisyearsofcareful attentionandguidance,histeaching,andforhisfriendship.IalsoProfessorsDavid MullinsandDonColladayfortheirinstructionandservicetomeasmembersofmy baccalaureatecommittee. IthankmyfriendsatNewCollegeforthreeyearsofsupportandsomanymemories thatIwillthinkoffondlythroughoutmylife.Thankstoyouall|Iamcertainlya healthierandhappierpersonbecauseofyourfriendship. Finally,Ithankmymotherandmysisterforbringingsomuchjoytomylife. i

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Contents Prefacei Abstractiv Chapter1.Introduction1 1.Explicitclasseldtheory1 2.Anapproachusingnoncommutativegeometry5 3.Localclasseldtheoryandthisthesis7 Chapter2.LocalFieldsandLocalClassFieldTheory9 1.The p -adicnumbers Q p 9 2.Nonarchimedeanlocalelds12 3.Statementsofclasseldtheory15 4.Explicitlocalclasseldtheory16 Chapter3. C -algebrasandQuantumStatisticalMechanics18 1.Introduction18 2.TheRiemannzetafunction19 3.Somephysics21 4.Hilbertspacesand C -algebras24 5. C -dynamicalsystems27 6.Group C -algebras28 7.Groupoidsandgroupoidalgebras30 8.vonNeumannalgebras32 ii

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Chapter4.ABost-ConnesSystemfor Q p 33 1.TheToeplitzalgebra p 33 2. ax + b groupsandthehomogeneousspace p 35 3.ThelocalBost-Connessystem C Q p 39 4.Representations48 5.Anactionof Z p 52 6.Futuredirections54 Bibliography55 iii

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ABOST-CONNESSYSTEMFOR Q p CodyGunton NewCollegeofFlorida,2012 Abstract Followingthe1995paperofBostandConnes,whichdenedtheBost-Connes system C Q ,wedeneandstudya C -dynamicalsystem C Q p relatedtotheclasseld theoryof Q p : Weconsiderquotientsofapair)]TJ/F80 11.9552 Tf 179.462 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 ofdiscretetwo-by-twomatrix groups,andshowthatthereisamapfromthecorrespondingHeckealgebra A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 tothealgebraofunitaryoperatorson ` 2 )]TJ/F80 11.9552 Tf 11.867 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : Extendingthismaptotheclosureof A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 inaregularrepresentationon ` 2 )]TJ/F80 11.9552 Tf 11.867 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 gives C Q p : Apresentationinterms oftwoclassesofgeneratorsisgiven,andisusedtondarepresentationof A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.315 -1.794 Td [(0 ; whichisconjecturedtoextendtoacovariantrepresentationof C Q p havingpartition functionequaltotheEulerfactoroftheRiemannzetafunctionat p: ProfessorPatrickMcDonald DivisionofNaturalSciences iv

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CHAPTER1 Introduction Thisintroductionbeginswithadescriptionoftherudimentsofalgebraicnumber theorynecessarytounderstandtheproblemofexplicitclasseldtheory.Inbrief,this istheproblemofgivingexplicitdescriptionsofobjectsthatariseinclasseldtheory butarenotconstructedinausefulwayintheproofs.Theintroductionthenbriey describesanewapproachtothisproblemusingnoncommutativegeometry,andits origininthe1995paper[ 3 ].Itendswithadescriptionofplaceofthisthesisinthat programofresearch.Generalpurposereferencesonthismaterialare[ 9 17 20 21 ]. 1.Explicitclasseldtheory Theclassicaltheoremsofalgebraicnumbertheoryarestatementsaboutnumber elds.Recallthatanumbereld K isaniteextensionoftherationalnumbers Q thatis,aeldthathasnitedimensionasavectorspaceover Q .Beinganite extension, K isnecessarilyalgebraic,henceisgeneratedbyattachingnitelymany rootsofpolynomialsto Q : Forexample,thequadraticeldsare Q p n = Q [ x ] = h x 2 )]TJ/F80 11.9552 Tf 11.956 0 Td [(n i .1 where0 ; 1 6 = n 2 Z : Thenotation h x 2 )]TJ/F80 11.9552 Tf 11.69 0 Td [(n i isusedfortheidealgeneratedby x 2 )]TJ/F80 11.9552 Tf 11.69 0 Td [(n Asaset, Q p n = f a + b p n : a;b 2 Q g : .2 Aeldextensionissometimeswrittenas K= Q toindicatethat K isbeingconsidered asanextensionoftherationals,ratherthananextensionofsomeintermediateeld. 1

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Havingextended Q ; itisnaturaltowonderifthereisacorrespondingextension oftheintegers Z : Thedenitionofsuchanextensionismotivatedbythefollowing: Proposition 1.1 Arationalnumber x 2 Q isthezeroofamonic 1 polynomial withcoecientsin Z ifandonlyif x 2 Z : Theifandonlyif"saysthatthisisanequivalentconditiontobeinganinteger. Hencewemakethefollowingdenition. Definition 1.1 Theintegers O K ofanumbereld K aretheelementsof K satisfyingamonicpolynomialwithcoecientsin Z : Ringsofalgebraicintegersdonot,ingeneral,haveallofthepropertiesofthe ring Z : Animportantdierenceisthatrationalintegersfactoruniquelyasproducts ofprimes,butringsofalgebraicintegersdon'tnecessarilyhavethisproperty.The standardexampleoffailureofuniquefactorizationisexpressedbytheequations + p )]TJ/F51 11.9552 Tf 9.298 0 Td [(5 )]TJ 11.956 9.975 Td [(p )]TJ/F51 11.9552 Tf 9.299 0 Td [(5=6=2 3asintegersin Q p )]TJ/F51 11.9552 Tf 9.298 0 Td [(5. Itturnsoutthatthisperceivedlossofuniquefactorizationisreallythelossofanother nicepropertyofthering Z ; Z isaprincipleidealdomain,butnotallringsofalgebraic integershavethisproperty.However,thefollowingisasuitablereplacement. Theorem 1.2 Thering O K isaDedekinddomain,andthusanyideal I of K determinesasetofexponents f e P : P isaprimeof K g ; onlynitely-manyof whicharenonzero,suchthat I = Y P P e P : .3 Itisconventionaltorefertoidealsof O K asidealsof K: Givensomeprime number p 2 Z ; onecanformaprimeideal p Z andtheideal p O K ; whichiseasilyseen 1 Apolynomial a 0 + + a n x n issaidtobemonicif a n =1. 2

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tocontain p Z : Theideal p O K neednotbeprime.Startingwithanumbereld k= Q andconsideringinsteadtherelativeextension K=k; onehasasimilarresult;if p is aprimeof K; then p O k isanidealof O K thatneednotbeprime.Afundamental questionmotivatingclasseldtheoryis Question 1.3 Givenanextension K=k ofnumbereldsandaprimeideal p O k ; howdoes p O K factorin O K ? Animportantstepintreatingthisquestionistoobservethatthedegreeof K itsdimensionasavectorspaceover k determines`howmuchgrowthaprimecan experiencewhenpromotedtoanidealof K .'Sinceanyprimeofanynumbereldis maximal,eachcorrespondingquotientisaeld,calledtheresidueeldof K atthat prime.When e P > 0inthefactorizationof p O K ; thenumber e P j p : = e P is calledtheramicationindexof P over p andthenumber f P j p : =# O K = P ; the dimensionoftheresidueeld,iscalledtheresidualdegree.Thesearerelatedbythe followingresult. Theorem 1.4 Supposethat K=k hasdegree n andthat p O K = Q P j p P e P j p Then n = P P j p e P j p f P j p : When K=k isGalois,allofthe e P j p areequal,say to e ,andallofthe f P j p areequal,sayto f; so n = efg where g isevidentlythe numberofprimefactorsof p O K : Theramicationandresidualindicesgiveawayofmeasuringhowfaranidealis frombeingprime.Therearethreeextremes: p O K canbeprime,so p O K = P with e P j p =1 ; whichforces f P j p = n ;itcanbetotallyramied,meaning p O K = P n with e P j p = n andthus f P j p =1 ; oritcansplit,havingmaximumallowable numberofprimefactor,sothat p O K = Q P j p p e P j p ; with e P j p = f P j p =1forall P j p : Classeldtheorysolvesthefollowingproblem: 3

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Problem 1.5 Givenaniteset S ofprimesof O k ,ndanabeliannumbereld K=k suchthattheprimesof k splittingin K areexactlythosein S: Notethehypothesisthat K=k beabelian,i.e.,thattheextensionisGaloiswith anabelianGaloisgroup.Thisisastrongrestriction,sayingthatgroupofsymmetries of K=k isofaparticularlysimpleclass.Theanalogousquestionforarbitrarynumber eldsisadicultopenproblem. Arststepinapproachingthisproblemistobundlealloftheabelianextensions of k intoasingleobject,themaximalabelianextension k ab =k: Asdescribedingreater detailinthenextchapter,oneofthemajorassertionsofglobalclasseldtheoryis theexistenceoftheArtinmap k : I k )167(! Gal k ab =k ; .4 ahomomorphismtakingtheideles I k of k cfDenition2.2intothesymmetry groupGal k ab =k : Thismapisshowntohavenicebehaviorunderrestrictiontonite subeldsof k ab ; or,equivalently,niteAbelianextensionsof k: Toparsethedenition oftheProblemofFabulousStatesgivenbelow,thereaderwillneedtoknowthatthe Artinmapcanbemodiedtoanisomorphismwithsourceequalto C k =C k ; a quotientoftheideleclassgroup C k : = I k =k ; byitsconnectedcomponentatthe identity. Theproofsofglobalclasseldtheoryarelongandabstract,relyingheavilyon groupcohomologycomputations 2 .Theproofsshow,forinstance,thattheArtinmap exists,butdonotgiveanexplicitdescriptionofthemath e.g., anitelistofrules forcomputingitsimage,anddonotgiveanexplicitdescriptionof k ab e.g., aset ofgeneratorsfor k ab =k .Theproblemofexplicitclasseldtheoryistogivemore tangibledescriptionsof k ab =k andtheArtinmap.Amorespecicvariant,calling forthegeneratorsofthemaximalabelianextensiontobegivenbysomeappropriate 2 NumbertheoristKevinBuzzardofImperialCollegeLondonontheproofsofglobalclasseldtheory:...youdon't wanttoreadtheproofs.Ididthispreciselyonceinmylifeandtheyareveryunilluminating."[ 2 ]. 4

