HNN Extensions of von Neumann Algebras

arXiv:math/0312439v3[math.OA]10Jan2005HNNEXTENSIONSOFVONNEUMANNALGEBRASYOSHIMICHIUEDAAbstract.ReducedHNNextensionsofvonNeumannalgebras(aswellasC∗-algebras)willbeintroduced,andtheirmodulartheory,factorialityandultraproductswillbediscussed.Inseveralconcretesettings,detailedanalysisonthemwillbealsocarriedout.1.IntroductionTherearetwofundamentalconstructionsincombinatorialorgeometricgrouptheory,whicharethoseoffreeproductswithamalgamationsandofHNN(G.Higman,B.H.NeumannandH.Neumann[13])extensions.Theinterestedreadermayconsult[16]asastandardreferenceonthetopics.EvenintheframeworkofvonNeumannalgebras(aswellasC∗-algebras),reducedfreeproductswithamalgamations([41][44][28]andalso[38])havebeenseriouslyinvestigatedsofarandplayedkeyrˆolesinseveralresolutionsof“existence”questionsinthetheoryofvonNeumannalgebras(see,e.g.[28][32][35][30]andalso[37]).However,HNNextensionshaveneverbeendiscussedsofarintheframework.Historically,manyideasingrouptheory,especiallypartofdealingwithcountablyinfinitediscretegroups,havebeenapplieddirectlyand/orindirectlytomanyaspectsinthetheoryofvonNeumannalgebras(aswellasC∗-algebras)sincethebeginningofthetheory.Infact,manyexplicitexamplesofvonNeumannalgebrasthatopenednewperspectivesinthetheorycamefromgrouptheory(seee.g.[18][17][11][3][4][5]andalsorecentbreakthroughs[29][22][23][24]),anditisstillexpectedtofindmuchmore“monsters”(i.e.,concreteexampleswithveryspecialproperties)livingintheworldofnon-amenablevonNeumannalgebras.Todoso,itseemsstilltobeoneoftheimportantguidingprinciplestoseekfornewideasingrouptheory.Followingthisprinciple,wewillintroducereducedHNNextensionsintheframeworkofvonNeumannalgebras(aswellasC∗-algebras)andtakeaveryfirststeptowardsseriousandsystematicinvestigationonthemwithaimingthattheirconstructionwillplayakeyrˆoleinfutureattemptsofconstructingnewmonstersintheworldofnon-amenablevonNeumannalgebras.Letusexplaintheorganizationofthisarticle.In§2,wewillreviewfreeproductswithamalga-mationsofvonNeumannalgebraswithspecialemphasisoftheadmissibilityofembeddingmapsofamalgamatedalgebrasintheconstruction.Althoughthisslightgeneralizationofthepreviouslyusedoneisofcourseafolklore,wewillbrieflyreviewittoavoidanyconfusionsincetheadmissibilityofembeddingmapsplaysakeyrˆoleinourconstructionofHNNextensions.In§3,reducedHNNextensionsofvonNeumannalgebraswillbeintroduced,andthentheircharacterization(ortheir“construction-free”definition)givenintermsofexpectedalgebraicrelationsand“moment-values”ofconditionalexpectationsasinthecaseoffreeproductswithamalgamations.Inthegroupsetting,onestandardwayofconstructingHNNextensionsistheuseof“shiftautomorphisms”on“infinitefreeproductswithamalgamations”overisomorphicbutnotnecessarycommonsubgroups(infact,twodifferentembeddingsofamalgamatedgroupsareneeded).Thisamalgamationprocedurebringsus“difficulty”inconstructing“shiftautomorphisms”inconnectionwithconditionalexpectationssincetheuniversalconstructionisnotapplicableinthevonNeumannalgebrasetting.Hence,adifferentideaisneededtoconstructthedesiredones,andindeeditisbasedonanobservationcomingfromourpreviouswork[39]onadifferenttopic.Roughlyspeaking,ourconstructioncanbeunderstoodasan“amalgam”(butnota“combination”)ofthoseofcovariantrepresentationswithoutunitarySupportedbyGrant-in-AidforYoungScientists(B)14740118.AMSsubjectclassification:46L10,46L05(primary),46L54,46L09(secondary).12Y.UEDAimplementationsinthecrossed-productconstruction(see[36,Vol.II;Eq.(10)inp.241])andoffreeproductswithamalgamations.Ourconstructionseemssomewhatnaturalfromthegrouptheoreticviewpoint.Infact,thenotionofHNNextensionsisknowntobenecessarytodescribeasubgroupofagivenfreeproductgroupwithamalgamationoveranon-trivialsubgroup.The§4willconcernmodulartheoreticalaspectsofreducedHNNextensions.Moreprecisely,wewillgiveacompletede-scriptionofmodularautomorphismsandalsoshowthatthecontinuouscoreofanyreducedHNNextensionbecomesagainareducedHNNextension.In§5,wewilldiscussthefactorialityandinves-tigatetheultraproductsofreducedHNNextensions.Theresultscorrespondtowhatweobtainedinourpreviouswork[40]onfreeproductswithamalgamations.In§6,wewillinvestigatereducedHNNextensionsofvonNeumannalgebrasinseveralconcretesettings.Thefirstoneisnaturallyarisenfromnon-commutative2-tori,thesecondfromthetensorproductoperation,andthethirdfromregularandsingularMASAsinthecrossed-productsby(non-commutative)Bernoullishifts.Thethirdoneseemsimportantforfurtherinvestigationbecauseanygivensurjective(partial)∗-isomorphismbe-tweenregularandsingularMASAsinquestioncanneverbeextendedtoanyglobal∗-automorphismonthegiven“base”algebras.In§7,reducedHNNextensionsofC∗-algebraswillbeintroducedinthesamemannerasinthevonNeumannalgebrasetting,andthensomebasicfactswillbegiven.Furtheranalysisonthemwillbepresentedelsewhere.Partofthisarticlewaspresentedintheconference“RecentAdvancesinvonNeumannAlgebras”celebratedtoProfessorMasamichiTakesaki’s70thbirthday,atUCLAinMay,2003.Wewouldliketoexpressoursincerethankstotheorganizers;ProfessorsYasuyukiKawahigashi,SorinPopa,andDimitriShlyakhtenko,whokindlygaveustheopportunitytopresentthisworkintheconferenc