JIANG-SU ALGEBRA AS A FRAI¨SSE´ LIMIT SHUHEIMASUMOTO Abstract. Inthispaper,wegiveaself-containedandquiteelementaryproofthat theclassofalldimensiondropalgebrastogetherwiththeirdistinguishedfaithful 6 tracesformsaFra¨ıssećlasswiththeJiang-Sualgebraasitslimit. Wealsoshow 1 0 thattheUHFalgebrascanberealizedasFra¨ısse´limitsofclassesofC*-algebras 2 ofmatrix-valuedcontinuousfunctionson[0,1]withfaithfultraces. n a J 0 1. Introduction 3 TheFra¨ısse´ theory wasoriginally invented byRolandFra¨ısse´ in[Fra54],where ] A a bijective correspondence between countable ultra-homogeneous structures and O classes with certain properties of finitely generated structures is established. The . classesandcorrespondingultra-homogeneousstructuresinquestionarecalledFra¨ısse´ h t classesandFra¨ısse´ limitsoftheclasses respectively. a m This theory has been, among the rest, a target of generalization to the setting [ of metric structures. For example, a general theory was developed in [Sch07], includingconnectionswithboundedcontinuouslogic. In[Yaa15],Ita¨ıBenYaacov 1 v conciselygaveaself-containedpresentationofageneraltheory,usingabrightidea 4 ofapproximate isomorphisms. 2 1 These attempts at generalization ended up successfully, and a number of met- 0 ric structures are recognized as Fra¨ısse´ limits. Ita¨ı Ben Yaacov [Yaa15] pointed 0 . out that the Urysohn universal space, the separable infinite dimensional Hilbert 2 space, and the atomless standard probability space are examples of Fra¨ısse´ limits 0 6 corresponding to suitable classes, and reconstructed discussion in [KS13] where 1 the Gurarij space had been implicitly shown to be a Fra¨ısse´ limit of the class of : v all finite dimensional Banach spaces. The latter result was quantized by Martino i X Lupini[Lup14]: itwasshownthatthenoncommutativeGurarijspaceistheFra¨ısse´ r limitoftheclassofallfinitedimensional 1-exactoperator spaces. a Amongthose instances are operator algebras. In [Eag+15], amore generalized version ofFra¨ısse´ theory formetricstructures waspresented, wheretheaxiomsof Fra¨ısse´ class were relaxed, and so the bijective correspondence established in the originaltheorynolongerholdsandthelimitstructureswouldhavelesshomogene- ity, though it is still powerful as a construction method. Using this version, the authors of the paper succeeded in realizing a family of AF algebras including the UHF algebras, the hyperfinite II factor and the Jiang-Su algebra as (generalized) 1 Fra¨ısse´limitsofaclassoffinitedimensionalC*-algebraswithdistinguishedtraces, 2010MathematicsSubjectClassification. Primary47L40;Secondary03C98. Keywordsandphrases. Fra¨ısse´theory;Jiang-Sualgebra;UHFalgebra. 1 2 S.MASUMOTO theclassoffinitedimensionalfactorsandtheclassofdimensiondropalgebraswith distinguished tracesrespectively. The Jiang-Su algebra was first constructed by Jiang and Su in [JS99] as the uniquesimplemonotracial C*-algebra amonginductive limitsofprimedimension dropalgebras,whichisKK-equivalenttothecomplexnumbersC. Oneofthemost important properties ofthisalgebra isthat itisstrongly self-absorbing, because of whichitplaysakeyroleintheElliott’sclassificationprogramofseparablenuclear