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Probability theory on semihypergroups

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Probability theory on semihypergroups
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Youmbi, Norbert
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Hypergroups
Completely simple semihypergroups
Rees convolution product
Haar measure
Multipliers
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Abstract:
ABSTRACT: Motivated by the work of Hognas and Mukherjea on semigroups,we study semihypergroups, which are structures closer to semigroups than hypergroups in the sense that they do not require an identity or an involution. A semihypergroup does not assume any algebraic operation on itself. To generalize results from semigroups to semihypergroups, we first put together the fundamental algebraic concept a semihypergroup inherits from its measure algebra.
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Thesis (Ph.D.)--University of South Florida, 2005.
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by Norbert Youmbi.
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ProbabilityTheoryonSemihypergroupsbyNorbertYoumbiAdissertationsubmittedinpartialfulllmentoftherequirementsforthedegreeofDoctorofPhilosophyDepartmentofMathematicsCollegeofArtsandSciencesUniversityofSouthFloridaMajorProfessor:ArunavaMukherjea,Ph.DAthanassiosKartsatos,Ph.DJogindarRatti,Ph.DYunchengYou,Ph.DDateofApproval:July19,2005Keywords:Hypergroups,Completelysimplesemihypergroups,Reezconvolutionproduct,Haarmeasure,MultiplierscCopyright2005,NorbertYoumbi

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Ipraiseyou,Father,Lordofheavenandearth,becauseyouhavehiddenthesethingsfromthewiseandlearned,andrevealedthemtolittlechildren.ThisdissertationisdedicatedtomyMotherwhoinvestedallshecouldtoseemesucceedinlife;myfatherforallthesacricehemadeformeandmysiblings.ToSuzanne,FrankandPerezfortheirpatienceduringmycontinualabsence.

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ACKNOWLEDGEMENTSIexpressmyprofoundgratitudetomysupervisorProfessorArunavaMukherjea,forhisguidanceandcontinuoussupportduringthiswork.Ihavebeenparticularlyinspiredbyhismathematicaleruditionandtherigorhealwaysappliedindoingmath-ematics.MywarmestappreciationsalsogoestoProfessorsAthanassiosKartsatos,Jogin-darRattiandYunchengYouforacceptingtobepartofthesupervisingcommittee.IwouldalsoliketothankProfessorSudeepSarkaroftheComputerScienceDepart-mentforkindlyacceptingtobethechairpersonofthedefensecommittee.IwishtoacknowledgemyprevioussupervisorsProfessorsL.S.O.LiverpoolBScandV.A.BabalolaMSc.IalsothankProfessorG.Hognasforhissuggestions.IthankthemathematicsdepartmentoftheUniversityofSouthFloridaforitsgenerousnancial,academic,andpersonalsupport.Ithankmyallmybrothersandsisterswhoputinallwhattheycouldtoseemegothroughhighereducation.InparticularmysistersJacquelineKwekam,AnneDjakam,SaraMbontem,JeannetteNgounouandMarlyseCheabo.MyspecialthanksgoestoBettyAllenandPrudenciaNchofang.IalsothanktheFriendsofInternationalsatUSF,theGozofamily,SusanBambo,andallmyfriends.FinallyIspeciallyrecognizethesupportofCerodieuseAcceuswasaspecialmentorandfriendhelpmegetthroughtheculturalshockIexperiencedinmyrsttrimesterinTampa-USA.

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TableofContentsAbstractiii1PRELIMINARIES11.1Introduction................................11.2BasicConcepts..............................21.2.1Notations.............................21.2.2TheMichaeltopology.......................31.3DenitionsofaHypergroups.......................31.3.1Dunkl'sDenitionofaHypergroup...............31.3.2Jewett'sDenitionofaConvos.................41.3.3Spector'sDenitionofaHypergroup..............51.4TheDJS-Hypergroup...........................72EXAMPLESOFSEMIHYPERGROUPSANDHYPERGROUPS102.1Somenite-elementsemihypergroups..................102.1.1Two-elementsemihypergroups..................102.1.2Three-elementhypergroups...................122.1.3Four-elementhypergroups....................142.2ProductFormula.............................172.2.1LegendrePolynomial.......................182.2.2PolynomialHypergroups.....................202.2.3Kingman'sHypergroup.....................21i

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2.2.4Chebli-Trimechehypergroups..................222.2.5SemihypergroupsandHypergroupsassociatedwithPartialdif-ferentialoperators........................233TopologicalSemihypergroup293.1Introduction................................293.2Preliminaries...............................293.3Idealsofsemihypergroups........................343.4ReesConvolutionProduct........................364ConvolutionProductsOnSemihypergroups444.1PreliminaryResults............................444.2ConvolutionEquation..........................494.3InvariantandIdempotentMeasures...................564.4WeakConvergenceofConvolutionProductsofProbabilityMeasuresonSemihypergroups...........................634.4.1ConcretizationforSemihypergroups...............634.4.2SequenceofConvolutionofMeasures..............665SemigroupsofMultipliersassociatedwithsemigroupsofOperatorsinLpH785.1Introduction................................785.2MultipliersonHypergroups.......................805.3SemigroupsofOperators,SemigroupsofMultipliersonLpH....92AbouttheAuthorEndPageii

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ProbabilityTheoryonSemihypergroupsNorbertYoumbiABSTRACTMotivatedbytheworkofHognasandMukherjeaonsemigroups[HM95],westudysemihypergroups,whicharestructuresclosertosemigroupsthanhypergroupsinthesensethattheydonotrequireanidentityoraninvolution.Dunkl[Du73]callsthemhypergroupswithoutinvolution,andJewett[Je75]callsthemsemiconvos.Asemi-hypergroupdoesnotassumeanyalgebraicoperationonitself.Togeneralizeresultsfromsemigroupstosemihypergroups,werstputtogetherthefundamentalalgebraicconceptasemihypergroupinheritsfromitsmeasurealgebra.Amongotherthings,wedenetheReesconvolutionproduct,andprovethatifX;Yarenon-emptysetsandHisahypergroup,thenwiththeReesconvolutionproduct,XHYisacompletelysimplesemihypergroupwhichhasallitsidempotentelementsinitscenter.Wealsopointoutstrikingdierencesbetweensemigroupsandsemihypergroups.Forinstance,weconstructanexampleofacommutativesimplesemihypergroup,whichisnotcom-pletelysimple.InacommutativesemihypergroupS,wesolvetheChoquetequation=,undercertainmildconditions.Wealsogivethemostgeneralresultforthenon-commutativecase.Wegiveanexampleofanidempotentmeasureonacommuta-tivesemihypergroupwhosesupportdoesnotcontainanidempotentelementandsocouldnotbecompletelysimple.Thisisincontrastwiththecontextofsemigroups,whereidempotentmeasureshavecompletelysimplesupports.iii

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TheresultsofHognasandMukherjea[HM95]ontheweakconvergenceofthesequenceofaveragesofconvolutionpowersofprobabilitymeasuresisgeneralizedtosemihypergroups.WeusetheseresultstogiveanalternativemethodofsolvingtheChoquetequationonhypergroupswhichwasinitiallysolvedin[BH95]withmanysteps.WeshowthatIfSisacompactsemihypergroupandisaprobabilitymeasurewithS= [S1n=1Suppn],thenforanyopensetGKwhereKisthekernelofSlimn)177(!1nG=1:Finally,weextendtohypergroupsbasictechniquesonmultiplierssetforthforgroupsin[HR70],namelypropositions5.2.1and5.2.2,wegiveaproofofanextendedversionofWendel'stheoremforlocallycompactcommutativehypergroupsandshowthatthisversionalsoholdsforcompactnon-commutativehypergroups.ForacompactcommutativehypergroupH,weestablishrelationshipsbetweensemigroupS=fT:>0gofoperatorsonLpH,1p<1,whichcommuteswithtranslations,andsemigroupM=fE:>0gofLpHmultipliers.Theseresultsgeneralizethoseof[HP57]forthecirclegroupsand[B074]forcompactabeliangroups.iv

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1PRELIMINARIES1.1IntroductionTheoriginofhypergroupcanbetracedbacktothetimeoftheriseofgrouptheoryin1900withtheworkofFrobenious.Inthemidthirties,FMarty[MA]andM.S.Wall[WA]introducedtheconceptofanalgebraichypergroup,mainlywithinthetheoryofnon-abeliangroupsandrelatedstructuresofspacesofconjugacyclassesanddoublecosets.ThetermhypergroupinatopologicalcontextwasrstusedbyDelsarte[DE]whenheintroducedthetheoryofgeneralizedtranslationoperators.HistheorywasusedbyLevitan[LE]andlateronBerzanski[BK63]intheirtheoryofhypercomplexsystem.In1956,I.I.Hirshman,Jr.[Hi56a],pointedoutthatthestructureforhar-monicanalysisexistsinasettingwherecertainorthogonalpolynomialscouldplaytheroleoftheexponentialsinclassicalFourieranalysis.AproductformulaonthesystemoforthogonalfunctionsisusedtodenedaconvolutiononthevectorspaceofRadonmeasures.I.M.Gelfand[Ge50],S.Bochner[B074],[Bo54]andI.I.HirshmanJr[Hi56a],obtainedproductformulasandanumberofveryinterestingconsequencesforultrasphericalpolynomialsand,inparticular,forLegendrepolynomialsandforBesselfunctions.ForJacobipolynomialsproductformulaswerefoundbyG.Gasper[GA70],[GA71],[GA72].About3decadesago,HarmonicAnalystsandProbabilityTheoristswerefacedwiththequestionofwhichtopologicalspaceshaveenoughstructuressothataconvo-lutionforallniteregularBorelmeasurescouldbedenedonthesespaces.CharlesDunkl[Du73],RobertJewett[Je75]andReneSpector[Sp78]independentlyaddressed1

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thisquestionandsetdownbasicaxioms,deningsuchastructure,whichhascometobecalledDJS-Hypergroups.Hypergroupsgeneralizeinmanywayslocallycompactsemigroups.Ahypergrouprequiresanidentityelementandaninvolutionwhichactsasthegroupinverse.Inthischapterweintroducedthepreliminarynotationsandbasicconcepts.Wewillalsopresenttheaxiomsofhypergroupsassetdownbyeachofthefounders.WewillendwithwhatistodaygenerallyacceptedastheDJS-hypegroupwhichwillbeourdenitionofreferencethroughoutthisdissertation.1.2BasicConcepts1.2.1NotationsWerstrecallsomestandardnotations.LetSbealocallycompactHausdorspace:i.CS:thespaceofcomplexcontinuousfunctionsonS,ii.CbS:thespaceofboundedelementsofCSiii.C0S:thespaceofelementsofCbSwhichtendsto0at1iv.CcS:thespaceofelementsofC0Swithcompactsupportv.C+cS:thespaceofnonnegativeelementsofCcS.vi.MSdenotethesetofniteregularBorelmeasures.vii.M+S:thespaceofallnonnegativeelementsofMSviii.M1Sdenotethesetofprobabilitymeasures.ix.If2MS,thenSupp=fx2S:ifVisanyopensetcontainingx,thenV>0gx.AnunspeciedtopologyonM+Sistheconetopology.xi.Ifx2Sthenxdenotesthepointmassatx2

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xii.B+1Sdenotestheextendednonnegativereal-valuedBorelfunctions.xiii.IfAisanysubsetofS, AistheclosureofA,andAcisthecomplementofA.1.2.2TheMichaeltopologyLetSbealocallycompactspaceandletCSbethespaceofallcompactsubsetsofS.ForA;BS,CAB=fC2CS:CA6=;andCBgThenCScanbegiventhetopologygeneratedbythesub-basisofallCUVforwhichUandVareopensubsetsofS.ThistopologywhichwasdevelopedbyMichael[Mi55]hasthefollowingproperties[Je75]Properties1.2.1i.IfSiscompact,thenCSiscompact.ii.CSisalocallycompactspace.iii.Themappingx7!fxgisahomeomorphismofSontoaclosedsubsetofCS.iv.ThecollectionofnonemptynitesubsetsofSisadensesubsetofCS.v.IfisacompactsubsetofCS,thenB=SfA:A2gisacompactsubsetofS.vi.IfSismetrizablewithmetricd,thentheMichaeltopologyonCSisstrongerthantheHausdortopologygivenbytheHausdormetricwhichforA;B2CSisdenedbyA;B=maxfhA;B;hB;AgwherehA;B=supfdx;B:x2Ag1.3DenitionsofaHypergroups1.3.1Dunkl'sDenitionofaHypergroupAlocallycompactspaceHiscalledahypergroupifthereisamap:HH!M1Hwiththefollowingproperties:3

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D1.Foreachf2CcHthemapx;y7)167(!ZHfdx;yisinCbHHandx7)167(!ZHfdx;yisinCbHforeachy2HD2.TheconvolutiononMHdenedimplicitlybyZHfd=ZHdxZHdyZHfdx;y;2MH,f2C0Hisassociative.D3.Thereisapointtheidentitye2Hsuchthatx;e=xx2HD4.ThehypergroupHissaidtobecommutativeifx;y=y;x8x;y2HRemark1.3.1Dunkldoesnotrequiretheexistenceofaninvolutioninhisdenition,ratherhecalledanyhypergroup,withaninvolution,whichpossessesaninvariantmeasure,a-hypergroup.HedoesnotrequirethatthesupportoftheconvolutionoftwopointmassesbecompactandconsequentlydoesnotusetheMichaeltopologyinhisdenition.1.3.2Jewett'sDenitionofaConvosApairK;willbecalledasemiconvoifthefollowingveconditionsaresatised:4

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J1.KisanonvoidlocallycompactHausdorspace.J2.ThesymboldenotesabinaryoperationonMKandwiththisoperationMKisacomplexassociativealgebra.J3.Thebilinearmapping;7)167(!ispositive-continuous.Thatis,0whenever0and0andtheconvolutionrestrictedtoM+SM+S)167(!M+Siscontinuous.J4.Ifx;y2Kthenxyisaprobabilitymeasurewithcompactsupport.J5.Themappingx;y7)167(!SuppxyfromKKtoCKwiththeMichaeltopologyiscontinuous.Ifinadditionwealsohave,J6.ThereexistsanecessarilyuniqueelementeofKsuchthatxe=ex=xforallx2KJ7.Thereexistsanecessarilyuniqueinvolutionx7)167(!x)]TJ/F15 11.955 Tf 12.91 -4.338 Td[(ofKsuchthatforx;y2Ktheelementeisinthesupportofxyifandonlyifx=y)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(,thesemiconvowillbecalledaconvo.Remark1.3.2Dunkl'sdenitionofhypergroupisacommutativesemiconvowithidentity.Jewett'scommutativeconvoisaccordingtoDunkl'sdenition,a-hypergroup.1.3.3Spector'sDenitionofaHypergroupAhypergroupisalocallycompactspaceX,togetherwithaconvolutionthatmakesMXaBanachalgebraandsatisfythefollowingproperties:S1.M1XM1XM1XS2.isseparatelycontinuousfromM1XM1XtoM1XwiththeweaktopologydenedbythedualitybetweenMXandC0XMX;C0.5

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S3.Themapx;y7)167(!xyiscontinuousfromXXontoM1XwiththeweaktopologyinducedbyMX;C0S4.Thereisanecessarilyuniquepointecalledthe"identityelementofthehyper-groupX",suchthateistheidentityelementoftheconvolution.S5.ThereisaninvolutivehomeomorphismofXontoX,denotedbyx7)167(!x)]TJ/F15 11.955 Tf 11.179 -4.338 Td[(withnaturalextensiontoMXsatisfying)]TJ/F15 11.955 Tf 10.817 -4.338 Td[(=)]TJ/F16 11.955 Tf 9.906 -4.338 Td[()]TJ/F15 11.955 Tf 7.084 -4.338 Td[(;inparticulare)]TJ/F15 11.955 Tf 10.817 -4.338 Td[(=ethishomeomorphismwillbecalledthe"symmetryofthehypergroup".S6.Foreveryx;y2X,e2Suppxyifandonlyifx=y)]TJ/F23 11.955 Tf -313.529 -35.969 Td[(S7.ForanycompactsubsetKofXandanyneighborhoodVofKthereexistsaneighborhoodUofesuchthatSuppKandSuppUimplySuppVandSuppVSuppKandSuppUcimplythatthesupportof)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(,)]TJ/F16 11.955 Tf 9.463 -4.339 Td[(,)]TJ/F15 11.955 Tf 7.085 -4.339 Td[(,and)]TJ/F16 11.955 Tf 9.741 -4.339 Td[(aredisjointwithU.Remark1.3.3Spectordoesnotrequirethesupportoftheconvolutionoftwopointmassestobecompact,whichleadssometimestosometechnicalcomplicationsasheacknowledgeshimself.Actuallyhealsoacknowledgesnothavinganysubstantialex-amplewherethisconditionfails.ConsequentlythereisnouseoftheMichaeltopologyinhisproofs.WenowgiveageneraldenitionofahypergroupwhichisnowcalledtheDJS-hypergroup.Tothisend,westartwiththedenitionofasemihypergroupandgivesimpleexamplesofsemihypergroupsandhypergroups.6

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1.4TheDJS-HypergroupAnonemptylocallycompactHausdorspaceSwillbecalledasemihypergroupifthefollowingconditionsaresatised:SH1MbS;+;isaBanachalgebra.SH2Forallx;y2S,xyisaprobabilitymeasurewithcompactsupportcontainedinS.SH3Themappingx;y7!xyofSSintoM1S,whereSShastheproducttopologyandM1Shastheweaktopology,iscontinuous.SH4Themappingx;y7!SuppxyofSSintoCSiscontinuous,whereCSisthespaceofcompactsubsetsofSendowedwiththeMichaeltopology,thatisthetopologygeneratedbythesubbasisofallCUV=fC2CS:CU6=;andCVgwhereUandVareopensubsetsofS.RemarkIfxy=yxforallx;y2S,thenwesaythatS;isacommutativesemihypergroup.If,inaddition,wealsohave:SH5thereexistse2Ssuchthatxe=ex=x8x2S,andSH6ThereexistsatopologicalinvolutionahomeomorphismfromSontoSsuchthatx)]TJ/F15 11.955 Tf 7.085 -4.339 Td[()]TJ/F15 11.955 Tf 10.405 -4.339 Td[(=x8x2S,withxy)]TJ/F15 11.955 Tf 10.405 -4.339 Td[(=y)]TJ/F16 11.955 Tf 9.12 -0.299 Td[(x)]TJ/F15 11.955 Tf 10.513 -0.299 Td[(ande2Suppxyifandonlyifx=y)]TJ/F15 11.955 Tf 10.986 -4.338 Td[(whereforanyBorelsetB,)]TJ/F15 11.955 Tf 7.085 -4.338 Td[(B=fx)]TJ/F15 11.955 Tf 10.406 -4.338 Td[(:x2Bg,thenS;iscalledahypergroupExample1.4.11.IfS;:isatopologicalsemigroup,whereSisalocallycompactHausdorspacethenwithconvolutiondenedbyxy=xy,S;isasemihypergroup.Alsoifasemihypergroupissuchthattheconvolutionoftwopointmassesisapointmass,thenitisasemigroup.7