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transcendentalcomplexfunction,isknownasHilbert's12thproblem,beingthat itwasthetwelfthstatedinhishighlyinuential1900addresstotheInternational CongressofMathematicians.ThisrenementoftheproblemisalsocalledKronecker's Jungendtraum,asitwasa youthfuldream ofhis. Theexplicitclasseldtheoryproblemisonlyconsideredsolvedintwoofthe simplestpossiblecases.Thesimplestisthecase k = Q ; wheretheKronecker-Weber theoremshowsthat Q ab = Q isgeneratedasaeldbytherootsofunity.Thenextsimplestcaseisthatwhere k isanimaginaryquadraticeld, i.e., where k = Q p )]TJ/F80 11.9552 Tf 9.298 0 Td [(n forsomepositivesquarefreeinteger n: Thisismuchmoredicultthantherst case.Here, k ab =k isgeneratedbyspecialvaluesofamodularformattachedtoan appropriateellipticcurvehavingaspecialsymmetrycalledcomplexmultiplication [ 20 ].Naturally,muchworkhasbeendoneonwhatshouldbenext-simplestnumber elds,therealquadraticelds,obtainedfrom Q byattachingthesquarerootofa positivesquarefreeinteger.Inthiscaseasetofgenerators,theStarknumbers,for themaximalabelianextensionhasbeenconjectured[ 19 ]. 2.Anapproachusingnoncommutativegeometry Intheir1995paper[ 3 ]HeckeAlgebras,TypeIIIFactorsandSpontaneousSymmetryBreakinginNumberTheory,"Jean-Beno^tBostandAlainConnesconstructa C -algebra C Q nowknownastheBost-Connessystem.Thisisstudiedasthealgebra ofobservablesofanensembleofquantummechanicalparticles,andsomeremarkable connectionstotheclasseldtheoryof Q arefound.Thatpaperprovesthefollowing: Theorem 1.6[ 3 ] Theequilibriumstatesof C Q areoftheform ; x : = Q )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 tr e e )]TJ/F23 7.9701 Tf 6.587 0 Td [(H ; .5 where e istheliftofarepresentation forsome 2 Gal Q ab = Q ofacertainHecke algebra A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 toits C -closure C Q ;H isanexplicitlydescribedHamiltonian, is 5

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arealparameterrepresentingthereciprocaltemperature,and Q istheRiemannzeta function.Inparticular,thepartitionfunctionof C Q istheRiemannzetafunction : MorewillbesaidabouttheRiemannzetafunctionlater.Fornow,wenotethatthe statesforxedinverse-temperature ofthesystem C Q arelabelledbytheelements ofthemaximalabelianextensionof Q : The2005paper[ 6 7 ]ofConnes,Marcolliand Ramachandranconstructsa C -algebraforanarbitraryimaginaryquadraticeldand provesananalogoustheorem.Thesetwopaperssuggestthatthetheoryofoperator algebrasmightshedsomelightontheexplicitclasseldtheoryproblem.Amore specicprogramofresearchfortheclasseldtheoryofrealquadraticeldswas proposedbyManinin[ 16 ].Thesolutiontotheexplicitclasseldtheoryproblemfor imaginaryquadraticeldsassociatestosuchaeldanellipticcurvetopologically, acomplextorusthathas`complexmultiplication,'meaningthatthecurvehasthe largestpossibleendomorphismgroup.Themaximalabelianextensioninthiscaseis generatedbyspecialvaluesofamodularformassociatedtotheellipticcurve.Manin's realmultiplicationprogramsuggestsdevelopingaparalleltheoryofnoncommutative toriwithspecialsymmetrytounderstandtheclasseldtheoryofrealquadraticelds. Somenoncommutativetoriwithrealmultiplicationhavealreadybeenstudiedin[ 18 ]. Theproblemofconstructingsuitablegeneralizationoftheconstructionsof[ 3 6 7 ]to othernumbereldsiscalledtheproblemoffabulousstates[ 17 ].Thoughsomeofthe objectsinvolvedhavenotbeenintroduced,astatementofthisproblemisgivenfor completeness. Problem 1.7TheProblemofFabulousStates Let k beanumbereld.The problemoffabulousstatesfor k isthatofconstructinga C -dynamicalsystem A; t withsubalgebra A Q satisfying: Thequotient C I k =C I k actson A assymmetriescompatiblewith t 6

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Letting E 1 denotethesetofextremalequilibriumstatesatzerotemperature, the 2 E 1 evaluatedon a 2 A Q ,satisfy a liesinanalgebraicclosureof k in C ; theelementsof f a : a 2 A;' 2 E 1 g generate k ab Theclasseldtheoryisomorphism e k : C I k =C I k )167(! Gal k ab =k intertwinestheactions, = e )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k ; forall 2 Gal k ab =k andforall 2 E 1 3.Localclasseldtheoryandthisthesis Thisthesisinitiatestheoperatoralgebrasstudyof local explicitclasseldtheory. Eachprimeidealofanumbereld k givesanabsolutevalue v on k: Thenonarchimedean,characteristic-zerolocaleldsarethecompletions k v with k anumber eldand v anabsolutevaluecomingfromaprimeidealof k: Localeldsareofgreat importanceinnumbertheory,providingsomethinglikeaworldparalleltothatof thenumberelds,whereoftenanideaaboutnumbereldsisstudiedintermsof localelds.Forinstance,thereisacompleteclasseldtheoryforlocaleldswhich providesalocalArtinmap k : k )167(! Gal k ab =k : Unlikeinthecontextofnumber elds,thereisacompletelyexplicittreatmentoflocalclasseldtheoryforarbitrary localelds,comprisingwhatisknownasLubin-Tatetheory. Chapter4constructsandstudiesa C -algebra C Q p thatisa p -adicanalogueof thesystemintroductedinthepaper[ 3 ]ofBostandConnes.Therststepinthe constructionistoassociatetoaprime p apairofdiscretematrixgroups)]TJ/F21 7.9701 Tf 170.246 -1.794 Td [(0 .The closureoftheassociatedHeckealgebra A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 initsleft-regularrepresentation on ` 2 )]TJ/F80 11.9552 Tf 11.866 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 givesasubalgebraof U \051 0 ; thecommutantoftheunitaryoperatorson 7

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` 2 )]TJ/F80 11.9552 Tf 11.867 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : Thissubalgebraisdenedtobe C Q p : Apresentationof C Q p iscomputed,and isusedtogiveacovariantrepresentationwithpartitionfunctionequaltotheEuler factorof Q at p: Finally,anactionon A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 byGal Q tot p = Q p isdeveloped.The thesisstopsshortofcompletelyconstructinga C -dynamicalsystemwithpartition functionequaltothefactoroftheRiemannzetafunctionat p; leavingasaconjecture atheoremregardingtheextensionofrepresentationsof A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 toits C -closure. 8

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CHAPTER2 LocalFieldsandLocalClassFieldTheory ThischaptergivesanaccountofthetheoryoflocaleldsandlocalclasseldtheoryrelevanttotheconstructionoftheBost-Connessystemfor Q p : Generalreferences forthischapterare[ 20 { 22 ]. 1.The p -adicnumbers Q p Recallthatanabsolutevalueisareal-valuedmap x 7)167(!j x j onaeld K satisfying thethreeconditions AV1 j x j 0withequalityifandonlyif x =0 AV2 j xy j = j x jj y j AV3 j x + y jj x j + j y j Theeld Q hasthefamiliarabsolutevalueoftherealnumberswhichxespositive numbersandmultipliesnegativenumbersby )]TJ/F51 11.9552 Tf 9.298 0 Td [(1 : Thereisalsoanabsolutevaluefor eachprimenumber p: Definition 2.1 Denethe p -adicabsolutevalueof r 2 Q by j r j p = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n ; where r = p n a b n 2 Z and p a;b Evidently,arationalnumberis p -adicallysmallerthananotherifitisdivisible byahigherpowerof p; andtwonumbersarecloseriftheirdierenceisdivisibleby ahigherpowerof p: Astartingpointinalgebraicnumbertheoryisthefollowing remarkabletheorem. Theorem 2.1 [Ostrowski]Anyabsolutevalueon Q isequivalenttotheabsolute valueoftherealsortosome p -adicabsolutevalue. 9

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Twoabsolutevaluesaresaidtobeequivalentifthecorrespondingmetrictopologiesarethesame.Theequivalenceisessentiallyachoiceofnameforeachopenball ofthemetricspace,andhenceisirrelevantinwhatfollows.Ostrowski'stheoremsuggestsanimportantchangeinperspective:thatweshouldregardabsolutevaluesas generalizationsofprimenumbers.Thoughthisperspectivedoesaordsomeuniformityinalgebraicnumbertheory,itisimportanttonotethatthe p -adicabsolutevalue is,inmanyways,drasticallydierentfromtheabsolutevalueofthereals,henceforth denotedby jj 1 .Thesedierencescomefromthefactthatthe jj p satisfy j x + y j p max fj x j p ; j y j p g ; .1 calledtheultrametricinequality,whichisstrongerthanthetriangleinequality. Recallthattheeldofrealnumbers R is,bydenition,thecompletionof Q at jj 1 : Theprocessofcompletionisvalidforanyabsolutevalueonavectorspace,and hencecanbeappliedto Q under jj p : Thisleadstotheelds Q p of p -adicnumbers, whicharecentralinwhatfollows. Theeld Q p isdenedasacompletion,soitmayfeelintangibletothosenot familiarwiththeprocedureofcompletion.Therestofthesectionwillbuildup Q p staringwithitsdiscretesubring Z ; sheddinglightontheviewof Q p asacompletion. Itisanelementaryfactthatanyintegercanbewrittenas a 0 = b 0 + c 0 p forsome 0 b 0

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Thatis,addinghighordertermsdoesnotchangethesizeofthesum,soanysuch innitesuminpowersof p converges.Thesetofsuchinnitesumsisalocally compacttopologicalring,theringof p -adicintegers,denotedby Z p ;asaset, Z p : = X i 0 a i p i : 0 a i

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that a=b 2 Z p if a 6 =0 ; soeach x j canbeexpandedinpowers p .Since f x j g converges, wehave j x k )]TJ/F80 11.9552 Tf 12.47 0 Td [(x j j p 0as j !1 for k>j: Thismeansthatthe p -expansionsof x k )]TJ/F80 11.9552 Tf 12.033 0 Td [(x j beginathigher-ordertermsforincreasing j; sothatforeachinteger n there isaninteger N suchthatall x j with j N havethesamerst n termsincommon. Thisshowsthat f x j g convergestoanelementof F .Thisprovesthat F = Q p : In viewofourabovedescriptionof Z p ; weseethattheelementsof Z p arethose p -adic numbersofnormnotexceeding1 ; i.e.,that Z p istheclosedunitballin Q p : Itisoftenusefultoconsideranelement x 2 Q p with 1 x = P i )]TJ/F23 7.9701 Tf 13.173 0 Td [(N x i p i asthesum x =[ x ]+ z ,where z : = P i 0 x i p i isevidentlyin Z p and[ x ] : = P 0 >i )]TJ/F23 7.9701 Tf 13.173 0 Td [(N x i p i isasum ofrationalnumbersof p -adicabsolutevalue > 1 : Therationalnumber[ x ]iscalledthe fractionalpartof x; and z theintegerpartof x: Itisclearthat x =[ x ]mod Z p isthen anelementoftheform a=p N with a notdivisibleby p .Everysuchrationalnumber correspondstoa p N -rootofunityunderthebijection a=p N exp ia=p N ; sowe identify Q p = Z p withthesetof p th-powerrootsofunityin C : Thesameconsiderations showthatthering Z p =p i Z p haselementsthatareinbijectionwith Z =p i Z : Proposition 2.3 Therearegroupisomorphisms Q p = Z p f setof p th-powerrootsofunity g S 1 C .3 Z p =p i Z p Z =p i Z : .4 2.Nonarchimedeanlocalelds Thefollowingtheoremisofdeepimportance. Theorem 2.4 Anylocaleldofcharacteristiczeroisaniteextensionofsome completionof Q : Toprovethis,onechoosesaHaarmeasure ontheeldandstudiesthefunction : K )167(! R determinedbytheequation xU = x U where U K isany 1 Whenevera p -adicnumberiswritteninthisform,itasassumedthat a )]TJ/F24 5.9776 Tf 5.756 0 Td [(N 6 =0 ; so j x j p = p N : 12