C*-algebras viaK-theoretic invariants [ET08]. Asispointedoutatthelastsection of[Eag+15],theproofthattheJiang-Sualgebrasatisfiesthispropertyisnontrivial, and there is a reasonable prospect that Fra¨ısse´ theoretic view of this algebra will giveashortcut. However,theproofgivenin[Eag+15]ofthefactthattheJiang-Su algebra is a Fra¨ısse´ limit was still “a bit unsatisfactory” in the authors’ phrase, as it used the existence of the Jiang-Su algebra itself and relied heavily on Robert’s theorem (see[Eag+15,Remark4.8andProblem7.2]). In this paper, we prove that the collection of all the dimension drop algebras together with their distinguished faithful traces forms a Fra¨ısse´ class. The impor- tance lies in that this proof is self-contained and quite elementary; in particular, it depends on neither the existence of the Jiang-Su algebra norRobert’s theorem, so that it can be considered as a solution to [Eag+15, Problem 7.2]. Also, we show that the UHF algebras are realized as a Fra¨ısse´ limit of a class of C*-algebras of matrix-valuedcontinuous functions ontheinterval[0,1]togetherwiththeirdistin- guished faithfultraces. Sincethisclassdiffersfromtheoneusedin[Eag+15],this resultimpliesadifferenthomogeneity oftheUHFalgebras. The paper consists of four sections. In the next section, we briefly introduce a version of Fra¨ısse´ theory for metric structures, which is essentially the same as the one in [Eag+15]. The third section contains the result on the UHF algebras. The argument included in this section is the basis of the fourth section, where the dimensiondropalgebrasandtheJiang-Sualgebraaredealtwith. 2. Fra¨ısse´ TheoryforMetricStructures In this section, we present a general theory of Fra¨ısse´ limits in the context of metricstructures, whichisalmostthesameastheonein[Eag+15,Section2]. The facts stated here are slight generalization of those of [Yaa15], and can be proved withtrivialmodification. Definition2.1. Alanguage Lconsists ofpredicate symbolsandfunction symbols. ToeachsymbolinLisassociatedanaturalnumbercalleditsarity. Weassumethat Lcontainsabinarypredicatesymbold. An L-structure isacomplete metric space M together withan interpretation of symbolsofL: • to each n-ary predicate symbol P is assigned a continuous map PM: Mn → R, where the distinguished binary predicate symbol d corresponds to the distance function; and • toeachn-aryfunction symbol f isassigned acontinuous map fM: Mn → M. JIANG-SUALGEBRAASAFRAÏSSE´ LIMIT 3 Anembedding ofanL-structure M intoanother L-structure N isamapϕsuchthat fN(ϕ(a ),...,ϕ(a )) = ϕ(fM(a ,...,a )) 1 n 1 n and PN(ϕ(a ),...,ϕ(a )) = PM(a ,...,a ) 1 n 1 n holdforanyfunctionsymbol f,anypredicatesymbolPandanyelementsa ,...,a 1 n in M. In this paper, we focus on unital C*-algebras with distinguished traces. We assumethatLconsistsofthebinarypredicatesymbold,anunarypredicatesymbol tr,binaryfunction symbols + and ·,anunaryfunction symbolsλforeachλ ∈ C which should be interpreted as multiplication by λ, an unary function symbol ∗, and 0-ary function symbols 0 and 1. Then every unital C*-algebra with trace is understood as an L-structure in the canonical manner. Note that an embedding in