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2.LetS=fx;ygwiththediscretetopology.ThenSisalocallycompactspacewecandenetheconvolutionofpointmassesbyxx=ax+byyy=b0x+a0yxy=px+p0yyx=qx+q0ywherea;b;a0;b0;p;p0;q;q0arenon-negativerealnumberssuchthata+b=a0+b0=p+p0=q+q0=1fortheconvolutionproductoftwopointmassestobeaprobabilitymeasureandbb0=pp0=qq0fortheconvolutionproducttobeassociative.ThenS;isasemihypergroup.3.LetS=fe;a;bg.Letebetheidentityelementandletusdeneaa=1 2a+1 2bbb=aab=ba=1 2e+1 2bThenS;isasemihypergroupandifwedenedaninvolutionbya0=bandb0=awehaveaa0=1 2a0+1 2b0=1 2b+1 2aButa0a0=bb=a6=1 2a+1 2bSoalthoughe2Suppabthisinvolutiondoesnotsatisfytheconditionab0=b0a0thissemihypergroupisalmostthoughnotahypergroupanditiscalledaregularsemihypergroup[On93].8

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4.LetH=fe;x;ygandletebetheidentityelement,theidentityfunctionisconsideredastheinvolution,andacommutativeconvolutionisdenedonHbyxx=ae+bx+cyyy=a0e+c0x+b0yxy=yx=qx+q0yThenH;isahypergroupprovideda+b+c=a0+b0+c0=q+q0=1fortheconvolutionoftwopointmassestobeaprobabilitymeasure,anda0c=aqforassociativityofconvolution.Otherexamplesofhypergroupsandsemihypergroupscouldbefoundin[Du73],[Je75],[Sp78].Inthenextsection,wewillgivesomeinterestingexamplesofsemihy-pergroupsandhypergroups.9

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2EXAMPLESOFSEMIHYPERGROUPSANDHYPERGROUPS2.1Somenite-elementsemihypergroupsInthissectionweconsidertwo-elementsemihypergroupsandtwo,threeandfour-elementhypergroups.Weshowthattherearenon-commutativetwo-elementsemi-hypergroupsbutfor1n<5everyn-elementhypergroupiscommutative.Theproofofthislatterresult,thoughwell-knownandpartofthefolklore,isnotavailableinprint,tothebestofourknowledge.2.1.1Two-elementsemihypergroupsLetX=fx;yg.ThemostgeneralconvolutionproductonelementsofXisgivenbyxx=ax+byyy=b0x+a0yxy=px+p0yyx=q0x+qywherea;b;a0;b0;p;p0;q;q0arenon-negativerealnumbers.Nowweobservethatfortheconvolutionproductoftwopointmassestobeaprobabilitymeasurewemusthavea+b=a0+b0=p+p0=q+q0=1.10

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Remark2.1.1Sincetheconvolutionisassociativewehavexxx=xxxButxxx=axx+bxyandxxx=axx+byxsothatwehavebxy=byxandifb6=0theconvolutioniscommutative.Inasimilarway,wecanshowthatifb06=0theconvolutioniscommutative.Sowecanonlyexpectcommutativitywheneitherborb0isnon-zero.Tohaveassociativitythefollowingrelationsalsoholdxyx=xyxxxy=xxyyxy=yxyyyx=yyxwhicharerespectivelyequivalentstothesystems8<:bp=bq0aq0+pq=ap+p0q0.18<:pp0=bb0bp+p02=ap0+ba0.211

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8<:b0p0=b0qa0q+p0q0=a0p0+pq.38<:bb0=qq0ab0+a0q0=q02+b0q.4whichareallequivalenttotherelationbb0=pp0=qq0Proposition2.1.1Ifatwo-elementsemihypergroupisnotcommutative,thenitisasemigroup.ProofFromtheremarkaboveifeitherborb0isnon-zerothesemihypergroupiscom-mutative.Nowletusassumethatbandb0arebothzero.Then,associativityofconvolutionimpliesthatpp0=qq0=0Sothatthesemihypergroupisactuallyasemigroupsinceoneofp,p0iszeroandoneofq,q0iszero.Thuswemosthavexx=x;yy=y;xy=y;yx=xwhichisanoncommutativesemigroup.RemarkWehaveobservedabovethatunlessatwo-elementsemihypergroupisasemi-group,itiscommutative.WeeasilyobservealsothatifX=fe;xgisatwoelementhypergroup,thenitiscommutative.Infacttheconvolutionproductwillbedenedbyxx=te+)]TJ/F23 11.955 Tf 11.955 0 Td[(txWheretisanypositiverealnumber,eistheidentityelementandtheinvolutionistheidentityfunction.Xisahermitianhypergroup.Wenowprovethesameresultforthreeandfour-elementhypergroups.2.1.2Three-elementhypergroupsLetH=fe;x;ygbeathree-elementhypergroupwithidentityelemente.IfHisHermitianthenitiscommutative.NowletsassumethattheinvolutiononHis12

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denedbyx)]TJ/F15 11.955 Tf 10.406 -4.338 Td[(=yandy)]TJ/F15 11.955 Tf 10.405 -4.338 Td[(=x.Theinvolutionpropertyxy)]TJ/F15 11.955 Tf 10.405 -4.338 Td[(=y)]TJ/F16 11.955 Tf 8.872 -0.299 Td[(x)]TJ/F15 11.955 Tf 10.392 -0.299 Td[(impliesthattheconvolutionofpointmassesaredenedbyxx=ax+byyy=bx+ayxy=me+px+yyx=m0e+p0x+yWherem;m0;a;b;p;p0arenonnegativerealnumbers.SinceHsodeneisahy-pergroup,theconvolutionoftwopointmassesisaprobabilitymeasure,sothata+b=m+2p=m0+2p0=1withmm06=0Theassociativityaxiomleadstothefollowingsystemofequations.8>>><>>>:bp=bp0m0p=mp0m+ap+pp0=m0+ap0+pp0.58>>><>>>:b2=m+p2am=mpap+ab=pb+p2.6Andthedualsystems8>>><>>>:bp0=pbmp0=m0pm0+pp0+ap0=m+pp0+ap.713

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8>>><>>>:b2=m0+p02am0=m0p0ap0+ab=p0b+p02.8Sincemm06=0andp;q;p0;q0arenonnegative,b6=0anditfollowsthatp=p0sothatm=m0.ThereforeHiscommutative.RemarkThisexamplealsoshowsthattherearenonHermitiannitehypergroups.2.1.3Four-elementhypergroupsLetH=fe;x;y;zgbeafour-elementhypergroupwithidentityelemente.IfHisHermitianthenitiscommutative.NowletsassumethattheinvolutiononHisdenedbyx)]TJ/F15 11.955 Tf 10.405 -4.338 Td[(=x,y)]TJ/F15 11.955 Tf 10.405 -4.338 Td[(=zandz)]TJ/F15 11.955 Tf 10.406 -4.338 Td[(=y.Theninvolutionpropertyxy)]TJ/F15 11.955 Tf 10.405 -4.338 Td[(=y)]TJ/F16 11.955 Tf 7.795 -0.299 Td[(x)]TJ/F15 11.955 Tf -426.622 -21.968 Td[(impliesthattheconvolutionofpointmassesaredenedbyxx=ae+bx+cy+zxy=px+qy+rzxz=sx+ty+uzyy=p0x+q0y+r0zyx=sx+uy+tzyz=a0e+b0x+c0y+zzz=p0x+r0y+q0zzx=px+ry+qzzy=a00e+b00x+c00y+z14

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where;a;b;c;a0;b0;c0;a00;b00;c00;p;q;r;p0;q0;r0;s;t;uarenonnegativerealnumbers.SinceHsodeneisahypergroup,theconvolutionoftwopointmassesisaprobabilitymeasure,sothata+b+2c=a0+b0+2c0=a00+b00+2c00=p+q+r=p0+q0+r0=s+t+u=1withaa0a006=0Fromassociativityproperty,xyx=xyxwhichimplies8>>>>>><>>>>>>:pa=aspb+qs+rp=bs+up+tspc+qu+r2=cs+uq+t2pc+qt+rq=cs+ur+tu.9Sinceaa0a006=0,p=ssothat2.9becomes8>>>>>><>>>>>>:p=spb+qp+rp=bp+up+tppc+qu+r2=cp+uq+t2pc+qt+rq=cp+ur+tu.10whichisequivalentto8>>>>>><>>>>>>:p=sqp+rp=up+tpr2=t2qt+rq=ur+tu.1115

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sothatr2=t2r=tsincer;tarenonnegative.butp+q+r=s+t+uandsincep=sandr=twehaveq=usothatxy=yxandalsoxz=zxFromassociativitywealsohaveyzy=yzywhichleadsto8>>>>>><>>>>>>:c0a00=c00a0pb0+c0p0+c0b00=b00s+c00p0+c00b0a0+b0q+c0q0+c0c00=a00+b00u+c00q0+c00c0b0r+c0r0+c0c00=b00t+c00r0+c0c00.12Wealsohaveyyz=yyzwhichleadstotheequation8>>>>>><>>>>>>:q0a0=c0a0p0c+q0b0+r0p0=b0s+c0p0+c0b0p0t+c0q0+r02=a0+b0u+c0q0+c02p0u+c0q0+r0q0=b0t+c0r0+c02.1316

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andzzy=zzyimplies8>>>>>><>>>>>>:q0a00=c0a00p0p+r0p0+q0b00=b00p+c00b00+c00p0p0q+r0q0+q0c00=b00r+c002+c00r0p0r+r02+q0c00=a00+b00q+c002+c00q0.14andsincea0a006=0,q06=0andwehaveq0=c0=c00alsofrom2.9butweknowthata0=a00a0+b0+2c0=a00+b00+2c00sob0=b00thereforeyz=zyHence,Hiscommutative.2.2ProductFormulaInthissectionwepresentexamplesofhypergroupsgeneratedbysimplefunctions,viasomeproductformula.WealsogiveanexampleofahypergroupgeneratedbyaSturmLiouvilleproblem,namely,theChebli-Trimechehypergroup.WeendwithanexampleofasemihypergroupgeneratedbycertainpartialdierentialoperatorsonthespaceXn=[0;+1][)]TJ/F23 11.955 Tf 9.299 0 Td[(n;n].Weshowthatunlessn=1thesemihypergroupisnotahypergroup.Denition2.2.1LetfPg2R,beafamilyoforthogonalfunctionsontherealintervalI.WesaythatfPg2Rhasaproductformulaifforeachs;t2I,thereisaBorelmeasures;twithSupps;tIsuchthatZPds;t=PsPtforevery2R17

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Example2.2.1LetX=[0;+1and'x=cosx,2[0;+1.Thenwehavetherelation'x'y=1 2['x+y+'x)]TJ/F23 11.955 Tf 11.956 0 Td[(y]forall2[0;+1.Since'isanevenfunction,thisrelationisequivalentto'x'y=1 2['x+y+'jx)]TJ/F23 11.955 Tf 11.955 0 Td[(yj]Letx;y=1 2[x+y+jx)]TJ/F24 7.97 Tf 6.586 0 Td[(yj].Thenf'gsatisestheproductformula'x'y=Z'zx;ydzNowgiventwoRadonmeasuresandonXwecandeneaconvolutionf=ZZZfzx;yzdxdyforallf2CcX.Withthisconvolution,MXisaBanachalgebra[Tr97].Noticethattaking=xand=y,thisgivesus:xy=1 2[x+y+jx)]TJ/F24 7.97 Tf 6.586 0 Td[(yj]SothatX;isahypergroupwithidentityelement0,theinvolutionherebeingtheidentityfunction.2.2.1LegendrePolynomialTheLegendrepolynomialsfPngn2N0areorthogonalwithrespecttotheLebesguemeasureonI=[)]TJ/F15 11.955 Tf 9.298 0 Td[(1;1]andarenormalizedbyrequiringthatPn=1.Theyalso18

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satisfyaproductformulaPnxPny=ZIKx;y;zPnzdz)]TJ/F15 11.955 Tf 9.298 0 Td[(1x;y1withKx;y;z=8<:)]TJ/F21 7.97 Tf 6.586 0 Td[(1)]TJ/F23 11.955 Tf 11.955 0 Td[(x2)]TJ/F23 11.955 Tf 11.955 0 Td[(y2)]TJ/F23 11.955 Tf 11.955 0 Td[(z2+2xyz)]TJ/F22 5.978 Tf 7.782 3.259 Td[(1 2if1)]TJ/F23 11.955 Tf 11.955 0 Td[(x2)]TJ/F23 11.955 Tf 11.955 0 Td[(y2)]TJ/F23 11.955 Tf 11.955 0 Td[(z2+2xyz>0;0otherwise:ObviouslyKx;y;z0andsinceP0x=1,itfollowsfromtheproductformulathatZIKx;y;zdz=1:Forf;g2L1I;dxdenefgz=ZIZIKx;y;zfxgydxdysothatZIfgxPnxdx=[ZIfxPnxdx]:[ZIgxPnxdx];anditfollowsthatL1;isaBanachalgebrawithrespecttothemeasurePnxdx.Theoperationiseasilyextendedtothepointmassesbydeningdxyz=Kx;y;zdz;)]TJ/F15 11.955 Tf 9.299 0 Td[(1
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forallf2CcI.HirschmanalsodiscussestheseconstructionsforultrasphericalpolynomialsfPnxgthatareorthogonalonIwithrespecttothemeasure)]TJ/F23 11.955 Tf -424.076 -21.668 Td[(x2)]TJ/F22 5.978 Tf 7.782 3.258 Td[(1 2dx;theLegendrepolynomialsareultrasphericalpolynomialswith=1 2.ThepolynomialsRnx=Pnx PnareusedinplaceoftheLegendrepolynomialPnx.InthatcaseKx;y;z=8<:21)]TJ/F22 5.978 Tf 5.756 0 Td[(2)]TJ/F24 7.97 Tf 6.587 0 Td[(x2)]TJ/F24 7.97 Tf 6.586 0 Td[(y2)]TJ/F24 7.97 Tf 6.587 0 Td[(z2+2xyz)]TJ/F22 5.978 Tf 5.756 0 Td[(1 )]TJ/F22 5.978 Tf 5.288 2.269 Td[(2[)]TJ/F24 7.97 Tf 6.586 0 Td[(x2)]TJ/F24 7.97 Tf 6.586 0 Td[(y2)]TJ/F24 7.97 Tf 6.586 0 Td[(z2])]TJ/F22 5.978 Tf 6.952 2.345 Td[(1 2if1)]TJ/F23 11.955 Tf 11.955 0 Td[(x2)]TJ/F23 11.955 Tf 11.955 0 Td[(y2)]TJ/F23 11.955 Tf 11.955 0 Td[(z2+2xyz>0;0otherwise:Soforeach)]TJ/F21 7.97 Tf 23.217 4.707 Td[(1 2Hirschman[Hi56a]obtainsameasurealgebrathatwedenotebyI;.Itisimportanttonotethatisadistinctconvolutionforeach)]TJ/F21 7.97 Tf 24.679 4.708 Td[(1 2,henceacontinuumofBanachalgebrasisbuiltonthesingleBanachspaceMI.ThealgebraicstructuredoesnotdependonanyarithmeticintheunderlyingspaceI.[See[Hi56a][Hi56b]]2.2.2PolynomialHypergroupsLetpn;qnandrnbethreesequencesofrealnumberssuchthatpn>0;rn0;qn+1>0;q0=0andpn+qn+rn=1foralln2N.ThepolynomialsdenedbyP01;P1x=xandxPnx=qnPn)]TJ/F21 7.97 Tf 6.587 0 Td[(1x+rnPnx+pnPn+1xn1.15areorthogonalpolynomialson[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]withrespecttosomemeasuredx.Iftheirlinearizationcoecientsarenonnegativei.e.form;nwehavePmxPnx=m+nXr=jm)]TJ/F24 7.97 Tf 6.586 0 Td[(njcm;n;rPrx20

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withcm;n;r0forallr,wecandeneanHermitianhypergroupstructureN;onNwithe=0andmn=m+nXr=jm)]TJ/F24 7.97 Tf 6.586 0 Td[(njcm;n;rr:.16Itiscalledapolynomialhypergroupwithparameterspn;qnandrn.TheHaarmeasureisgivenby!=P1n=0!nnwith!n=p0p1:::pn)]TJ/F21 7.97 Tf 6.587 0 Td[(1 q1q2:::qn.17n1!=1.ThecharactersarethefunctionsonN:n7)167(!Pnxwithx2[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1]see[BH95]2.2.3Kingman'sHypergroupConsiderapairofindependentrandomvariablesX;YinR2,withlengthsXandY,butwithdirectionsuniformlydistributed.ThesumZ=X+Yalsohasuniformlydistributeddirection,butitslengthZ=jZjisarandomnumberwiththerangejX)]TJ/F23 11.955 Tf 12.002 0 Td[(YjZX+Y.Ingeneral,ifXandYareindependentrandomvariablesinR2withuniformlydistributeddirection,butwithlengthsXandYhavingprobabilitydistributions;2M1R+,thenZisarandomvariableinR+withaprobabilitydistributiondependingonand,denotedby,andwewriteZ=XY.TheoperationcanbeextendedtoallofMR+sothatMR+;becomesahypergroupmeasurealgebrathatisisometricallyisomorphictothesubalgebraofthegroupconvolutionalgebraMR2;,consistingofthemeasuresinvariantwithrespecttorotationsoftheplane.ThecharactersareindexedbyR+andgivenbyyx=J0xywhereJistheBesselfunctionoftherstkindoforder.ThesesatisfyaproductformulathatyieldsZR+yd=[ZR+yd][ZR+yd]21

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sothattheusefulsubstituteforthecharacteristicfunctionoftherandomvariableXisXy=RR+yd.TheproductformulafortheBesselfunctionsalsoensuresthefundamentalpropertyofcharacteristicequationsXY=XYwhenXandYareindependentrandomvariablesinR+.KingmanactuallydescribesacontinuumofHermitianhypergroupsR+;ofcourse,heneverusestheword"hypergroup".Theidentityelementis0,andthecharactersaregivenbyyx=Jyx=2\050+1yx)]TJ/F24 7.97 Tf 6.586 0 Td[(Jyxfory2R+.Whenn=2+2isaninteger,R+;isisometricallyisomorphictothealge-brasofrotationinvariantmeasuresonRn.Thereisagainnousefulalgebraicstructureintheunderlyingspaces.NeverthelessKingmanisabletodenerandomwalkandBrownianmotion,andobtainalawoflargenumbers,acentrallimittheorem,arecur-rencetheorem,andcharacterizationsofinnitelydivisibleandstabledistributions.Whenn=2+2isaninteger,allofthisisaninheritancefromthegroupstructureonRn,butKingmanobtainshisresultsforallreal)]TJ/F21 7.97 Tf 23.498 4.707 Td[(1 2withnoreferencetothegroupcaseexceptforinspiration[see[Ki63]].2.2.4Chebli-TrimechehypergroupsLetAbeanincreasingunboundedrealvaluedfunctiononR+suchthatA=0.WesupposeAdierentiable,A0=Anon-increasingonR+,limx!+1A0x=Ax=20andA0x=Ax==x+Bxinaneighborhoodofzero,with>0;BaninnitelydierentiableoddfunctiononR.Letusconsidertheoperator=d2 dx2+A0x Axd dx:22

