TOEPLITZ ALGEBRAS ON DISCRETE GROUPS AND THEIR NATURAL MORPHISMS

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【作者】 J.LORCH 

【出版日期】2005-01-25

【摘要】<正>Let G be a discrete group, E1 and E2 be two subsets of G with E1(?)E2, and e ∈E2. Denote by TE1 and TE2 the associated Toeplitz algebras. In this paper, it is proved that the natural morphism γE2,E1 from TE1 to TE2 exists as a C*-morphism if and only if E2 is finitely covariant-lifted by E1 Based on this necessary and sufficient condition, some applications are made.

【刊名】Chinese Annals of Mathematics

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互1 .Illtroduetion The objeet of the Present PaPer 15 to study the natural morPhisms between ToePlitz algebras.If G 15 a diserete group and E gG,one may form the assoeiated ToePlitz algebra丁E.Given two subsets刀1 and EZ with EI互E2,there 15 a naturaJ morphism 俨2,E‘:丁E‘升丁EZ .In some eases,this morphism fails to be aC*一morphism,or even fails to be well defined.Our main task 15 to Put forth a neeessary and suffieient eondition under whieh守EZ,E‘exists as aC*一morphism. Toward this end,a teehnique initiated by E.park in【2」and generalized in【6}yielded the finite deeomposition eondition.However,as shown in[4〕,while this eondition 15 suffieient for the existenee of守E,,E‘,it 15 not neeessary.In this paper we put forth a more natural eondition,ealled the finite eovariant一lift eondition.w七will show that this latter eondition 15 not only suffieient,but also neeessary(see Theorem 2.1).Based on this new eondition, some aPplieations also have been made. The PaPer 15 organized in the followingw盯.In SeetionZ,we give the Preeise definition of the finite eovariant一lift eondition,and show that it 15 both neeessary and suffieient for the existenee of守E,,E‘.In Seetion3,we show that肠eplitz algebras assoeiated to quasi一ordered groups have a eertain universal proPerty(see Corollary 3.1).In Seetion4,we extend eertain results from【4}to the nonabelian ease.These results eoneern the natural morphism between ToePlitz algebras eorresPonding to a quasi一ordered grouP and its indueed partially一ordered grouP.Using the finite eovariant一lift eondition,our arguments beeome rnueh simPler than those in!41.Finally,in Seetion 5 we turn to study Toeplitz algebras on diserete abelian groups.In!1」,G.Murphy proved that if(G,G+)15 a diserete abelian ordered group,then 14斗 J .L ORCH&XU, Q.X 舫 tlle’孙(·plitz al隽ebra丁‘了+has a 11niversal property fori、onletrie representations ofG+ a箕e:leralizatiorl,it was proved in!5}that the same asso(、iate〔It()diserete abelian quasi一ordered grouPs. (、)nversei、al、()true(S、,e Theorem 5.1). PrOPerty holdsf()rT()eplitz alg。、bra、 Ir一this seetion,we will show that tl一。、 互2 .The Natural MorPhisms Between 11、〕ePlitz Algebras on Diserete GrouPs L(·t Gb。、a di、〔‘r。、te隽roup and{百。}g〔G}be theu、ual orthonor。