thesenseofDefinition2.1isaninjectivetrace-preserving∗-homomorphisminthis case,whichweshallcallamorphisminthesequel. Remark2.2. Thedefinitionoflanguagesandmetricstructuresvariesbypaper(see [Yaa15,Remark2.2]),andtheoneweadopted hereisthesameas[Yaa15,Defini- tion2.1]. Somevariants suchas[Eag+15,Definition2.1]requirethatallthemaps which appear should be bounded or uniformly continuous, in which case the lan- guage carries additional informations. A C*-algebra is seemingly not an instance of a metric structure in these cases, because it is apparently unbounded and the multiplication is not uniformly continuous. Indeed, this can be easily overcome byusing the unit ball asits representative, asin[Eag+15]. Anyway, the results of Fra¨ısse´ theoryinbothperspectives canbeeasilytranslatedtoeachother,soweare inthesamelineas[Eag+15]. Definition2.3. AclassK of L-structures issaidtosatisfy • the joint embedding property (JEP) if for any A,B ∈ K there exists C ∈ K suchthatboth Aand BcanbeembeddedinC. • the near amalgamation property (NAP) if for any A,B ,B ∈ K , any embed- 1 2 dingsϕ : A → B,anyfinitesubsetG ⊆ Aandanyε> 0,thereexistembeddings i i ψ of B into someC ∈ K such that d(ψ ◦ϕ (a), ψ ◦ϕ (a)) isless than ε for i i 1 1 2 2 alla ∈G. An L-structure A is said to be finitely generated if there exists a tuple ~a = (a ,...,a ) ∈ An such that the smallest substructure of A containing all a ,...,a 1 n 1 n is A, for some n ∈ N. (Note that we assumed L-structures to be necessarily complete.) AswefocusonunitalC*-algebras withdistinguished traces, thisdefinition coincides with the usual one, that is, a (unital) C*-algebra (with its distinguished trace)isfinitelygenerated ifthereexists afinitesubset suchthatitsclosure byad- dition, multiplication, scalar multiplication and ∗-operation is dense in the whole C*-algebra. LetK be aclass offinitely generated L-structures. Foreach n ∈ N,wedenote by K the class of all the pairs (A,~a), where A is a member of K and~a ∈ An is n 4 S.MASUMOTO agenerator of A. If K satisfies JEP and NAP,then wecan define apseudometric dK onK by n dK (A,~a), (B,~b) := infmaxd(f(a), g(b)), i i i (cid:0) (cid:1) where ~a = (a ,...,a ),~b = (b ,...,b ) and the infimum is taken over all the 1 n 1 n embeddings f,gof A,BintosomeC inK . Definition2.4. AclassK offinitelygenerated L-structures withJEPandNAPis saidtosatisfy • the weak Polish property (WPP) if K is separable with respect to the pseudo- n K metricd foralln. • theCouchyContinuityProperty(CCP)if (1) for any n-ary predicate symbol P, the map A,(~a,~b) 7→ PA(~a) from Kn+m intoRsendsCauchysequences toCauchyseque(cid:0)nces; and(cid:1) (2) foranyn-aryfunctionsymbol f,themap A,(~a,~b) 7→ A,(~a,~b, fA(~a)) from Kn+m intoKn+m+1 sendsCauchysequences t(cid:0)oCauchy(cid:1) seq(cid:0)uences. (cid:1) Remark 2.5. CCP implies that dK (A,~a),(B,~b) = 0 holds if and only if there is an isomorphism between A and B(cid:0) sending ~a (cid:1)to ~b ([Yaa15, Remark 2.13 (i)]). Note that if K is a class of finitely generated unital C*-algebras with traces and ifitsatisfiesJEPandNAP,thenitalsosatisfiesCCPautomatically, becauseallthe relevantfunctions are1-Lipschitz