C -ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS ∗ GRAPHS 5 0 0 2 MARIUSIONESCUANDANDYASUOWATATANI n a Abstract. A Mauldin-Williams graph M is a generalization of an iterated J functionsystem byadirectedgraph. Itsinvariantset K plays theroleofthe 9 self-similarset. We associate a C∗-algebraOM(K)withaMauldin-Williams 1 graph M and the invariant set K laying emphasis on the singular points. We assume that the underlying graph G has no sinks and no sources. If M ] satisfies the open set condition in K and G is irreducible and is not a cyclic A permutation, then the associated C∗-algebra OM(K) is simple and purely O infinite. Wecalculate theK-groupsforsomeexamples includingthe inflation ruleofthePenrosetilings. . h t a m 1. Introduction [ Self-similar sets are often constructed as the invariant set of iterated function 2 v systems[2],[13],[19]. Manyotherexamples,suchastheinflationruleofthePenrose 0 tilings,showthatagraphdirectedgeneralizationofiteratedfunctionsystemsisalso 8 interestingandhasbeendevelopedasMauldin-Williamsgraphs[3],[11],[23]. Since 4 we can regard graph directed iterated function systems as dynamical systems, we 0 expect that there exist fruitful connections between Mauldin-Williams graphs and 1 4 C∗-algebras. 0 In [14] the first-named author defined a C -correspondence for a Mauldin- ∗ / Williams graph and showedthat the Cuntz-Pimsner algebra X (see [12], [29]) h M OX t isisomorphictothe Cuntz-Kriegeralgebra G ([8])associatedwiththe underlying a O graphG. Inparticular,iftheMauldin-WilliamsgraphhasonevertexandN edges, m the Cuntz-Pimsner algebra is isomorphic to the Cuntz algebra ([7]), which N : OX O v recoversaresultin[30]. The constructionis usefulbecause itgivesmanyexamples i ofdifferentnon-self-adjointalgebraswhicharenotcompletelyisometricallyisomor- X phic, but have the same C -envelope ([25]) as shown by the first-named author in ∗ r a [15]. On the other hand, Kajiwara and the second-named author introduced C - ∗ algebras associated with rational functions in [17] including the singular points (i.e. branched points). They developed the idea to associate C -algebras with ∗ self-similar sets considering the singular points in [16]. In this paper we associate another C -algebra (K) with a Mauldin-Williams graph and its invariant ∗ OM M set K putting emphasis on the singular points as above. We show that the associ- atedC -algebra (K)isnotisomorphictotheCuntz-Kriegeralgebra forthe ∗ G OM O underlying graph G in general. This comes from the fact that the singular points cause the failure of the injectivity of the coding by the Markov shift for G. We assume that the underlying graph G has no sinks and no sources. We show that the associated C -algebra (K) is simple and purely infinite if satisfies the ∗ OM M opensetconditioninK andifGis irreducibleandisnotacyclicpermutation. We 1 C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 2 calculatetheK-groupsforsomeexamplesincludingtheinflationruleofthePenrose