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ForeveryinnitelydierentiableevenfunctionfonR,thesolutionuonR+R+oftheCauchyhyperbolicproblemxu=yu,withinitialconditionux;0=fxand@ @yux;0=0,canbewrittenintheformux;y=Z+10ftxydt;wherexy2M1R+isauniqueprobabilitymeasurewithsupporttheinterval[jx)]TJ/F23 11.955 Tf -424.076 -21.669 Td[(yj;x+y].Nowletxy=xy,theinvolutionbedenedbyx)]TJ/F15 11.955 Tf 10.406 -4.338 Td[(=x,andtheidentityelemente=0.ThenR+;withtheusualtopologyiscalledtheChebli-TrimechehypergroupwithfunctionA.TheHaarmeasureis!dx=Axdxandcharactersarethefunctions'2Cthataresolutionsoftheeigenvalueproblem'=)]TJ/F15 11.955 Tf 9.298 0 Td[(2+2';'=1;'0=1Moreoverthedual^R+consistsofcharacters'with2R+[i[0;][see[BH95]].2.2.5SemihypergroupsandHypergroupsassociatedwithPar-tialdierentialoperatorsForxedn2NwedenoteUn=8<:)]TJ/F23 11.955 Tf 9.299 0 Td[(n;0[;n;ifn2N;Rifn=0:VCn=8<:Z=n;ifn2N;Cifn=0:LetXn=[0;+1] Un.ConsiderthefollowingpartialdierentialoperatorsD1=@ @23

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D2=@2 @y2+[+1cothy+tanhy]@ @y)]TJ/F15 11.955 Tf 27.743 8.087 Td[(1 cosh2y@2 @2++12wherey;2;+1Unand0.Theuniquesolutionofthesystem8>>><>>>:D1u=iu;2VCnD2u=)]TJ/F23 11.955 Tf 9.298 0 Td[(2u;2Cu;0=1;@u @y;=0forall2Un.18denotedby';y;isgivenby';y;=eicoshy';y.19where';isaJacobifunction,thatis,theuniquesolutionoftheequation8<:;';x=)]TJ/F15 11.955 Tf 9.299 0 Td[(2+2';x';=1;d dx';=0;.20where;istheJacobidierentialoperator;=1 A;xd dx[A;xd dx]withA;x=22sinhx2+1coshx2+1and=++1.Thefunction';;;2VCnC,satisesthefollowingproductformulasi.if>0,thenforally;;t;2Xn';y;';t;= ZD';[coshycoshtei++sinhysinht]jj2)]TJ/F21 7.97 Tf 6.586 0 Td[(1dmwhereDistheopenunitdiscofCcenterat0anddm1+i2=d1d2.24

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ii.If=0,thenforally;;t;2Xn';y;';t;=1 2Z20';[coshycoshtei++sinhysinhtei]d:NowletCXndenotethespaceofcontinuousfunctionsonR Unevenwithrespecttotherstvariableandsuchthatforn6=0,thefunction7)167(!fy;is2n-periodiconR.Letf2CXn,wedenedtheconvolutionoftwopointmassesofXnbyi.if>0,forally;;t;2Xny;t;f= ZDf[coshycoshtei++sinhysinht])-222(jj2)]TJ/F21 7.97 Tf 6.587 0 Td[(1dmwhereDistheopenunitdiscofCcenterat0anddm1+i2=d1d2.ii.If=0,thenforally;;t;2Xny;t;f=1 2Z20f[coshycoshtei++sinhysinhtei]dTheconvolutionofpointmasseshasthefollowingproperties:i.Forall2Un;t;2Xn;t;=t;+ii.forally;;t;2Xny;t;=t;y;iii.Forally;2Xny;;0=y;So;0istheidentityelement.Remark2.2.125

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LetW,>0bethefunctiondenedonXnXnCbyWy;;t;;z=8>>><>>>: 22+1[1)]TJ/F15 11.955 Tf 11.955 0 Td[(cosh2y)]TJ/F15 11.955 Tf 11.955 0 Td[(cosh2t)-222(jzj2+2Recoshycoshtei+z])]TJ/F21 7.97 Tf 6.587 0 Td[(1;ifz2Dy;;t;;0ifz=2Dy;;t;:whereDy;;t;isthediscofCcenteredatcoshycoshtei+withradiussinhysinht.Thenforally;;t;2Xni.if>0y;t;f=ZDy;;t;fzWy;;t;;zdmnzwheredmnz=22+1x2+y2)]TJ/F15 11.955 Tf 11.955 0 Td[(1dxdyifz=x+iyii.if=0y;t;f=ZCy;;t;fzW0y;;t;;dzwhereCy;;t;isthediscofCcenteredatcoshycoshtei+withradiussinhysinhtandW0y;;t;;dzthemeasuregivenbyW0y;;t;;dz=dz z)]TJ/F15 11.955 Tf 11.955 0 Td[(coshycoshtei+Remark2.2.2Forally;;t;2Xnsatisfyingy;t6=0wehavei.ThefunctionWy;;t;;zispositiveandwehaveZDy;;t;Wy;;t;;zdmnz=126

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ii.wealsohaveZCy;;t;W0y;;t;;dz=1iii.Themeasuredmncanbewrittenforz=coshrei!withr0,!2Un,intheformdmnr;!=22+1sinhr2+1coshrdrdn!wheredn!=8>>><>>>:1 2nd!;ifn2N;d!ifn=0:Wenowstatethefollowingtheorem[Tr97],[Si95]Theorem2.2.1Withconvolutiondenedbyf=ZXnZXny;t;fdy;dt;forall;2MXnandforallf2CbXn,MXn;isacommutativeBanachalgebrawithidentityelement;0andwithaninvolutiondeneonXnbyy;)]TJ/F15 11.955 Tf 10.879 -4.338 Td[(=y;)]TJ/F23 11.955 Tf 9.299 0 Td[(andtheHaarmeasureismn.Thenextpropositionisfrom[Tr97][Si95]Proposition2.2.1Forally;;t;2Xn,wehavei.Suppy;t;=8>>><>>>:fr;s2Xn:coshreis2Dy;;t;g;for>0;fr;s2Xn:coshreis2Cy;;t;gotherwise:ii.Suppy;t;iscompactifandonlyifn2N.iii.;02Suppy;t;ifandonlyify=tandcos+=127

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Remark2.2.3i.Ifn=0,X0;isnotasemihypergroupbecausethesupportoftheconvolutionoftwopointmassesisnotcompact.ii.Ifn=1thenX1;isahypergroupcalledthehypergroupoftheexterioroftheunitdisciii.Ifn2,thenXn;isasemihypergroupwhichisnotahypergroup.Infactalltheaxiomsofahypergrouparesatisedexcepttheproperty;02Suppy;t;ifandonlyify;=t;)]TJ/F17 11.955 Tf 7.084 -4.338 Td[(.Infactfromproposition2.2.1iii,;02Suppy;t;ifandonlyify=tandcos+=1,thatis+=2nforalln2Nsoforn=1+=0andforn2,wecanhave+6=0.28

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3TopologicalSemihypergroup3.1IntroductionAsobservedinthepreviouschapter,asemihypergroupdoesnotassumeanyalgebraicoperationonitself.Sotobeabletoworkonsemihypergroups,weinducedfromtheconvolutiondenedonitsmeasurealgebraanalgebraicoperation,whichenablesustogeneralizemanyresultsfromsemigroupstosemihypergroups.Wesetdowninthissectionbasicresultsnecessarytodoharmonicanalysisorprobabilitytheoryonsemihypergroups.3.2PreliminariesDenition3.2.11.Anelemente2SiscalledaleftrightidentityelementofSifex=xxe=xforeveryx2S.AnelementeiscalledatwosidedidentityofSorsimplyanidentityofS,ifitisbothaleftandrightidentity.Theidentity,whenitexists,isunique.2.Anelementz2SiscalledaleftrightzeroelementofSifzx=zxz=zforallx2S.Ifzisbothleftandrightzero,wesimplycallitthezeroofS.Asemihypergrouphasatmostonezero.3.Anelementa2SiscalledanidempotentelementofSifaa=aRemark3.2.1Theonlyidempotentelementinahypergroupistheidentityelement.Forifthereisanidempotentelement,itspointmasswouldbeanidempotentmeasure29

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anditssupportasingletonsubhypergroup[Je75]10.2E.Denition3.2.2LetS;beasemihypergroup1.Ifx2SandA,BaresubsetsofSwedeneAx=[y2ASuppyxxA=[y2ASuppxyAB=[x2A;y2BSuppxyRemark3.2.2AclosednonemptysubsetFofScanbeveriedtobeasubsemihy-pergroupofSifandonlyifFFFThenextlemmaisfrom[Je75]Lemma3.2.1LetSbeasemihypergroupandA;B;CS.Theni.AB AB.ii.IfAandBarecompactthenABiscompactiii.ConvolutionisacontinuousoperationonCSiv.IfAandBarecompactandUisanopensetcontainingAB,thenthereexistopensetsVandWsuchthatAV,BWandVWUv.ABC=ABCCSwith*sodenedisatopologicalsemigroup.Remark3.2.3Thefollowingremarkisfrom[Je75]1.IffxgisanetinahypergroupS,thentheexpressionx!1meansthatx2S)]TJ/F23 11.955 Tf 11.955 0 Td[(AeventuallyforeachcompactsubsetAofS.30

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2.IffAgisanetinCS,thentheexpressionA!f1gmeansthatAS)]TJ/F23 11.955 Tf 10.419 0 Td[(AeventuallyforeachcompactsubsetAofS.NotethatA!1andA!f1ghavedierentmeanings.Thenextlemmaisstatedwithoutproofin[Je75].Wegivehereadetailedproof.Lemma3.2.2[Jewett]IfHisahypergroupandA;B;CaresubsetsofH,theni.e2A)]TJ/F16 11.955 Tf 9.741 -4.339 Td[(BifandonlyifAB6=;;alsoe2AB)]TJ/F17 11.955 Tf 11.269 -4.339 Td[(ifandonlyifAB6=;ii.ABC6=;ifandonlyifBA)]TJ/F16 11.955 Tf 9.741 -4.338 Td[(C6=;ifandonlyifACB)]TJ/F15 11.955 Tf 7.084 -4.338 Td[(6=;iii.IfBisopen,thenABisopenandAB=ABiv.IfAiscompactandBisclosed,thenABisclosed.Proofi.Supposee2A)]TJ/F16 11.955 Tf 7.405 -4.339 Td[(B.Thenthereexistsx2Aandy2Bsuchthate2Suppx)]TJ/F16 11.955 Tf 7.073 -0.299 Td[(ywhichimpliesx=yfromSH6,soAB6=;.NowifAB6=;thenthereexistsx2AB,andsoe2Suppx)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(x.Therefore,e2A)]TJ/F16 11.955 Tf 9.741 -4.338 Td[(Bii.ABC6=;ifandonlyife2AB)]TJ/F16 11.955 Tf 9.473 -4.339 Td[(Cifandonlyife2B)]TJ/F16 11.955 Tf 9.474 -4.339 Td[(A)]TJ/F16 11.955 Tf 9.473 -4.339 Td[(CifandonlyifBA)]TJ/F16 11.955 Tf 9.793 -4.338 Td[(C6=;ifandonlyife2BC)]TJ/F16 11.955 Tf 9.792 -4.338 Td[(A=BC)]TJ/F15 11.955 Tf 7.084 -4.338 Td[(AifandonlyifACB)]TJ/F15 11.955 Tf 7.085 -4.338 Td[(6=;iii.SupposeBisopen.Leta2A,thenx2fagBifandonlyifBfa)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gfxg6=;fromiiabove.Sincethemapx7)167(!fa)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gfxgiscontinuousfromSH4,thesetCBHisanopensetintheMichaeltopologywhichcontainsfa)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfxgbecauseBfa)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfxg6=;andfa)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfxgHsoitsinverseimagebythecontinuousfunctionx7)167(!fa)]TJ/F16 11.955 Tf 7.084 -4.338 Td[(gfxgisopen,whichis,fy2H:fa)]TJ/F16 11.955 Tf 7.084 -4.338 Td[(gfygB6=;g=fagB.ThusfagBisanopensubsetofH.31

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iv.LetxnbeasequenceofelementsofABconvergingtoanelementx2S.ThentherearesequencesanAandbnBsuchthatxn2fangfbngforeachn.Thisisequivalenttobn2fa)]TJ/F24 7.97 Tf 0 -7.294 Td[(ngfxngforeachnfromiiaboveseealsotheremarkibelow.SinceAiscompact,thesequenceanhasaconvergentsubsequencesay,aksuchthatbk2fa)]TJ/F24 7.97 Tf 0 -8.278 Td[(kgfxngforeachk.Furthermorea)]TJ/F24 7.97 Tf 0 -8.277 Td[(kandxkarerelativelycompactAsconvergentsequences.Sobkhasaconvergentsubsequenceconvergingtoapointb2BsinceBisclosed.NowfromSH4ifak)167(!a2Athenfa)]TJ/F24 7.97 Tf 0 -8.278 Td[(kgfxkg)167(!fa)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gfxg.Sob2fa)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gfxgsincebk2fa)]TJ/F24 7.97 Tf 0 -8.278 Td[(kgfxgforallk.Andagainfromiiaboveb2fa)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gfxgifandonlyifx2fagfbgAB.ThusABisclosed.Remark3.2.4i.Fromiiabovewealsohavez2Suppxyifandonlyify2Suppx)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(zifandonlyifx2Suppzy)]TJ/F15 11.955 Tf 6.752 -0.299 Td[(ii.IfHisacompacthypergroup,thenHA=AH=HforallA2CHsoHisthezeroofCH;.ButifHisnotcompact,thenHisnotanelementofCH.Thisresultisnotalwaystrueforcompactsemihypergroupsaswewillseebelowwiththedenitionofidealsinsemihypergroups.Denition3.2.31.Ahomomorphismofsemihypergroupsisdenedviameasurealgebraasfollows:LetSandTbetwosemihypergroups.AmappingfromSintoTiscalledasemihypergrouphomomorphismifandonlyif:M1S;)167(!M1T;isasemigrouphomomorphism.Thatis,=,8;2M1S,suchthatxisapointmassinM1T,8x2S.Ifinadditionisonetooneandonto,itisreferredtoasanisomorphism.2.Productofsemihypergroups.LetS;;T;betwosemihypergroups.ThesetSTwiththeproducttopologyisalocallycompactspace,andthiscanbemadeintoasemihypergroupbydeningx;ys;t=xsyt32

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wherex;y;s;t2STandx;ys;tisaproductmeasureonST.Denition3.2.4LetSbealocallycompactsemihypergroup.ThecenterofSisdenedbyZS=fx2S:Suppxyisasingleton,forally2SgRemark3.2.5ForahypergroupHthecenteristhemaximumsubgroupdenedbyJewettasZH=fx2H:xx)]TJ/F15 11.955 Tf 12.414 -0.299 Td[(=x)]TJ/F16 11.955 Tf 10.345 -0.299 Td[(x=egToseethis,supposethatxx)]TJ/F15 11.955 Tf 11.089 -0.298 Td[(=x)]TJ/F16 11.955 Tf 9.815 -0.298 Td[(x=eandlety2Hbearbitrarilychosen,andassumethata;b2Suppxythensincea2fxgfygfromlemma3.2.21.1.7iifxgfygfag6=;whichisequivalenttoy2fx)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfag;similarly,y2fx)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gfbgwhichmeansthatfx)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gfagfx)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gfbg6=;andthisisequivalenttofagfxgfx)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gfbg6=;andsincexx)]TJ/F15 11.955 Tf 10.585 -0.298 Td[(=x)]TJ/F16 11.955 Tf 9.614 -0.298 Td[(x=eitfollowsthatfagfbg6=;thatisa=bsothatSuppxyisasingleton,forally2HConverselysupposeanelementxissuchthatSuppxyisasingleton,forally2HthenSuppxx)]TJ/F15 11.955 Tf 6.752 -0.299 Td[(isasingletonandsincebydenitionitcontainsewehavexx)]TJ/F15 11.955 Tf 10.073 -0.298 Td[(=eExample3.2.1i.Everysemigroupisasemihypergroupanditscenteristheentiresemigroup.Alsoeverygroupisahypergroupwhichisthemaximumsubgroupequivalentlythecenterofitself.ii.IfHisahypergroup,thene2Hsothecenterofahypergroupisnonempty.WhenZH=feg,thecenterissaidtobetrivial.iii.LetS=fx;ygwithconvolutiondenedbyxx=yyy=1 4x+3 4yxy=yx=1 2x+1 2y33

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fromexample1.4.1iiSisasemihypergroupwithavoidcenteriii.Considerthesegment[0;1]withconvolutiondenedbyrs=1 2jr)]TJ/F24 7.97 Tf 6.587 0 Td[(sj+1 21j1)]TJ/F24 7.97 Tf 6.586 0 Td[(r)]TJ/F24 7.97 Tf 6.586 0 Td[(sjforallr;s2[0;1]Zeuner[Ze89]provedthat[0;1];isahypergroupwithanontrivialcenterf0;1g3.3IdealsofsemihypergroupsDenition3.3.1Ideals1.AsubsemihypergroupLRofasemihypergroupSiscalledaleftrightidealofSifSLLRSR;IiscalledanidealofSifandonlyifitisbotharightandleftideal.2.Siscalled,leftrightsimpleifitcontainsnoproperleftrightideal.Sissaidtobesimpleifitcontainsnoproperideal.Aleftrightidealissaidtobeaprincipalleftrightidealifitisoftheformfag[Safag[aSforsomea2SRecallthatwewriteSatomeanSfag.3.8a;b2Swesaythattheequationxa=bissolvableifandonlyifthereexistsx02Ssuchthatb2Suppx0aProposition3.3.1Sisleftsimpleifandonlyif8a;b2Stheequationxa=bissolvable.Proof:First,assumeSisleftsimple.Then8a2S,SaisaleftidealofSandsinceSisleftsimpleS=Saanditfollowsthat8b2S,9x02Ssuchthatb2Suppx0asoxa=bissoluble.Nowassumethatxa=bissolubleforalla;b2S,andLisaleft34

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idealofS.Thengivena2L;SaL.Alsogivenb2Stheequationxa=bissolubleso9x02Ssuchthatb2Suppx0awhichisasubsetofSa,soSSaLthereforeS=LandsoSisleftsimple.Wecanalsomakeasimilarargumentforrightideals.Remark3.3.1i.EveryleftrightidealcontainsaleftrightidealoftheformSaaSforsomea2S.ForifLisaleftidealthenforanya2L,SaisaleftidealcontaininL.Asimilarstatementholdsforrightideals.ii.Asemihypergroupcanbeleftandrightsimplewithoutbeingahypergroup.Anex-ampleisthefollowingsemihypergroup.LetS=fx;ygwithconvolutiondenedbyxx=yyy=1 4x+3 4yxy=yx=1 2x+1 2yFromexample1.1.4iiSsodenedisasemihypergroupwithnoproperidealbutisnotahypergroupsinceithasnoidentityelement.Denition3.3.21.AnidempotentelementinasemihypergroupSissaidtobeaprimitiveidempotentelementifitisinthecenterofthesemihypergroupandisminimalwithrespecttothepartialorderonESthesetofidempotentelementsofS,denedbyefef=fe=e2.Acompletelysimplesemihypergroupisasimplesemihypergroupwhichcontainsaprimitiveidempotentelement.Remark3.3.2TheorderdenedonESusesconvolutionofpointmassestocom-pareidempotentelementsofS.NotethatifaisaprimitiveidempotentofS,aisnot35

