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A DIRECT APPROACH TO CO-UNIVERSAL ALGEBRAS ASSOCIATED TO DIRECTED GRAPHS 0 1 AIDANSIMSANDSAMUELB.G.WEBSTER 0 2 n a J Abstract. We prove directly that if E is a directed graph in which every cyclehasanentrance,thenthereexistsaC∗-algebrawhichisco-universalfor 2 Toeplitz-Cuntz-Krieger E-families. In particular, our proof does not invoke 1 ideal-structuretheoryforgraphalgebras,nordoesitinvolveuseofthegauge actionoritsfixedpointalgebra. ] A O 1. Introduction . In recent years there has been a great deal of interest in graph C∗-algebras and h t their generalisations (see [3] for a survey). To associate C∗-algebras to a given a generalisation of directed graphs, one assigns partial isometries to the edges of m the graph in a way which encodes connectivity in the graph. One then aims to [ identify relations amongst the partial isometries so that the C∗-algebra universal 2 for these relations satisfies a version of the Cuntz-Krieger uniqueness theorem. v For directed graphs, this theorem states that if every cycle has an entrance, then 3 any representation of its Cuntz-Krieger algebra which is nonzero on generators is 0 faithful. Intryingtoidentifyappropriaterelations,thereistypicallysomeanalogue 3 3 of a left-regular representation which points to a natural notion of an abstract . representation;inthecaseofdirectedgraphs,eachgraphcanberepresentedonthe 2 Hilbertspacewithorthonormalbasisindexedbyfinite pathsinthegraph,andthis 1 9 gives rise to the notion of a Toeplitz-Cuntz-Krieger family. However,the universal 0 C∗-algebra for such representations is typically too big to satisfy a version of the : Cuntz-Krieger uniqueness theorem, and one has to identify an additional relation v i to correct this. X In [2], in the much more general context of Cuntz-Pimsner algebras associated r to Hilbert bimodules, Katsura developed a very elegant solution to this problem. a Fordirectedgraphs,hisresultssaythatgivenanyToeplitz-Cuntz-KriegerE-family consisting of nonzero partial isometries, if the C∗-algebra B it generates carries a circle action compatible with the canonical gauge action on the Toeplitz algebra, then there is a canonical homomorphism from B onto the Cuntz-Krieger algebra. We call this a co-universal property: the Cuntz-Krieger algebra is co-universal for Toeplitz-Cuntz-Krieger families consisting of nonzero partial isometries and compatible with the gauge action. Katsura proved this theorem a posteriori: the Cuntz-Krieger algebra had already been defined in terms of a universal property. However,he pointed out that this theorem implies that the Cuntz-Krieger algebra Date:December10,2009. 1991 Mathematics Subject Classification. Primary46L05. Key words and phrases. Graphalgebra,co-universalproperty. ThisresearchwassupportedbytheAustralianResearchCouncil. 