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openset.Thisthesisrestrictsattentiontothecaseofcharacteristic0,butitshould bementionedthatthelocaleldsofpositivecharacteristic,whichareexactlythe niteextensionsoftheelds F p T offormalpowerseriesoverniteelds,have provedfruitfulgroundsforformulatingconjecturesandtestingideasaboutnumber elds.ItisbecauseofTheorem2.4andthefactthatfunctioneldshaveanotionof 1 comingfromprojectivegeometrythatweusethenotation jj 1 ,andsometimes refertothisasthe`primeatinnity.' Anabsolutevalueonaeld K restrictstoanabsolutevalueonanysubeldof K: Thus,byTheorems2.4and2.1,thelocaleldsaredeterminedbythepossible extensionsoftheabsolutevalues jj p and jj 1 or,equivalently,theyaredetermined bytheextensionsof Q p and R .Theextensionsofthe Q p andthecorresponding absolutevaluesarecallednonarchimedean,sincetheydonothavethearchimedean propertythatforeach B 2 R andany x intheeldthereareonlynitely-many integers n suchthat j nx j B: Fortherestofthissection, K denotesanonarchimedeanlocaleld.Thering ofintegers O K of K isdenedtobetheclosedunitballin K; thesetofelements ofnormnotexceeding1.Itisafactthatthereisanelement k 2O K uniqueup tomultiplicationbyaunit,suchthateveryelementof K isoftheform u n K with u 2O K and n 2 Z : Theintegers O K arethenthose u n with n 0[ 21 ].There arenotionsoframicationandinertiainanalogywiththosefornumberelds,as discussedbrieyintheintroduction.Theramicationindexof K=k ofdegree n is denedtobetheinteger e K=k suchthatforsome u 2O K ; e K=k K = u k ; or,equivalently, O e K=k K = O k : .5 13

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Theresidualindex f K=k isdenedtobethedegreeoftheextensionofresidue elds f K=k : =deg h O K = K O k = k i .6 Thetheoryofnonarchimedeanlocaleldsandnumbereldsttogetherasfollows. Eachnumbereld k determinesacollectionofprimeidealsoftheringofintegers O k : Eachsuchprime p determinesanabsolutevalue v on k; andthecorresponding completion k v isalocaleld.Bytheabove, k v containssome p -adiceld Q p asa subeld,andtheabsolutevalue v issaidtolieover,ortodivide,the p -adicabsolute value jj p : Inadiagram, p o o / / v k / / k v p Z p o o / / jj p Q / / ? O O Q p ? O O .7 Moregenerally,if K=k isaniteextensionofanumbereld k ,thenanyabsolute value w on K liesoversomeabsolutevalue v of k; givinganextensionoflocalelds K w =k v : Inadiagram, P o o / / w K / / K w p o o / / v k / / ? O O k v ? O O .8 Startingwithanideal a of O k ; therearenitelymanyprimesof O k dividing a : Eachof theseprimescanbeextendedin K=k ,andthecorrespondingextensionofcompletions 14

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givesalloftheinformationabouttheextensionofthatprimein K=k: Inadiagram, p i O K D D D D D D D D D / / P j / / O K / / K P j o o / / w j o o / / K w j p i / / O k ? O O / / k ? O O p i o o / / v i o o / / k v i ? O O a = = z z z z z z z z .9 e K w j =k v i = e P j j p i .10 f K w j =k v i = f P j j p i .11 3.Statementsofclasseldtheory Oneoftheassertionsoflocalclasseldtheoryisthefollowing. Theorem 2.5 Foranynonarchimedeanlocaleld k thereexistsamap k ,the localArtinmap,suchthatanyniteextensionabelianextension K=k givesacommutativediagram k k / / Gal k ab =k res : k = Nm K K=k / / Gal K=k ; .12 where K=k isinducedby k and Nm K isacertainsubgroupof k : Forvariousreasons,itisoftenconvenienttoreplaceanumbereld k byitsring ofadeles A k ; andtheunits k of k byitsgroupofideles I k : Definition 2.2 Let M k denotethesetofabsolutevaluesof k: Thering A k of adelesof k isthesubring A k Y v 2 M k k v .13 15

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consistingofthoseelements ;x v ; suchthat x v 2O k v forallbutnitely-many v: Theidelesof k istheunitsgroup I k = A k ; whichiseasilyseentobethesubgroup I k Y v 2 M k k v .14 consistingofthoseelements ;x v ; suchthat x v 2O k v forallbutnitely-many v: Oneofthemainassertionsofglobalclasseldtheoryisthefollowing: Theorem 2.6 Foranynumbereld k thereexistsauniquecontinuoushomomorphism k : I k )167(! Gal k ab =k .15 suchthatanyniteabelianextension K=k withanychoiceofaprime w of K lying overaprime v of k givesacommutativediagram I k a 7! k a j L / / Gal K=k k v k v / / ? O O Gal K w =k v ? O O .16 Deeperconsiderationsleadonetoanisomorphism C k =C k )167(! Gal k ab =k ; .17 where C k : = I k =k istheideleclassgroupof k and C k istheconnectedcomponentof C k attheidentityinatopologywhichneedn'tbeconsidered. 4.Explicitlocalclasseldtheory Asmentionedabove,thereisanexplicitapproachtolocalclasseldtheorycalled Lubin-Tatetheory.Thefollowingtheoremisasummary.Itmakesuseoftwoinnitedimensionalextensionsof K; obtainedinmuchthesamewayas K ab : Oneofthese 16

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is K un ; thesmallestextensionof K containingalloftheunramiedextensionsof K: Theotheris K tot ; thesmallestextensionof K containingallofthetotallyramied extensionsof K: Theorem 2.7Lubin-TateTheory[ 20 ] Let K= Q p beanonarchimedeanlocal eld. Themaximalunramiedextension K un =K of K isgeneratedbytherootsof unityofordernotdivisibleby p: Thetotallyramiedextension K tot =K isgeneratedbytheunionofallofthe sets n ; where n consistsoftherootsinanalgebraicclosureof K ofthe n foldcomposition f f foranychoiceofformalpowerseries f 2O K [[ X ]] satisfyingtwoadditionalproperties. Themaximalabelianextension K ab =K isthesmallesteldcontaining K un and K tot ; i.e.,thecompositumofthesetwoelds. Forany a 2 K ; thereisanexplicitformulafortheactionof K a on K un and K tot ; andthusonthecompositum. TheresultsofChapter4ofthisthesisonlyconcernthelocaleld Q p ; which satises Q tot p Z p ; Q un p b Z ; .18 where Z p istheunitsgroupoftheringof p -adicintegersand b Z isthepronite completionoftheintegers,consistingof ;x j +1 ;x j ; 2 Y j 0 Z =n j Z with n j j n j +1 .19 suchthat x j = x j +1 mod n j Z ; where f n j : j 0 g isanyofintegerssuchthat n j j n j +1 forall j 0 : 17

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CHAPTER3 C -algebrasandQuantumStatisticalMechanics 1.Introduction Thischaptergivesthebackgroundinphysicsandthetheoryofoperatoralgebras necessarytomotivateanddescribetheworkof[ 3 ]andthenalchapterofthisthesis. Itdoesnotprovideacomplete,balancedaccountofanyofthistheory,focusing insteadongivingaminimalpresentationofthoseideasthatwillhelpthereader understandtheformulationoftheproblemoffabulousstatesdescribedabove.An excellentintroductiontooperatoralgebrasisgivenin[ 10 ].Otherusefulreferences are[ 8 15 23 ]. ItisperhapssurprisingthattheworkofBost-Connes,whichcallsuponmuch modernabstractmathematicalmachinery,hasitsrootsinthefollowingdown-toearthobservation: TheRiemannzetafunctionlookslikeapartitionfunction : TheRiemannzetafunctionissometimesdenotedby Q toemphasizethatitisintrinsicallyassociatedtothenumbereld Q ; asdescribedlater.Theaboveobservation leadsnaturallytothefollowingproblem: Problem 3.1 Describeastatisticalmechanicalsystemwith Q asitspartition function. Thischapterwillmakeprecisethequestionofwhatitmeanstoconstructsuch asystem,anddescribetherelatedproblemofconstructingastatisticalmechanical systemforeachprimeof Q : 18

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2.TheRiemannzetafunction TheRiemannzetafunctionisdenedas[ 11 ] s : = X n 1 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(s ; .1 whichweseeisconvergentfor < s> 1bythecomparison P n ; R dn: Wetakethe stancethattheRiemannzetafunctionisofsucientinterestonitsowntojustify interestinProblem3.1.Considerthefollowing. Proposition 3.2Eulerfactorization TheRiemannzetafunctionfactorsasas productoverallprimes Q s = Y p 1 1 )]TJ/F80 11.9552 Tf 11.955 0 Td [(p )]TJ/F23 7.9701 Tf 6.586 0 Td [(s ; .2 andthisfactorizationisequivalenttotheuniquefactorizationofintgersintoprimes. Proof. Assumeuniquefactorizationofintegersintoprimes.Then,organizing theintegersintermsoftheirdivisibilitybyaprime p; Q s = X n 1 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(s .3 = X j 0 0 @ X p ~ n p j ~ n )]TJ/F23 7.9701 Tf 6.586 0 Td [(s 1 A .4 = X p ~ n ~ n )]TJ/F23 7.9701 Tf 6.586 0 Td [(s X j 0 p )]TJ/F23 7.9701 Tf 6.586 0 Td [(s j .5 = )]TJ/F80 11.9552 Tf 11.956 0 Td [(p )]TJ/F23 7.9701 Tf 6.586 0 Td [(s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 X p ~ n ~ n )]TJ/F23 7.9701 Tf 6.586 0 Td [(s .6 TheEulerfactorizationfollowsbyinduction. NowassumetheEulerfactorizationof Q : Foranytwoprimes p;`; )]TJ/F80 11.9552 Tf 11.955 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [(s )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F80 11.9552 Tf 11.955 0 Td [(` )]TJ/F23 7.9701 Tf 6.587 0 Td [(s )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = X j 0 p )]TJ/F23 7.9701 Tf 6.587 0 Td [(sj X k 0 ` )]TJ/F23 7.9701 Tf 6.587 0 Td [(sk = X j;k 0 p j ` k )]TJ/F23 7.9701 Tf 6.587 0 Td [(s : .7 19