tilings. The C -algebras associated with tilings were first studied by Connes in [6] ∗ and also discussed by Mingo [24] and Anderson-Putnam [1]. But we do not know the exact relatition between their constructions and ours. Ourconstructionhasacommonflavorwithseveraltopologicalgeneralizationsof graphC -algebras[21]byBrenken[5],Katsura[18],MuhlyandSolel[26]andMuhly ∗ and Tomforde [27]. Since graph directed iterated function systems are sometimes obtained as continuous cross sections of expanding maps, C -algebras associated ∗ withintervalmapsintroducedbyDeaconuandShultzin[9]arecloselyrelatedwith our construction. In a different point of view, Bratteli and Jorgensen studied a relationbetweeniterated function systems andrepresentationofCuntz algebrasin [4]. TheauthorswouldliketothankP.Muhly forsuggestingtheircollaborationand giving them the chance to discuss this work at a conference at the University of Iowa. 2. Mauldin-Williams graphs and the associated C -correspondence ∗ ByaMauldin-Williams graph wemeanasystem =(G, T ,ρ , φ ), v v v E0 e e E1 where G = (E0,E1,r,s) is a graph with a finite sMet of vert{ices E0}, a∈ fini{te s}et∈of edgesE1,arange maprandasource maps,andwhere T ,ρ and φ v v v E0 e e E1 { } ∈ { } ∈ are families such that: (1) Each T is a compact metric space with a prescribed metric ρ , v E0. v v ∈ (2) For e E, φ is a continuous map from T to T such that e r(e) s(e) ∈ cρ (x,y) ρ (φ (x),φ (y)) cρ (x,y) ′ r(e) s(e) e e r(e) ≤ ≤ for some constants c,c satisfying 0<c <c<1 (independent of e) and all ′ ′ x,y T . r(e) ∈ We shall assume, too, that the source map s and the range map r are surjective. Thus, we assume that there are no sinks and no sources in the graph G. In the particular case when we have only one vertex and N edges, we obtain a so called iterated function system (K, φ ). i i=1,...,N { } Aninvariantlist associatedwithaMauldin-Williamsgraph =(G, T ,ρ , φ ) v v v E0 e e E1 is a family (K ) of compact sets such that K T for Mall v E0{, and }su∈ch { } ∈ v v E0 v v that ∈ ⊂ ∈ K = φ (K ). v e r(e) e[∈E1 s(e)=v Since eachφ is a contraction, hasa unique invariantlist(see [23, Theorem1]). e M We set T := T and K := K and we call K the invariant set of the v E0 v v E0 v Mauldin-William∈s graph. ∈ S S Intheparticularcasewhenwehaveonevertexvandnedges,i.e. inthesettingof aniteratedfunction system,theinvariantsetistheuniquecompactsubsetK :=K v of T =T such that v K =φ (K) φ (K). 1 n ∪···∪ In this paper we forget about the ambient space T. That is, we consider that the Mauldin-Williamsgraphis =(G, K ,ρ , φ ),withK = K being the invariant set. M { v v}v∈E0 { e}e∈E1 v∈E0 v S C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 3 We say that a graph G=(E0,E1,r,s) is irreducible (or totally connected) if for every v ,v E0 there is a finite path w such that s(w) = v and r(w) = v . We 1 2 1 2 ∈ will assume in this paper that the graph G=(E0,E1,r,s) is irreducible and not a cyclicpermutation. Thatis thereexists avertexv E0 andthere existtwoedges ∗ ∈ e =e such that s(e )=s(e )=v . 