1 2 A.SIMSANDS.B.G.WEBSTER could be defined to be the algebra co-universalfor nonzero Cuntz-Kriegerfamilies; one would then have to work to prove that such a co-universalalgebra exists. When every cycle in the graph E has an entrance, it is a consequence of the Toeplitz-Cuntz-KriegeruniquenesstheoremthateveryCuntz-KriegerE-familycon- sisting of nonzero partial isometries is automatically compatible with the gauge action. In particular, in this case the use of the gauge action in Katsura’s analysis should not be necessary. In this article we show that this is indeed the case: we presentadirectargumentthatforanarbitrarydirectedgraphinwhicheverycycle hasanentrance,thereexistsaC∗-algebrawhichisco-universalforToeplitz-Cuntz- Krieger families consisting of nonzero partial isometries. In particular, we do not firstidentify the Cuntz-Kriegerrelationorthe correspondinguniversalC∗-algebra. We also do not proceed via the machinery of ideal structure of Toeplitz algebras of directed graphs, and we do not deal with the gauge-action of the circle or with an analysis of its fixed-point algebra. Instead we work directly with the abelian subalgebra generated by range projections associated to paths in the graph. 2. Preliminaries A directed graph E = (E0,E1,r,s) consists of a countable set E0 of vertices, a countable set E1 of edges, and maps r,s : E1 → E0 indicating the direction of the edges. We will follow the conventions of [3] so that a path is a sequence α = α α ...α of edges such that s(α ) = r(α ) for all i. We write |α| for n, and if 1 2 n i i+1 wewanttoindicateasegmentofapath,weshalldenoteitα =α α ...α . [p,q] p+1 p+2 q If n = ∞ (so that α is actually a right-infinite string), then we call α an infinite path. The range of an infinite path α=α α ... is r(α):=r(α ). 1 2 1 For n∈N, we write En for the collection of paths of length n. We write E∗ for the category of finite paths in E (we regard vertices as paths of length zero), and E∞ for the collection of all infinite paths. Given α ∈ E∗ and X ⊂ E∗∪E∞, we denote {αµ: µ∈X,r(µ) =s(α)} by αX. Given v ∈E0, a set X ⊂vE∗ is said to be exhaustive if, for every λ∈vE∗ there exists α∈X such that either λ=αλ′ or α=λα′. Given a directed graphE, a Toeplitz-Cuntz-Krieger E-family in a C∗-algebraB is a pair (t,q) where t : e 7→ t assigns to each edge a partial isometry in B, and e q :v 7→q assigns to each vertex a projection in B such that v (TCK1) the q are mutually orthogonal v (TCK2) each t∗t =q , and e e s(e) (TCK3) for each v ∈E0 and each finite subset F ⊂vE1, we have q ≥ t t∗. v e∈F e e There is a C∗-algebraTC∗(E) generatedby a Toeplitz-Cuntz-Kriegerfamily (s,p) P whichisuniversalinthesensethatgivenanyToeplitz-Cuntz-Kriegerfamily(t,q)in aC∗-algebraBthereisahomomorphismπ :TC∗(E)→Bsuchthatπ (s )=t t,q t,q e e and π (p )=q for all e∈E1 and v ∈E0. t,q v v Given a path α ∈ E∗ and a Toeplitz-Cuntz-Krieger E-family (t,q), we write t α for t t ...t . α1 α2 α|α| 3. The co-universal algebra Theorem 3.1. Let E be a directed graph, and suppose that every cycle in E has an entrance. There exists a Toeplitz-Cuntz-Krieger E-family (Sap,Pap) consisting of nonzero partial isometries such that C∗ (E) := C∗(Sap,Pap) is co-universal min in the