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Iffollowsthat Q = Y p )]TJ/F80 11.9552 Tf 11.955 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [(s )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = X k 0 X primes p 1 ;:::;p k X e 2 Z k p e 1 1 p e k k )]TJ/F23 7.9701 Tf 6.586 0 Td [(s : .8 Then,byassumption, Q s = X n 1 n )]TJ/F23 7.9701 Tf 6.587 0 Td [(s = X k 0 X primes p 1 ;:::;p k X 0
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log 0 : 80 : Thelargestcontributiontothe`errorterm' x )]TJ/F80 11.9552 Tf 11.605 0 Td [(x comesfromthe zerosof ; whichthereforecontaindeepinformationabouttheadditivestructureof theprimes. Justas Q containsinformationabouttheprimesof Z Q ; theDedekindzeta function K ofanumbereld K containsinformationabouttheprimesof O K K with O K asinDenitionDenition1.1.Itisasum 1 overtheidealsof O K denedby K s : = X a N a )]TJ/F23 7.9701 Tf 6.586 0 Td [(s ; where N a : =# O K = a .11 Sinceidealsfactoruniquelyasproductsofprimesin O K ; theproofoftheEuler factorizationof Q canbeadjustedtoshowthat K s = Y p 1 1 )]TJ/F51 11.9552 Tf 11.956 0 Td [( N p )]TJ/F23 7.9701 Tf 6.586 0 Td [(s : .12 Along-termobjectiveoftheauthoristoextendtheworkofthisthesistoEuler factors )]TJ/F80 11.9552 Tf 12.435 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [(s )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ofarbitraryglobaleldsinawayreectingthelocalclasseld theoryofthecorrespondingcompletion. 3.Somephysics Referenceforthissectionare[ 14 25 26 ]. Instatisticalmechanics,onebeginswithacollectionofstatesofasystemandthe collectionoftheirenergies f E s g andassemblesthepartitionfunction Z : = X s e )]TJ/F23 7.9701 Tf 6.586 0 Td [(E s ; .13 where istheinversetemperature,whicharisesnaturallyfromthefundamental assumptionofstatisticalmechanicsandisusedtodenetemperature =1 =: Itis assumedthroughoutthisthesisthattheenergiesaresuchthatthepartitionfunction isaconvergentsum.Saythatthereissomediscretecollectionofstatesofdenite energies E s : Supposethattheseindextheorthonormalbasis s ofaHilbertspace 1 See[ 21 ]foraproofthatthissumconverges. 21

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equippedwithaHamiltonianoperator H suchthat H" s = E s s : Recallthatthe exponentialofanoperatorisdenedbythepowerseries e T : = X n 0 T n n ; .14 whenthisexpressionconvergesintheHilbert-Schmidtnorm,andalsorecallthatthe traceof T isdenedby tr T = X h T" s ;" s i : .15 Usingthefactthattheeigenvaluesof e T areexactlythesetof e where isan eigenvalueof T; tr e )]TJ/F23 7.9701 Tf 6.587 0 Td [(H = X s h e )]TJ/F23 7.9701 Tf 6.587 0 Td [(H s ;" s i = X s e )]TJ/F23 7.9701 Tf 6.587 0 Td [(E s h s ;" s i = X s e )]TJ/F23 7.9701 Tf 6.586 0 Td [(E s = Z : .16 Comparing Z withthezetafunction Q = X n 1 n )]TJ/F23 7.9701 Tf 6.586 0 Td [( ; .17 oneseesthat Z = Q if E s =log s forinteger s 1 : GivenaclassicalHamiltonian H ona2 n -dimensionalphasespacewithlocalcoordinates p i ;q i for1 i n onanopenset U ,Hamilton'sequationsare d dt p i = )]TJ/F80 11.9552 Tf 10.494 8.088 Td [(@H @q i ; d dt q i = @H @p i : .18 ThevectoreldwhoseowisdeterminedbyHamilton'sequationsistheHamiltonian vectoreld X H ,anditsow g t ; givenby g t p;q = p t ;q t ; iscalledtheHamiltonianphaseow.Thisisusedtoevolveintimeaclassicalobservable f t ,denedto beasmoothreal-valuedfunctiononphasespace,bytherule f t p;q = f g t p;q : .19 22

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Thenthetimederivativeof f t is df t dt = df t + s ds s =0 = d f t g s ds s =0 = X H f t : .20 Writing X f g = f f;g g denesthePoissonbracketoftheclassicalobservables f;g: ThisturnsthealgebraofobservablesintoaPoissonalgebra,aLiealgebraforwhich thebracketisaderivation.Insum,inclassicalmechanics d dt f t = f H;f t g .21 Inthequantumcase,wetakeaboverelationasapostulate,usingthedenition f ; g ~ : = i ~ [ ; ] : .22 Hence d dt A t = i ~ [ H;A t ] : .23 Letting U t = e itH= ~ ; weverifythat A t = U t AU t )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 isasolutionto.23,where A 0 = A .Since A = A 0 doesnotdependontime, d dt A t = d dt U t AU t )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + U t A d dt U t )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = i ~ HU t AU t )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + U t A )]TJ/F80 11.9552 Tf 11.727 8.088 Td [(i ~ HU t )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 : Viewing U t asapowerseriesin iH= ~ ; itisclearthat[ H;U t ]=0 ; so d dt A t = i ~ U t [ A;H ] U t )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = f H;A t g ~ : .24 Thisshowsthatthetimeevolutionofanobservable A isgivenbyconjugatingitby e iHt= ~ : Thetheoryof C -algebrasabstractsallofthis.Inthatframework,asystemof particlesisreplacedbya C -algebra A whosesubalgebraofself-adjointelementsis takentobetheobservablesofthesystem.Thetimeevolutionisarepresentation 23

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of R asautomorphismsof A ,and,wheneverpossible,thisisimplementedusinga representationof A andaHamiltonianoperatorforwhichastraightforwardanalogue ofEquation.21holds. 4.Hilbertspacesand C -algebras ABanachspace A; jjjj isavectorspaceofarbitrarydimensionover R or C completeunderthenorm jjjj : If jj x jj = h x;x i 1 = 2 forsomehermitianinnerproduct on A ,then A iscalledaHilbertspace. AnoperatoronaHilbertspace H isalinearmapof H intoitself,anelementof End H : Themap H )167(! R givenby jj T jj B H : =sup fj Tf j H : j f j H =1 g .25 isanormonthethesubspaceforwhichthesupremumisnite.Theelementsofthis subspace B H arecalledboundedoperators. TheclassofHilbertspacesknownas ` 2 -spaces,denedbelow,arecentralto thisthesis.Thesespacesareimportantbecauseofthefollowingfact:anyinnitedimensionalHilbertspacewithacountableorthonormalbasisisisometricallyisomorphictoevery ` 2 -space.The ` 2 spacesconsistoffunctionsonsomecountableset S withsuitabledecayatinnity. Definition 3.1 Foracountableset S; let ` 2 S bethesubspaceofthecomplex vectorspacewithbasis f s : s 2 S g indexedby S givenby ` 2 S : = X s 2 S c s s : X c 2 s < 1 ; .26 ThisisaBanachspaceundertheobviousadditionandscalingby C : 24

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Weoftenview P c s s 2 ` 2 S asthefunction f : 8 > < > : S )167(! C s 7)167(! c s : .27 Fromthisperspective,the ` 2 norm j f j ` 2 = P j f s j 2 lookslikethenorm j f j L 2 = R X j f j 2 d with X ameasurespace. IfaBanachspace A hasanassociativemultiplicationand jj xy jjjj x jjjj y jj ; then A iscalledaBanachalgebra.A C -algebraisaBanachalgebrawithanabstract notionofadjointness. Definition 3.2 AninvolutiononaBanachalgebra A isanendomorphism x 7)167(! x of A satisfying .28 x + y = x + y xy = y x x = x x = x: Werefertoanalgebranotnecessarilycompletewithaninvolutionasaninvolutivealgebra.A C -algebraisaBanachalgebrawithaninvolutionsatisfyingthe C -identity jj x x jj = jj x jj 2 : C -identity Amorphism ofa C -algebraisamorphismofBanachalgebrasi.e.,acontinuous, linearandmultiplicativemapsuchthat a = a : A C -algebra A neednothaveamultiplicativeidentity,butitisalwayspossible toenlarge A toanalgebrawithaunitcalledaunitalalgebrabytakingthedirect sumwith C anddeningmultiplicationappropriately.Theprototypical C -algebra is B H ; thealgebraofboundedoperatorsontheHilbertspace H withinvolution givenbytheadjoint. 25

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Definition 3.3 Let A beaunital C -algebraandlet a 2 A: Denethespectrum of a by a : = f 2 C : a )]TJ/F80 11.9552 Tf 11.955 0 Td [( 1 isnotinvertible g : .29 Denethespectralradiusof a by a : =sup fj j : 2 a g : .30 Itcanbeshownthat a =lim jj a n jj 1 =n : The C -identityisasurprisinglyrigid constraint,connectingthenormofelementsof A withthespectralradiusandconstrainingtherepresentationsof A: Firstofall,if x 2 A isself-adjointi.e., x = x then jj x jj 2 = jj x x jj = jj x 2 jj : Aninductionargumentshowsthat jj x jj 2 n = jj x 2 n jj ; sothat jj x jj = jj x 2 n jj 1 = 2 n forall n: Itfollowsfromthespectralradiusformulathat jj x jj = x : Sinceany a a 2 A isself-adjoint, jj a jj 2 = jj a a jj = a a : .31 Thisshowsthefollowing. Theorem 3.3 Let A beanalgebrawithaninvolution.Thenthereisatmostone C -algebracontaining A asasubalgebra. Proof. Thenormof a 2 A foranylarger C -algebra e A A isdeterminedby a ; whichdependsonlyon A: Consideranymorphism : A )167(! B ofunital C -algebras.If a )]TJ/F80 11.9552 Tf 12.252 0 Td [( 1isnot invertible,then a )]TJ/F80 11.9552 Tf 11.985 0 Td [( 1isnotinvertible.Thisshowsthat a )]TJ/F80 11.9552 Tf 11.985 0 Td [( 1isnotinvertible, i.e.,that a a : Thisgivesthefollowing. 26

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Proposition 3.4 Anymorphismof C -algebrasisacontraction. Proof. Thisproofrestrictstotheunitalcase.Thentheaboveshows a a : Sincethespectralradiusisthenormonself-adjointelements, jj a a jj = a a a a = jj a a jj ; .32 so jj a jjjj a jjjj a a jjjj a jj 2 ; .33 Sinceinvolutionsareisometries,theleft-handsideisequalto jj a jj 2 : Weseethat isacontraction. Arepresentationofa C -algebraisamorphisminto B H .Fromtheprevioustwo propositions,weseethatanyinvolutivealgebra A canbecompletedtoa C -algebra inthenorm jj f jj C : =sup fjj f jj : isaninvolutiverepresentationof A g : .34 Definition 3.4 Theuniversal C -envelopingalgebra C A ofacomplexinvolutivealgebra A istheclosureof A in jj f jj C : Itmaybethatthisalgebracoincideswiththenormofaparticularrepresentation of A: Theorem19of[ 3 ]showsthatthisisthecaseforacertainHeckealgebra,proving thattheBost-Connessystem C Q coincideswiththeclosureofthatalgebraundera regular ` 2 representation. 5. C -dynamicalsystems IntheHeisenbergformalismdiscussedabove,theobservablesofasystemare operatorsthatevolveintime.Torealize A asanalgebraofobservablesrequires twoadditionalpiecesofinformation:aone-parametergroupofautomorphismsand acovariantrepresentationof A: 27