1 2 1 2 ∗ 6 Foranaturalnumberm,wedefineEm := w =(w ,...,w ):w E1 and r(w )= 1 m i i { ∈ s(w ) for all i = 1,...,m 1 . An element w Em is called a path of length i+1 − } ∈ m. We extend s and r to Em by s(w)=s(w ) and r(w) =r(w ) for all w Em. 1 m ∈ We set E = Em and denote the length of a path w by l(w). We define also ∗ m 1 Em(v):= w E≥m : s(w)=v to be the set of paths of length m starting at the { S∈ } vertex v and E (v)= Em(v) to be the set of finite paths starting at v. ∗ m 1 Theinfinitepathspace≥isE = w=(w ) : r(w )=s(w ) for all n 1 ∞ n n 1 n n+1 S { ≥ ≥ } and the space of infinite paths starting at a vertex v is E (v) = w E : ∞ ∞ { ∈ s(w)=s(w )=v . OnE (v)wedefinethe metricδ (α,β)=cl(α β) ifα=β and 1 ∞ v ∧ } 6 0 otherwise, where α β is the longest common prefix of α and β (see [10, Page ∧ 116]). Then E (v) is a compact metric space, and, since E equals the disjoint ∞ ∞ union of the spaces E (v), E is a compact metric space in a natural way. For ∞ ∞ w Em, let φ = φ φ and K = φ (K ). Then for any infinite path ∈ w w1 ◦··· wm w w r(w) mαa=p(παn:)En∈N, Kn≥b1yK(απ1(,α...),αn=) containKs only one. pSoinincte.πT(Ehere)foisrealwsoe acannindveafirniaenat ∞ →T { } n 1 (α1,...,αn) ∞ set, we have that π(E ) = K. Th≥us π is a continuous, onto map. Moreover, for ∞ anyy K andanyneighborhoTodU K ofy,thereexistn Nandw En(v ) ∈ v0 ⊂ v0 ∈ ∈ 0 such that y φ (K ) U. w r(w) ∈ ⊂ WesaythataMauldin-Williamsgraph satisfiestheopen setcondition in K if M thereexists afamily ofnon-emptysets(V ) , suchthatV K forallv E0, v v E0 v v and such that ∈ ⊂ ∈ φ (V ) V for all v E0, e r(e) v ⊂ ∈ e[∈E1 s(e)=v and φ (V ) φ (V )= if e=f. e r(e) f r(f) ∅ 6 Then V := v E0Vv is an open d\ense subset of K. Moreover, for n ∈ N and w,v En, if w∈=v and r(w)=r(v) then φ (V ) φ (V )= . w r(w) v r(v) ∈ S 6 ∅ For e E1, we define the cograph of φ to be the set e ∈ T cograph(φ )= (x,y) K K : x=φ (y) K K. e s(e) r(e) e { ∈ × }⊂ × We shall consider the union = ( φ : e E1 ):= cograph(φ ). e e G G { ∈ } e E1 ∈[ ConsidertheC -algebraA=C(K)andletX =C( ). ThenXisaC -correspondence ∗ ∗ G over A with the structure defined by the formulae: (a ξ b)(x,y) = a(x)ξ(x,y)b(y), · · ξ,η (y) = ξ(φ (y),y)η(φ (y),y), A e e h i eX∈E1 y∈Kr(e) for all a,b A, ξ,η X, (x,y) and y K. It is clear that the A-valued inner ∈ ∈ ∈G ∈ product is well defined. The left multiplication is given by the -homomorphism ∗ C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 4 Φ : A (X) such that (Φ(a)ξ)(x,y) = a(x)ξ(x,y) for a A and ξ X. Put → L ∈ ∈ ξ = ξ,ξ 1/2. 2 A k kFor aknhy niatku∞ral number n, we define = ( φ : w En ) and a C - n w ∗ G G { ∈ } correspondence X =C( ) similarly. We also define a path space of length n n n n G P by = (φ (y),φ (y),...,φ (y),y) Kn+1 : Pn { w1,···,wn w2,···,wn wn ∈ w =(w ,...,w ) En,y K . 