sense that given any other Toeplitz-Cuntz-Krieger E family (t,q) in which CO-UNIVERSAL ALGEBRAS OF DIRECTED GRAPHS 3 each q is nonzero, there is a homomorphism ψ : C∗(t,q) → C∗ (E) such that v t,q min ψ (t )=Sap and ψ (q )=Pap for all e∈E1 and v ∈E0. t,q e e t,q v v OurproofreliesonunderstandingthestructuretheC∗-algebrageneratedbythe projections {t t∗ :λ∈E∗} for a Toeplitz-Cuntz-Krieger E-family (t,q). We begin λ λ with the following definition. Definition 3.2. Let E be a directed graph. A boolean representation of E in a C∗-algebra B is a map p:λ7→p from E∗ to B such that each p is a projection, λ λ and p if ν =µν′ ν p p = p if µ=νµ′ µ ν  µ 0 otherwise. IfpisabooleanrepresentationofE,thenthepλ commute,sospan{pλ :λ∈E∗} is a commutative C∗-algebra. Lemma 3.3. Let E be a directed graph, let p be a boolean representation of E, and fix a finite subset F ⊂E∗. For µ∈F, define qF :=p (p −p ). µ µ µ µµ′ µµ′∈YF\{µ} Then the qF are mutually orthogonal projections and for each µ∈E∗, µ (3.1) p = qF µ µµ′ µµ′∈F X Proof. We proceed by induction on |F|. If |F| = 1 then (3.1) is trivial. Suppose (3.1) holds whenever |F| < n, and fix F with |F| = n. Let λ ∈ F be of maximal length, and let G=F \{λ}. Then qF =p , and for µ∈G, λ λ qG if λ6=µµ′ qF = µ µ (qµG−qµGpλ if λ=µµ′. Fixµ∈F. Ifλ6=µµ′,thentheinductivehypothesisimpliesthat qF = µµ′∈F µµ′ qG =p . If λ=µµ′, then µµ′∈G µµ′ µ P P qF = (qG−qGp )+qF µµ′ µ µ λ λ µµ′∈F µµ′∈G X X = qG− qGp +p µ µ λ λ µµ′∈G µµ′∈G X X =p −p p +p by the inductive hypothesis µ µ λ λ =p −p +p since λ=µµ′ µ λ λ =p . (cid:3) µ Given a directedgraphE, we define the set of aperiodic boundary paths in E by ∂Eap := λ∈E∗ :|s(λ)E1|∈{0,∞} ⊔ E∞\ λµ∞ :s(λ)=r(µ)=s(µ) . Let H:=ℓ2(cid:8)(∂Eap), with orthonormal ba(cid:9)sis {ξx :x(cid:8)∈∂Eap}, and define {Pλap(cid:9):λ∈ E∗}⊂B(H) by ξ if x∈λ∂Eap Papξ = x λ x (0 otherwise. 4 A.SIMSANDS.B.G.WEBSTER Since each Pap = proj , it is straightforward to check that Pap is a λ span{ξx:x∈λ∂Eap} boolean representation of E. Suppose that every cycle in E has an entrance. We claim that each Pap is λ nonzero. Indeed, fix λ ∈ E∗. Since every cycle in E has an entrance, there exists an x∈∂Eap with r(x)=s(λ). Then λx∈∂Eap so Papξ =ξ 6=0. λ λx λx Lemma 3.4. Let E be a directed graph. Let λ∈E∗, and suppose that F ⊂s(λ)E∗ is finite and exhaustive. Then (Pap−Pap)=0. λ λµ µ∈F Y Proof. Let x∈∂Eap. We seek µ∈F such that (Pap−Pap)ξ =0. λ λµ x If Papξ = 0, then any µ ∈ F will suffice, so suppose that Papξ 6= 0. Then λ x λ x x(0,|λ|) = λ. If there exists µ ∈ F such that x = λµx′, then (Pap −Pap)ξ = λ λµ x ξ −ξ = 0, so we suppose that x 6= λµx′ for all µ ∈ F and seek a contradiction. x x SinceF isexhaustive,thereexistsµ∈F suchthatλµ=xµ′ forsomeµ′ ofnonzero length. Inparticular,xisafinitepathandsinceµ′ hasnonzerolength,s(x)E1 6=∅, so x∈∂Eap forces |s(x)E1|=∞. But since no initial segment of x belongs to λF, that F is exhaustive implies that for each e ∈ s(x)E1, there exists µ ∈ F such e that λµ =xex′ for some x ∈E∗, and this contradicts that F is a finite set. (cid:3) e