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necessarilyaprimitiveidempotentinM1S,accordingtothedenitionofprimitiveidempotentsinthesemigroupwithrespecttoconvolutionM1S.Lemma3.3.1LetSbeacompactsemihypergroup.TheneachleftrightidealofScontainsatleastoneminimalleftrightideal,andeachminimalleftrightidealisprincipalthatis,oftheformSaaSforsomea2S.Everycompactsemihypergrouphasaminimaltwosidedideal.Proof:GivenaleftidealI,denethefamilyF=fJ:JisaleftidealofS,JIg.Foranya2I,SaisaleftidealofSandhenceanelementofF.TheusualinclusionrelationisapartialorderonF.Furthermore,anylinearlyorderedsubfamilyofFhasaminimalelementsincesetsinFarecompact.ByZorn'slemma,thereexistsatleastoneminimalelementwithrespecttoinclusioninF.CallthisminimalleftidealI0.Clearly,foranyxinI0,Sx=I0.Proposition3.3.2LetSbeacompactsemihypergroup.ThenShasakernelK,thatis,aminimaltwosidedideal.Proof:BythecompactnessofS,thereisaminimal2-sidedidealK0.LetK=SfxS:xSisaminimalrightidealofSg.NoticethatK0SisalsoanidealcontainsinK0.andtherefore,K0=K0S.ThereforeK0SfyS:y2K0g.SincexSK0K0xS,K0xSisarightidealcontainsinxSthereforexSK0,anditfollowsthatK0=K.WewillnowdenetheReesconvolutionproductwhichwillbeusedtoconstructaclassofcompletelysimplesemihypergroupswithnonemptyandinnitecenter.3.4ReesConvolutionProductLetH;beahypergroupwithcenterZandX,Ybetwononemptysets.Let:YX)167(!Zbeamapping.LetusdeneaconvolutiononpointmassesofXHY36

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byx;h;yx0;h0;y0=xhy;x0h0y0ThisproductwillbereferredtoastheReesconvolutionproduct.Proposition3.4.1IfHisahypergroup,andXandYaretwononemptylocallycompactHausdorspaces,thenthespaceXHYisasemihypergroupwiththeReesconvolutionproduct,asdenedabove.Proof:LetK=XHYandx;h;y;x0;h0;y0betwopointsinK.Then[x;h;yx0;h0;y0]K=[xhy;x0h0y0]K=xX[hy;x0h0H]y0Y=1Sincehy;x0isaprobabilitymeasurewithcompactsupportinH,hy;x0h0isaprobabilitymeasurewithcompactsupportinHanditfollowsthatxhy;x00hy0isaprobabilitymeasurewithcompactsupportinK.Nextwehavetoshowthatisassociative.Letx;h;y;x0;h0;y0andx00;h00;y00bethreearbitraryelementsofKthen[x;h;yx0;h0;y0]x00;h00;y00=[xhy;x0h0y0]x00;h00;y00=xhy;x0h0y0;x00h00y00Andx;h;y[x0;h0;y0x00;h00;y00]=x;h;y[x0h0y0;x00h00y00=37

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xhy;x0h0y0;x00h00y00Andwecaneasilyseethat[x;h;yx0;h0;y0]x00;h00;y00=x;h;y[x0;h0;y0x00;h00;y00]ThisshowsthatK;isasemihypergroup.Uptothispointwehaveconsidered:YX)167(!HandhavenotusedthefactthatmapsYXintoZ,thecenterofH.Wewillrequirethisinwhatfollows.Lemma3.4.1Anelementx;h;y2Kisanidempotentelementifandonlyifh=y;x)]TJ/F17 11.955 Tf 7.084 -4.339 Td[(.Furthermore,idempotentelementsofKareinitscenter.Proof:Letx;h;ybeanidempotentelementofK.Then,wehave:x;h;yx;h;y=xhy;xhy=xhyThatis,hy;xh=hMultiplyingbothsidesoftheequalityabovebyy;xontheleft,wehavey;xhy;xh=y;xhThisshowsthaty;xhisanidempotentelementofthehyperpgroupHandsoistheidentityofH,therefore,h=y;x)]TJ/F15 11.955 Tf -197.63 -26.007 Td[(Wenoteherethatifwedidnotassumethaty;xwasinthecenterofHthisresultwillstillholdasy;xhwillbeconsideredanidempotentprobabilitymeasureandsoitssupportisasubhypergroup[Je75]10.2EofHcontainingthe38

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identitysothath=y;x)]TJ/F15 11.955 Tf 7.085 -4.338 Td[(,byaxiomSH6inthedenitionofahypergroup.Nextweneedtoshowthat8x2Xandy2Yx;y;x)]TJ/F23 11.955 Tf 7.085 -4.339 Td[(;yisanidempotentelementofKforx;y;x)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;yx;y;x)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;y=xy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(y;xy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(y=xy;x)]TJ/F16 11.955 Tf 9.409 -0.298 Td[(y=x;y;x)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;ysincey;x)]TJ/F15 11.955 Tf 9.561 -0.299 Td[(isinthecenterofHthisisthersttimewehaveusedthecenterpropertyofZ,letx;y;x)]TJ/F23 11.955 Tf 7.084 -5.812 Td[(;yandx0;h0;y0betwoarbitraryelementsofK.Then,x;y;x)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;yx0;h0;y0=xy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(y;x0h0y0NoticethatbythecenterpropertyofZ,y;x)]TJ/F16 11.955 Tf 9.487 -0.299 Td[(y;x0h0isapointmass.Thus,x;y;x)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;yisinthecenterofK.Proposition3.4.2IfHisahypergroup,andXandYaretwononemptylocallycompactHausdorspaces,thenthesemihypergroupK=XHYwiththeReesconvolutionproduct,asdenedabove,iscompletelysimple.Proof:First,weneedtoshowthatKissimple.LetIbeanidealofKandletx;h;y2KbeapointinK.Wewillshowthatx;h;y2IwhichshowsthatK=I.Todothis,letx1;h1;y1beanypointofI.Thenthesupportoftheprobabilitymeasurex;h;yx1;h1;y1x;h;yisasubsetofI.Wewillprovethatthepointx;h;y2I.BydenitionoftheconvolutionproductonKx;h;yx1;h1;y1x;h;y=xhy;x1h1y1;xhy39

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AndobservethatSuppx;h;yx1;h1;y1x;h;y=fxgSupphy;x1h1y1;xhfygThuswheneverx;h;y2K,fxgSupphy;x1h1y1;xhfygISincey;x1h1y1;x=kforsomek2H,wehavex;k)]TJ/F23 11.955 Tf 7.084 -4.339 Td[(;y2fxgSuppk)]TJ/F16 11.955 Tf 7.839 -0.299 Td[(y;x1h1y1;xk)]TJ/F15 11.955 Tf 6.752 -0.298 Td[(fygIforsomek2H.Nowifk)]TJ/F16 11.955 Tf 9.926 -0.298 Td[(y;x=u,thenx;u)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;y2Kandx;e;y2fx;k)]TJ/F23 11.955 Tf 7.084 -4.338 Td[(;ygfx;u)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;ygI.Nowforanyh2H,x;fy;x)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfhg;y2Kandwehavex;e;yx;fy;x)]TJ/F16 11.955 Tf 7.085 -4.936 Td[(gfhg;y=x;h;y2I:ThisshowsthatI=K,andthusKissimple.NextweneedtoshowthatKcontainsaprimitiveidempotentelement.Nowsupposex;y;x)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;yandx0;y0;x0)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;y0aretwoidempotentelementsofKsuchthatx;y;x)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;yx0;y0;x0)]TJ/F23 11.955 Tf 7.084 -4.338 Td[(;y0thenx;y;x)]TJ/F24 7.97 Tf 6.255 -2.269 Td[(;yx0;y0;x0)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;y0=x;y;x)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;ywhichisequivalenttoxy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(y;x0y0;x0)]TJ/F16 11.955 Tf 9.41 -0.299 Td[(0y=xy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(ysothaty0=yAndx0;y0;x0)]TJ/F24 7.97 Tf 6.255 -2.269 Td[(;y0x;y;x)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;y=x;y;x)]TJ/F24 7.97 Tf 6.255 -2.269 Td[(;ywhichisequivalentto40

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x0y0;x0)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(y0;xy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(y=xy;x)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(ysothatx0=x.Combiningthesetworesultsweseethatx;y;x)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;y=x0;y0;x0)]TJ/F23 11.955 Tf 7.084 -4.338 Td[(;y0Thatisx0;y0;x0)]TJ/F23 11.955 Tf 7.085 -4.339 Td[(;y0isaminimalidempotentelement.Andsimilarlywecanshowthatx;y;x)]TJ/F23 11.955 Tf 7.085 -4.339 Td[(;yisaminimalidempotentelement.SoallidempotentelementsofKareprimitives,soKisacompletelysimplesemihypergroup.Remark3.4.1Iftheoperationdenedaboveiscommutativethen,XandYareeachasingletonsetandinthiscaseKisahypergroup.Thisisprovedinthecorollarybelow.Corallory3.4.1LetH;beahypergroupands;ttwoelements.ThenfsgHftgwiththeReesconvolutionproductisacellhypergroupwithidentityelements;t;s)]TJ/F23 11.955 Tf 7.084 -4.339 Td[(;tandtheinvolutiondenedbys;h;t_=s;h0;tifandonlyifh0=t;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(h)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t;s)]TJ/F20 11.955 Tf -259.456 -39.9 Td[(Proof:Firstweneedtoshowthats;t;s)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;tistheidentityoffsgHftgLeth2Hthens;t;s)]TJ/F24 7.97 Tf 6.254 -2.269 Td[(;ts;h;t=st;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t;sht=shtAndsincet;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t;sistheidentityinHthisequalityhold.Nextweneedtoshowthatforallh2Hs;h;t__=s;h;t,s;t;s)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;t2Supps;h;ts;h0;tifandonlyifs;h;t_=s;h0;t.41

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Supposes;h;t_=s;h0;twhereh0=t;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(h)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t;s)]TJ/F15 11.955 Tf -277.015 -39.9 Td[(Supposealsothats;h0;t_=s;h00;twhereh00=t;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(h0)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t;s)]TJ/F15 11.955 Tf -261.891 -39.9 Td[(thenh0)]TJ/F15 11.955 Tf 10.073 -0.299 Td[(=t;sht;sSothath00=t;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t;sht;st;s)]TJ/F15 11.955 Tf 10.073 -0.299 Td[(=hSoh=h00andtherefores;h;t__=s;h;tNextsupposes;t;s)]TJ/F23 11.955 Tf 7.084 -4.338 Td[(;t2Supps;h;ts;h0;t,thatist;s)]TJ/F16 11.955 Tf 14.547 -4.338 Td[(2fhgft;sgfh0gwhichisequivalenttoh02ft;s)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfh)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gft;s)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gbutft;s)]TJ/F16 11.955 Tf 7.085 -4.339 Td[(gfh)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gft;s)]TJ/F16 11.955 Tf 7.085 -4.338 Td[(gisasingletonast;s)]TJ/F15 11.955 Tf 10.716 -4.338 Td[(isinthecenterofHsoh0=t;s)]TJ/F16 11.955 Tf 8.857 -0.298 Td[(h)]TJ/F16 11.955 Tf 8.857 -0.298 Td[(t;s)]TJ/F15 11.955 Tf 10.655 -0.299 Td[(whichshowsthats;h;t_=s;h0;t.Nowsupposes;h;t_=s;h0;tthenh0=t;s)]TJ/F16 11.955 Tf 9.863 -0.299 Td[(h)]TJ/F16 11.955 Tf 9.863 -0.299 Td[(t;s)]TJ/F15 11.955 Tf 11.32 -0.299 Td[(whichimpliesthath02t;s)]TJ/F16 11.955 Tf 9.134 -4.339 Td[(fh)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gft;s)]TJ/F16 11.955 Tf 7.084 -4.339 Td[(gwhichisequivalenttot;s)]TJ/F16 11.955 Tf 10.405 -4.339 Td[(2fhgft;sgfh0gwhichshowsthats;t;s)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(;t2fsgfhgft;sgfh0gftgThatiss;t;s)]TJ/F23 11.955 Tf 7.084 -4.338 Td[(;t2Supps;h;ts;h0;t.Nextweneedtoshowthats;h;ts;g;t_=s;g;t_s;h;t_NotethatbythedenitionofinvolutiononthefsgHftg,if2MHthenst_=st;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[()]TJ/F16 11.955 Tf 9.741 -4.936 Td[(t;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t42

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Nows;h;ts;g;t_=sht;sgt_=st;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(ht;sg)]TJ/F23 11.955 Tf 7.085 -4.936 Td[(t;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t=st;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[()]TJ/F24 7.97 Tf -0.448 -7.892 Td[(gt;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[()]TJ/F24 7.97 Tf -0.448 -8.278 Td[(ht;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t=st;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[()]TJ/F24 7.97 Tf -0.448 -7.892 Td[(gt;s)]TJ/F16 11.955 Tf 9.41 -0.299 Td[(t;st;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[()]TJ/F24 7.97 Tf -0.448 -8.277 Td[(ht;s)]TJ/F16 11.955 Tf 9.409 -0.299 Td[(t=sg0t;sh0t=s;g;t_s;h;t_Whichcompletestheproof.43

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4ConvolutionProductsOnSemihypergroups4.1PreliminaryResultsInthissectionSwilldenotealocallycompactHausdorsecond-countablesemihy-pergroup.Oftenassertionsinvariousresultsofthissectionarevalidinmoregeneraltopologicalstructures;howeverthisisnotpointedoutexplicitly.WerecallthatfromBanach-Alaoglu'stheoreminfunctionalanalysisthattheunitballinthedualofCcSisweak*compactthesetBSf:2MS+withS1giscompactintheweak*topology.Recall:AnetinBS,wconvergestoinBSifandonlyifforeveryfinCcS,Rfd!Rfd.However,PSf2BS:S=1gneednotbeweak*compact,unlessSiscompact.NotethatinPS,weak*compactnessisequivalenttoweakcompactness,andthusPSisweak*compactifandonlyifSiscompact.Forasubset)]TJ/F16 11.955 Tf 146.6 0 Td[(PS,theweak*closureof)-376(inPSisweak*compact,if)-376(istight;thatis,given>0,thereisacompactsubsetKSsuchthat2)]TJ/F16 11.955 Tf 10.635 0 Td[(K>1)]TJ/F23 11.955 Tf 11.955 0 Td[(Thereasonforthisisobvioussince2w-closureof)-267(and)-266(istightonlyif2PSandsinceBSisw-compact.Denition4.1.1IffisaBorelfunctiononSandx;y2S,thenwedenefxyfxyfyx=ZSfdxy44

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Ifthisintegralexists,evenwhenitisnotnite,fxiscalledthelefttranslationoffandfxiscalledtherighttranslationoff.Thenexttwolemmasareprovedin[Je75]Lemma4.1.1LetfbeacontinuousfunctiononSandletx2Si.Themappingx;y7!fxyisacontinuousfunctiononSSii.fxandfxarecontinuousfunctionsonS.Lemma4.1.2Letf2B1S,;2M+Sandx;y;z2Si.Themappingx;y7!fxyisaBorelfunctiononSSii.fxandfxareBorelfunctionsinSiii.RSfd=RSRSfxydxdyiv.RSfxd=RSfdxv.fxyz=fzxyNotation4.1.1LetSbealocallycompactsemihypergroup.Then8x2S;2M1S,andf2CS,wewrite:xf=ZSfxdfx,sayandalso,xf=fxDenition4.1.2LetSbealocallycompactsemihypergroupandBbeaBorelsubsetofS.ThenBx)]TJ/F15 11.955 Tf 10.406 -4.936 Td[(=fy2S:SuppyxB6=;g45

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Similarly,x)]TJ/F23 11.955 Tf 7.085 -4.936 Td[(B=fy2S:SuppxyB6=;gProposition4.1.1LetBbeaBorelsubsetofasemihypergroupS.Thenforanyx2S,thesetsBx)]TJ/F17 11.955 Tf 11.269 -4.339 Td[(andx)]TJ/F23 11.955 Tf 7.084 -4.339 Td[(BarealsoBorelsubsetsofS.Proof:WeonlyprovethatBx)]TJ/F15 11.955 Tf 11.616 -4.339 Td[(isBorelwheneverBisBorel.Tothisend,rstnoticethatifBisopen,thenwehave:SuppyxB6=;impliesthatyxB>0.Sincethemapx;y7)167(!yxisacontinuousmapwithrespecttoweaktopologyinM1SbyaxiomSH3,thereisanopensubsetNycontainingysuchthatforeachy02Ny,y0xB>0.ThismeansthatSuppy0xB6=;foreachy02NysothatNyBx)]TJ/F15 11.955 Tf 7.085 -4.339 Td[(;consequently,Bx)]TJ/F15 11.955 Tf 11.624 -4.339 Td[(isopenwheneverBisopen.LetusnowsupposethatBisaclosedsubsetofS.Letx2Sandy2Bx)]TJ/F15 11.955 Tf 7.085 -4.339 Td[(c.Thenwehave:SuppyxB=;sothatSuppyx,whichiscompact,iscontainedintheopensetBc.SincebySH4,themapx;y7)167(!yxiscontinuouswithrespecttotheproducttopologyinthedomainandtheMichaeltopologyforthecompactsubsetsintherange,thesetfy0:Suppy0xBcgisanopensetcontainingy;inotherwords,Bx)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(cisopen,andthismeansthatBx)]TJ/F15 11.955 Tf -426.29 -26.007 Td[(isclosedwheneverBisclosed.NowletusdenetheclassFby46

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F=fB:Bx)]TJ/F15 11.955 Tf 11.751 -4.338 Td[(isBorelwheneverBisBorelandx2Sg.ThenFcontainsallopenandallclosedsubsetsofS.Furthermore,ifVisanopensetandWisaclosedset,thensinceSislocallycompactHausdorsecondcountable,thereisasequencefFngofclosedsetssuchthatV=S1n=1Fn.[VWx)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(]c=fy:SuppyxVW=;g=fy:SuppyxWcg[fy:SuppyxWVcg=fy:SuppyxWcg[[1n=1fy:SuppyxWFcng]=fy:SuppyxWcg[[1n=1ffy:SuppyxFcn[Wcgfy:SuppyxWcgg]Nowthemapping:SS)167(!CS:y;x7)167(!Suppyxiscontinuous,andsincethesetsCSWc,CSFcn[WcareopensetsintheMichaeltopology,)]TJ/F21 7.97 Tf 6.586 0 Td[(1CSWc=fy:SuppyxWcgand)]TJ/F21 7.97 Tf 6.586 0 Td[(1CSFcn[Wc=fy:SuppyxFcn[Wcgareopen,soareBorelsets.Itfollowsthat[VWx)]TJ/F15 11.955 Tf 7.084 -4.338 Td[(]cisaBorelset.Therefore,VWx)]TJ/F15 11.955 Tf 10.987 -4.339 Td[(isaBorelset.ThismeansthatthealgebraAniteintersectionsandcomplementsgeneratedbyallopensubsetsofSiscontainedinF.Itisalsoclearthat1[n=1Bnx)]TJ/F15 11.955 Tf 10.405 -4.936 Td[(=1[n=1Bnx)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(wheneverBn2F;n1;andx2S.ThismeansthatthemonotoneclassgeneratedbyA,whichisa-algebraandwhichcontainsallBorelsubsetsofS,iscontainedinF.Lemma4.1.3LetSbealocallycompactsemihypergroup,BSandx2S.ThenBBxx)]TJ/F15 11.955 Tf -30.982 -52.059 Td[(47