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Definition 3.5 A C -dynamicalsystemisapair A; ,where A isa C -algebra and isarepresentationof R asautomorphismsofa C -algebra A: Amorecommondenitionlets bearepresentationofanarbitrarylocallycompact group G; butDenition3.5restrictsterminologytothecasewhere G = R ; wherethe actionisinterpretedastheowingoftime. Definition 3.6 Acovariantrepresentationofa C -dynamicalsystem A; over R isapair ;H consistingofanondegenerateinvolutiverepresentation of A onaHilbertspace H andanoperator H on H suchthat t a = e itH a e )]TJ/F23 7.9701 Tf 6.587 0 Td [(itH forall a 2 A;t 2 R : .35 Thisdenitionisextendedtoaninvolutivealgebrawithaone-parameterautomorphismgroupinthenaturalway. 6.Group C -algebras Following[ 13 ],thissectiondescribesthecanonicalassociationofa C -algebra C G toalocallycompactgroup. Givenacommutativering R andadiscretegroup G ,therearetwoequivalent viewsofthegroupalgebra R [ G ].Itcanbethoughtofasthesetofformalsums P g 2 G x g g with x g 2 R and x g =0foralmostall g; andwithmultiplicationgivenby xy = X g x g g X h y h h : = X g;h x g y h gh .36 whichcanberewrittenusingtheobviouschangesofvariablesas xy = X g X h x gh )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 y h g = X g X h x h y h )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 g g .37 28

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Onecanalsoview R [ G ]asthe R -valuedfunctionson G denedby x g = x g : Inthis view,thealgebramultiplicationistheconvolutionproduct xy g = X h x h y h )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g : .38 The C -algebra C G ofadiscretegroup G isdenedusingthegroupalgebra R [ G ] with R = C : Inthiscase,thereisalwaystheinvolutiongivenby x g = x g )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 : To thatend,therearetwoconstructionsyieldinga C -algebrawhichareconnectedby theconceptofamenability,bothofwhichinvolveclosingtheimageof C [ G ]under representations.Itisalwayspossibletoputa C -algebrastructureon C [ G ]byclosing itinthenorm.34.Ontheotherhand,thereisamuchmoretractablenorm comingfromtheliftoftheleftregularrepresentation of G on ` 2 G to C [ G ];the representation : 8 > < > : G )167(! End ` 2 G g 7)167(! X c h h 7)167(! X c h gh .39 liftstoarepresentation ~ : 8 > < > : C [ G ] )167(! End ` 2 G x = X x g g 7)167(! X c h h x 7)167(! XX c h x g gh ; .40 andtheclosureoftheimageof C [ G ]intheoperatornorm kk B ` 2 G on ` 2 G jj f jj r : = e f B ` 2 G .41 iscalledthereducedgroup C -algebra,andisdenotedby C r G : Thegroupsfor whichtheclosureof C [ G ]in.34isthesameas C r G areexactlytheamenable groups[ 13 ]. Thoughitwillnotbeusedinwhatfollows,itisworthmentioningthatgroup C -algebrasareexamplesofthemoregeneralconstructionofwhatareknownas crossedproducts,sincethedevelopmentofBost-Connessytemsintermsofgroupoid 29

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algebrasofequivalencerelationsleadstodescriptionsofthosesystemascrossedproducts.Crossedproductsareanalogoustosemidirectproducts,wheremultiplication istwistedinoneoftwocomponentsbytheactionoftheothercomponentasautomorphisms.Whentheautomorphismscomefroma C -dynamicalsystem,there isacorrespondencebetweenrepresentationsofthecrossedproductandcovariant representationsofthesystem. 7.Groupoidsandgroupoidalgebras Recalltheobjects A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 and C Q p mentionedinthenalsectionoftheintroduction,tobedenedinChapter4.WithguidancefromtheproofofTheorem 19of[ 3 ],theauthorbelievesthatConjecture4.8canbeprovedusingthenotion ofanamenablegroupoidtoshowthat C Q p istheuniversal C -envelopingalgebra of A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 ; sothatallrepresentationsof A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 liftuniquelytorepresentations of C Q p : Thistheoremwouldshowthataparticularcovariantrepresentation,thatof Proposition4.7,canbeextendedto C Q p ,showingthatthepartitionfunctionof C Q p istheEulerfactorat p of Q : Referencesonthismaterialare[ 1 ],[ 13 ]. Agroupoid G isasmallcategoryallofwhosemorphismsareisomorphisms.A smallcategoryconsistsofaset G ; whoseelementsarecalledtheobjectsof G; and acollectionofarrowsbetweenelementsofOb G : Eacharrow a hasawell-dened source s a andawell-denedtarget t a ,asin s a = a )391()222()222()391(! = t a ; whichareclassicallyreferredtoasthedomainandcodomainor,bysome,range when a isafunctionbetweensets.OnewritesHom x;y forthesetofarrowswith s a = x;t a = y: Thearrowsofacategorymustbecomposablewhenitispossible todoso,inthesensethat a 2 Hom x;y ;b 2 Hom y;z = thereexists b a 2 Hom x;z : .42 30

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Theconditionthateacharrowbeanisomorphismssaysthatif a 2 Hom x;y then thereis a )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 Hom y;x suchthat a )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 a =id x and a a )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 =id y : Forgroupoids, itisconvenienttoconsider G asthecollectionofallofitsarrows,embeddingOb G into G asloopsfrom x 2 Ob G toitself. Whenthesetofmorphisms,writtenas G; ofagroupoid G isalocally-compact topologicalspaceandthesource,targetandinversionmapsarecontinuous,oneassociatesaconvolutionalgebrato G andagivensystemofmeasuresonthebersofthe targetmapasfollows.Denotethebersby G x : = t )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 x ; sothat G x isthesetofall arrowspointingto x: Givenasystemofmeasures t a onthe G t a thatiscontinuous andinvariantintheappropriatesense,denetheconvolutionalgebraon C c G : = f f : G )167(! C : f iscontinuousandhascompactsupport g .43 by f 1 f 2 g : = Z G t g f 1 h f 2 h )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g d t g h : .44 Asinthecaseofgroups,thereisauniversal C -envelopingalgebra C G attached to G; theclosureunderthenorm jj f jj C : =sup fjj f jj : isaninvolutiverepresentationof C c G g : .45 Foreachber G x ,thereisarepresentationof C c G ontheHilbertspace L 2 G x ;d x givenbyz x : 8 > > < > > : C c G )167(! End L 2 G x ;d x f 7)167(! g x f 7)167(! Z G t g f h h )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g d t g h : .46 Thesegiveanormdenedby jj f jj r : =esssup fjj x f jj L 2 G x ;d x : x 2 Ob G g : .47 31

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Inanalogywiththespecialcasewhere G isagroup,wecallthecompletionof C c G in thenormthereduced C -algebraof G; denotedby C r G : Thetheoryofamenability forgroupoids,describedin[ 1 ],thenshowsthat C G = C r G when G isamenable. 8.vonNeumannalgebras ThoughthenotionofavonNeumannalgebrawillnotbeusedinthisthesis,itis worthbrieymentioningafewofthebasicideashere,sincethesystem C Q p liesinside ofavonNeumannalgebra M andisactedonbytherestrictionofanactionof Z p on M: ThetheoryofvonNeumannalgebrasisoneofthemajorareasofcontemporary mathematicalresearch.AvonNeumannalgebraisbydenitionaunital -algebra ofsome B H satisfyingthreeequivalentconditions.Twooftheseareclosureunder topologieson B H ; andthethirdistherequirementthat M isequaltoitsdouble commutant,wherethecommutantofasubalgebra M B H is M 0 : = f T 2 B H : TS = ST forall S 2 M g .48 andthedoublecommutantis M 00 : = M 0 0 : Itisatheoremthatif M isalready acommutant,than M isavonNeumannalgebra i.e., M 000 = M 0 always.von Neumannalgebrasenjoymanyniceproperties.Chiefamongtheseisthefactthat suchanalgebraisgeneratedbyitsprojections,andthatvonNeumannalgebrascan beclassiedbythestructureofanorderrelationonthesetofprojections.Thereis extensivestructuretheoryforvonNeumannalgebras,sooneishopefulaboutstudying arithemticobjectsinthisnew,highly-developedcontext.. 32

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CHAPTER4 ABost-ConnesSystemfor Q p 1.TheToeplitzalgebra p AsnotedinSection3ofChapter3,theoperator H =log T on k 0 C k with T" k = p k p k satisestrexp )]TJ/F80 11.9552 Tf 9.299 0 Td [(H = )]TJ/F80 11.9552 Tf 10.659 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [( )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 : Thischapterbeginswithanalgebra p forwhich H determinesatimeevolutioninthesenseofHamiltonianmechanics. Later,thisisenlargedtocontainasubalgebraisomorphicto C Q p = Z p : DenetheToeplitzalgebra p tobethethecyclicalgebraononegenerator p ; theset p : = n X c n n p : c n 2 C andthesumisnite o .1 withadditionandmultiplicationdenedintheobviousway.Thepair ;e itH ; with H =log T; givenby .2 : 8 > < > : p )167(! End ` 2 p Z > 0 n p 7)167(! k 7)167(! k + n ;T : 8 > < > : ` 2 p Z > 0 )167(! ` 2 p Z > 0 k 7)167(! p k k ; givesacovariantrepresentationof p : Toseethis,onecomputes t n k = T it p n T )]TJ/F23 7.9701 Tf 6.586 0 Td [(it p k .3 = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(itk T it p n k .4 = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(itk T it p k + n .5 = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(itk p it k + n k + n .6 = p itn k + n .7 = p itn n k : .8 33

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Dening t n = p itn n ; thisshowsthat t = t : Fromhereitisclearthat thepairgivesacovariantrepresentation. ThepaperofBostandConnesdescribesanappealingandsimpleobservationthat motivatedtheirinvestigation.Themodicationofthisobservationprovidedbelow showsthattheHamiltonian H hasaninterpretationcomingfrom`secondBosonic quantization,'afunctordenedasfollows.GivenaHilbertspace H ; denoteby S n H theringoforder n symmetrictensors.Denethefunctor S onHilbertspacesand endomorphismby .9 S : 8 > > < > > : H 7)167(! S H : = M n 0 S n H T 7)167(! S T : 1 n 7)167(! T 1 T n Foranyself-adjointoperator T onaHilbertspace,andrecallthat T denotesthe spectrumof T: Asimpleresultof[ 3 ]is T = f primes p 2 Z g S T = Z > 0 : .10 Ourstartingpointisthefollowing: Proposition 4.1 Let T beself-adjointon H andlet p beaprimenumber.Then, T = f p g S T = f p n : n 1 g ; .11 whereforthe = implicationtoholditisnecessarytoassumethateveryeigenvalue of S T hasmultiplicityone. Proof. The= directionholdsforanyeigenvalueof S T isaproductofeigenvaluesof T: Thesameobservationshowsthat p 2 T ; since T S T and S T contains p: If p n 2 T ; then p n hasmultiplicity1asaneigenvalueof S T if andonlyif n =1 : Thisshowsthat T = f p g : 34