ontheunitball. Definition 2.6. A class K of finitely generated L-structures is called a Fra¨ısse´ class if itsatisfies JEP, NAP,WPPand CCP.A Fra¨ısse´ limit of a Fra¨ısse´ class K isaseparable L-structure M whichis (1) a K -structure: for any finite subset F of M and any ε > 0, there exists an embedding ϕ of a member of K such that the ε-neighborhood of the image of ϕ includes F. (2) K -universal: everymemberofK canbeembeddedinto M. (3) approximately K -homogeneous: if A is a member of K and a ,...,a are 1 n elements of A, then for any embeddings ϕ,ψ of A into M and any ε > 0, there existsanautomorphism αof M withd(α◦ϕ(a), ψ(a)) < εfori= 1,...,n. i i The definition here is more relaxed than that of [Yaa15] and close to [Eag+15, Definition 2.6]: our Fra¨ısse´ class is incomplete and lacks the hereditary property (see [Yaa15, Definitions 2.5 (ii) and 2.12]). Consequently, we cannot establish a bijective correspondence between Fra¨ısse´ classes and separable structures with homogeneity, which is a part of the main result of Fra¨ısse´ theory. The following theorem summarizeswhatremainsinourframework. Theorem2.7. EveryFra¨ısse´ classK admitsauniquelimit. Moreover,foranyL- structureA inK ,thereexistsasequenceofembeddingsA −−ϕ→0 A −−ϕ→1 A −−ϕ→2 ··· 0 0 1 2 suchthatA belongstoK forallianditsinductivelimitcoincideswiththeFra¨ısse´ i limitofK . JIANG-SUALGEBRAASAFRAÏSSE´ LIMIT 5 3. UHFAlgebras Asupernatural number isaformalproduct ν = pnp, p:Yprime where n is either a non-negative integer or ∞ for each p such that n = ∞. p p p In [Gli60, Theorem 1.12] it was proved that to each UHF algebra isPassociated a supernatural number as its complete invariant. Now, given a supernatural number ν, wedenote by N(ν)the set of all natural numbers which formally divides ν, and by K (ν) the class of all the pairs hC[0,1]⊗M ,τi, where n is in N(ν) and τ is a n faithful trace on the C*-algebra C[0,1]⊗M . Our goal in this section is to show n that K (ν) is a Fra¨ısse´ class the limit of which is the UHF algebra with ν as its associated supernatural number. First, note that C[0,1] ⊗ M is canonically isomorphic to C([0,1], M ), the n n C*-algebra of allthe continuous M -valued functions onthe interval [0,1]. In the n sequel, we shall denote this C*-algebra by A for simplicity. Next, let τ be a n probability Radon measure on [0,1], whichisidentified withastate onC[0,1] by integration. Thenτ⊗trisclearly atraceonA ,wheretristheunique normalized n traceonM . ItiseasytoseethateverytraceonA isofthisform,soaprobability n n Radon measure on [0,1] mayalso beidentified with atrace on A . In the sequel, n we simply write τ instead of τ⊗tr and use adjectives for measures and traces in common. Forexample,ameasureissaidtobefaithful ifitscorresponding traceis faithful. Also, all the measures are assumed to be probability Radon measures so thattheyalwayscorrespond totraces. A measure is said to be diffuse or atomless if any measurable set of nonzero measure is parted into two measurable sets of nonzero measure. The following is oftenusedinthesequelwithoutreferring. Lemma3.1. Letσ,τbefaithfulmeasures. Ifσisdiffuse,thenthereexistsaunique non-decreasing