e e Lemma 3.5. Let E be a directed graph and p bea boolean representation of E such that p 6=0 for each λ∈E∗. If {α′ :αα′ ∈F} is not exhaustive, then qF 6=0. λ α Proof. Since {α′ : αα′ ∈ F} is not exhaustive, there exists τ ∈ E∗ such that τ 6= α′α′′ and α′ 6=ττ′ for each α′ such that αα′ ∈F. In particular, each p p =0, αα′ ατ and hence qFp =p 6=0, whence qF 6=0. (cid:3) α ατ ατ α Proposition 3.6. Let E be a directed graph and let p be a boolean representation of E such that p 6= 0 for each λ ∈ E∗. Then there is a homomorphism ψ : λ p span{p : λ∈ E∗} → span{Pap : λ ∈ E∗} satisfying ψ (p ) = Pap for all λ ∈ E∗. λ λ p λ λ Moreover, ψ is injective if and only if (p −p ) = 0 for all λ ∈ E∗ and p µ∈F λ λµ finite exhaustive F ⊂s(λ)E∗. Q Proof. Forthe firstassertionitsuffices to showthatfor everyfinite subsetF ofE∗ and every collection of scalars {a :λ∈F}, λ (3.2) a Pap ≤ a p λ λ λ λ (cid:13)λX∈F (cid:13) (cid:13)λX∈F (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Fix a finite subset F ⊂ E∗(cid:13), and for α(cid:13)∈ F(cid:13), define QF(cid:13):= Pap (Pap − α α αα′∈F\{α} α Pap ). For λ∈E∗, Lemma 3.3 gives αα′ Q a Pap = a QF, and a p = a qF. λ λ µ α λ λ µ α λX∈F αX∈F(cid:16) µX∈F (cid:17) λX∈F αX∈F(cid:16) µX∈F (cid:17) α=µµ′ α=µµ′ By Lemmas 3.4 and 3.5, we have {α∈F :QF 6=0}⊂{α∈F :qF 6=0}. Hence α α a Pap = max a ≤ max a = a p , λ λ µ µ λ λ QF6=0 qF6=0 (cid:13)λX∈F (cid:13) α (cid:12) µX∈F (cid:12) α (cid:12) µX∈F (cid:12) (cid:13)λX∈F (cid:13) (cid:13) (cid:13) (cid:12)α=µµ′ (cid:12) (cid:12)α=µµ′ (cid:12) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12) (cid:12) (cid:12) (cid:13) (cid:13) and the first assertion follows. CO-UNIVERSAL ALGEBRAS OF DIRECTED GRAPHS 5 For the second assertion, note that if (p − p ) = 0 for all λ ∈ E∗ µ∈F λ λµ and finite exhaustive F ⊂ s(λ)E∗, then for each finite exhaustive F, we have Q {α ∈ F : QF 6= 0} = {α ∈ F : qF 6= 0} so the calculation above shows that ψ is α α p isometric. (cid:3) We now show that the C∗-algebra generated by any Toeplitz-Cuntz-Krieger E- familyinwhichallthe partialisometriesarenonzeroadmits aconditionalexpecta- tion onto the subalgebra spanned by the range projections s s∗. Recall that given λ λ a Toeplitz-Cuntz-KriegerE-family (t,q), we write π for the canonicalhomomor- t,q phism from TC∗(E) to C∗(t,q) induced by the universal property of the former. We first need a technical lemma. Lemma 3.7. Suppose λ,µ,ν ∈E satisfy |λ|≥|ν|>|µ| and (3.3) t t∗t t∗t t∗ 6=0. λ λ µ ν λ λ Then λ=νν′ =µµ′ν′ for some µ′,ν′, and t t∗t t∗t t∗ =t t∗ λ λ µ ν λ λ λ λρ for some cycle ρ∈E. Proof. Equation 3.3 forces t t∗t t∗ 6= 0 and t t∗t t∗ 6= 0. Hence λ = νν′ and λ λ µ µ ν ν λ λ λ = µα for some ν′ and α, and since |µ| < |ν|, this forces ν = µµ′ and hence λ=µµ′ν′ for some µ′. We have (3.4) 06=t t∗t t∗t t∗ =t t∗t (t∗ t∗)(t t t )t∗ =t t∗t t t∗, λ λ µ ν λ λ λ λ µ µ′ µ µ µ′ ν′ λ λ λ µ ν′ λ forcing s(µ) = r(ν′). Since r(µ′) = s(µ) and s(µ′) = r(ν′), µ′ is a cycle, and has nonzero length since |ν|>|µ|. Furthermore, continuing from (3.4) 06=t t∗t