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Considerthemultiplication-byp operator T onthetrivialone-dimensionalHilbert space ` 2 f p g = C p : Wehave ` 2 f p g 7)167(! S ` 2 f p g = ` 2 p Z > 0 .12 T : 0 7)167(! p" 0 7)167(! S T : p n 7)167(! p n p n 2 End ` 2 p Z > 0 : .13 Thatis, H = S T: Notethattheprime p labelsthesingletonsetthatindexesthebase ofthe ` 2 -space. 2. ax + b groupsandthehomogeneousspace p Theenlargementof p willbetheanalgebraofoperatorsonahomogeneousspace p ,whichwenowdevelop.Givenacommutativering R; denoteby P R the ax + b group P R : = 1 b a : a 2 R ;b 2 R : .14 Thisiswrittenasasemidirectproduct P R R o R ; .15 wherethemap : R )167(! Aut R requiredforthedenitionofthesemidirect producttakes r 2 R tothecorrespondingmultiplicationmap. Let p = P Q p =P Z p : Thefunctionsonthisspacearecentraltowhatfollows.Recall thatagroup G issaidtoactonatopologicalspace X ontheleftifthereisamap G X 3 g;x 7)167(! g x 2 X .16 suchthat g 1 g 2 x = g 1 g 2 x forall g 1 ;g 2 ;x: Agroupactionissaidtobetransitive ifforeachpairofpoints x;y 2 X thereissome g 2 G suchthat g x = y: The subgroupof G ofelementsxingsomespecied`basepoint' iscalledtheisotropy groupof : 35

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Theorem 4.2[ 4 ],III.4.2Proposition4 Let G beatopologicalgroupacting properlyonatopologicalspaceandlet beapointof X: Let G: denotetheorbitof ; andlet G denotetheisotropysubgroupof : Thenthecanonicalmappingof G=G onto G: isahomeomorphism. Toapplythistheorem,werstlet )]TJ 10.635 0.415 Td [(: = 1 Q Q > 0 ; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : = 1 Z p Z p > 0 : .17 Proposition 4.3 Thegroup )]TJ/F93 11.9552 Tf 11.243 0 Td [(actstransitivelyon p : Theisotropysubgroupof thebasepoint P Z p is )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 : Proof. Weshowthat P Q p =)]TJ/F80 11.9552 Tf 21.057 0 Td [(P Z p : Let[ 1 a k ] 2 P Q p : Welookfor r;s;u;z such that 1 a k = 1 r s 1 z u 2 1 Q Q > 0 1 Z p Z p : .18 Theequalityis 1 a k = 1 z + ru su : .19 Itmustbethat j k j p = j s j p and j a j p = j r j p : Itispossibletochoose z;r and u sothat z + ru = a infact,wemayassume z =0andthentochoose s sothat su = k Therefore P Q p =)]TJ/F80 11.9552 Tf 20.432 0 Td [(P Z p : Fromthisitisclearthat)-361(actstransitivelyon P Q p ;writing x;y in P Q p as x = z and y = 0 z 0 in)]TJ/F80 11.9552 Tf 21.258 0 Td [(P Z p ,weseethat 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 x = y: Theisotropy groupof P Z p is )]TJ/F81 11.9552 Tf 9.971 0 Td [( P Q p = QoQ > 0 Z p oZ p = Z p o Z p > 0 : .20 Theactioniswell-behaved,sothetheoremapplies;therefore p )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 astopologicalspaces.Wenowstudytheactionof)]TJ/F21 7.9701 Tf 229.594 -1.793 Td [(0 on p : 36

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Doublecosets. Throughoutthissectionwelet : = 1 a k 2 )]TJ/F80 11.9552 Tf 7.315 0 Td [(: .21 Thedoublecosetof is )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 1 a k )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 = 1 Z p Z p > 0 1 a k 1 Z p Z p > 0 = 1 a Z p > 0 + k Z p + Z p k Z p > 0 .22 Itisclearthatthiscosetdependsonlyon j a j p and j k j p : Rightquotients. Therightcoset)]TJ/F21 7.9701 Tf 88.996 -1.793 Td [(0 is )]TJ/F21 7.9701 Tf 7.315 -1.794 Td [(0 : = f 0 : 0 2 )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 g = 1 a + z 0 k s 0 k : z 0 2 Z p ;s 0 2 Z p > 0 = 1 a + k Z p k Z p > 0 .23 Theactionof)]TJ/F21 7.9701 Tf 79.826 -1.793 Td [(0 ontherightofthiscosethasorbit O R a;k = 1 z + as + k Z p k Z p > 0 : z 2 Z p ;s 2 Z p > 0 .24 Itisclearthattheorbitsarenite.Wenowdetermineasetofrepresentativesforthe rightcosetsandndhowthesearepermutedundertheactionof)]TJ/F21 7.9701 Tf 347.935 -1.793 Td [(0 ontherightof )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 : Weseeimmediatelythatwemaytake j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p asthe2,2-entryofarepresentative. Fixingthenormof k; theclassesarethendeterminedbythe1,2-entry.Wedetermine S R suchthat )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 1 a k )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 = G 2 S R )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 1 j k j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 p : .25 Realizingthatthesizeof k determineshowmanydigitsinthe p -expansionof z + as arefree,wendthefollowing. 37

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i1 j a j p ; j k j p Since z + as 2 Z p S R = Z p = j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p Z p : .26 ii j a j p > 1, j a j p j k j p Theproper j a j p -throotsofunity S R = n : )]TJ/F80 11.9552 Tf 5.48 -9.684 Td [(e 2 i N =1if N = j a j p 6 =1if N< j a j p o j a j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p Z p = j k j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 p Z p : .27 iiiOtherwise, S R = f 0 g : .28 Leftquotients. Theleftcoset )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 is )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 : = f 0 : 0 2 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 g = 1 z 0 + as 0 s 0 k : z 0 2 Z p ;s 0 2 Z p > 0 = 1 a Z p > 0 + Z p k Z p > 0 .29 Theactionof)]TJ/F21 7.9701 Tf 79.826 -1.793 Td [(0 ontherightofthiscosethasorbit O L a;k = 1 a + zk Z p > 0 + Z p k Z p > 0 : z 2 Z p ;s 2 Z p > 0 .30 Inthesameveinasabove,wedetermine S L suchthat )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 1 a k )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 = G 2 S L 1 j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : .31 i z;k 2 Z p S L = f 0 g : .32 ii j a j p > 1 j k j p S L = a + Z p : .33 38

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iii j a j p j k j p > 1 S L = j k j p [ j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p a ]+ j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p Z p = Z p : .34 withbracketsdenotingthefractionalpart. iv j k j p > 1 ; j k j p j a j p S L = j k j )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 p Z p = Z p : .35 Writing S L = a + j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p Z p forthesetofall a + r with r 2j k j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 p Z p consideredas elementsof Q p = Z p givesauniformnotationforalloftheabovefourcases. 3.ThelocalBost-Connessystem C Q p Thereisunitaryrepresentationof)-327(on ` 2 )]TJ/F80 11.9552 Tf 11.866 0 Td [(= )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 givenby .36 : 8 > < > : )]TJ/F81 11.9552 Tf 10.635 0 Td [()167(! Aut ` 2 )]TJ/F80 11.9552 Tf 11.866 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 7)167(! u : x 7)167(! x : Let U \051 : = f u : 2 )]TJ/F81 11.9552 Tf 7.314 0 Td [(g : Foreachelementofthecommutant T 2U \051 0 thereisa C -valuedfunctionon)]TJ/F80 11.9552 Tf 145.861 -35.543 Td [(f T : 7)167(!h T" e ; )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 e i ; .37 where e istheidentityof)]TJ/F80 11.9552 Tf 95.824 0 Td [(: Thus f T = t )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 ; where T" e = P x 2 )]TJ/F23 7.9701 Tf 5.288 0 Td [(= )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 t x x ; so T" e = X x 2 )]TJ/F23 7.9701 Tf 5.289 0 Td [(= )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 f T x )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 x .38 39

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and,since T" x = u x T" e ;T actson a = P a x x 2 ` 2 )]TJ/F80 11.9552 Tf 11.867 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 ; as Ta = X a u T" e .39 = X X x a u f T x )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 x .40 = X x X a f T x )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 x : .41 Thusitisseenthatthefunction f T issucienttoexpress T: Proposition 4.4 Theadjointof u is u )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 : Thefunctions f T are )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 -bi-invariant, i.e., f T 0 = f T = f T 0 forany 2 )]TJ/F80 11.9552 Tf 7.314 0 Td [(; 0 2 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : .42 Proof. With a : = P a x x and b : = P b x x ; h a;u b i = X x X y a x b y h x ;" y i = X x a x b )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 x = X x X y a x b y h )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 x ;" y i = h u )]TJ/F22 5.9776 Tf 5.757 0 Td [(1 a;b i : .43 Tosee)]TJ/F21 7.9701 Tf 43.473 -1.794 Td [(0 -bi-invariance,notethat f T = f T whenever ; 2 )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 ; as, f T = h T" e ;u )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 e i = h Tu e ;u )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 u )]TJ/F22 5.9776 Tf 5.756 0 Td [(1 e i = h T" e ;u )]TJ/F22 5.9776 Tf 5.757 0 Td [(1 e i = f T : .44 The)]TJ/F21 7.9701 Tf 31.379 -1.793 Td [(0 -bi-invariantfunctionson)-326(areexactlythosefunctions ~ f on)-326(liftedfrom functions f on)]TJ/F21 7.9701 Tf 22.837 -1.793 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 bytherule ~ f x = f X where x isanyelementinthedouble coset X 2 )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : Henceeach f T canbeexpressedintermsofabasisofindicator 40