continuousfunctionβfrom[0,1]onto[0,1]withβ (σ)= τ. More- ∗ over,τisdiffuseifandonlyifβisahomeomorphism. Proof. We first assume σ is equal to the Lebesgue measure λ and set α(t) := τ([0,t)). Note that α is a strictly increasing upper semi-continuous function from [0,1]into[0,1]. Letβbetheuniquenon-decreasing functionextendingα−1. Then β (λ)([0,t)) = λ β−1([0,t)) = λ [0,α(t)) = α(t) = τ([0,t)), ∗ (cid:0) (cid:1) (cid:0) (cid:1) soβ (λ)isequaltoτ. Also,ifτisdiffuse,thenαiscontinuous, whenceβ= α−1 is ∗ ahomeomorphism. Forthegeneralcase,letβ ,β besuchthat(β ) (λ) = σand(β ) (λ) = τ. Then σ τ σ ∗ τ ∗ β:= β ◦β−1 satisfiesβ (σ)= τ,whichcompletestheproof. (cid:3) τ σ ∗ Thenextpropositions areimmediatecorollariesoftheprecedinglemma. Recall that amorphism between elements of K (ν)is an injective unital trace-preserving ∗-homomorphism. 6 S.MASUMOTO Proposition3.2. LetτbeafaithfultraceonA . Thenforanyfaithfuldiffusetrace n σonA ,thereisamorphismϕ: hA ,τi → hA ,σi. n n n Proof. Let β be the non-decreasing continuous function as in Lemma 3.1. Then ϕ := β∗ isthedesiredmorphism. (cid:3) Proposition 3.3. TheclassK (ν)satisfiesJEP. Proof. Suppose n ,n are in N(ν) and put n := gcd(n ,n ). Then n is also in 1 2 1 2 N(ν). Also,hA ,λiisclearly embeddable intohA ,λibyamplification. Thisfact ni n together withProposition 3.2impliesthatK (ν)satisfiesJEP. (cid:3) Next, weshall show that the class K (ν) satisfies NAP.Forthis, we begin with provingthatallthemorphismsofK (ν)areapproximately diagonalizable. Definition3.4. Amorphism ϕ: hA ,τi → hA ,σiissaidtobediagonalizable if n m there are a unitary u ∈ A and continuous maps ξ ,...,ξ : [0,1] → [0,1] such m 1 k that f ◦ξ 0 1 (1) ϕ(f) = u  ...  u∗ forall f ∈ A .  0 f ◦ξk n Inthispaper,weshallcallEq.(1)adiagonalexpressionofϕ,anduandξ ,...,ξ 1 k itsassociated unitaryandmaps. Notethattheunionoftheimagesofthemapsas- sociated to a diagonal expression is equal to [0,1], as morphisms are necessarily faithful. Also,compositionsofdiagonalizablemorphismsareagaindiagonalizable. Proposition 3.5. Let ϕ: hA ,τi → hA ,σi be a morphism. Then for any fi- n m nite subset G ⊆ A and any ε > 0, there exists a diagonalizable morphism n ψ: hA ,τi → hA ,σi with kϕ(g) − ψ(g)k < ε for all g ∈ G. Moreover, we can n m takeψsothatthemapsξ ,...,ξ associatedtoadiagonalexpressionofψsatisfies 1 k ξ ≤ ··· ≤ ξ . 1 k Proof. Fort ∈ [0,1],letev : A → M betheevaluation∗-homomorphism. Then t m m ev ◦ϕ isaunital ∗-homomorphism from A to the finitedimensional C*-algebra t n M ,sothere existaunitary v ∈ M andrealnumbers st,...,st ∈ [0,1]suchthat m t m 1 k theequation ev ◦ϕ(f) = Ad(v) diag(f(st),..., f(st)) t t 1 k holds for all f ∈ A . Note that {{st,.(cid:0)..,st}} coincides with(cid:1)the spectra of ev ◦ n 1 k t ϕ(id ⊗1 )asmultisets. Bycontinuity, ift andt areclosetoeachother,then [0,1] Mn 1 2 soarethespectrum ofev ◦ϕ(id ⊗1 )andev ◦ϕ(id ⊗1 )withrespect t1 [0,1] Mn t2 [0,1] Mn totheHausdorffdistance. Therefore, ifwedefine ξ (t) := max{{st,...,st}}; 1 1 k ξ(t) := max{{st,...,st}}\{{ξ (t),...,ξ (t)}}, i 1 k 1 i−1 thenobviously ξ ,...,ξ arecontinuous functions from[0,1]into[0,1]satisfying 1 k ξ ≤ ··· ≤ ξ . 