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Proof:Bxx)]TJ/F15 11.955 Tf 11.513 -4.339 Td[(=fy2S:SuppyxBx6=;gSinceBx=Sb2BSuppbx,Ify2B,thenSuppyxBx;therefore,SuppyxBx6=;soy2Bxx)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(,whichimpliesBBxx)]TJ/F20 11.955 Tf -130.569 -36.779 Td[(Lemma4.1.4LetSbealocallycompactsemihypergroupandCbeacompactsubsetofS.IfBS,B)]TJ/F23 11.955 Tf 11.955 0 Td[(Cxx)]TJ/F16 11.955 Tf 10.406 -4.937 Td[(Bx)]TJ/F16 11.955 Tf 9.742 -4.937 Td[()]TJ/F23 11.955 Tf 11.955 0 Td[(CProof:Ify2B)]TJ/F23 11.955 Tf 10.398 0 Td[(Cxx)]TJ/F15 11.955 Tf 10.224 -4.338 Td[(thenSuppyxB)]TJ/F23 11.955 Tf 10.399 0 Td[(Cx6=;=SuppyxB6=;andSuppyxCxc6=;=y2Bx)]TJ/F15 11.955 Tf 11.172 -4.338 Td[(andSuppyxisnotentirelyinCxthatisy=2Cforify2CthenSuppyxCx=y2Bx)]TJ/F16 11.955 Tf 9.215 -4.339 Td[()]TJ/F23 11.955 Tf 11.429 0 Td[(C=B)]TJ/F23 11.955 Tf 11.429 0 Td[(Cxx)]TJ/F16 11.955 Tf 10.406 -4.339 Td[(Bx)]TJ/F16 11.955 Tf 9.741 -4.338 Td[()]TJ/F23 11.955 Tf 11.955 0 Td[(C.Thenextlemmawasprovedforhypergroupsin[BH95].Thesameresultholdsforsemihypergroupswiththesameproofwhichwereproducehere.Lemma4.1.5LetSbealocallycompactspaceand2M1S.Then8x2SandcompactCSxCx)]TJ/F23 11.955 Tf 7.085 -4.936 Td[(CProof:Bydenition,x)]TJ/F23 11.955 Tf 7.085 -4.937 Td[(C=fy2S:SuppxyC6=;gSoy2x)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(CifandonlyifSuppxyC6=;.Thus,xC=ZSxyCdy=Zx)]TJ/F24 7.97 Tf 6.255 -2.269 Td[(CxyCdyx)]TJ/F23 11.955 Tf 7.084 -4.936 Td[(CsincexyC1RemarkAspointedoutin[BH95]wecannotexpectequalityhereevenwhenSiscompact.48

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4.2ConvolutionEquationProposition4.2.1LetSbealocallycompactHausdorsemihypergroup.Letf2CcSand2M1S.Considerthefunctiongdenedbygx=xf.Thenforeverysuchfandeach2M1S,thefunctiong2C0SifandonlyiftheconvolutiononM1Sisseparatelycontinuousontherightintheweak-startopologythatis,ifasequenceofprobabilitymeasuresnontheBorelsubsetsofSweak-starconvergestoanonnegativenotnecessarilyaprobabilitymeasure0,thenforanyprobabilitymeasure,thesequencenweak-starconvergesto0.Proof:TheonlyifpartWeassumethatallfunctionsofthetypegx,asdescribedaboveintheproposition,vanishatinnityonS.Letnbeasequenceofprobabilitymeasuresw-convergingtothemeasure0.LetbeanygivenprobabilitymeasureonS.Letf2C+cS.NoticethatthefunctiondenedbyZfxydy=xf;asafunctionofxonS,vanishesatinnity,byourhypothesisinthe"onlyif"part.Thus,Zxfndx)167(!Zxf0dxasn)167(!1.ButnoticethatZfundu=Z[Zfxydy]ndx=ZxfndxandsimilarlyZfu0dx=Zxf0dx:Thedesiredseparatecontinuityofconvolutionfollowsandtheproofofthispartisnowcomplete.The"ifpart"49

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Supposethatconvolutionisseparatelycontinuousinthesensdescribedabove.LetfbeacontinuousfunctiononSwithcompactsupport.LetbeanyprobabilitymeasureonSandgx=xf.WeclaimthatgvanishesatinnityonS.Ifnot,thentheremustexistb>0suchthatthesetfx:gx>bgisnotrelativelycompact.ThismeansthatthereexistsasequenceofelementsxninSsuchthatforeachn,gxn>bandthesequencexnofunitmassesatxnw-convergencestothe0)]TJ/F15 11.955 Tf 9.299 0 Td[(measure.Butthen,bytheassumptionofseparatecontinuityofconvolution,itfollowsthatforeachy2SZfuxnydu)167(!0asn)167(!1.Thiswouldthenmeanthatgxn=Z[Zfuxnydu]dymustalsoconvergetozeroasngoestoinnitybythedominatedconvergencethe-orem.Butthisisacontradictionsinceeachgxn>b>0.Theproofofthepropositionisnowcomplete.Theproofofthenextpropositionisadaptedfromtheproofforsemigroups[HM95].Proposition4.2.2Let2M1S,BaBorelsubsetofSandVanopenclosedorcompactsubsetofS.Theni.gx=xVisaloweruppersemi-continuousfunctionofxandii.gx=xBisBorelmeasurable.Proof:Fromproposition4.2.1iff2CSand2M1Sthengx=xfiscontinuous.MoreoverxV=supfRfyxdy:f2CS;0f1;f=0onS)]TJ/F23 11.955 Tf 11.962 0 Td[(Vg.Thisimpliesthatgx=xVislowersemi-continuoussee[[HR70]theorem11.10].ThisalsoimpliesthatF=fB2B:xBisaBorelmeasurable50

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functionofxgcontainsallopensubsetsandallclosedsubsetsofS,furthermoreFisamonotoneclasscontainingtheclassF=fVW:VisanopensetinB,andWisaclosedsetinBg.HereBisthefamilyofBorelsubsetsofS.WeobservethatFisclosedunderniteintersectionsandthecomplementofanysetinFisanitedisjointunionofsetsinF,infactVWc=Vc[Wc)]TJ/F23 11.955 Tf 12.264 0 Td[(VcisanitedisjointunionofsetsinF.ThusitbelongstoF.SinceFcontainsthealgebrageneratedbyF,andBisthesmallest-algebrageneratedbyopensets,BF.Remark4.2.1LetSbealocallycompactsemihypergroupand;2M1Sandf2CS,denegbygx=xf.Thenxg=xf.Sincexg=Zgyxdy=Zyfxdy=ZZfyuduxdy=xfTheorem4.2.1LetSbeacommutativesemihypergroupwhereconvolutionissepa-ratelycontinuousintheweak*-topology,and;2M1Sthen,wehave:==ifandonlyif=y=y8y2[Supp]Proof:Supposey=8y2Supp.Letf2CcS.Thenf=ZZfxydxdy=ZZfxydydx=Zxfdx=Zfdx=fSincethisistrueforanyf2CcS,=.Conversely,suppose=andletf2C+cS.Letgbedenedbygx=xfforallx2S.Then,fromremark4.2.1,xg=xf=xf=gx.Nowsinceg2C0Sbyproposition4.2.1,and[Supp]isclosed,thereexists51

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b2[Supp]suchthatkgk[Supp]=gb=maxfx;x2[Supp]g.However,gb=bg=ZZgbydy.1Alsoforally2Supp,gby=Zgubydukgk[Supp]=maxfx;x2[Supp]g=gbsothatforanyy2Supp,gbygb:.2Itfollowsfrom4.1and4.2thatgb=gby.3forally2Supp.Sincegby=Rgubydu,bythesameargumentasabovewehavegu=gbforallu2Suppbyandhenceforallu2fbgSupp.Wealsoknowthat==n=forallnsothattheargumentaboveremainsvalidwhenisreplacedbyn.Thismeansthatgy=gb.4forally2S1n=1fbgSuppn.LetH= S1n=1Suppn.ThenHisnonemptyandHHHsothatHisasemihypergroupcontainingSupp.Furthermore,Hisclosedanditfollowsfromthecontinuityofgthatgb=gyforally2fbg[Supp].Moreover,b2[Supp],sofbg[Supp][Supp].Noticethatsofarwehavenotusedcommutativity.Now=impliesthatforeachn1,n=.Ifthesequencenisnottight,thensincethesetf:isanonnegativeBorelmeasureonSandS1gisacompactintheweak*-topology,thereisasubsequencenkofpositiveintegerssuchthatnkweak*-convergestosomemeasure.Bytheassumptionofcontinuityofconvolutionintheweak*topology,=.Butthis52

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impliesthatmustbeaprobabilitymeasureonS.Inotherwords,thesequencenmustbetight.Thus,theweaklimit0ofthesequence1 nPnk=1kexists,02M1S,andfurthermore,00=0andSupp0=[Supp].Since0=,followingthesameargumentasbefore,itfollowsthatforanyyandzin[Supp],gy=gzsince[Supp]issimpleby[Du73]theorem1.13.Nowfrom4.3gb=Zgubydu=Zufbydu=ZZfuxdxbydu=byfforally2Supp,thatisgb=byforally2[Supp]andfrom4.4wehavethatgx=xyfforalmostallx;y2[Supp],thatisxf=xyfsothatf=f=ZZfxudxdu=Zxfdx=Zxyfdx=Zfxuydudx=yf=yfforally2[Supp]:ThenextresultconsidertheChoquetequationfornoncommutativesemihyper-group.Proposition4.2.3LetSbeasemihypergroup,;2M1Ssuchthat==.Thenforx2Supp;y2Suppwehavex=yx53

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andx=xyProof:Since==forallf2CS,f=f=f.Letx2SuppandKbeanycompactsubsetofS.LetxK=aand>0begiven.Sincexisregular,thereexistopensetsGandWandaclosedsetCsuchthatKWCGsuchthatxGa+.NotethatthesetA=fs2S:sC0sincex2Supp.DenethefunctiongonSbygs=maxfsC)]TJ/F23 11.955 Tf 11.956 0 Td[(a)]TJ/F23 11.955 Tf 11.955 0 Td[(;0gthenZgtdt=ZZgstdsdtsothatZ[Zgstds)]TJ/F23 11.955 Tf 11.956 0 Td[(gt]dt=0.5Usingthefactthat==wehavefort2S;tC=tCRecallB=RsBds,forallBorelsetB.Nowifht=tCthenht=tC=tC=ZsCtds=Zhstds=Zhstds=thAndagainsincegs=maxfht)]TJ/F23 11.955 Tf 11.955 0 Td[(a)]TJ/F23 11.955 Tf 11.955 0 Td[(;0gand0ht)]TJ/F23 11.955 Tf 11.955 0 Td[(a)]TJ/F23 11.955 Tf 11.955 0 Td[(=Z[hst)]TJ/F23 11.955 Tf 11.955 0 Td[(a)]TJ/F23 11.955 Tf 11.955 0 Td[(]ds54

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Zgstds0sothatfort2SgtRgstdsthatisZgstds)]TJ/F23 11.955 Tf 11.955 0 Td[(gt0.6Combining4.5and4.6wehavethatgt=Rgstds-almostsurely.Fort2A;ht)]TJ/F23 11.955 Tf 11.253 0 Td[(a)]TJ/F23 11.955 Tf 11.253 0 Td[(<0sogt=0thatisRgstds=0sogst=0foralmostalltwithrespecttoinA.Alsohst=Zhxstdx=ZxCstdx=ZuxCdustdx=stCSogst=0=hsta+=stCa+whichimpliesstWa+asWCandsinceWisopenthefunctionss7!stWandt7!stWarebothlowersemi-continuous,thenitfollowsthatforallt2Aands2SuppwemusthavestWa+.Sinceagainx2AandKW,sxKsxWa+Buta=xKsosxKxK+SosxKxKforalls2Supp.55

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Nowletsdenefx=xKsince=,fx=xK=ZsKxds=Zfsxds=Zfsxds:Andsincefsx=sxKwehavefx=Zfsxds=ZsxKdsWhichimpliesZ[fx)]TJ/F23 11.955 Tf 11.956 0 Td[(sxK]ds=0ItfollowsthatforallswithrespecttoxK=sxKFromtheuppersemi-continuityofthefunctions7!sxKwehavexK=sxKwheneverx2Suppands2Supp.Sinceisregular,x;sxarealsoregularandwehavexB=sxBforx2Supp;s2SuppandBanyBorelset.Thereforex=sxforx2Supp;s2Supp.Thesecondinequalityfollowsinthesamemanner.4.3InvariantandIdempotentMeasuresDenition4.3.1LetSbealocallycompactsemihypergroup.AmeasuremonSnotnecessarilyboundedwillbecalledleftsubinvariantifxmisdenedandxmmforallx2S.Ifwehavexm=m,mwillbecalledaleftinvariantmeasureonS.Rightinvariantmeasuresaredenedthesameway.Example4.3.1i.ThespaceR;+isalocallycompactgroupsoisahypergroupwiththeappropriateconvolutionandhasaleftinvariantmeasurewhichisthe56

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Lebesguemeasure.ii.TheSturmLiouvillehypergroupisdenedfromaSturmLiouvilleproblemwhoseeigenfunctionsareorthogonalwithrespecttoaweightfunctiondmx=xdx,thismeasureistheinvariantmeasureoftheSturmLiouvilleHypergroup.iii.ThesemihypergroupS=fe;x;ygwhereeisconsideredtheneutralelementandconvolutionisdenedbyxx=yyy=1 4x+3 4yxy=yx=1 2x+1 2yHasinvariantmeasurem=1 3x+2 3ywecanalsoobservethatSuppm6=Sal-thoughSuppmisasimpleidealwhichisnotcompletelysimpleitcontainsnoidempotentelementthisisincontrastwithsemigroupswherethesupportoftheinvariantmeasureofanabeliansemigroupisagroup.ForcompactcommutativesemihypergroupwehavethefollowingtheoremprovedbyDunkl.Thenexttheoremisfrom[Du73]Theorem4.3.1IfSisacompactcommutativesemihypergroupthenShasauniqueinvariantmeasurem,thesupportofmisasimplesubsemihypergroup.Proof:SinceSiscompact,thesetM1SofprobabilitymeasuresonSisweak*compactandconvex.FurtherM1Sactsasacommutativesemigroupofweak*continuouslinearoperatorsonitselfbyconvolution[[Du73]proposition1.8]sobytheMarkov-KakutaniTheorem,thereexistsm2M1Ssuchthatm=mforall2M1Sinparticularmx=xm=m.Nowsupposethatissuchthatx=x2S,thenRfdm=RdmSRfxd=Rfdsom=butm=msom=.NextweobservethatifI1andI2areidealsofSthenI1I2I1I2andsinceSiscompact,ShasaminimalidealsayI.Forx2I,mx=mwhichimpliesthat57

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Suppm=fxgSI.Converselyforanyx2SfxgSuppmSuppmsoSuppmisaclosedidealcontainedinIandsinceIisminimal,Suppm=I.Remark4.3.1i.Jewett[Je75]andSpector[Sp78]provedthateverylocallycom-pacthypergroupHhasaleftsubinvariantmeasuremandSuppm=H.ForalocallycompactgrouptheexistenceofasubinvariantmeasureimpliesthatofaninvariantmeasureinfactifGisagroupandmissuchthatxmmthengivenanyBorelsetA,xmAmAbutmA=emA=xx)]TJ/F16 11.955 Tf 9.304 -0.299 Td[(mAxmAIngroupsxx)]TJ/F15 11.955 Tf 11.747 -0.299 Td[(=eandwehavex)]TJ/F16 11.955 Tf 10.079 -0.299 Td[(mAmAanditfollowsthatxm=m.ThisisnotthecaseforhypergroupseeanexampleofNaimarkin[Je75]9.5thoughitiseasytoprovethatwhenacompacthypergrouphasasubinvariantmeasureitisalsoinvariant.Bothauthorsalsoprovedtheexistenceofinvariantmeasuresfordiscretehypergroups.Spector[Sp78]provedthatifahypergroupiscommutativeithasaninvariantmeasure.ii.Jewett'sconjecture[Je75]thatthereexistaleftinvariantmeasureonalllocallycompacthypergroupisyettobeproved.iii.Onipchuk[On93]announcedtheproofofthisconjecturebutinreadingthroughitwerealizedthatheisusingcommutativityimplicitlyinhisassumptions.PreciselytheenvelopingalgebraAA0isnotinvolutiveunlessthesemihypergroupiscommutative.Thenexttworesults,generalizetosemihypergroupsresultsgivenby[BH95]forhy-pergroup.Proposition4.3.1LetCbeacompactsubsetofthesemihypergroupSandz2C.IfCisasubsemihypergroupofS,thenthereexists2M1SwithSuppC,z=and=.Proof:58

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Foralln1,denen=1 nz+zz+:::+nz.7Thenkzn)]TJ/F23 11.955 Tf 11.955 0 Td[(nk=1 nkz)]TJ/F23 11.955 Tf 11.956 0 Td[(nzk2 nNowbyassumption,wehaveSuppCforalln2Nwhichjustsaysthatnisuniformlytight.BytheProhorov'sTheoremnisrelativelycompactintheweaktopologyandhencethereexists2M1Swithn)167(!w.ClearlySuppCandz=.Byequation4.7wehavethatn=foralln2Nandhence=Denition4.3.2LetSbealocallycompactsemihypergroup.Aprobabilitymeasureissaidtobeidempotentifandonlyif=Example4.3.2ConsiderthesemihypergroupS=fe;x;ygwhereeisconsideredtheneutralelementandconvolutionisdenedbyxx=yyy=1 4x+3 4yxy=yx=1 2x+1 2yThemeasure=1 3x+2 3yisanidempotentmeasureasaninvariantmeasurewithSupp=fx;yg.Theorem4.3.2LetSbeacommutativesemihypergroupsuchthattheconvolutionisseparatelycontinuousintheweak*-topology.For2MbSletL=fx2S:x=gThenLisacompactsubsemihypergroupofS.Proof:59