1 n r(w) ∈ ∈ } Then Y :=C( ) is a C -correspondence over A with an A-valued inner product n n ∗ P defined by ξ,η (y)= ξ(φ (y),...,φ (y),y)η(φ (y),...,φ (y),y), h iA w1,...,wn wn w1,...,wn wn w∈En y∈XKr(w) for all ξ,η Y and y K. n ∈ ∈ Proposition 2.1. Let = (G, K ,ρ , φ ) be a Mauldin-Williams v v v E0 e e E1 graph. Let K be the invMariant set{. Then}X∈ ={C(})∈is a full C -correspondence ∗ G over A = C(K) without completion. The left action Φ : A (X) is unital and → L faithful. Similar statements hold for Y =C( ). n n P Proof. For any ξ X we have ∈ 1/2 ξ ξ 2 =sup ξ(φe(y),y)2 √N ξ , k k∞ ≤k k y K | | ≤ k k∞  ∈ y∈eX∈KEr(1e)    where N is the number of edges in E1. Therefore the norms and are 2 k k k k∞ equivalent. Since C( ) is complete with respect to , it is also complete with G k k∞ respect to . 2 kk Let ξ X be defined by the formula ∈ 1 ξ(x,y)= for all (x,y) . #(e:y K ) ∈G r(e) ∈ Then ξ,ξ (y)=1,hence pX,X containsthe identity ofA. ThereforeX is full. A A h i h i If a A is not zero, then there exists x K such that a(x )=0. Since K is the 0 0 ∈ ∈ 6 invariant set of the Mauldin-Williams graph, there exists e E1 and y K 0 r(e) ∈ ∈ suchthatx K andφ (y )=x . Chooseξ X suchthatξ(x ,y )=0. Then 0 s(e) e 0 0 0 0 Φ(a)ξ =0, he∈nce Φ is faithful. The statements f∈or Y are similarly prov6ed. (cid:3) n 6 Definition 2.2. Let = (G, K ,ρ , φ ) be a Mauldin-Williams v v v E0 e e E1 graph with the invariaMnt set K.{ We a}ss∈ociat{e a}C∈ -algebra (K) to as ∗ OM M the Cuntz-Pimsner algebra of the C -correspondence X = C( ) over the C - X ∗ ∗ O G algebra A=C(K). As in [16], we denote by alg the -algebrageneratedalgebraicallyby A andS with ξ X. The gauge actiOonXis γ :∗R Aut defined by γ (S )=eitS for alξl X t ξ ξ ∈ → O ξ X, and γ (a)=a for all a A. t ∈ ∈ C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 5 Proposition 2.3. Let = (G, K ,ρ , φ ) be a Mauldin-Williams v v v E0 e e E1 graph. Assume that K iMs the inva{riant se}t o∈f the{gr}ap∈h. Then there is an isomor- phism ϕ :X n C( ) as C -correspondences over A such that n ⊗ n ∗ → P (ϕ (ξ ξ ))(φ (y),...,φ (y),y) n 1⊗···⊗ n w1,...,wn wn =ξ (φ (y),φ (y))ξ (φ (y),φ (y))...ξ (φ (y),y) 1 w1,...,wn w2,...,wn 2 w2,...,wn w3,...,wn n wn for all ξ ,...,ξ X, y K, and w = (w ,...,w ) En such that y K . 1 n ∈ ∈ 1 n ∈ ∈ r(wn) Moreover, let ρ : be an onto continuous map such that n n n P →G ρ (φ (y),...,φ (y),y)=(φ (y),y). n w1,...,wn wn w1,...,wn Then ρ : C( ) f f ρ C( ) is an embedding as a Hilbert submodule ∗n Gn ∋ → ◦ n ∈ Pn preserving inner product. Proof. Itiseasyto seethatϕ is well-definedandabimodule morphism. Weshow n that ϕ preserves inner product. Consider the case when n = 2 for simplicity of n the notation. Let ξ ,ξ ,η ,η X. We have 1 2 1 2 ∈ ξ ξ ,η η (y) = ξ , ξ ,η η (y) 1 2 1 2 A 2 1 1 A 2 A h ⊗ ⊗ i h h i · i = ξ (φ (y),y) ξ ,η (φ (y))η (φ (y),y) 2 e 1 1 A e 2 e h i e∈E y∈XKr(e) = ξ (φ (y),φ (y))ξ (φ (y),y)η (φ (y),φ (y))η (φ (y),y) 1 fe e 2 e 1 fe e 2 e yf∈XeK∈rE(e2) = ϕ (ξ ξ ),ϕ (η η ) (y). 