t t∗ =(t t )(t∗ t∗)t t (t∗ t∗ )=t t t∗ t t∗ t∗ , λ λ µ ν′ λ µ µ′ν′ µ′ν′ µ µ ν′ ν′ µµ′ µ µ′ν′ µ′ν′ ν′ ν′ µµ′ so t∗ t′ is nonzero, forcing µ′ν′ ν (3.5) µ′ν′ =ν′ρ for some ρ∈E. We claim ρ is a cycle. We proceedby induction on |ν′|. As a base case, suppose that |ν′|≤|µ′|. Then ν′ =µ′ , so [0,|ν′|] µ′ν′ =ν′ρ =⇒ ρ=µ′ µ′ , [|ν′|,|µ′|] [0,|ν′|] whence r(ρ)=s(ρ)=s(ν′). Now suppose as an inductive hypothesis that ρ is a cycle whenever (3.5) holds with |ν′| < n for some n > |µ′|, and fix ν′ with |ν′| = n satisfying (3.5). In particular, |ν′| > |µ′|, so µ′ = ν′ and µ′ν′ = ν′ = ν′ ρ. Since [0,|µ′|] [|µ′|,|ν′|] [|µ′|,|ν′|] |ν′ |=|ν′|−|µ′|<n, the inductive hypothesis now implies that ρ is a cycle. [|µ′|,|ν′|] Finally, from (3.4), 06=t t∗t t∗t t∗ =t t∗t t t∗ λ λ µ ν λ λ λ λ µ ν′ λ =t (t∗ t∗)t t t∗ =t t∗ t t∗ =t t∗t∗ t t∗ =t t∗t∗, (cid:3) λ µ′ν′ µ µ ν′ λ λ ν′ρ ν′ λ λ ρ ν′ ν′ λ λ ρ λ Given a directed graph E and e∈E1, define Sap ∈B(ℓ2(∂Eap)) by e ξ if s(e)=r(x) Sapξ = ex e x (0 otherwise. 6 A.SIMSANDS.B.G.WEBSTER Then with {Pap :v ∈E0} as on page 3, (Sap,Pap) is a Toeplitz-Cuntz-Krieger E- v family, and Sap(Sap)∗ =Pap for all λ∈E∗. We denote C∗(Sap,Pap) by C∗ (E). λ λ λ min Proposition 3.8. Let E be a directed graph, and suppose that every cycle in E has an entrance. Let (t,q) be a Toeplitz-Cuntz-Krieger E-family. Then q : λ 7→ t t∗ λ λ is a boolean representation of E, and there exists a conditional expectation Φ : t,q C∗(t,q)→span{q :λ∈E∗} satisfying Φ (t t∗)=δ q . In particular we have λ t,q µ ν µ,ν µ (3.6) ψ ◦Φ ◦π =Φ ◦π . q t,q t,q Sap,Pap Sap,Pap Proof. It is standard that the q form a boolean representation of E. λ Claim 1. Given a finite subset F of E∗ and a collection {a : µ,ν ∈ F} of µ,ν scalars, a q ≤ a t t∗ . µ,µ µ µ,ν µ ν (cid:13)µX∈F (cid:13) (cid:13)µX,ν∈F (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) By enlarging F (and set(cid:13)ting the extr(cid:13)a sc(cid:13)alars equal to z(cid:13)ero), we may assume that F is closed under initial segments in the sense that if µν ∈F then µ∈F. For each λ ∈ F, let TF := {λ′ ∈ s(λ)E∗ : λλ′ ∈ F,|λ′| > 0}. For each λ ∈ F λ suchthat TF is notexhaustive,fix apathαλ suchthat αλ 6=µµ′ andµ6=αλα′ for λ allµ∈TF. Since everycycle inE has anentrance,[3,Lemma 3.7]implies thatfor λ each v such that v =s(αλ) for some λ, there exists τv ∈vE∗ such that either: (1) s(τv)E1 = ∅; or (2) |τv| > max{|λ| : λ ∈ F,TF is not exhaustive}, and τv 6= τv λ k |τv| for all k <|τv|. We write τλ for τs(αλ). For each λ∈F we define q if TF is not exhaustive φF := λαλτλ λ λ (qλF otherwise. By definition we have each φF ≤qF, and since the qF are mutually orthogonal, it λ λ λ follows that the φF are also. Hence λ a t t∗ ≥ φF a t t∗ φF . µ,ν µ ν λ µ,ν µ ν λ (cid:13)µX,ν∈F (cid:13) (cid:13)λX∈F (cid:16)µX,ν∈F (cid:17) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) Fix λ,µ,ν ∈F(cid:13). We claim tha(cid:13)t (cid:13) (cid:13) φF if µ=ν and λ=µλ′ for some λ′ (3.7) φFt t∗φF = λ λ µ ν λ (0 otherwise. Toseethis,supposefirstthatµ=ν. Ifλ=µλ′ thenφF ≤t t∗ ≤t t∗ bydefinition