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Letf2C+cSbexedandletgbesuchthatgx=xfforallx2S.Then8x2Lgx=f.Thus,gisconstantonLandsinceg2C0S,Liscompact.Next8x;y2Lxy=x=.Thisimpliesthat=xyisaprobabilitymeasuresatisfying=with=xy.SinceSiscommutative,forallz2Suppxy,z=seetheChoquetequationtheorem4.2.1whichimpliesz2LthatisSuppxyLforallx;y2L,soLisasubsemihypergroupofS.RemarkEveryprobabilitymeasureonasemihypergroupinvariantonitssupportisanidempotentmeasure.Theconverseisnotalwaystrue.Jewett[Je75]provedinthecaseofahypergrouphavinganinvariantmeasurethefollowingtheoremwhichwegivewithoutproof.Theorem4.3.3LetHbeahypergroupwithaninvariantmeasure.If2M+H6=0and=isanidempotentmeasureonH,then)]TJ/F15 11.955 Tf 12.609 -4.338 Td[(=,thesetG=SuppisacompactsubhypergroupofH,andisthenormalizedinvariantmeasureonG.Remark4.3.2WhenSisacommutativesemihypergroup.Dunkl[Du73]provedthatanidempotentmeasureisinvariantonitssupportwhichisacompactsimplesemi-hypergroup.Onipchuk[On89]provedDunkl'sresultforcompactnon-commutativesemihypergroup,withtheadditionalconditionthatforanyidempotentmeasure,x=xforallx2Supp.WeprovebelowthatDunkl'sresultextendstolocallycompactsemihypergroupnotnecessarilycommutativewithamorerelaxedcondition.Theorem4.3.4LetSbealocallycompactsemihypergroupwiththeconditionthatforanyidempotentmeasure,Suppfxg=fxgSuppforallx2Supp.Supposealsothatconvolutionisseparatelycontinuousintheweak*-topology.If260

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M1Sisanidempotentmeasure,thenSuppisacompactsubsemihypergroupofSwithnotwosidedproperideals.Furthermore,isinvariantonitssupport.Proof:LetK=Suppthensince=weobviouslyhaveKKKsoK;isasubsemihypergroupofS.Claimiff2C+cSandgx=xftheniffisnotidenticallyzeroonK,gz>0forsomez2K.Toseethissupposegz=0forallz2KthenRfzydy=0forallz2Kwhichimpliesthatfzy=0forallz;y2Ksothatf0onKK=KacontradictionsoRfzydy>0forsomez2K.Nextletf2C+cS,fnotidenticallyzeroonS.Letgbedenedbygx=xfthenfromproposition4.2.1g2C0S.Furtherg0,andisnotidenticallyzeroonS.WehavefromRemark4.2.1thatxg=xf=gxSinceg2C0SandKisaclosedsubsetofSthereexistsx02Ksuchthatgx0=x0g=Rgx0ydyforally2Kwhichinturnsimpliesthatgz=gx0forallz2x0K.Sincex0KisanidealinSandgisconstantonx0K,itisacompacttwosidedidealofKsinceg2C0Sandgisnotidenticallyzero.FurthersinceSuppfxg=fxgSuppforallx2K,KhasaminimumcompactnonemptyidealIx0Kandforeachf2C+cS;gdenedasaboveisconstantonIwithvaluekgkK.NowsupposeI6=Kthenthereexistsz2K,z=2I,hencethereexistsf2C0Ssuchthatfz6=0andfI=0.Ifwedeneafunctiongasabove,g2C0Sandthereexistsx02Ksuchthatgx0=Supx2Kjgxjandsincegx0=x0gwehavegx0=gx0yforally2K.Ify2ISuppx0yI.Sogx0y=Zgdx0y=61

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Zgux0ydu=Zufx0ydu=ZZfuvdux0yduButu2IsoSuppuvIsogx0y=0whichimpliesgx0=0butfisnotidenticallyzeroonKthisisacontradictionsoK=IthereforeKisacompactsimplesemihypergroup.Finallyletf2C+cSbegivenandstilldenethefunctiongasgx=xfforallx2S.Thenaswehaveseenabovethereexistsx02Ksuchthatgx0=Supx2KjgxjandgisconstantonK.Moreoverif2M1Kf=ZKxfdx=ZKgxdx=gx=xfSincegisconstantonKandisaprobabilitymeasureonK.Thisshowsthatf=xfforallx2Supp.Sincewasarbitrarilychosen,inparticularfor=wewillhave==xsoisinvariantonK=Supp.Remark4.3.3ItiswellknownthatifSisacommutativesemigroupandisidem-potentinS,thenSuppisagroup.Thisresultfailinsemihypergroupsingeneral.Inexample4.3.2wehavea3pointscommutativesemihypergroupwithandidempo-tentmeasurewhosesupportdoesnotcontainanidempotentelementandsocannotbeahypergroup.62

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4.4WeakConvergenceofConvolutionProductsofProbabil-ityMeasuresonSemihypergroups4.4.1ConcretizationforSemihypergroupsDenition4.4.1AtripletX;;consistingofacompactspaceX,aprobabilitymeasure2M1X,andaBorel-measurablemapping:SSX!SiscalledaconcretizationofthesemihypergroupS;iffz2X:x;y;z2Ag=xyAForallx;y2SandA2BS.Example4.4.11.LetGbealocallycompactgroupwithmultiplication,acon-volutionandaneutralelemente.ThetripletX;;denedbyX=feg,=eandx;y;e:=xyforallx;y2GisaconcretizationofG.2.ConsiderthehypergroupK=R+withconvolutiondenedbyxy=1 2jx)]TJ/F24 7.97 Tf 6.587 0 Td[(yj+1 2x+yforallx;y2KweobtaintheconcretizationX;;whereX=f)]TJ/F15 11.955 Tf 15.276 0 Td[(1;1g;=1 2)]TJ/F21 7.97 Tf 6.586 0 Td[(1+1 21andx;y;)]TJ/F15 11.955 Tf 9.298 0 Td[(1=jx)]TJ/F23 11.955 Tf 11.955 0 Td[(yjx;y;1=x+ySinceisBorelmeasurablewejustneedtocheckthatfz2X:x;y;z2Ag=xyA63

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Actuallyfz2X:x;y;z2Ag=xyA=1 2jx)]TJ/F24 7.97 Tf 6.587 0 Td[(yjA+1 2x+yAAndsinceX=f)]TJ/F15 11.955 Tf 15.276 0 Td[(1;1g,thenifx;y;z=2A8z2f)]TJ/F15 11.955 Tf 27.278 0 Td[(1;1g,thenjx)]TJ/F23 11.955 Tf 11.955 0 Td[(yj=2Aandx+y=2Asothatfz2X:x;y;z2Ag=0andxyA=0.Ifx;y;)]TJ/F15 11.955 Tf 9.298 0 Td[(12Aandx;y;1=2A,thenf)]TJ/F15 11.955 Tf 15.276 0 Td[(1g=1 2andxyA=1 2jx)]TJ/F24 7.97 Tf 6.587 0 Td[(yjA=1 2andifx;y;)]TJ/F15 11.955 Tf 9.298 0 Td[(1=2Aandx;y;12Athenf1g=1 2andxyA=1 2x+yA=1 2.Finallyifx;y;)]TJ/F15 11.955 Tf 9.298 0 Td[(12Aandx;y;12Athenf)]TJ/F15 11.955 Tf 15.276 0 Td[(1;1g=1andxyA=1.SowehaveX;;asdenedaboveisaconcretizationofR+;.Thenexttheoremisfrom[BH95]itisalsovalidforsemihypergroupswiththesameproof.Theorem4.4.1LetSbeasecondcountablesemihypergroup.Thereexistsameasur-ablemappingfromSS[0;1]intoSsuchthat[0;1];[0;1];isaconcretizationofS.Remark4.4.1InthespecialcaseofonedimensionalsemihypergroupS=R+wemayassumewithoutlossofgeneralitythatminsuppxy=jx)]TJ/F23 11.955 Tf 11.956 0 Td[(yjmaxsuppxy=x+ywheneverx;y2KThemeasurablemapping:SS[0;1])167(!SestablishedinTheorem4.4.1alsosatisesthefollowingveproperties:1.x;y;0=jx)]TJ/F23 11.955 Tf 11.955 0 Td[(yj2.x;y;1=x+y64

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3.x;y;t=y;x;t8t2[0;1]4.;x;t=x;0;t=x8t2]0;1]5.Themapping:;:;t:SS)167(!Sislowersemicontinuous.NowletSbeasemihypergroupwithaxedconcretizationX;;and;A;Pdenoteanarbitraryprobabilityspace.Denition4.4.2ForanyS-valuedrandomvariablesXandYonXnn1andanauxiliaryX-valuedrandomvariableon;A;PsuchthatisstochasticallyindependentofXYandhasdistributionP=wedenetherandomizedsumofXwithYbyX^+Y=X;Y;.Remark4.4.2ThisdenitioncanbeextendedtosequencesXnn1ofX-valuedrandomvariableson;A;PprovidedallrandomvariablesoccurringinthesequenceXnn1andnn1areindependentandPn:=foralln1infactbytherecurrence0Xj=1^Xj:=enXj=1^Xj:=Xn^+n)]TJ/F21 7.97 Tf 6.587 0 Td[(1Xj=1^Xj;n1therandomizedsumsSn=Pnj=1^Xj,n1areintroducedagainasS-valuedrandomvariableson;A;P,whichformanonhomogeneousMarkovchainSnn0withcorrespondingsequenceNnn1oftransitionkernelsonS;BSsatisfyingNnx;A=PXnxA=PSn2A:Sn)]TJ/F21 7.97 Tf 6.587 0 Td[(1=xForPSn)]TJ/F22 5.978 Tf 5.756 0 Td[(1-almostallx2S;A2BSandn165

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Proposition4.4.1LetXandYbeS-valuedrandomvariablesandletbeanX-valuedrandomvariableon;A;PwithPn:=suchthatX;Y;areindependentthenPX^+Y=PXPYProof:8A2S;BSPX^+YA=PX;Y;2A=ZZPX;Y;2APXdxPYdyZZ[X;Y;2A]PXdxPYdyZZxyAPXdxPYdyPXPYASoPX^+Y=PXPYRemark4.4.3Formingrandomizedsumsisgenerallynotanassociativeoperationalthoughconvolutionobviouslyis.WhilerandomizedsumX^+YclearlydependsontheparticularchoiceoftheunderlyingconcretizationofSthejointdistributionoftherandomvariablesX;YandX^+Ydoesnot.4.4.2SequenceofConvolutionofMeasuresTheorem4.4.2LetSbealocallycompactsemihypergroup.Assume2M1Sandsupposethatthesequencenistight.SupposealsothatS= [1[n=1Suppn]letK=f2M1S:isaweaklimitpointofthesequencengalsoletusdeneS0=[fSupp:2Kg66

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andS1=S0thenthesequence1 nnXk=1kconvergesweaklytoaprobabilitymeasuresuchthat===andSuppistheclosedminimalidealofS.Proof:Writen=1 nPnk=1kthenfork1,kn=nkandlimn!1kn)]TJ/F23 11.955 Tf 11.955 0 Td[(knk=0.8itsfollows,sincethesequencenistight,thatthesequencenisalsotightsothatfn:n1gisweaklyrelativelycompact.Let1and2betwolimitpointsofnthenby4.8k1=1k=1k2=2k=2Itfollowsthat1 nnXk=1k1=1 nnXk=11k=1and1 nnXk=1k2=1 nnXk=12k=2Thatisn1=1n=1n2=2n=2whichthenimpliesthat1=2and===andsinceisanidempotentmeasureitisasimplesemihypergroupandsince SuppSupp= SuppSupp=Supp,SuppistheminimalidealofS= [S1n=1Suppn]Remark67

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IfnconvergesweaklythenliminfSuppnisnonempty.Toseethissupposen)167(!wthenclaimSuppliminfSuppnforletx2SuppthenforeveryneighborhoodUofx,U>0butUliminfnUsoliminfnU>0whichimpliesthatx2liminfSuppnwhichimpliesthatSuppliminfSuppnthereforeliminfSuppn6=;.WenowsolvetheChoquetequationfornotnecessarilycommutativehypergroupsanalternativeproofcanalsobefoundin[BH95]butrequiredlotsofsteps.Corallory4.4.1SupposeHisahypergroupwithaninvariantmeasureand;2M1H.Then=ifandonlyif=xforallx2[Supp]thesmallestsubhypergroupofHcontainingSuppProof:Theifpartistrivial.Nowsupposethat=then=n.Given>0,letKbeacompactsubsetofHsuchthatK>1)]TJ/F23 11.955 Tf 11.955 0 Td[(Then1)]TJ/F23 11.955 Tf 11.955 0 Td[(
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f2CcHanddenegbygx=xfforallx2Hthenxg=Zgyxdy=Zyfxdy=ZZfzydzxdy=xf=xf=gxSinceg2C0HandSuppiscompact,thereexistsx02Suppsuchthatgx0=kgkSupp=Supx2SuppjgxjNowx0g=gx0sothatgx0=Rgyx0dywhichimpliesgx0=gyx0forally2Suppwhichimpliesgx0=gyx0=Zguyx0du=Zufyx0du=ZZfzudzyx0du=yx0fSincegx0=gyx0forally2SuppgisconstantonSuppx0SuppwhichisarightidealofSuppsocontainstheneutralelemente.Sowehavege=yef=yfandsincege=efwehaveef=yf69

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sothatf=yfforallf2CcH.Therefore=yforally2SuppandsinceSuppSuppwehavethat=xforallx2[Supp]Corallory4.4.2LetSbeasemihypergroupand2M1SbesuchthatthesequencenistightandS= [S1n=1Suppn].Let2M1Ssuchthat=Thenthefollowingassertionsarevalid.i.ShasasimpleidealK=Supp0,where0istheweaklimitof1 nPnk=1kand0=0=0ii.SuppKand=Proof:Assertionifollowsfromtheorem4.4.2.Supposenowthat=forsome2M1S.Then1 nnXk=1k=;n1anditfollowsthat0=andSupp= SuppSupp0Supp0=KNowletx2Suppandf2CbHthen0=impliesxf=x0f=Zxy0fdy=Zx0fdy=x0fWehavexy0=x0sinceSuppSupp0and0=00byproposition4.2.3.Anditfollowsthatf=Zxfdx=Zx0fdx=0f=fSothat=.isanidempotentmeasuresoSuppisasimplesubsemihypergroupofK=Supp070

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Corallory4.4.3LetSbeasemihypergroupand2M1S,S= [S1n=1Suppn].SupposethatSsatisesthefollowingcompactnessconditionKiscompact,x2S=x)]TJ/F23 11.955 Tf 7.085 -4.338 Td[(KiscompactLet2M1Ssuchthat=then1 nPnk=1kconvergesweaklyto02M1,andconsequentlyalltheresultsincorollary4.4.3remainvalid.Proof:Letbeaweak*limitpointsofthesequencen=1 nnXk=1kIfallsuchweak*limitpointsareprobabilitymeasures,thenitfollowsfromtheorem4.4.2,thatthesequence1 nPnk=1kconvergesweaklytosome0inM1S,andtherestofcorollary4.4.3thenfollowsexactlyasincorollary4.4.2.Thusitsucestoshowthat2M1S.Letf2CcSandx2S.Thenfx2CcS.Letnkbethesubsequencesuchthatnkweak*convergesto.Thenletusdenethefunctiongkandgbygkx=xnkfandgx=xfSinceconvolutionisseparatelycontinuousxnk!wx,sogkx)167(!gxask)167(!1thereforebytheboundedconvergencetheorem,forf2CcSwehavef=Zfxdx=nkf=Zxnkfdx=Zgkxdx)167(!Zgxdx=71

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Zxfdx=fSothat=.ThatisS=SSwhichimpliesthatS=1so2M1S.Proposition4.4.2SupposeSisacompactsemihypergroup,withacontinuouscon-cretization,2M1SandS= [S1n=1Suppn]thenforanyopensetGcontainingthekernelKofS,limn!1nG=1Proof:LetKG,Gopen,sinceKSG,S;Karecompact,thereexistsanopensetVcontainingKsuchthatVSG.Noticethatiflimk!1nkV=1;.9then8>0thereexistsk0suchthatm>nk0impliesmGnk0Vm)]TJ/F24 7.97 Tf 6.586 0 Td[(nk0S>1)]TJ/F23 11.955 Tf 11.955 0 Td[(whichmeansthatlimn!1nG=1Thereforeitisenoughtoestablished4.9forsomesubsequencenk.Tothisendletx2KthensinceSxSKVthereexistsanopensetWsuchthatx2WandSWSVsincex2WS= [S1n=1Suppn]thereexistsm>0suchthatmW>0.LetXnbeasequenceofindependentS-valuedrandomvariableeachwithdis-tributionm.ThenwehaveXPXn2W=172

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andbyBorelCantellilemmawehavePXn2W;i:o=1SincefXn2Wgareindependent,8>09m0suchthatPm0[n=0fXn2Wg>1)]TJ/F23 11.955 Tf 11.955 0 Td[(:Nowifx2m0[n=0fXn2Wg:9n0suchthatXn0x2W,letSn=Pnk=1^Xknm0.NotethatifXandYaretworandomvariablessuchthatXisA-valuedandYisB-valuedthenX^+YisAB-valued.ForX^+Y=X;Y;whenis[0;1]-valuedsothatX^+Yx=Xx;Yx;t.SetXx=z;Yx=y;t=sClaim:z;y;s2SuppzyABforalls2[0;1].Toseethissupposex2A;y2BletVbeanopensetcontainingz;y;s,s2[0;1]thenxyV=fs:z;y;s2Vgandsinceiscontinuous,fs:z;y;s2Vg>0sothatxyV>0,thatisz;y;s2SuppzyAB.SoX^+YisAB-valuedandbythedenitionoftherandomizedsumSn=X1^+X2^+X3^+:::^+XnSinceXn0isK-valuedSnwillbeV-valuedsothatm0[n=0fXn2WgfX1^+X2^+X3^+:::^+Xn2Vg;nm073

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SinceSWSV.NowasX1^+X2^+X3^+:::^+Xnhasdistributionmn,itisclearthatfornm0mn>m0sothatmnV>1)]TJ/F23 11.955 Tf 11.955 0 Td[(forall>0.SomnV=1.Proposition4.4.3LetIbeaBorelsetthatisanidealofS.Supposethatforsomepositiveintegerm,mI>0forsome2M1S.ThenthesequencenImonotonicallyincreasesto1.Proof:ISIson+1InIS=nIForallpositiveintegern.NowtheprovefollowsasabovesinceSISI.Proposition4.4.4LetnbeasequenceinM1Ssuchthatthesubsequence0;ntwherek;n=k+1:::nhasatleastoneweak*limitpointinM1S.SupposethatShasthepropertysuchthatconvolutionasamapfromM1SBS)167(!BSiscontinuousintheweak*sense.Thenthereisasequenceptntsuchthatforeachpositiveintegerkk;pt)167(!wk2M1Spt)167(!w2M1Sk=kWhereBS=f:isanonnegativeregularBorelmeasurewithS1gProof:Suppose0;nt)167(!w02M1S.Notethatw*-convergenceisweakconvergencewhenthelimitisinM1S.Nowforeachpositiveintegertynt0;nt;1;nt;:::;nt)]TJ/F22 5.978 Tf 5.757 0 Td[(1;nt;0;0;0;:::74