2 1 2 2 1 2 A h ⊗ ⊗ i Since ϕ preservesthe inner product, it is one to one. Using the Stone-Weierstrass n Theorem, one can show that ϕ is also onto. The rest is clear. (cid:3) n We let i : C( ) C( ) be the natural inner-product preserving embed- n,m n m P → P ding, for m n. ≥ Definition 2.4. Consider a covering map π : K defined by π(x,y) = y for G → (x,y) . Define the set ∈G B( ):= x K : x=φ (y)=φ (y) for some y K and e=f . e f M { ∈ ∈ 6 } The set B( ) will be described by the ideal I :=Φ 1( (X)) of A. We define a X − M K branch index e(x,y) at (x,y) by ∈G e(x,y):=# e E1 : φ (y)=x . e { ∈ } Hence x B( ) if and only if there exists some y K with e(x,y) 2. For ∈ M ∈ ≥ x K we define ∈ I(x):= e E1 : there exists y K such that x=φ (y) . e { ∈ ∈ } Lemma 2.5. In the above situation, if x K B( ), then there exists an open ∈ \ M neighborhood U of x satisfying the following: x (1) U B( )= . x M ∅ (2) If e I(x), then φ (φ 1(U )) U = for e=f, such that r(e)=r(f). T∈ f −e x x ∅ 6 (3) If e / I(x), then U φ (K )= . x e r(e) ∈ T ∅ T C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 6 Proof. Let x K B( ). Let v E0 such that x K . Since B( ) and ∈ \ M 0 ∈ ∈ v0 M φ (K ) are closed and x is not in either of them, there exists an open e/I(x) e r(e) nei∈ghbourhood W K of x such that S x ⊂ v0 W (B( ) φ (K ))= . x e r(e) M ∪ ∅ \ e∈/[I(x) For each e I(x) there exists a unique y K with x = φ (y ), since x / B( ). e e e ∈ ∈ ∈ M For f E1, if r(e) = r(f) and f = e then φ (y ) = φ (y ) = x. Therefore there f e e e ∈ 6 6 exists anopen neighborhoodVe ofx such that φ (φ 1(Ve)) Ve = if f =e and x f −e x x ∅ 6 r(f)=r(e). Let U :=W ( Ve). Then U is an open neighborhood of x and satisfies all thexrequirxementse∈.I(x) x x T (cid:3) T T Proposition 2.6. Let = (G, K ,ρ , φ ) be a Mauldin-Williams v v v E0 e e E1 graph. Assume that theMsystem {satisfie}s∈the o{pen}s∈et condition in K. Then M I = a A=C(K) : a vanishes on B( ) . X { ∈ M } Proof. The proof requiresonly minor modifications fromthe proof of [16, Proposi- tion 2.4]. (cid:3) Corollary 2.7. #B( )=dim(A/I ). X M Corollary 2.8. The closed set B( ) is empty if and only if Φ(A) is contained in M (X) if and only if X is finitely generated projective right A module. K 3. Simplicity and Pure Infinitness Let = (G, K ,ρ , φ ) be a Mauldin-Williams graph. Let A = v v v E0 e e E1 C(K) aMnd X =C{( ). Fo}r∈e E{1 d}e∈fine an endomorphism β :A A by e G ∈ → a(φ (y)) if y K (β (a))(y):= e ∈ r(e) e (0 otherwise, for all a A and y K. We also define a unital completely positive map E : ∈ ∈ M A A by → 1 (E (a))(y): = a(φ (y)) M # e E1 :y Kr(e) e { ∈ ∈ } eX∈E1 y∈Kr(e) 1 = β (a)(y), # e E1 :y K e r(e) { ∈ ∈ } eX∈E1 y∈Kr(e) for a A, y K. For the function ξ X defined by the formula 0 ∈ ∈ ∈ 1 ξ (x,y)= 0 # e E1 :y K r(e) { ∈ ∈ } we have p E (a)= ξ ,Φ(a)ξ andE (I)= ξ ,ξ =I. 0 0 A 0 0 A M h i M h i Let D :=S (K). ξ0 ∈OM Lemma 3.1. In the above situation, for a A, we have that D aD =E (a) and ∗ in particular D D =I. ∈ M ∗ Proof. The same with [16, Lemma 3.1]. (cid:3) C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 7 Definition 3.2. Let = (G, K ,ρ , φ ) be a Mauldin-Williams v v v E0 e e E1 graph. We say that anMelement a{ A=C}(∈K) i{s ( } ,∈En)-invariant if ∈ M a(φ (y))=a(φ (y))for anyy K andα,β En such thaty K =K . α β r(α) r(β) ∈ ∈ ∈ If a A is ( ,En)-invariant then a is also ( ,En 1)-invariant. Then, similar − ∈ M M like in [16, Definition on page 11], if a is ( ,En)-invariant, we can define M βk(a)(y):=a(φ ...