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function f 1 X : X 2 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 g where 1 X g = 8 > > < > > : 1if g 2 X; i.e.,if X =)]TJ/F21 7.9701 Tf 19.74 -1.793 Td [(0 g )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 0otherwise. .45 Assuming f T issupportedinnitely-manycosets,onehas f T = P X f T X 1 X 2 ` 2 )]TJ/F80 11.9552 Tf 11.867 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 : Thisbasisallowsonetoreversetheassociation T 7! f T : Definition 4.1 DenetheHeckealgebra A = A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 tobethesetoffunctions f : )]TJ/F81 11.9552 Tf 10.635 0 Td [()167(! C suchthat f 0 = f 0 = f forall 2 )]TJ/F80 11.9552 Tf 7.314 0 Td [(; 0 2 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 f =0 unless )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 = X j forsomenitecollection f X j g ofdoublecosets. Thishasthestructureofaninvolutivealgebraundertheconvolutionandinvolution denedby f 1 f 2 g : = X h 2 )]TJ/F23 7.9701 Tf 5.288 0 Td [(= )]TJ/F22 5.9776 Tf 5.288 -1.107 Td [(0 f 1 h f 2 h )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g .46 f g : = f g )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 : .47 Thealgebra A isreferredtoastheHeckealgebraofthepair)]TJ/F80 11.9552 Tf 247.892 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 ; andshould bethoughtofasthecompactly-supportedfunctionsonthespaceoforbitsofthe actionof)]TJ/F21 7.9701 Tf 55.764 -1.793 Td [(0 on)]TJ/F21 7.9701 Tf 23.573 -1.793 Td [(0 n )-326(and)]TJ/F80 11.9552 Tf 41.292 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : Proposition 4.5 Let f 2 A .Thenthereisauniqueelement r f 2U \051 0 such that f = h r f e ; )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 e i : .48 Proof. Toeachdoublecoset X weassociate T X denedby h T X 1 e ; 2 e i = 1 X )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 1 : .49 41

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The T X areinthecommutantsince,for 2 )]TJ/F80 11.9552 Tf 7.314 0 Td [(; h T X 1 e ; 2 e i = 1 X )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 1 .50 = 1 X )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 .51 = h T X 1 e ; )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 e i .52 = h T X 1 e ; 2 e i .53 forany 1 ; 2 : Thelefthandexpressionisequalto h T X e ; )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 2 1 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 e i ; sowesee that T X e = P x 1 X x x .Theoperator r f isconstructedasfollows.If f X j g is thecollectionofdoublecosetsonwhich f isnonzero,thendene r f by r f : = P X j f X j T X j : Thereisaninvolutiverepresentationof A on ` 2 )]TJ/F80 11.9552 Tf 11.867 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 givenby : 8 > > > > < > > > > : A )167(! End ` 2 )]TJ/F80 11.9552 Tf 11.866 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 f 7)167(! 0 @ g f 7)167(! X h 2 )]TJ/F23 7.9701 Tf 5.289 0 Td [(= )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 f h h )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g 1 A : .54 Theclosure A oftheimageof A under withtheobviousinvolutionisa C -algebra. Themap r : A )167(!U \051 0 oftheprecedingpropositionextendsbycontinuitytoa mapdenedon A : Definition 4.2 Thealgebrageneratedbythe r f for f 2 A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 istheBostConnessystemfor Q p ; denotedby C Q p : Presentation. Anyfunction T 2 A operateson A by T : f 7)167(! Tf : = T f: Considerthecasewhere T istheindicatorfunction 1 X 2 A ofthedoubleclass X =)]TJ/F21 7.9701 Tf 20.421 -1.793 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 : From.22, X =)]TJ/F21 7.9701 Tf 20.421 -1.793 Td [(0 1 p )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 forsome 2 Q p = Z p andsome 2 Z : 42

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With S L oneofthesetsoccurringin.31, 1 X f g = X h 2 )]TJ/F23 7.9701 Tf 5.289 0 Td [(= )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 1 X h f h )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g = X 2 S L f 1 p )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g = X 2 S L f 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(p )]TJ/F23 7.9701 Tf 6.586 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [( g : .55 Similarly,observingthat h )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 rangesthrough)]TJ/F21 7.9701 Tf 90.411 -1.793 Td [(0 n )-426(as h rangesthrough)]TJ/F80 11.9552 Tf 90.411 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 ; one nds,with S R equaltooneofthesetsoccurringin.25,that 1 X f g = X h 2 )]TJ/F23 7.9701 Tf 5.289 0 Td [(= )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 1 X h )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 f h )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g = X h 2 )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 n )]TJ/F79 11.9552 Tf 7.281 14.546 Td [(1 X h f hg = X 2 S R f 1 p g ; .56 Denetwoclassesofelementsof A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 :for n 2 Z > 0 ; n = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n= 2 1 X n ; where X n istheclassof 1 p n in)]TJ/F21 7.9701 Tf 20.971 -1.794 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 .57 and,for 2 Q p = Z p ; e = 1 X ; where X istheclassof 1 1 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 in)]TJ/F21 7.9701 Tf 20.971 -1.793 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 ..58 Proposition 4.6 Thealgebra G generatedbytheelements n and e isequal totheHeckealgebra A C Q p : Thefollowingrelationsgiveapresentationof A : a n n =id b n + m = n m c e = e )]TJ/F80 11.9552 Tf 9.299 0 Td [( d e + = e e e e e =id f e n = n e p n g n e n = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n P p n = e Proof. Let X beadoublecoset X : =)]TJ/F21 7.9701 Tf 19.739 -1.793 Td [(0 1 p )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 ; with 2 Q p = Z p ; 2 Z ..59 43

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Itwillbeshownthat 1 X 2 G : Itisconvenienttomakesomepreliminarycalculations. First,since j a j p =1and j k j p 1 ; .31shows)]TJ/F21 7.9701 Tf 76.009 -1.793 Td [(0 X n )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 = X n )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 ; sothat S L = f 0 g and.55gives n f g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n= 2 X 2 S L f 1 p n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n= 2 f 1 p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n g .60 Similarly,)]TJ/F21 7.9701 Tf 59.341 -1.793 Td [(0 X )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 = X )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 ; sothat S L = f g and e f g = X 2 S L f 1 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 g = f 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( 1 g : .61 Equations.31and.56leadto n f g = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n= 2 X 2 Z =p n Z f 1 p n g .62 Compute n e m f g = n e p )]TJ/F23 7.9701 Tf 6.587 0 Td [(m= 2 X 2 Z p =p m f 1 p m g .63 = n p )]TJ/F23 7.9701 Tf 6.586 0 Td [(m= 2 X 2 Z p =p m f 1 p m 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( 1 g .64 = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(m= 2 p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n= 2 X 2 Z p =p m f 1 p m 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( 1 1 p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n g .65 = p )]TJ/F21 7.9701 Tf 6.587 0 Td [( m + n = 2 X 2 Z p =p m f 1 )]TJ/F80 11.9552 Tf 11.956 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n p m )]TJ/F23 7.9701 Tf 6.586 0 Td [(n g : .66 Letting n = and m =0sothat m = 1 id =id ; thisis e f g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(= 2 f 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [( p )]TJ/F23 7.9701 Tf 6.586 0 Td [( g : .67 Letting m = and n =0sothat n =id ; thisis e f g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(= 2 X 2 Z p =p f 1 )]TJ/F80 11.9552 Tf 11.955 0 Td [( p g = p = 2 f 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [( p g : .68 44

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Consider.55.Itwasshownabovethat S L = + p Z p = Z p withtheconventionthat p Z p = Z p : = Z p when > 0 : When p 2 Z p ; 1 X f g = X = + Z p f 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [( p )]TJ/F23 7.9701 Tf 6.586 0 Td [( g = f 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [( g ; .69 so 1 X = p = 2 e 0 : .70 When p 62 Z p ; 1 X f g = X 2 + p Z p = Z p f 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(p )]TJ/F23 7.9701 Tf 6.587 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [( g : .71 Each p )]TJ/F23 7.9701 Tf 6.587 0 Td [( with 2 + p Z p = Z p isequivalentto in Q p = Z p ; so 1 X f g = p )]TJ/F23 7.9701 Tf 6.587 0 Td [( f 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( p )]TJ/F23 7.9701 Tf 6.586 0 Td [( g : .72 Hence, 1 X = p = 2 e 0 : .73 Thishasshownthat 1 X 2 G foranydoublecoset X: Sinceeveryfunctionin A canbe writtenasanitelinearcombinationofindicatorfunctions,itmustbethat A = G : Toprovetheidentity a ; rstcompute m n f g = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n= 2 m f 1 p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n g .74 = p )]TJ/F21 7.9701 Tf 6.587 0 Td [( n + m = 2 X 2 Z p =p m f 1 p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n 1 p m g .75 = p )]TJ/F21 7.9701 Tf 6.587 0 Td [( n + m = 2 X 2 Z p =p m f 1 p m )]TJ/F23 7.9701 Tf 6.586 0 Td [(n g : .76 45

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When m = n; thisbecomes n n f g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n X 2 Z p =p n f 1 1 g .77 = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n X 2 Z p =p n f g .78 = f g .79 since[ 1 1 ] 2 )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 for 2 Z p : Therefore n actsastheidentity. For b ; n m f g = f 1 p )]TJ/F23 7.9701 Tf 6.586 0 Td [(m 1 p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n g = f 1 p )]TJ/F21 7.9701 Tf 6.587 0 Td [( n + m g = n + m f g : .80 For c ; e f g = X h 2 S R 1 X h f hg .81 where S R isinthecase ii of.25with j k j p =1.Noteherethat X isthedouble classnotofthematrix 1 1 ,butoftherightclass 1 1 )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 : From.29,itisclear thatthereisonlyoneclass,theclassof 1 1 : Hence e f g = f 1 1 g = e )]TJ/F80 11.9552 Tf 9.299 0 Td [( f g : .82 For d ; e e f g = e f 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( 1 g = f 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( 1 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [( 1 g = e + f g .83 For e ; since e istheidentity, e e = e )]TJ/F80 11.9552 Tf 9.299 0 Td [( e = e )]TJ/F80 11.9552 Tf 11.955 0 Td [( = e =id : .84 46

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For f ; .67gives n e p n f g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n= 2 f 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [( p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n g .85 = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n= 2 f 1 p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n 1 )]TJ/F80 11.9552 Tf 9.298 0 Td [( 1 g .86 = e n f g : .87 For g ; from.63with n = m; n e n f g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n X 2 Z p =p n f 1 )]TJ/F80 11.9552 Tf 11.955 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n 1 g .88 = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n X 2 Z p =p n f 1 )]TJ/F80 11.9552 Tf 11.955 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n 1 g : .89 Thenumbers )]TJ/F80 11.9552 Tf 12.953 0 Td [( p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n with 2 Z p =p n areexactlythose ~ 2 Q p = Z p suchthat p n ~ = )]TJ/F80 11.9552 Tf 9.298 0 Td [(; so n e n f g = p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n X p n = f 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [( 1 g = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n X p n = e f g : .90 Thecomputationoftheorbitsfortheactionsof)]TJ/F21 7.9701 Tf 254.783 -1.793 Td [(0 on)]TJ/F21 7.9701 Tf 23.271 -1.793 Td [(0 n )-301(and)]TJ/F80 11.9552 Tf 40.689 0 Td [(= )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 showsthat )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 isanalmost-normal 1 subgroupof)]TJ/F80 11.9552 Tf 71.096 0 Td [(; andthusthatProposition4of[ 3 ]appliesto showthatthereisaone-parametergroup t ofautomorphismsof A t f g = L g R g )]TJ/F23 7.9701 Tf 6.586 0 Td [(it f ; .91 where L resp. R denotesthenumberofleftresp.rightcosetsoccurringinthe decompositionof)]TJ/F21 7.9701 Tf 100.123 -1.794 Td [(0 g )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 : Theabovecomputationsofthedecompositionsofdouble 1 Asubgroup)]TJ/F22 5.9776 Tf 51.768 -1.107 Td [(0 )-389(issaidtobealmost-normalif)]TJ/F22 5.9776 Tf 127.021 -1.107 Td [(0 )]TJ/F22 5.9776 Tf 5.289 -1.107 Td [(0 canbewrittenasaunionofnitely-manyrightorleft cosetsof)-354(forall 2 47