1 k JIANG-SUALGEBRAASAFRAÏSSE´ LIMIT 7 Next, fix t ∈ [0,1]. We claim that there exists δ(t ) > 0 with the following 0 0 property: if|t−t | < δ(t ),thenthereexistsaunitaryw ∈ M withkv −w k < ε 0 0 t0 m t t0 suchthattheequation ev ◦ϕ(f)= Ad(w ) diag(f(st0),..., f(st0)) t0 t0 1 k (cid:0) (cid:1) holds for all f ∈ A . To see this, let s ,...,s be distinct eigenvalues of ev ◦ n 1 l t0 ϕ(id ⊗ 1 ) and take mutually orthogonal non-negative continuous functions [0,1] Mn f ,..., f suchthat f isconstantly equal to1onsomeneighborhood of s foreach 1 l i i i. Notethatif{e }isamatrixunitofM ,then{ev ◦ϕ(f ⊗e )} formsamatrix p,q n t0 i p,q i,p,q unitofIm(ev ◦ϕ),andiftissufficientlyclosetot ,then{ev ◦ϕ(f ⊗e )} isa t0 0 t i p,q i,p,q matrixunitofasubalgebra ofIm(ev ◦ϕ)whichiscloseto{ev ◦ϕ(f ⊗e )} . t t0 i p,q i,p,q Hence, as in the proof of [Dav96, Lemma III.3.2], we can find a unitary w with kw−1k < εsuchthat w(ev ◦ϕ(f ⊗e ))w∗ = ev ◦ϕ(f ⊗e ), t0 i p,q t i p,q andw := vwhasthedesiredproperty. t0 t Nowtakeδ > 0sufficiently smallsothattheinequalities 0 kg◦ξ(s)−gξ(t)k < ε, kev ◦ϕ(g)−ev ◦ϕ(g)k < ε i i s t holdforallg∈G whenever|s−t| < ε,andconsider anopencovering U := U (t) |t ∈ [0,1]&δ < min{δ(t),δ } δ 0 (cid:8) (cid:9) of [0,1], where U (t) denotes the open ball of radius δ and center t. Since [0,1] δ iscompact, thereexists afinitesubcovering, say{I ,...,I }. Wedenote thecenter 1 r of I by c , and without loss of generality, we may assume c < ··· < c and j j 1 r Ij∩Ij+1 , ∅forall j. Takeη > 0andbj ∈ Ij∩Ij+1∩(cj+η,cj+1−η)foreach j, andfindaunitaryu ∈ A suchthat m • u(b )isequaltov forall j; j bj • theimageofuon[cj+η,cj+1 −η]isincluded intheε-ballofcenteru(bj);and • theimageofuon[c −η,c +η]isincluded inthepath-connected subset j j w ev ◦ϕ(f) = Ad(w) diag(f ◦ξ (c ),..., f ◦ξ (c )) cj 1 j k j n (cid:12)(cid:12) (cid:0) (cid:1)o ofunitaries, (cid:12) (cid:12) whichispossible bytheclaimweprovedinthepreviousparagraph. Weshallset ψ(f) := Ad(u) diag(f ◦ξ ,..., f ◦ξ ) 1 k andshowthatthisψhasthedesiredp(cid:0)roperty. First,itisclear(cid:1)fromthedefinitionof ξithatψistrace-preserving. Now,supposet ∈ [cj+η,cj+1−η]andg ∈G. Without lossofgenerality, wemayassumethatthenormofgislessthan1. Thenwehave ev ◦ψ(g) = Ad(u(t)) diag(g◦ξ (t),...,g◦ξ (t)) t 1 k ∼ Ad(u(b(cid:0) )) diag(g◦ξ (b ),...,g◦ξ(cid:1)(b )) = ev ◦ϕ(g) 3ε j 1 j k j bj ∼ ev ◦ϕ(g).(cid:0) (cid:1) ε t 8 S.MASUMOTO Ontheotherhand,ift ∈ [c −η,c +η],then j j ev ◦ψ(g) = Ad(u(t)) diag(g◦ξ (t),...,g◦ξ (t)) t 1 k ∼ Ad(u(t)(cid:0)) diag(g◦ξ (c ),...,g◦ξ (c(cid:1) )) = ev ◦ϕ(g) ε 1 j k j cj ∼ ev ◦ϕ(g(cid:0)). (cid:1) ε t Consequently, itfollowsthatkϕ(g)−ψ(g)k < 4εforallg ∈G,whichcompletesthe proof. (cid:3) Thefollowing proposition isalso animmediate corollary of Lemma3.1. How- ever, as a preparation to the next section, we shall present a slightly ineffective proof. Proposition 3.6. Let τ,σ be faithful diffuse measures on [0,1]. Then for any n ∈ N(ν) and any ε > 0, there exist m ∈ N(ν) and a diagonalizable morphism ϕ: hA ,τi → hA ,σi such that the images of the maps associated to a diagonal n m expression ofϕhavediameterslessthanε. Proof. Sinceτisdiffuse,thereexistsδ> 0suchthatτ([t ,t ]) < δimplies|t −t | < 1 2 2 1 1/6. Takek ∈ Nso that m := nk is inN(ν) and 1/k is smaller than δ. Then there 1 are t ∈ (1/2,2/3) and r ∈ N with τ([0,t ]) = r/k. We set τ := kτ| and 0 0 1 r [0,t0] τ := k τ , soτ = rτ + k−rτ . ByLemma3.1, one canfind increasing maps 2 k−r [t0,1] k 1 k 2 η : [0,1] → [0,t ] and η : [0,1] → [t ,1] such that τ is equal to (η) (σ) for 1 0 2 0 i i ∗ i= 1,2. Weset η if j= 1,...,r, ξ1 := 1 j ( η if j= r+1,...,k, 2 anddefineϕ : A → A by 1 n m1 ϕ (f) = diag(f ◦ξ1,..., f ◦ξ1). 