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areelementsintheproductspaceY=1Yi=1Xi;Xi=BSwithweak*topology,whereYhastheproducttopologyandisthereforecompact,sinceBSisw*compact.SinceYiscompactandrstcountable,thereisasubsequencemtntsuchthatymt)167(!y2Y,inthetopologyofY.Thismeansthatforeachk0,thereexistsk2BSsuchthatk;mt)167(!kSinceconvolutioniscontinuousasamapfromM1SBS)167(!BSitfollowsthatforeachk10;mt=12:::mukk+1:::mt=0;kk;mt)167(!0;kkintheweak*senseandthismeansthat0;kk=0;k1since0;mt)167(!0Howeversince02M1Sthisimpliesthatk2M1Sforeachk1.Letptmtbeasubsequencesuchthatpt)167(!2BSintheweak*sens.Nowforxedintegersandt>ssuchthatps>k,wehavek;psps;pt=k;ptAgainbythecontinuityofconvolution,itfollowsthatgivenk0foreachssuchthatps>kk;psps=k75

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whichinturnimpliesthatk=k,k1,sincek2M1S.Thelastequationimpliesthat=.Proposition4.4.5SupposeSsatisesthefollowingcompactnesscondition.Kcompactandx2Simpliesx)]TJ/F23 11.955 Tf 7.085 -4.339 Td[(Kiscompact.Ifn)167(!weaklyinM1Sandn)167(!2BSintheweak*sensewithn2M1Sthennn)167(!intheweak*topology.Proof:Letf2CcS.Thenforeachs2S,t7!fstisinCcS.HenceifgnsZfstndtgsZfstdtThenlimn!1gns=gsSinceisaregularmeasure,itiseasilyseenthatgisaboundedcontinuousfunctiononS.AlsobyEgoro'stheoreminanalysis,given>0thereexistsacompactsetKsuchthatK
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becauseZgndn)]TJ/F28 11.955 Tf 11.955 16.273 Td[(Zgd=Zgndn)]TJ/F28 11.955 Tf 11.955 16.273 Td[(Zgdn+Zgdn)]TJ/F28 11.955 Tf 11.955 16.273 Td[(Zgd=ZKgndn+ZKcgndn)]TJ/F28 11.955 Tf 11.955 16.273 Td[(ZKgdn)]TJ/F28 11.955 Tf 11.955 16.273 Td[(ZKcgdn+Zgn)]TJ/F28 11.955 Tf 11.955 16.273 Td[(Zgd:=ZKgndn)]TJ/F28 11.955 Tf 11.955 16.272 Td[(ZKgdn+ZKcgn)]TJ/F23 11.955 Tf 11.955 0 Td[(gdn+Zgdn)]TJ/F28 11.955 Tf 11.955 16.272 Td[(ZgdSolimn!1Zgndn=ZgdThismeansthatZfdnn=ZZfstndsndt=Z[Zfstndt]nds=Zgnsnds)167(!Zgd=ZfdSonn)167(!77

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5SemigroupsofMultipliersassociatedwithsemigroupsofOperatorsinLpH5.1IntroductionInthischapterweextendtohypergroupsbasictechniquesonmultiplierssetforthforgroupsin[HR70],namelypropositions5.2.1and5.2.2,aswellasanextendedversionofWendel'stheorem.LetHbeacompactcommutativehypergroupand^Hitsdualspace.WeestablishrelationshipsbetweensemigroupsS=fT:>0gofoperatorsonLpH,1p<1,whichcommuteswithtranslations,andsemigroupsM=fE:>0gofLpHmultipliers.Theseresultsgeneralizetohypergroupsthoseof[HP57]forthecirclegroupsand[B074]forcompactabeliangroups.Remark5.1.1LetHbealocallycompacthypergroup.Then8x2H;2MH,andf2CHxf=ZHfxdfx,sayandsimilarlyxf=fxDenition5.1.1AcharacteronahypergroupHisacontinuouscomplex-valued78

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functiononHwhichisnotidenticallyzeroandsatisesZHdxy=xyforallx;y2H.AcharacterissaidtobeHermitianifandonlyifx)]TJ/F15 11.955 Tf 7.084 -4.339 Td[(= x.Remark5.1.2Forcommutativehypergroups,thedual^HofHisthesetofallHermitiancharactersonH.^Hisahypergroupprovideditisahypergroupwithrespecttopointwisemultiplication.Thatis,withaconvolutiondenedsuchthat12x=1x2xforallx2H.Unlikeinthegroupcase,^Hisnotalwaysahypergroupeveninthecommutativecase[Je75].Alsoif^Hisahypergroup,^^Hmaynotnecessarilybeahypergroup.IfHadmitsadualhypergroupstructure,itiscalledahypergroupoftypeD.If^Hisahypergroup,H^^Hinanaturalmanner,andifinaddition,H=^^Hholds,wecallHastronghypergroup.Wewillassumethroughoutthischapterthat^Hissupportedbyitsinvariantmeasure.Denition5.1.2LetH;beacommutativehypergroupwithinvariantmeasurem.TheFourierStieljestransform^ofisdenedon^Hby^=ZHdAndforallf2L1H,theFouriertransform^foffwithrespecttomisdenedby^f=ZHfdmAlsotheconvolutionofafunctionfwithameasureisgivenbyfx=ZHZHfdxy)]TJ/F23 11.955 Tf 6.753 -0.299 Td[(dy79

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5.2MultipliersonHypergroupsDenition5.2.1LetHbealocallycompacthypergroupwithinvariantmeasurem,andLpHhasitsusualmeaning1p1.LetU=CHorLpH.GivenAU,wedene,by^A,thesetofallFouriertransforms^foff2A.Acomplexvaluedfunction'onthedualspace^HofHiscalledanA;B-multiplierifandonlyif'^f2^B'^fisapointwisemultiplicationforeachf2AwhereA;BaresubsetsofU.ThesetofallA;B-multiplierisdenotedbyMA;B.Bymultiplierweherereallymeanaleftmultiplier.Rightmultipliersaredenedinasimilarway.Remark5.2.1Firstwepointoutthatinthecaseofacompactnoncommutativehypergroup,^HdenotethedualobjectofH,thatis,thesetofcontinuousunitaryrepresentationsUofH.SupposeU2^HandfjgdUj=1isanorthonormalbasisforHUtheHilbertspaceassociatedwithUwithdimensiondU.WedenecoordinatefunctionsforUasin[Vr79]byujkx=where1j;kdU.Fordetailsaboutrepresentationsoncompacthypergroups,see[Vr79].TrigUHisthelinearspanofcoordinatefunctionsofUandTrigH=SfTrigUH:U2^Hg.Furtherthe)]TJ/F17 11.955 Tf 15.277 0 Td[(algebraQU2^HBHUwillbedenotedbyE^H;scalarmultiplication,addition,multiplicationandadjointofanelementaredenedcoordinatewise.LetE=EUbeanelementofE^H.For1p<1wedenekEkp=XU2^HkUkEUkp'p1 pandkEk1=supfkEUk'1gThenormsk:k'Paretheoperatornormsof[[HR70]D.37,D.36e]andthenotations80

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Ep^H,E00^H,andE0^Hisasin[[HR70]28.24].Asinthegroupcase[HR70],manyoftheargumentusedinthetheoryofmultipliersarebasedontheclosedgraphtheorem.Thenextresultputintooneplaceallthoseclosedgraphtheoremargumentsandalsolistallthespacesthatwedealwithinthissection.Proposition5.2.1LetHbeacompacthypergroup.LetUandBbeanyofthespacesi.Ep^H,p1,E0^H,ii.LpH,p1,CbH,MHwhere^HdenotethedualobjectofH.LetEbeaU;B-multiplier.DenethemappingT:U)167(!Bbythefollowingrulesiii.Tg=Egforg2UifUandBarechosenfromiiv.dTg=Egforg2U,UchosenfromiandBfromiiv.Tf=E^fforf2U[orT=E^for2MH]ifUischosenfromiiandBfromivi.[Tf=E^fforf2U[or[T=E^for2MH]ifUandBarechosenfromii.IfUandBaregiventheirusualnorm,thenTisaboundedlineartransformationfromUtoBProof:Theproofisadaptedfrom[[HR70]35.2]forthegroupcase.First,weneedtoshowthatTiswell-denediniii,iv,v,vi.81

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Foriii,sinceUandBarechosenfromi,bythedenitionofaU;B-multiplier,forallg2U,Eg2BuniquelysothatTiswelldened.InvUisfromiisothat^UisasubsetofasetiniandEisa^U;B-multipliersoforallf2U,^f2UandE^f2Buniquely,thereforeTiswelldenedas^fisuniquelydened.ForivUischosenfromiandBfromiisoEisaU;^B-multiplierso8g2U,Eg2B.Nowif[Tg=Eg,bytheuniquenessoftheFouriertransform[[Je75]7.3E],Tgiswelldened.SimilarlyinviforU;BfromiiEisa^U;^B-multiplierthatis8f2U,^f2^UandE^f2Bsoif[Tf=E^fthenbytheuniquenessoftheFourierStieltjestransformTfisunique,Tisthenwelldened.NowifUischosenfromithenwehaveUE1^Hand[[HR70]28.32iv]showsthatforallg2Ukgk1kgkU.1IfUischosenfromiiUMHandasMHisisomorphicwithE1^H[[Vr79]3.2]andsincetheisomorphismisnorm-decreasingwehavek^j1kkkkUForallg2^UThisisobtainedbywritingk^k^UforkkUinthepreviousinequality,wehavekgk1kgk^U.2Relations5.1and5.2showsthatinallcasesUcanberegardedasasubspaceofE1^Hforwhich5.1holds.ThesameremarkevidentlyholdsforB.ThuswemayconsiderUandBaslinearsubspacesofE1^Hwithcompletenormk:kUandk:kBsatisfyingtheinequality5.1.SowewilljustprovethatthemappingTdenedfromUtoBbyTg=EgisaboundedlineartransformationcarryingUtoBforallsubspacesofE1^Hhavingcompletenormsthatsatisfy5.1.SinceE;g2E1^Hwehaveforg1;g22U,g1;g22E1^HandEg1+g2=Eg1+Eg2soTg=Egis82

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alineartransformation.Nowletg2Uandfgng1n=1beasequenceinUsuchthatlimn!1kgn)]TJ/F23 11.955 Tf 11.955 0 Td[(gkU=0.3Supposethatg0isalimitpointofBsuchthatlimn!1kTgn)]TJ/F23 11.955 Tf 11.955 0 Td[(g0kB=0.4Thenfrom5.1appliedtoBand5.4limn!1kEgn)]TJ/F23 11.955 Tf 11.955 0 Td[(g0k1=limn!1kTgn)]TJ/F23 11.955 Tf 11.955 0 Td[(g0k1=0.5ForeachU2^H,5.5showsthatlimn!1kEUgnU)]TJ/F23 11.955 Tf 11.955 0 Td[(g0Uk'1=0.6andfrom[HR70]D.52iand5.1kEUgnU)]TJ/F23 11.955 Tf 11.955 0 Td[(EUgUk'1kEUk'1kgnU)]TJ/F23 11.955 Tf 11.956 0 Td[(gUk'1From5.3wehavelimn!1kEUgnU)]TJ/F23 11.955 Tf 11.955 0 Td[(EUgUk'1=0.7ForeachU2^H.Theinequalities5.6and5.7implythatEg=g0,thatis,ThasaclosedgraphinUB.andfromtheclosedgraphtheoremTiscontinuous.Ournextresultdescribesthedualitypropertiesofmultipliers.Itprovidesusefulshortcutsincomputationsinvolvingmultipliers.Proposition5.2.2SupposethatUandBandalsotheirconjugatespacesU,Bare83

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amongthespaceslistedinProposition5.2.1[i,ii][ThusneitherUnorBcanbeE1^H,L1HorMH]ThenwehaveMU;B=MB;U~Proof:ItisenoughtoshowthatE2MU;BimpliesE~2MB;UWewillexaminefourcasesaccordingtowhetherUandBarechosenfromProposition5.2.1iorProposition5.2.1ii.Throughoutthisprove,letTbeasdenedinProposition5.2.1fortheelementE2MU;B.FirstsupposethatbothUandBarechosenfromProposition5.2.1i,consideraxedB2B.ForeachA2UdeneA=where<;>isdenedasin[[HR70]28.28i].Holder'sinequality[[HR70]28.28ii]andtheboundednessofTshowthatjAj=jj=kTAkBkBkBkTkkAkUkBkBForallA2U.ThespaceE00^HiscontainedinUandsoforA2E1^Hwehave===ItfollowsthatE~B=CandconsequentlythatE~2MB;U~NextsupposethatUischosenfromProposition5.2.1iandBfromProposition5.2.1ii.Consideraxedbutarbitraryf2B.ForeachA2U,TAbelongstoMH[IncaseTAisafunctionC2Lp0HwemeanbythisthatTAisa84

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measuresuchthatdTA=gdm],wedeneonUbyA=ZfdTA.8forallA2UthenkAkkfkBkTAkkfkBkTkkAksoisboundedbecauseTisbounded.beingaboundedlinearfunctionalonU,thereisaC2UforwhichA=.9forallA2U.SinceE00^HU,TA0isdenedforA0E00^H.TheelementEA0isinE00^HandthedenitionProposition5.2.1ivofTshowsthatdTA0=gdmwhereg2TrigHand^g=EA0.Applying5.8,5.9and[[HR70]34.33]weobtain=ZfdTA0=Zfgdm=<^g;^f>==:.10SincefisarbitraryinBandA0isarbitraryinE00^H,5.10showsthatE~carries^BintoU;thatisE~isinMB;U.ThirdsupposethatUischosenfromProposition5.2.1iiandBfromProposition5.2.1i,foraxedbutarbitraryB2BdeneonUbyf=forallU.Asdenedbeforeisaboundedlinearfunctional,applying[[HR70]14.10],ifU=CH]and[[HR70]12.18]ifU=LpH],wedeneameasurewhichhas85

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theformgdmifU=LpHsuchthat f=Zfdforallf2Uforeachf2TrigHUwehave<^f;^>= <^;^f>=ZHfd=f===<^f;E~B>andhenceE~B=^.ThusagainE~belongstoMB;U.SupposenallythatUandBarechosenfromProposition5.2.1ii,andconsiderg2B.Forf2Udenef=RgdTfasinthepreviouscase,thereisa2Usuchthat f=Rfdforf2U.Forf2TrigHwehavedTf=hdmwhereh2TrigHand^h=E^f.Hencewecanwrite<^f;^>=f=Z^gdTf=Zghdm=<^h;^g>==<^f;E~^g>onceagainE~^g=^andE~isinMB;UWenowconsideraversionofWendel'stheorem.ThistheoremtellsuswhenboundedlinearoperatorsonL1Hcommutewithtranslationoperators.Itwasstatedandprovedforlocallycompactabeliangroupsin[Lr71].Astatementofthistheoremforlocallycompactcommutativehypergroupsisin[Ls82].Wegivehereacompleteproof.Theorem5.2.1LetHbealocallycompactcommutativehypergroup.SupposeT:L1H!L1Hisaboundedlineartransformation.Thenthefollowingstatementsareequivalent:86

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i.TcommuteswithrighttranslationoperatorthatisTfs=Tfsforalls2Hii.Tfg=Tfgforeachf;g2L1Hiii.Thereexistsauniquetransformation'on^Hsuchthat[Tf='^fforeachf2L1H.iv.Thereexistsauniquemeasure2MHsuchthat[Tf=^^fforeachf2L1Hv.Thereexistsauniquemeasure2MHsuchthatTf=fforeachf2L1HProof:iimpliesiiSupposeTcommutewithrighttranslation,letk2L1HthenthemappingdenedonL1Hbyf7!ZTftkt)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmtisalinearfunctionalonL1HmoreoverkZTftkt)]TJ/F15 11.955 Tf 7.084 -4.936 Td[(dmtkkkk1kTfk1kkk1kTkkfk1wherekTkdenotestheusualoperatornormofT.Consequentlythereexistsafunctionh2L1HsuchthatZTftkt)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmt=Zftht)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmt.11byvirtueof[HK75]20.20.Iff,g2L1HwehaveZ[Tfg]tkt)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmt=Z[ZTftsgs)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dms]kt)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmt=Z[Tfstgs)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dms]kt)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmt=87

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Z[ZTfstgs)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dms]kt)]TJ/F15 11.955 Tf 7.084 -4.936 Td[(dmtandfromFubini'stheorem=Zgs)]TJ/F15 11.955 Tf 7.084 -4.936 Td[(ZTfstkt)]TJ/F15 11.955 Tf 7.084 -4.936 Td[(dmtdmsandfrom5.11Zgs)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(Zfstht)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmtdms=Zht)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(Zfstgs)]TJ/F15 11.955 Tf 7.085 -4.936 Td[(dmsdmt=Zfgtht)]TJ/F15 11.955 Tf 7.084 -4.937 Td[(dmt=ZTfgtkt)]TJ/F15 11.955 Tf 7.085 -4.937 Td[(dmt:AndsincekwasarbitrarilychoseninL1HitfollowsthatTfg=Tfgforallf;g2L1H.AtthispointcommutativityisnotassumeandwillbeassumenowiiimpliesiiiSupposeTfg=Tfgforallf;g2L1H.ThensinceHiscommutative,L1Hiscommutativethatisfg=gfforallf;g2L1HsothatTfg=TgfandwehaveTfg=Tfg=Tgf=TgfInparticularforallf;g2L1HwehaveTfg=Tgf[Tf^g=[Tg^fNowforall2^Hchooseg2L1Hsuchthat^g6=0seeHilleandPhilips[HP57]4.15fortheexistenceofsuchgdene'=Tg ^gthentheequation[Tf^g=^f[Tg[Tf ^f=[Tg ^g88

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Therefore'isindependentofgandwehave[Tf^g=^f[Tgwhichimplies[Tf=^f[Tg ^g='^f='^fTherefore[Tf='^fiiiimpliesivSupposethat[Tf='^fforallf2L1H.Thatis'^f2L1H.Itfollowsthat'^fisaFouriertransformofTfandsince'2C^H['^fiscontinuous]'isaFourierStieltjestransform[[Ls82]Theorem2.1.3],thatis,thereexists2MHsuchthat'=^so[Tf=^^fivimpliesv[Tf=^^f=[fNowTf)]TJ/F23 11.955 Tf 11.955 0 Td[(f^=0impliesTf=fFinallyvimpliesiSincefs2L1Hthereexists2MHsuchthatTfs=fs=fs)]TJ/F15 11.955 Tf 6.753 -0.299 Td[(=fs)]TJ/F15 11.955 Tf -320.054 -39.9 Td[(butf=fsothatTfs=fs)]TJ/F15 11.955 Tf 10.073 -0.299 Td[(=fs=TfsSoTfs=Tfs.ThenexttheoremisareducedformofWendel'stheoremforlocallycompactnon-commutativehypergroupsItisstatedwithoutproofin[[BH95]Theorem1.6.24.]Theorem5.2.2SupposeHisalocallycompactnotnecessarilycommutativeandT89