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cosetsshowthat t n g = )]TJ/F80 11.9552 Tf 5.479 -9.684 Td [(p p n )]TJ/F23 7.9701 Tf 6.586 0 Td [(it n g .92 t e g = p )]TJ/F51 11.9552 Tf 11.955 0 Td [(1 p j j p )]TJ/F23 7.9701 Tf 6.586 0 Td [(it e : .93 Herethereisanimportantdierencebetween C Q and C Q p : In[ 3 ],theisotropy groupusedinthedecompositionofthehomogeneousspaceis P + Q P A f = Zo 1 : .94 Inthisthesis,theisotropygroupusedinthedecompositionofthehomogeneousspace p is P + Q P Z p = Z p o Z p > 0 : .95 Thefactthattherstsemidirectproductinvolvesatrivialgroupismanifestedinthe timeevolutionofthesystems;in C Q p ; theisometriesindexedby Q p = Z p evolveintime while,in C Q ; theisometriesindexedby Q = Z arexedintime.Thuscompletionof Q atitsnonarchimedeanplacesleadstonewdynamics. 4.Representations ThepresentationgiveninProposition4.6isusedtondacovariantrepresentation of A .Thisisconjecturedtoextendtoarepresentationof C Q p : Proposition 4.7 Foreach 2 Gal Q tot p = Q p Z p ; thereisacovariantrepresentation ;H of A on ` 2 p Z > 0 satisfying n k = n + k .96 e k = e 2 ip k k .97 H" k =log p k k : .98 48

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Proof. Weconsider = id ; whereidisidentityelementofGal Q tot p = Q p : Dene on n and e asinthestatementoftheproposition,andextendittoamorphism ofinvolutivealgebrasinthenaturalway.Theaboveproofofthepresentationshowed thatany T 2 C Q p isalinearcombinationofnitely-manyindicatorfunctions f 1 X j g : Sinceforany f 2 ` 2 p Z > 0 j Tf j ` 2 = X c j 1 X j f ` 2 X j c j 1 X j f j 2 ; .99 toshowthat T isbounded,i.e.,that j Tf j isniteforany f; itsucestoconsider T = 1 X forany X 2 )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 n )]TJ/F80 11.9552 Tf 7.314 0 Td [(= )]TJ/F21 7.9701 Tf 7.315 -1.793 Td [(0 : Equations.70and.73showthat 1 X = n e m forsome n;m; and withoneof n;m equaltozero.Usingthepropertiesofnorms andthefactthattheinvolutionon B H preservesnorms, jj n e m jjjj n jjjj e jjjj m jj : .100 intheoperatornorm jj T jj B ` 2 p Z > 0 : =sup fj Tf j ` 2 p Z > 0 : j f j ` 2 p Z > 0 =1 g : .101 Itsucestoshowthat e and n arebounded.Butthisisimmediate,as j e f j ` 2 = e X f k k ` 2 .102 = X f k e 2 ip k k ` 2 .103 = X j f k e 2 ip k j 2 .104 = X j f k j 2 .105 = j f j ` 2 .106 and j n f j ` 2 = n X f k k ` 2 = X f k n + k ` 2 = X j f k j 2 = j f j ` 2 .107 49

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Thus isa -homomorphismof A into B H : Theremainderoftheproofshows that n and e satisfytherelationsofProposition4.6. For a ; wecompute h n n j ;" k i = h n j ; n k i = h j + n ;" k + n i = h j ;" k i = h id j ;" k i : .108 Therefore n n actsastheidentity. For b ; n + m k = k + n + m = m k + n = m n k : .109 For c ; sincetheinnerproductishermitian, h e j ;" k i = h j ; e k i = e )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 ip k h j ;" k i : .110 Thisiszerounless j = k; inwhichcase h e j ;" k i = h e )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 ip k j ;" k i = h e )]TJ/F80 11.9552 Tf 9.299 0 Td [( j ;" k i : .111 Therelation d followsimmediatelyfromthefactthat e a + b = e a e b : Forrelation e ; e e k = e )]TJ/F80 11.9552 Tf 9.298 0 Td [( e 2 ip k k = e 2 ip k e )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 ip k k = k =id k : .112 Forrelation f ; e n k = e 2 ip n + k k + n = n e 2 ip k p n k = n e p n k : .113 For g ; rstnotethat,using a n n =id= n n n + k = n n n k = n k = n + k = n n =id : .114 50

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Therelation f gives n e n k = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n n n k = p )]TJ/F23 7.9701 Tf 6.586 0 Td [(n k = e 2 ip k )]TJ/F24 5.9776 Tf 5.756 0 Td [(n k : .115 Relation g followssincethereare p n -many 2 Q p = Z p suchthat p )]TJ/F23 7.9701 Tf 6.587 0 Td [(n = Carefullytrackingtheabovedetails,itisclearthatthisproofcanbemodiedto thegeneralcaseof : Conjecture 4.8 Anyrepresentationof A )]TJ/F80 11.9552 Tf 11.867 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 admitsauniquelifttoarepresentationoftheclosure A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.793 Td [(0 = C Q p : Itsucestoshowthat jj f jj C = jj f jj C Q p ; sinceanyrepresentationof A liftstoa representationoftheclosureunder jjjj C : Theinequality jj f jj C jj f jj C r obviously holds,sothegoalistoprovethereverseinequality.TheexamplegiveninTheorem 19of[ 3 ]suggestsstudyingtheconvolutionalgebrasassociatedtothegroupoid G = f z;r 2 Z p Q > 0 : zr 2 Z p g .116 ofarrowsbetween p -adicintegersthatdierbyapositiverationalscalar,withcompositiongivenby z;r 1 z;r 2 = z;r 1 r 2 : Theauthorintendstoshowthat G ismeasurewiseamenable,andthentoapplyTheorem6.1.8of[ 1 ]toshowthat C r G = C G : Then,theauthorwillndanembedding A C c G suchthat jj f jj r =esssup fjj f jj : isaninvolutiverepresentationof C c G g = jj f jj C Q p .117 for f 2 A ; where jj f jj r isthenormofequation.47.Thiswouldprovethetheorem. 51

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ConditionalTheorem 4.9 AssumingConjecture4.8,thepartitionfunctionof C Q p is E p : Proof. Theprecedingtworesultsgiveacovariantrepresentationof C Q p with Hamiltonian H equaltothatoftheToeplitzalgebra p introducedatthebeginning ofthechapter. 5.Anactionof Z p Thehomogeneousspace p P Q p =P Z p isofcourseactedonby P Q p ; andthus alsobyanyofitsquotients.Considerthesubgroup P + Q p : = Q p oQ > 0 ; whichacts transitivelyon p becauseitcontains P + Q : Thedecomposition Q p = Q > 0 Z p gives P Q p =P + Q p = 1 Q p Q p 1 Q p Q > 0 1 Z p Z p : .118 Thereisamap : 8 > < > : Z p )167(! Aut U P + Q 0 u 7)167(! x 7)167(! uxu x 2 P + Q 0 : .119 Toshowthat reallydoestake Z p toautomoprhismsof U P + Q 0 ; rstlet u : = 1 a r 2 P Q p T : = 1 s t 2 P + Q : .120 Usingthefactthat u = u )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ,compute u x T = 1 a r x 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(ar )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 s t .121 = 1 a r x 1 rs + a )]TJ/F80 11.9552 Tf 11.955 0 Td [(at t 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(ar )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 .122 = 1 a r 1 rs + a )]TJ/F80 11.9552 Tf 11.955 0 Td [(at t x 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(ar )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 .123 = 1 s t 1 a r x 1 )]TJ/F80 11.9552 Tf 9.299 0 Td [(ar )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 r )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 .124 = T u x : .125 52

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Thesubalgebra C p Z > 0 C Q p P + Q 0 isxedby : u n = u n u )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 .126 = 1 u 1 p n 1 u )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 .127 = 1 p n .128 = n ; .129 andso commuteswith t onthissubalgebra. Therestrictionof to C Q p = Z p C Q p P + Q 0 isdeterminedby u e = ue u )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 .130 = 1 u 1 1 1 u )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 .131 = 1 u )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 .132 = e u )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 : .133 Sincetheactionof Z p on Q p isnorm-preserving,itisclearfrom.92that commuteswiththerestrictionofthetimeevolutionto C Q p = Z p : Since C Q p isgenerated bythe e andthe n ,thisshowsthat C Q p isstableunder ; andthatthetime evolutioncommuteswith ; i.e.,thatthedynamicsof C Q p preservea Z p symmetry. Recallingtheisomorphism Z p Gal Q tot p = Q p ; .134 oneidentiesasymmetryof C Q p withtheramiedpartofGal Q ab p = Q p : 53

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PREFACE54 6.Futuredirections Thisthesishasintroducedapairofgroups)]TJ/F21 7.9701 Tf 243.784 -1.794 Td [(0 )-460(forwhichthereisstrong evidenceofarelationshipbetweentheHeckealgebra A andtheclasseldtheory of Q p : Thereaderisawarethatakeyresultinthistheoryisleftasaconjecture;a rststepinadvancingthisworkistoproveConjecture4.8.Agoodstartingplacefor thisistoshowthatthegroupoid.116isamenableandusethistoshowthat C Q p coincideswiththeuniversal C -envelopingalgebraof A )]TJ/F80 11.9552 Tf 11.866 0 Td [(; )]TJ/F21 7.9701 Tf 7.314 -1.794 Td [(0 : Beyondthistherearemanyotherpossiblecontinuationsofthiswork.Since [ 6 7 ],mostpapersintheareahavedevelopedBost-Connessystemsas C -algebras ofgroupoidsofequivalencerelations.Thisviewpointwassolidiedin[ 12 ],which giveswhathasbecometheaccepteddenitionforBost-Connessystemsforarbitrary numberelds.Theauthorhopestoshowthat C Q p canbeobtainedinthisframework. Thepaper[ 5 ]developsa p -adictheoryofBost-Connessystems,butthisseems toonlyconnectwith Q un p : Apossiblenextstepistoconsider C Q p astheramied counterpartofthat. Anotherpossiblecontinuation,discussedbrieyintheintroduction,istodevelop aBost-Connessystemforarbitrarynumberelds.Thoughthereisasatisfactory explicitclasseldtheoryforlocalelds,itisnotobvioustotheauthorhowthe constructionsofthisthesiswillgeneralize.Itmaybethattheframeworkofgroupoid C -algebraswillmakethisgeneralizationeasier. Finally,thereisthequestionofstudyingandclassifyingthestatesof C Q p andof whatevergeneralizationsmaybefound.Tothatend,therearemanyresultsof[ 3 ]to provideguidance.

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