λ λ λ µ µ of φF. If λ6=µλ′, then either µ=λµ′ in which case φF ≤(t t∗ −t t∗ )⊥t t∗, λ λ λ λ λµ′ λµ′ µ µ or else µ6=λµ′ in which case φF ≤t t∗ ⊥t t∗. λ λ λ µ µ Now suppose that µ 6= ν; by symmetry under adjoints, we may assume that |µ| <|ν|. We must show that φFt t∗φF =0. Since φF ≤t t∗, if t t∗t t∗t t∗ = 0 λ µ ν λ λ λ λ λ λ µ ν λ λ then we are done, so we may assume that t t∗t t∗t t∗ 6= 0. Then Lemma 3.7 λ λ µ ν λ λ implies that λ = νν′ = µµ′ν′ and that t t∗t t∗t t∗ = t t∗ for some cycle ρ ∈ E. λ λ µ ν λ λ λ λρ Hence φF ≤t t∗ forces λ λ λ φFt t∗φF =φFt t∗ φF. λ µ ν λ λ λ λρ λ We consider two cases: TF is exhaustive, or it is not. λ CO-UNIVERSAL ALGEBRAS OF DIRECTED GRAPHS 7 FirstsupposethatTF isexhaustive. Fixnsuchthatn|ρ|>max{|λ′|:λ′ ∈TF}. λ λ Then ρn = λ′β for some λ′ ∈ TF and β ∈ E∗. In particular, λ′ = ρn , forcing λ [0,|λ′|] ρ =λ′. Since λλ′ ∈F and F is closed under initial segments, λρ ∈F and then 1 1 1 t t∗ φF ≤t∗ t∗ (t t∗ −t t∗ )=0, λρ λρ λ λρ λρ λ λ λρ1 λρ1 giving φFs s∗φF =0. λ µ ν λ Now suppose that TF is not exhaustive. Then φF =q . We then have λ λ λαλτλ φFt t∗φF =t t∗ t t∗ . λ µ ν λ λαλτλ µ′ν′αλτλ ν′αλτλ λαλτλ This is nonzero only if µ′ν′αλτλ = ν′αλτλζ for some ζ. This is impossible if s(τλ)E1 = ∅, so suppose that s(τλ)E1 6= ∅. By choice of τλ, we have |τλ| > |µ′|. Let m=|ν′αλτλ|. Then (µ′ν′αλτλ) =τλ 6=τλ =(ν′αλτλ) , m |τλ|−|µ′| |τλ| m so µ′ν′αλτλ 6=ν′αλτλζ for all ζ, and hence φFt t∗φF =0, establishing (3.7). λ µ ν λ We now have a q = a qF µ,µ µ µ[0,n],µ[0,n] µ (cid:13)µX∈F (cid:13) (cid:13)µX∈F(cid:16)nX≤|µ| (cid:17) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)=m(cid:13)ax a (cid:13) µ∈F µ[0,n],µ[0,n] (cid:12)nX≤|µ| (cid:12) (cid:12) (cid:12) = (cid:12) a (cid:12) φF µ[0,n],µ[0,n] µ (cid:13)µX∈F(cid:16)nX≤|µ| (cid:17) (cid:13) (cid:13) (cid:13) =(cid:13) φF a t t∗ φF(cid:13) by (3.7) λ µ,ν µ ν λ (cid:13)λX∈F (cid:16)µX,ν∈F (cid:17) (cid:13) (cid:13) (cid:13) ≤(cid:13) a t t∗ (cid:13) µ,ν µ ν (cid:13)µX,ν∈F (cid:13) (cid:13) (cid:13) completing the proof of Claim(cid:13) 1. (cid:13) Claim1impliesthattheformulat t∗ 7→δ t t∗ extendstoawell-definedlinear µ ν µ,ν µ µ map Φ from C∗(t,q) to span{q : λ ∈ E∗}. This Φ is a linear idempotent of t,q λ t,q norm1,andhenceaconditionalexpectation(seeforexample[1,DefinitionII.6.10.2 and Theorem II.6.10.2]). The final statement is straightforward since the two maps in question agree on spanning elements. (cid:3) Lemma 3.9. Let E be a directed graph in which every cycle has an entrance. Then the expectation Φ : C∗ (E) → span{Pap : λ ∈ E∗} obtained from Sap,Pap min λ Proposition 3.8 is faithful on positive elements. Proof. It suffices to show that for a∈C∗ (E), Φ (a) is equal to the strong- min Sap,Pap operator limit aξ |ξ )θ for all a ∈ C∗ (E), where the ξ are the x∈∂Eap x x ξx,ξx min x canonicalorthonormalbasisforℓ2(∂Eap)andθ istherank-oneprojectiononto Cξ . Fix µ,ν ∈PE∗ and x(cid:0)∈∂Eap. We have ξx,ξx x 1 if x=νy =µy for some y Sap(Sap)∗ξ ξ = (Sap)∗ξ Sap)∗ξ = µ ν x x ν x µ x (0 otherwise. (cid:0) (cid:12) (cid:1) (cid:0) (cid:12)(cid:0) (cid:1) Since