φ (y)), for anyw Ek, such thaty K . w1 wn ∈ ∈ r(wn) Then, for any ξ ,...,ξ X and a A ( ,En)-invariant , we have the relation: 1 n ∈ ∈ M aS ...S =S ...S βn(a). ξ1 ξn ξ1 ξn Lemma 3.3. Let =(G, K ,ρ , φ ) be a Mauldin-Williams graph v v v E0 e e E1 such that (E0,E1,Mr,s) is an{irreduc}ib∈le gr{aph}. ∈For any non-zero positive element a A and for every ε > 0 there exists n N and ξ X n with ξ,ξ = 1 such ⊗ A ∈ ∈ ∈ h i that a ε S aS a . k k− ≤ ξ∗ ξ ≤k k Proof. Letx K besuchthat a =a(x ). Letv E0suchthatx K . Then 0 ∈ k k 0 0 ∈ 0 ∈ v0 thereexistsanopenneighborhoodU ofx inK suchthatforanyx U wehave 0 0 v0 ∈ 0 a ε a(x) a . Let U anopen neighborhoodof x in K and K compact k k− ≤ ≤k k 1 0 v0 1 suchthatU K U . Sincethemapπ :E K isontoandcontinuous,there 1 1 0 ∞ exists some n⊂1 N⊂and α En1(v0) such that→φα(Kr(α)) U1. For any vertex ∈ ∈ ⊂ v V, since the graph G is irreducible, there exists a path w E from r(α) to v ∗ ∈ ∈ v. Then φ (K ) U . Hence, for each v V, there is α En(v)(v ), for some αwv v ⊂ 1 ∈ v ∈ 0 n(v) n , suchthat φ (K ) U . For each v V, define the closed subsets F ≥ 1 αv v ⊂ 1 ∈ 1,v and F of K K by 2,v × F = (x,y) K K : x=φ (y),x K ,y K ,α En(v)(v ) 1,v α 1 v 0 { ∈ × ∈ ∈ ∈ } F = (x,y) K K : x=φ (y),x Uc,y K ,α En(v)(v ) . 2,v { ∈ × α ∈ 0 ∈ v ∈ 0 } Since F F = for all w V and F F = if v = w, there exists 1,v 2,w 1,v 1,w ∩ ∅ ∈ ∩ ∅ 6 g C( ) such that 0 g (x,y) 1 and v n(v) v ∈ G ≤ ≤ 1 if (x,y) F 1,v g (x,y)= ∈ v (0 if (x,y) w E0F2,w w=vF1,w. ∈∪ ∈ ∪ 6 Since φαv(Kv)⊂U1 for eachy ∈Kv, there exists xy ∈SU1 suchthatxy =φαv(y)∈ U K . Therefore 1 1 ⊂ (3.1) g ,g (y)= g (φ (y),y)2 g (x ,y)2 =1 v v A v α v y h i | | ≥| | α∈XEn(v) y∈Kr(α) for all y K . Let n =max n(v) : v E0 . We identify X n with C( ) as in v ⊗ n ∈ { ∈ } P Proposition 2.3. We denote i (ρ (g )) C( ) also by g for each v V. n(v),n ∗n(v) v ∈ Pn v ∈ Let g := g C( ). Then g,g (y) 1 for all y K. Thus b := g,g v V v ∈ Pn h iA ≥ ∈ h iA is positive an∈d invertible. Let ξ :=gb 1/2 X n. Then P − ∈ ⊗ ξ,ξ = gb 1/2,gb 1/2 =b 1/2 g,g b 1/2 =1 . A − − A − A − A h i h i h i For anyy K and any α En suchthat y K , let x=φ (y). If x U , then r(α) α 0 ∈ ∈ ∈ ∈ a ε a(x), and, if x Uc, then k k− ≤ ∈ 0 ξ(φα1,...,αn(y),...,φαn(y),y)=g(x,y)b−1/2(y)=0. C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 8 Therefore a ε = ( a ε) ξ,ξ (y) A k k− k k− h i = ( a ε) ξ(φ (y),...,φ (y),y)2 k k− | α1,...,αn αn | α∈En y∈XKr(α) a(φ (y))ξ(φ (y),...,φ (y),y)2 ≤ α | α1,...,αn αn | α∈En y∈XKr(α) = hξ,aξiA(y)=Sξ∗aSξ(y). We also have that Sξ∗aSξ =hξ,aξiA ≤kakhξ,ξiA =kak. (cid:3) Lemma 3.4. Let =(G, K ,ρ , φ ) be a Mauldin-Williams graph. v v v E0 e e E1 Assume that K is tMhe invari{ant set }of∈the g{rap}h.