1 1 k Thenitcanbe easily verified thatϕ isamorphism from hA ,τitohA ,σi,and 1 n m1 thattheimagesofthemapsξ1,...,ξ1 areeither[0,t ]or[t ,1],sotheirdiameters 1 k 0 0 arelessthan2/3. Nowtaked ∈ Nlargeenoughsothat(2/3)d islessthanε,andrepeattheproce- dureaboveford timestoobtainasequence hA ,τi−−ϕ−→1 hA ,σi −−ϕ−→2 ···−ϕ−d−−→1 hA ,σi. n m1 md Thenϕ := ϕ ◦···◦ϕ hasthedesired property. (cid:3) d−1 1 Proposition 3.7. TheclassK (ν)satisfiesNAP. Proof. Let ϕ1,ϕ2 be morphisms from hAn0,τ0i into hAm′,σ′i, hAm′′,σ′′i respectively,andGbeafinitesubsetofA . Ourgoalistoshowthatgivenε > 0,wecan n0 find morphisms ψ1 and ψ2 from hAm′,σ′i and hAm′′,σ′′i respectively into some hA ,τ i ∈ K (ν)withkψ ◦ϕ (g)−ψ ◦ϕ (g)k < εforallg ∈G. Toseethis,we n2 2 1 1 2 2 may assume that m′ = m′′ =: n and σ′ = σ′′ =: τ , by Proposition 3.3; and that 1 1 bothϕ andϕ arediagonalizable, byProposition 3.5. 1 2 Letζi,...,ζi bethemapsassociated toadiagonal expression ofϕ. Takeδ > 0 1 l i sothat|s−t| < δimplies|g(s)−g(t)| < εforanyg∈G,andapplyProposition3.6to JIANG-SUALGEBRAASAFRAÏSSE´ LIMIT 9 obtainamorphismρfromhA ,τ iintosomehA ,τ isuchthattheimagesofthe n1 1 n2 2 mapsassociated toadiagonalexpressionofρ◦ϕ˜ havediameterslessthanδ/3for i eachi. ThenapplyingProposition3.3,findadiagonalizablemorphismΦ suchthat i theinequaily kρ◦ϕ(g)−Φ(g)k < εholds for g ∈ G, andthat themaps ξi,...,ξi i i 1 k associated to a diagonal expression of Φ satisfies ξi ≤ ··· ≤ ξi. Recalling the i 1 k proof of Proposition 3.5, one can easily check that the diameters of the images of ξi isstilllessthanδ/3. j Weclaimthattheinequalitykξ1−ξ2k< δholdsforall j. Supposeonthecontrary j j thatξ1(t) ≥ ξ2(t)+δatsomepointt ∈ [0,1],andsetc := maxξ2,d := minξ1 . (If j j j j+1 jisequaltok,thensetd := 1instead.) Thenitfollowsthat • theimageofξ2 isincludedin[0,c]if1 ≤ l ≤ j;and l • iftheimageofξ1 intersects with[0,d),thenlislessthanorequalto j. l Sincedislargerthancbyatleastδ/3,andsinceΦ istrace-preserving, wehave i j j j= τ (ξ2)−1[0,c] ≤ n τ [0,c] < n τ [0,d) ≤ τ (ξ1)−1[0,1] = j, 2 l 2 0 2 0 1 l Xl=1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Xl=1 (cid:0) (cid:1) whichisacontradiction. Therefore, kξ1−ξ2kmustbesmallerthanδ,asdesired. j j Now,letu beaunitarysuchthattheequality i Φ(f) = Ad(u) diag(f ◦ξi,..., f ◦ξi) i i 1 k holdsforall f ∈ A ,andputψ :=(cid:0)ρandψ := Ad(u u∗)◦(cid:1)ρ. Thenforg ∈G,we n0 1 2 1 2 have ψ ◦ϕ (g) = Ad(u u∗)◦ρ◦ϕ (g) 2 2 1 2 2 ∼ Ad(u u∗)◦Φ (g) ε 1 2 2 = Ad(u ) diag(g◦ξ2,...,g◦ξ2) 1 1 k ∼ Ad(u (cid:0)) diag(g◦ξ1,...,g◦ξ1(cid:1)) ε 1 1 k = Φ (g) (cid:0) (cid:1) 1 ∼ ψ ◦ϕ (g), ε 1 1 whichcompletestheproof. (cid:3) Theorem3.8. TheclassK (ν)isaFra¨ısse´ class. Proof. WehavealreadyshownthatK (ν)satisfiesJEPandNAPinPropositions3.3 