,ai basis for若2(G). wll(、r(、 fl, 百、(h)=弋 LU, if夕=h, ()tlzerwise f’()r。,l;任G .For any夕任G,we define a unitary operator。,on‘2(G)by。,(占,‘)二占。, l’()rj,任G .l:()r al,y、ubset E ofe,let迢2(百)be tl,e elosed subspaee of沼2(G)gelleratod}), {占、}。任E},and letp厂denote the projeetion from沼2(e)onto迢2(君). Definition 2.1.TheC‘一alge乙ra。enora‘ed乙;{呀:二,E。。,El。任G},、de了‘o‘。d6。 “,‘d 15 eallod thoTo印百加alge占m切乞艺h二印eet to E. 补 Definition 2.2.LetE,andEZ乙e toos、占、ets ojG。,云thE,互EZE:z、 ji了‘,北。l军。ooa:乞aot一l价ed乙鲜El ifjo:a。夕万。乞tes。乙、et F OjG,th。二ex坛stsg*任G fo:。yg〔F .9任凡扩ando耐y扩g·g,任E:. said to石‘· ,?止ch that Remark 2.1.(i)LetE,and EZ be two subsets ofGwithEI红EZ.ThenEZisfi,iitol冬 。()variant一lifted by EI if and only if the following three eonditions are satisfied: (l)FOr any two non一empty subsets Flg场and凡gG\EZ,there exists夕,任G suel、 thatFI·g,gE:and凡·夕,gG\E:.A diagram illustrating sueh a eondition 15 as foll〔)ws: FI gE:凡互G\场 丰牛 Fl·g,gE:凡·g,gG\El; (2)F()r any finite non一empty subset FI 9 EZ,there exists夕1任G sueh thatFI·夕:互E,: (3)凡,any finite non一empty subset凡互G\场,there exists 92任Gs::eht王lat凡·穿:g 召\E 1. (11)IfC任EZ and凡15 finitely eovariant一lifted byEI,then upon replaeing the 61、it‘、 、ubset F byF口{。},we see that夕,m即be ehosen in EI .Furthermore,ifC任EI,thenw。 mas. ehooseg:=。,50 the eondition(3)above 15 satisfied automatieally. Theorem 2.1.Let G be ad短serete gro即and Er,肠互G withE:互 Tll。,‘the,*atoral 00印h乞s。守E,,E,:丁E‘斗丁E,,二人乞e人、at葱项e、年E,,E,( anyg任G,e劣艺sts as aC水一叨orPh饭sm prooL Suppose守石,,E‘exists朋 扩aod on勿扩凡乞、万n坛te匆cooaoaot- 肠a几d‘任及. 呀‘)=丫2 j()r l价ed 6yE:. aC牢一algebra morphism.FOr a eontradietion,、uppos,· that EZ isn()t finitely eovariant一lifted by Ei .Then Remark 2.1 irnplies that one of tlx。 foll()wing three eases must oceur: Case 1.There exist two finite non一empty subsets Flg凡and凡互G\E:sueh th*、t fof any夕,〔El。 ifFI夕二互El,then(凡·g*)自E:并 0.(22) ALGEBRAS ON DISCRETE GROUPS AND THEIR NAI,URAL MORPHISMS LetFI={夕1,夕2,…,夕。}and凡={hl,hZ,…,h。}.Set 145 T一(旦(‘一货1嘴”) /1-T。万;二E,\ 、二几g‘J./ 泛=1 Then 俨2 OE!(T)一(11(‘- 夕二1 冷心)(县冷创 By(2 .1)we know thatT=o,50俨么E‘(T)=o,But eleajrly俨2,E‘(T)占。=占。笋。,yielding a eontradietion. Case 2.There exists a finite subsetF={91,92,二,,g。}gEZ,such that fora叮 g*任El,there existsg,。〔F,sueh thatg‘。·g,彭El .Let T一n孚华 ThenT=0,but守EZ,E,(T)d。=几尹o,whieh 15 a contradietion. Fhj0 Case 3.There exists a finite subset g。〔E1,there existsh,。〔F,suCh that ={hl,hZ,…,h。}gG\场,sueh that for any ·g,〔Ei .Let T一县“一嚼1货’· ThenT=O,but守E么E‘(T)占。=占。尹0,whieh 15 a eontradietion. Now,for the reverse direetion,suPPose that场15 finitely eovariant一lifted饰El .LetT be an operator in丁E‘of the form ”乞几云 T=艺氛n票 =1了=1 Then 7刀几试 :场,E1(T)一艺乐n嗡for乐。C,*,。G. i=1 TO show that甲EZ,E‘15 well defined and 了=1 ean be extended韶aC*一morphism,it suffiees to show that}}守EZ,E‘(T川兰】IT】},as these operators Given£>o,there exists苟任迢2(场)wi

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