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isaboundedlineartransformationofL1Hintoitself.Thenthefollowingstatementsareequivalenti.TcommuteswithrighttranslationoperatorthatisTfs=Tfsforalls2Hii.Tfg=Tfgforeachf;g2L1Hiii.Thereexistsauniquemeasure2MHsuchthatTf=fforeachf2L1HProof:iimpliesiiandiiiimpliesifollowexactlyasaboveandiiimpliesiiifollowsasin[[HR70]Theorem35.5].WenowgiveaproofofTheorem5.2.1forcompactnotnecessarilycommutativehypergroups.Theorem5.2.3LetHbeacompactnotnecessarilycommutativehypergroup.Sup-poseT:L1H!L1Hisaboundedlineartransformation.Thenthefollowingstatementsareequivalent:i.TcommuteswithrighttranslationoperatorsthatisTfs=Tfsforalls2Hii.Tfg=Tfgforeachf;g2L1Hiii.Thereexistsauniquetransformation'on^Hsuchthat[Tf='^fforeachf2L1H.iv.Thereexistsauniquemeasure2MHsuchthat[Tf=^^fforeachf2L1Hv.Thereexistsauniquemeasure2MHsuchthatTf=fforeachf2L1HProof:FromTheorem5.2.2i,ii,vareequivalent.Wenowshowthatiiandiiiareequivalent.90

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iiimpliesiiiLet^HbethedualobjectofH,supposeU2^HandfjgdUj=1isanorthonormalbasisforHUTheHilbertspaceassociatedwithUwithdimensiondU.WithcoordinatefunctionsdenedforUbyujkx=where1j;kdUthenifU;V2^H,thereexistsaconstantkUwithkUdUsuchthatZujkvrsdm=8<:k)]TJ/F21 7.97 Tf 6.586 0 Td[(1uwhenU=V,j=r,k=s;0otherwise:moreoverifHisacompacthypergroupthenkU=dU[Vr79]theorem2.6.NowletU=k)]TJ/F21 7.97 Tf 6.586 0 Td[(1UIUwherekUdU.ThenUisinthecenterZL1HofL1HthatisfU=UfbecausefU=^f^U=^fk)]TJ/F21 7.97 Tf 6.586 0 Td[(1U^IU=k)]TJ/F21 7.97 Tf 6.587 0 Td[(1U^IU^fk)]TJ/F21 7.97 Tf 6.586 0 Td[(1UIU^f=[k)]TJ/F21 7.97 Tf 6.587 0 Td[(1UIUfb]=UfSofU=UfWecannowdene'U=TkUUUcTfU=cTfU^IUU=cTfUkUUU=[TfkUUb]U=[TfkUUb]U=[TkUUfb]U=[TkUUfb]U=TkUUU^fU='U^fUthatisTf^U='U^fU='^fUwhichimpliesTf='^f.91

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Nowiiiimpliesii,assumeTfg=Tfgforeachf;g2L1HthenThereexistsauniquetransformation'on^Hsuchthat[Tf='^fTfg='[fg='^f^g=[Tf^g=TfgsothatTfg=TfgTheequivalenceofivandvisobtainedfromtheisomorphismoftheFouriertransform.RemarkWhenHisacompactcommutativehypergroupand10gofoperatorsinBXiscalledasemigroupofoperatorsonXifandonlyifT1+2=T1T2forall1;2>0TheinnitesimaloperatorA0ofSisdenedasthelimitinnormas!0+ofAx=1 [T)]TJ/F23 11.955 Tf 11.955 0 Td[(I]x92

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wheneveritexists.IngeneralA0isanunboundedlinearoperator;howeverthedomainofA0isdenseintheunionoftherangespacesoffT;>0g.TheoperatorA0isingeneralnotclosed;itsclosureA,whenitexists,willbecalledtheinnitesimalgeneratorofS..AcomprehensiveaccountofsemigroupsofoperatorsonBanachspacescanbefoundinHilleandPhillips[HP57],whereallundenedtermsusedinthisworkinconnectionwithsuchsemigroupsareexplained.Theorem5.3.1LetS=fT:>0gbeasemigroupofboundedlinearoperatorsonU=LpH.Supposethatforeach>0,theoperatorTcommutewithtrans-lations.ThenSdenesasemigroupM=fE:>0gofU;U-multiplierssuchthati.Foreach>0,E^f=Tfforeachf2U;andii.E1+2=E1E2,1;2>0and2^HIfmoreover,Tisweaklymeasurable,thenthereexistsasubset^H0of^Handamapping':7!'of^H0intoCsuchthatE=8<:e'if2^H0;0if=2^H0:foreach>0Proof:i.Forall>0,TisacontinuouslinearoperatoronUwhichcommuteswithtranslationandfromTheorem5.2.1,thereisauniqueEon^HsuchthatTf=E^fforallf2U93

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ii.Nowforall1;2>0T1+2f=E1+2^fButT1+2f=[T1T2]f=E1T2f=E1E2^fSincefisarbitrary,E1+2=E1E2ThatisE1E2=E1+2whichprovesii.SupposenowthatTisweaklymeasurable.Thenforeachcontinuouslinearfunc-tionalonUandforeachf2U,7!TffromR!CisLebesguemeasurable.Inparticularifforeach2^Hwedenebyf=^f,f2U,thenisacontinuouslinearoperatoronUsuchthatthemapping7!T=E^ismeasurable.Itfollowsthatforeach,Eismeasurable.SinceE1+2=E1E2Eisameasurablecharacterandfrom[HilleandPhillips[HP57]corollarytotheorem4.17.3]itfollowsthatfornontrivialcharactersE=e'forsomecomplexnumbers'.Nowwecanset^H0=f2^H:E6=0g94

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and':7!':^H0!CisawelldenedmappingsuchthatE=8<:e'if2^H0;0if=2^H0:foreach>0.Whichendstheproof.Nowlet^H0beaxedsubsetof^HandletE=8<:e'if2^H0;0if=2^H0:foreach>0AssumethatEasdenedhereisaU;U-multiplier,thenwehaveTheorem5.3.2Foreach>0,deneamappingTofUintoitselfbyTf=E^f,f2Utheni.S=fT:>0gdenesasemigroupofboundedlinearoperatorsonU,theelementsofwhichcommutewithtranslationsandarecontinuousinthestrongoperatortopologyfor>0.iiForeachf2DA0and=2^H0wehave^f=0whereA0denotesthein-nitesimaloperatorofSandDA0isthedomainofA0.Moreover,'isaDA0;U-multipliersincedA0f='^fforallf2DA0iiiIfSisofclassA, ^H0=^HandDA=ff2U:'^f2^UgThatis'isaDA;U-multiplierandmoreovercAf='^fforallf2DA,whereAistheinnitesimalgeneratorofS95

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Proof:i.Fromproposition5.2.1vi,Tisaboundedlinearoperatorforeach>0,moreoverwehaveT1+2f=E1+2^f=E1E2^f=E1[T2f=T1T2fSoT1+2=T1T2andS=fT:>0gisasemigroupofboundedlinearoperatorsonU.AndfromthedenitionofTeachoftheoperatorsTcommuteswithtranslation.NowtoprovethatTiscontinuousinthestrongoperatortopologyfor>0,rstsupposethatt2IH,thesetofallnitecomplexlinearcombinationofcontinuouscharactersonH.Thustisoftheformt=Pni=1ii,theorthogo-nalityofIHimpliesTisdenedby[Ttx=nXi=1ie'ixx2HthenwehavekTt)]TJ/F23 11.955 Tf 11.955 0 Td[(T0tk=knXi=1[ie'ii)]TJ/F23 11.955 Tf 11.955 0 Td[(ie'i0i]knXi=1jijje'i)]TJ/F23 11.955 Tf 11.955 0 Td[(e'i0j!0as!0SupposenowthatfisarbitrarilychoseninUandlet>0begiven.Then,96

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thereexistst2IHsuchthatkf)]TJ/F23 11.955 Tf 11.955 0 Td[(tk<"sincekTf)]TJ/F23 11.955 Tf 11.955 0 Td[(T0tkkTkkf)]TJ/F23 11.955 Tf 11.956 0 Td[(tkforeach>0,Tisstronglymeasurable[HP57]3.5.4.Henceby[HP57]10.2.3,Tiscontinuousinthestrongoperatortopologyfor>0.Thiscompletetheproofforiii.Letf2DA0,thenlim!0+1 Tf)]TJ/F23 11.955 Tf 11.955 0 Td[(fexists.Letthislimitbeg,theng=A0f,thelimitbeingtakeninthenormtopology.Foreach1 [Tf)]TJ/F15 11.955 Tf 14.503 3.154 Td[(^f]!^gandsinceTf=E^fwehave1 [E)]TJ/F15 11.955 Tf 11.956 0 Td[(1]^f!^gas!0+butE=8<:e'if2^H0;0if=2^H0:sothat^f=lim!0+^g=0if=2^H0,thatis^f=0andE=e'if2^H0,solim!0+1 [E)]TJ/F15 11.955 Tf 11.955 0 Td[(1]=lim!0+1 [e')]TJ/F15 11.955 Tf 11.956 0 Td[(1]='97

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Andwehavelim!0+1 [E)]TJ/F15 11.955 Tf 11.955 0 Td[(1]^f='^fThatisdA0f='^f.iii.SupposethatfT:>0gisofclassAwithinnitesimalgeneratorA=A0,thesmallestclosedextensionofitsinnitesimaloperatorA0.ThenU0=fTf:f2U;>0gandDA0aredenseinU.Supposethereexists02^Hsuchthat0=2^H0choosef2Usuchthat^f06=0,thengiven">0thereexistsanf02DA0suchthatkf0)]TJ/F23 11.955 Tf 11.955 0 Td[(fk<"thenj^f00)]TJ/F15 11.955 Tf 14.503 3.155 Td[(^f0jkf0)]TJ/F23 11.955 Tf 11.955 0 Td[(fk<"andsincethisistrueforall>0,^f00=^f0=0acontradiction.Hence^H0=^H.Finallylet!0=inf1 logkTk=lim!11 logkTkThatisSisoftype!0.ForwithRe>!0,letR:AdenotetheresolventoftheinnitesimalgeneratorAofSthenthereexistsa!1>!0suchthatR:Af=Z10e)]TJ/F24 7.97 Tf 6.586 0 Td[(Tfdf2U0,Re>!1since82^H,themappingf7)167(!^fisaboundedlinearfunctionalonU,wehaveforallf2U0R:Af=Z10e)]TJ/F24 7.97 Tf 6.587 0 Td[(Tfd=Z10e)]TJ/F24 7.97 Tf 6.586 0 Td[(e'^fd=98

PAGE 106

1 ')]TJ/F23 11.955 Tf 11.956 0 Td[(e')]TJ/F24 7.97 Tf 6.586 0 Td[(^fj=1=0=)]TJ/F23 11.955 Tf 11.955 0 Td[(')]TJ/F21 7.97 Tf 6.586 0 Td[(1^fforeach2^H.SinceU0isdenseinU,wehaveR;Af=)]TJ/F23 11.955 Tf 11.955 0 Td[(')]TJ/F21 7.97 Tf 6.587 0 Td[(1^f5.12forallf2UwithRe>!1.Let>!1bexedandsupposethatf2DA.Thenthereexistsag2Usuchthatf=R;Agandwehaveforeach2^HcAf=[R;Ag)]TJ/F23 11.955 Tf 11.955 0 Td[(gb]=[R;Agb])]TJ/F15 11.955 Tf 12.37 0 Td[(^g=)]TJ/F23 11.955 Tf 11.955 0 Td[(')]TJ/F21 7.97 Tf 6.586 0 Td[(1^g)]TJ/F15 11.955 Tf 12.371 0 Td[(^g= )]TJ/F23 11.955 Tf 11.955 0 Td[(')]TJ/F15 11.955 Tf 11.955 0 Td[(1^g)]TJ/F23 11.955 Tf 11.956 0 Td[(+' )]TJ/F23 11.955 Tf 11.955 0 Td[('^g='R;Ag='^fThuswheneverf2DA,'^f2^U.Conversely,supposethatfisanelementofUsuchthat'^f2^U.Thismeansthatthereexistsanh2Usuchthat'^f=^hforall2^H.Theng=f)]TJ/F23 11.955 Tf 11.956 0 Td[(h2Uandforall2^H[R;Agb]=)]TJ/F23 11.955 Tf 11.955 0 Td[(')]TJ/F21 7.97 Tf 6.586 0 Td[(1^g=[)]TJ/F23 11.955 Tf 11.955 0 Td[('])]TJ/F21 7.97 Tf 6.587 0 Td[(1[^f)]TJ/F23 11.955 Tf 11.955 0 Td[('^f]=[)]TJ/F23 11.955 Tf 11.956 0 Td[('])]TJ/F21 7.97 Tf 6.586 0 Td[(1[)]TJ/F23 11.955 Tf 11.956 0 Td[(']^f=^fWhichimpliesthatR;Ag=fwhichalsoimpliesthatf2DA99

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References[B074]BabalolaV.A.andOlubummoA.,Semigroupsofoperatorscommutingwithtranslations,Colloq.Math.3174,241-246.[BK63]BerezanskiiYu,M.andKrein,S.G.,Hypercomplexsystemswithcontin-uousbasis.Amer.Math.Soc.Transl.16358{364.46.00[BH95]BloomW.RandHeyerH.,TheHarmonicAnalysisofProbabilityMea-suresonHypergroups,deGruyterStud.Math.,vol.20,deGruyter,BerlinandHawthorne,NewYork,1995.[Bo54]BochnerS.Positivezonalfunctionsonthespheres,Proc.Nat.acad.Sci.U.S.A.Vol.40,1954,p.1141-1147.[Bo56]BochnerS.SturmLiouvilleandheatequationswhoseeigenfunctionsareultrasphericalpolynomialsorassociatedBesselfunctions.,Proceedingsoftheconferenceondierentialequations.UniversityofMarylandBookStore,CollegePark,Maryland,1956,p.23-48.[DE]Delsartes,J.,Hypergroupesetoperateursdepermutationetdetransmu-tation.French1956Latheoriedesequationsauxderiveespartielles.ColloquesInternationauxduCentreNationaldelaRechercheScien-tique,LXXICentreNationaldelaRechercheScientique,ParisRe-viewer:J.B.Diaz35.00Nancy,9-15avril1956pp.29{45.[Du73]DunklC.F.,ThemeasureAlgebraofaLocallyCompactHypergroup,Trans.Amer.Math.Soc.179,331-348.[GA70]GasperG.,LinearizationoftheproductformulaofJacobipolynomialsI,II.,Canad.J.Math.22,1970,p.171-175andp.582-593.100

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[GA71]GasperG.Positivityandtheconvolutionstructureforjacobiseries.,Ann.Math.,93,,1971,p.112-118.[GA72]GasperG.,BanachalgebraofJacobiSeriesandpositivityofakernel.,Ann.Math.,95,1972,p.261-280.[Ge50]GelfandI.M.SphericalfunctionsinsymmetricRiemannspacesRussian.,Dok.Akad.Nauk.SSSRNS,Vol.70,1950,p.5-8.[HK75]HewittE.andKarlS.,RealandAbstractAnalysis,ThirdprintingGrad-uateTextinmath.25SpringerVerlag,NewYork-Heidelberg.[HR70]HewittE.andRossA.K.,AbstractHarmonicAnalysis,Vol1,2Springer-Verlag,BerlinandNewYork,1970.MR417378.[HP57]HilleE.andPhillipsR.S.,FunctionalAnalysisandSemigroups,Amer.Math.Soc.Colloq.Publ.Vol.31Amer.Math.Soc.Providence,R,I.,MR19664.[Hi56a]HirschmanI.I.Jr,Harmonicanalysisandultrasphericalpolynomials.,SymposiumoftheconferenceonHarmonicAnalysis,Vol.1,Cornell,1956.[Hi56b]HirschmanI.I.Jr,Surlespolynomesultraspheriques.,C.R.Acad.Sci.Paris,242:2212-2214,1956.[HM95]HognasG.andMukherjeaA.ProbabilityMeasuresonSemigroups,Con-volutionProducts,RandomWalks,andRandommatricesPlenumPress,NewYorkandLondon.[Je75]JewettR.I.,Spaceswithanabstractconvolutionofmeasures,AdvancesinMath.18,1-101.[Ki63]KingmanJ.F.C.Randomwalkswithsphericalsymmetry.,ActaMath.,109:11-53,1963.101

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[Lr71]LarsenR.,AnintroductiontotheTheoryofMultipliers,Springer-VerlagNewYorkHeidelbergBerlin.[Ls82]LasserR.,Fourier-Stieljestransformonhypergroups,Analysis,2281-303.[LE]Levitan,B.M.Generalizedshiftoperatorsandsomeoftheirapplica-tions.Gosudarstv.Izdat.Fiz.-Mat.Lit.,Moscow,323pp.Reviewer:R.C.Gilbert47.25.90.[MA]Marty,F.Sureunegeneralizationdelanotiondegroups.Huitiemecon-gresdesmathematiciensscandinaves.Stockholm,45-49[Mi55]Michael,E.A.Topologiesonthespacesofsubsets.Trans.Amer.Math.Soc.71,152-182.[On89]OnipchukS.V.,IdempotentMeasuresonCompactSemihypergroups,UkrainskiimatematicheskiiZhurnal,Vol.41,No91244-1247.[On93]OnipchukS.V.,Regularsemihypergroups,RussianAcad.Sci.Sb.Math.Vol.76No1-164[Si95]SiM.,Centrallimittheoremandinnitelydivisibleprobabilitiesasoci-atedwithpartialdierentialoperators.,JournalofTheoreticalprobabil-ity,Vol.8No3,p.475-499.[Sp78]SpectorR.,Mesuresinvariantessurleshypergroupes,Trans.Amer.Soc.239-165.[Tr97]TrimecheK.,GeneralizedWaveletsandHypergroups,GordonandBreachSciencePublishers,.[Vr79]VremR.C.,HarmonicAnalysisonCompactHypergroups,PacicJ.Math.vol.85-251.102

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[WA]Walls,H.S.HypergroupsAmer.J.Math,59[Ze89]ZeunerH.OnedimensionalhypergroupsAdv.math.,76:1-18.103

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AbouttheAuthorNorbertYoumbiwasbornin1969inMeiganga-Cameroon.HereceivedaBachelorofSciencehonorsattheUniversityofJos-Nigeria,andaMastersofScienceinMathematicsattheUniversityofIbadan-Nigeria.HereceivedtheUniversityofJosprizeforthebestgraduatingstudentineachfaculty.From1997to1999MrYoumbiwasanassistantLecturerattheuniversityofIbadan-Nigeria.From1999to2001hewasanassistantLecturerattheuniversityofNgaoundere-Cameroon.IntheFallof2001hewasadmittedintothePhDprograminmathematicsattheUniversityofSouthFloridaUSF,inTampawhereheworkedunderthesupervisionofProfessorArunavaMukherjea.HisscholarlyinterestsareinMeasureandIntegrationtheory,FunctionalAnalysisandProbabilitytheory.


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Probability theory on semihypergroups
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ABSTRACT: Motivated by the work of Hognas and Mukherjea on semigroups,we study semihypergroups, which are structures closer to semigroups than hypergroups in the sense that they do not require an identity or an involution. A semihypergroup does not assume any algebraic operation on itself. To generalize results from semigroups to semihypergroups, we first put together the fundamental algebraic concept a semihypergroup inherits from its measure algebra.
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Haar measure.
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