and3.7. Also,itcanbeeasilyverifiedformtheproofofProposition3.2thatK (ν) satisfies WPP.SinceK (ν)automatically satisfies CCP,asisnotedinRemark2.5, itfollowsthatK (ν)isaFra¨ısse´ class. (cid:3) We close this section by showing that the Fra¨ısse´ limit of K (ν) is the unique UHF algebra M corresponding to the supernatural number ν. The following ν lemmawillbeneededfortheproof. Lemma3.9. LethA ,τibeamemberofK (ν). Thenforanyfinitesubset F ⊆ A n n andanyε > 0,thereexistamorphismfromhA ,τiintosomehA ,σi ∈ K (ν)and n m 10 S.MASUMOTO afinite dimensional C*-subalgebra B ⊆ A such that the image ϕ[F]is included m intheε-neighborhood ofB. Proof. Wemay assume F = {id ⊗1 }∪{1 ⊗e | i, j = 1,...,n} where [0,1] Mn C[0,1] i,j {e } is the standard matrix unit of M , because this set generates A . Also, we i,j n n mayassumethatτisdiffusebyProposition3.2. Now,letϕbeasinProposition3.6. Then ϕ(id ⊗1 ) = diag(ξ ,...,ξ ) [0,1] Mn 1 k ∼ 1 ⊗diag(ξ (0),...,ξ (0)), ε C[0,1] 1 k soϕ[F]isincluded intheε-neighborhood ofthefinitedimensional C*-subalgebra 1 ⊗M ,asdesired. (cid:3) C[0,1] m Theorem 3.10. TheFra¨ısse´ limitofK (ν)ishM ,tri,where tristhe unique trace ν onM . ν Proof. LethA,θibetheFra¨ısse´ limitofK . ByK -universality andTheorem2.7, it is clear that A and M have the same K-theory. Therefore, it suffices to show ν thatAisanAFalgebra. Forthis,letF beasubsetofA. Thengivenε > 0,wecan findamorphismϕofsomehA ,τi ∈ K (ν)intohA,θiandafinitesubset F′ ⊆ A n n such that F is included in the ε-neighborhood of ϕ[F′]. On the other hand, by Lemma3.9, there is a morphism ψ from hA ,τi into some hA ,σi ∈ K (ν) such n m thatψ[F′]isincludedintheε-neighborhood ofafinitedimensionalC*-subalgebra ofA . SincehA,θiisK -universal andapproximately K -homogeneous, thereis m a morphism ι: hA ,σi → hA,θi such that d(ϕ(f),ι ◦ψ(f)) is less than ε for all m f ∈ F′. ItfollowsthatF isincludedinthe3ε-neighborhoodofafinitedimensional C*-subalgebra of A, so by [Dav96, Theorem III.3.4], A is an AF algebra, which completestheproof. (cid:3) 4. TheJiang-SuAlgebra Let p,q be natural numbers. We shall begin with the well-known observation that if {eij}i,j=1 and {fkl}k,l=1 are the standard matrix units of Mp and Mq respectively, then {e ⊗ f } isa matrix unit ofM ⊗M , soM ⊗M is canonically ij kl i,j,k,l p q p q identifiedwithM . Now,thedimensiondropalgebra Z isdefinedby pq p,q Z := {f ∈ A | f(0) ∈ M ⊗1 & f(1) ∈ 1 ⊗M }, p,q pq p Mq Mp q where we took over the notation A = C([0,1],M ) from Section 3. It is said to n n beprimeif pandqareco-prime. Wedenote byK theclassofallpairshZ ,τi, p,q whereZ isaprimedimensiondropalgebra andτisafaithfultraceonit. p,q In[JS99],JiangandSuconstructedtheJiang-Sualgebraasaninductivelimitof primedimensiondropalgebras,andprovedthatitistheuniquemonotracialsimple C*-algebra among such inductive limits. Our goal here is to show that the Jiang- Su algebra together with its unique trace is the Fra¨ısse´ limit of the class K . The directionoftheproofisthesameasthatofSection3,butweneedsomeadditional observations becauseofthepinching condition.

Jiang-Su Algebra as a Fra\"iss\'e Limit

Similar Jiang-Su Algebra as a Fra\"iss\'e Limit