∈For any non-zero positive element a A and for any ε > 0 with 0 < ε < a , there exists n N and u X n such ⊗ ∈ k k ∈ ∈ that kuk2 ≤(kak−ε)−1/2 andSu∗aSu =1. Proof. The proof is identical with the proof of [16, Lemma 3.4]. (cid:3) Lemma 3.5. Let =(G, K ,ρ , φ ) be a Mauldin-Williams graph. v v v E0 e e E1 Suppose that the grMaph G ha{s no sin}ks∈and{no}so∈urces and it is irreducible and not a cyclic permutation. Assume that K is the invariant set of the graph and that satisfies the open set condition in K. For any n N, for any T (X n), and fMor ⊗ ∈ ∈L any ε>0 there exists a positive element a A such that a is ( ,En)-invariant, ∈ M Φ(a)T 2 T 2 ε, k k ≥k k − and βp(a)βq(a)=0 for p,q =1,...,n with p=q. 6 Proof. Let n N, let T (X n), and let ε>0. Then there exists ξ X n such ⊗ ⊗ ∈ ∈L ∈ that ξ =1 and T 2 Tξ 2> T 2 ε. Hence thereexistsy K forsome k k2 k k ≥k k2 k k − 0 ∈ v0 v V such that 0 ∈ Tξ 2= (Tξ)(φ (y ),...,φ (y ),y )2 > T 2 ε. k k2 | α1,...,αn 0 αn 0 0 | k k − α∈En r(Xα)=v0 Then there exists an open neighborhood U of y in K such that for any y U 0 0 v0 ∈ 0 (Tξ)(φ (y),...,φ (y),y)2 > T 2 ε. | α1,...,αn αn | k k − α∈En r(Xα)=v0 Since satisfies the open set condition in K, there exists a family of non-empty M sets (V ) , such that V K , for all v E0, v v E0 v v ∈ ⊂ ∈ φ (V ) V for all v E0, e r(e) v ⊂ ∈ e[∈E1 s(e)=v and φ (V ) φ (V )= if e=f. e r(e) f r(f) ∅ 6 MThoerneotvheerr,etehxeirsetsisy1so∈mVevk0∩′ U0Nanadnda\n(eo1p,e.n..n,eeikg′h)borEhko′o(vd0U)1suocfhy1thwaitthU1 ⊂V∩U0. ∈ ∈ φ (V ) U V U . e1,...,ek′ r(ek′) ⊂ 1 ⊂ v0 ∩ 0 C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 9 Since the graph G is not a cyclic permutation, there is a vertex v E0 and two ∗ ∈ edges e,e E1 such that e = e . Since the graph G is irreducible, there exists ′ ′′ ′ ′′ ∈ 6 a path from r(ek′) to v∗. Hence we have a path (e1,...,ek) Ek(v0) for some k N, k k , such that r(e )=v and φ (V ) U ∈V U . Then we ∈ ≥ ′ k ∗ e1,...,ek r(ek) ⊂ 1 ⊂ v0∩ 0 can find a path (e ,...,e ) En(v ) such that e = e if i = 1. To see k+1 k+n ∗ k+1 k+i ∈ 6 6 this, let e = e. If r(e ) = v , take e = e . If r(e ) = v , since G has k+1 ′ k+1 ∗ k+2 ′′ k+1 ∗ 6 no sinks, there is an edge e E1 such that s(e) = r(e ). Then e = e . Let k+1 k+1 ∈ 6 e = e. If r(e ) = v , take e = e ; if r(e ) = v , take e to be any k+2 k+2 ∗ k+3 ′′ k+2 ∗ k+3 6 edge such that s(e ) = r(e ). Therefore e = e . Inductively, we obtain k+3 k+2 k+3 k+1 6 the path (e ,...,e ) En(v ) with the desired property. Then k+1 k+n ∗ ∈ =φ (V ) U V U . ∅6 e1,...,ek+n r(ek+n) ⊂ 1 ⊂ v0 ∩ 0 There exist y U , an open neighborhood U of y in K and a compact set L 2 ∈ 1 2 2 v0 such that y U L φ (V ) U V U . 2 ∈ 2 ⊂ ⊂ e1,...,ek+n r(ek+n) ⊂ 1 ⊂ v0 ∩ 0 Let b A such that 0 b 1, b(y2) = 1 and bUc = 0. For α En such that ∈ ≤ ≤ | 2 ∈ r(α)=v , we have 0 φ (y ) φ (U ) φ (L) φ (V ). α 2 ∈ α 2 ⊆ α ⊆ α v0 Moreover,for α,β En such that r(α)=r(β)=v , by the open set condition, 0 ∈ φ (L) φ (L)= ifα=β. α β ∩ ∅ 6 We define a positive function a on K by the formula b(φ 1(x)) ifx φ (L), α En such thatr(α)=v a(x)= −α ∈ α ∈ 0 (0 otherwise. ThenaiscontinuousonK,hencea A. Byconstruction,ais( ,En)-invariant. ∈ M Let p n be a natural number. Let (α ,...,α ) Ep. If there is no path 1 p ≤ ∈ (α ,...,α ) En p(r(α )) such that r(α ) = v , then β ...