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

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by  Shuhei Masumoto

JIANG-SU ALGEBRA AS A FRAI¨SSE´ LIMIT
SHUHEIMASUMOTO
Abstract. Inthispaper,wegiveaself-containedandquiteelementaryproofthat
theclassofalldimensiondropalgebrastogetherwiththeirdistinguishedfaithful
6
tracesformsaFra¨ısse´classwiththeJiang-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-SUALGEBRAASAFRA¨ISSE´ 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 com-
plete.) 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-SUALGEBRAASAFRA¨ISSE´ 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-SUALGEBRAASAFRA¨ISSE´ 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 respec-
tively,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-SUALGEBRAASAFRA¨ISSE´ 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 respec-
tively, 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.