x ∈ ∂Eap,(cid:12)we have y 6= ρ∞(cid:12)for any cycle ρ, so µy = νy forces |µ| = |ν|, and hence µ=ν. 8 A.SIMSANDS.B.G.WEBSTER Hence Sap(Sap)∗ξ |ξ )θ =δ proj =δ Pap µ ν x x ξx,ξx µ,ν span{ξx:x∈µ∂Eap} µ,ν µ x∈∂Eap X (cid:0) as required. (cid:3) We now have the tools we need to prove the main theorem. Proof of Theorem 3.1. We showed that the Pap, and in particular the Pap are λ v nonzero immediately subsequent to their definition. FixaToeplitz-Cuntz-KriegerE-family(t,q)witheachq nonzero. Wewillshow v that ker(π )⊂ker(π ) in TC∗(E), and hence that π descends to the t,q Sap,Pap Sap,Pap desired homomorphism ψ :C∗(t,q)→C∗ (E). t,q min Since eacht∗t =q , eachq is nonzero,soProposition3.6 implies that there λ λ s(λ) λ is a homomorphism ψ : span{q : λ ∈ E∗} → span{Pap : λ ∈ E∗} taking each q q λ λ λ to Pap. λ We calculate π (a)=0 ⇐⇒ π (a∗a)=0 t,q t,q (3.8) =⇒ ψ ◦Φ ◦π (a∗a)=0 q t,q t,q ⇐⇒ Φ (π (a∗a))=0 by (3.6) Sap,Pap Sap,Pap ⇐⇒ π (a∗a) by Lemma 3.9 Sap,Pap ⇐⇒ π (a)=0. (cid:3) Sap,Pap Itis,ofcourse,interestingtoknowwhenthehomomorphismψ ofTheorem3.1 t,q is injective. Theorem 3.10. Let E be a directed graph in which every cycle has an entrance. (i) If (t,q) is a Toeplitz-Cuntz-Krieger family with each q nonzero, then v the homomorphism ψ of Theorem 3.1 is injective if and only if (a) t,q (q −q ) = 0 whenever v ∈ E0 and F ⊂ vE∗ is finite exhaustive; λ∈F v λ and (b) the expectation Φ is faithful. t,q Q (ii) If π is homomorphism from C∗ (E) to a C∗-algebra C such that each min πPvap is nonzero, then π is injective. Proof. (i) Since conditions (a) and (b) hold in C∗ (E), the “only if” implication min is trivial. For the “if” implication, note that given a Toeplitz-Cuntz-Krieger E- family (t,q), we have ker(π ) = ker(π ) whenever the implication (3.8) is t,q Sap,Pap equivalence. Condition (a) implies that ψ is faithful by the final statement of q Proposition 3.6, and this combined with (b) implies that ψ ◦Φ is faithful on q t,q positive elements, giving π (a∗a)=0 ⇐⇒ ψ ◦Φ ◦π (a∗a)=0 t,q q t,q t,q as required. (ii)Define aToeplitz-Cuntz-KriegerE-family byt :=π(Sap)andq :=π(Pap). e e v v Theorem 3.1 supplies a homomorphism ψ : C∗(t,q) → C∗ (E) which is an t,q min inverse for π. (cid:3) CO-UNIVERSAL ALGEBRAS OF DIRECTED GRAPHS 9 References [1] B. Blackadar, Operator algebras, Springer-Verlag, Berlin, Theory of C∗-algebras and von Neumannalgebras,OperatorAlgebrasandNon-commutative Geometry,III,2006, xx+517. [2] T.Katsura,AclassofC∗-algebrasgeneralizingbothgraphalgebrasandhomeomorphismC∗- algebras III. Ideal structures,ErgodicTheoryDynam.Systems26(2006), no.6,1805–1854. [3] I.Raeburn,Graph algebras,CBMSRegionalConferenceSeriesinMathematics 103(Amer. Math.Soc.,Providence,RI,2005). Aidan Sims, School of Mathematics and Applied Statistics, Austin Keane building (15),University of Wollongong, NSW2522,AUSTRALIA E-mail address: [email protected] SamuelWebster,SchoolofMathematicsandAppliedStatistics,UniversityofWol- longong,NSW2522,AUSTRALIA E-mail address: [email protected]