β (a) = 0. If p+1 n ∈ − p n 0 αp α1 thereis atleastone path(α ,...,α ) En p(r(α ))suchthatr(α )=v ,then p+1 n − p n 0 ∈ supp(β ...β (a)) φ (supp(b)). αp α1 ⊆ αp+1...αn (αp+1,...,αn[)∈En−p(r(αp)) r(αn)=v0 Since supp(b) L φ (V ) we have that ⊆ ⊂ e1...ek+n r(ek+n) supp(β ...β (a)) φ φ (V ). αp α1 ⊆ αp+1...αn e1...ek+n r(ek+n) (αp+1,...,αn[)∈En−p(r(αp)) r(αn)=v0 Then, for 1 p<q n we have that ≤ ≤ supp(βp(a)) φ φ (V ) ⊆ αp+1...αn e1...ek+n r(ek+n) (αp+1,...[,αn)∈En−p r(αn)=v0 and supp(βq(a)) φ φ (V ). ⊆ αq+1...αn e1...ek+n r(ek+n) (αq+1,...[,αn)∈En−q r(αn)=v0 Since (n p)+(k+1)-th subsuffixes are different as e = e , we have k+1 k+1+(q p) that supp−(βp(a)) supp(βq(a))= . Hence βp(a)βq(a)=0.6 − ∩ ∅ C∗-ALGEBRAS ASSOCIATED WITH MAULDIN-WILLIAMS GRAPHS 10 Moreover,we have Φ(a)Tξ 2 = sup (a(φ (y))(Tξ)(φ (y),...,φ (y),y) 2 k k2 | α α αn | y∈K y∈αXK∈Er(nα) = sup (b(y))(Tξ)(φ (y),...,φ (y),y) 2 | α αn | y∈Ly∈αXK∈Er(nα) (Tξ)(φ (y ),...,φ (y ),y ) 2 ≥ | α 2 αn 2 2 | α∈En y2∈XKr(α) > T 2 ε, k k − because y U . Thus Φ(a)T 2 T 2 ε. (cid:3) 2 0 ∈ k k ≥k k − As in [16], we let be the C -subalgebra of generated by (X k), k = n ∗ X ⊗ F F K 0,1,...,n and B be the C -subalgebra of generated by n ∗ X O n {Sx1···SxkSy∗k...Sy∗1 : x1,...,xk,y1,...,yk ∈X}∪A. i=1 [ We will also use the isomorphism ϕ: B such that n n F → ϕ(θx1⊗···⊗xk,y1⊗···⊗yk)=Sx1...SxkSy∗k...Sy∗1. Lemma 3.6. In the above situation, let b=c c for some c alg. We decompose ∗ ∈OX b = b with γ (b ) = eijtb . For any ε > 0 there exists P A with 0 P I j j t j j ∈ ≤ ≤ satisfying the following: P (1) Pb P =0 (j =0). j 6 (2) Pb P b ε. 0 0 k k≥k k− Proof. The proof requires only small modifications from the proof of [16, Lemma 3.6]. (cid:3) Having proved the equivalent of the [16, Lemma 3.1-Lemma 3.6], we obtain, usingthesameproofas[16,Theorem3.7],thecorrespondingresultfortheMauldin- Williams graph: Theorem3.7. Let =(G, K ,ρ , φ )beaMauldin-Williamsgraph. v v v E0 e e E1 Suppose that the graMph G has{no sink}s∈and{no }so∈urces, it is irreducible, and is not a cyclic permutation. Assume that K is the invariant set of the Mauldin-Williams graph and that satisfies the open set condition in K. Then the associated C - ∗ M algebra (K) is simple and purely infinite. OM Using the same argument as in [16, Proposition3.8] one can show that (K) OM is separableandnuclear,andsatisfiesthe UniversalCoefficientTheorem. Thus, by the classificationtheorem of Kirchbergand Phillips [20, 28], the isomorphismclass of (K) is completely determined by the K-theory with the class of the unit. OM 4. Examples We will compute the K-groups of the C -algebra associated with a graph G = ∗ (E0,E1,r,s) using the fact that K (C (G)) is isomorphic to the kernelof 1 At : 1 ∗ − G