NOTES ON CUNTZ KRIEGER UNIQUENESS THEOREMS AND C -ALGEBRAS OF LABELLED GRAPHS

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1 NOTES ON CUNTZ KRIEGER UNIQUENESS THEOREMS AND C -ALGEBRAS OF LABELLED GRAPHS BERNHARD BURGSTALLER Abstract. In this note we deal with Cuntz-Krieger uniqueness theorems and extend the class of algebras introduced in [3]. We use this analysis to show two Cuntz-Krieger uniqueness theorems for C -algebras of labelled graphs. 1. Introduction One of the aims of this note is to extend the family of generalized Cuntz-Krieger algebras introduced in [3]. For convenience of the reader and to establish notation we briefly recall the basic definitions of [3] at first. We consider an alphabet A, the free nonunital -algebra F over C generated by A, a self-adjoint two-sided ideal I in F, and a compact subgroup H T A, where T denotes the torus. Given such a system (A, F, I, H) of generators and relations we denote by W = { a 1... a n F/I a i A A } the set of words in the quotient F/I. (For convenience we write a 1... a n rather than a 1... a n + I if this is clear from the context.) Let bal : W \{0} Ĥ be the balance function of [3, Lemma 3.1] which is determined by the formula bal(a)(λ) = λ a, for all a A and λ = (λ b ) b A H, and by the identities bal(xy) = bal(x)bal(y) and bal(z ) = bal(z) Mathematics Subject Classification. 46L05. Key words and phrases. Cuntz-Krieger algebra, uniqueness, labelled graph, C - algebra. This research was supported by the Austrian Research Foundation (FWF) project S8308 and the EU-project Quantum Space and Noncommutative Geometry, HPRN-CT

2 2 B. BURGSTALLER for all x, y, z W such that xy 0 and z 0. A nonzero word x W is called zero-balanced if bal(x) = 1 Ĥ (and otherwise called nonzero-balanced). Let W 0 = { x W \{0} bal(x) = 1 } be the set of all zero-balanced words. Let A be the -subalgebra generated by W 0, (1) A = Alg (W 0 ) = lin(w 0 ) F/I, and denote by P the set of nonzero projections of A. Projections in F/I are always understood to be idempotent and self-adjoint, and we have a natural order p q iff pq = p. If we want to emphasize that A refers to the system (A, F, I, H) then we write A (A,F,I,H). The system (A, F, I, H) may satisfy the following properties. (A) For all λ = (λ a ) a A H the ideal I is invariant under the automorphism t λ : F F determined by t λ (a) = λ a a for all a A. (B) A is the inductively ordered union of finite dimensional sub- C -algebras of A. This is equivalent to saying that A is the algebraic direct limit of finite dimensional C -algebras, that is, A is locally matricial. (C) For all words x W \W 0 and all projections e, e 1, e 2 P there exist projections p, p 1, p 2 P such that p e, p 1 e 1, p 2 e 2 and pxp = 0 and p 1 xp 2 = 0. Theorem 1.1 ([3], Theorem 3.3). If the system (A, F, I, H) satisfies the properties (A), (B) and (C), and π 1 : F/I A 1 and π 2 : F/I A 2 are -homomorphisms with dense images in C -algebras A 1, A 2, and π 1 is injective on A, then there exists a -homomorphism σ : A 1 A 2 such that σπ 1 = π 2. The map σ is an isomorphism if π 2 is also injective on A. As explained in [3], the aim of the last theorem is to check Cuntz- Krieger uniqueness theorems for C -algebras induced by generators and relations, as it was done for classical Cuntz-Krieger algebras [5], graph C -algebras [9, 8, 10], Robertson-Steger algebras [11], or higher rank graph C -algebras [7]. In section 2 we will show that Theorem 1.1 still

3 CUNTZ KRIEGER UNIQUENESS THEOREMS 3 holds if we replace condition (C) by a weaker and more easily checkable condition (C ): (C ) For all words x W \W 0 and all projections e P there exists a projection p P such that p e and pxp = 0. All Exel-Laca algebras satisfying the no-terminal-loop condition satisfy the conditions (A), (B) and (C ) (this is not true for condition (C), see [3]). In section 3 we will show that we can replace the set P appearing in property (C ) by a much smaller set P 0 if, roughly speaking, all words are partial isometries with commuting range projections. In the last section 4 we use the analysis of the previous sections to show two Cuntz-Krieger uniqueness theorems (Theorems 4.3 and 4.4) for C -algebras of (cancellable) labelled graphs [1] (including Tomforde s ultragraph, Exel-Laca, and Matsumoto algebras). 2. Condition (C ) The aim of this section is to show that we can replace property (C) by property (C ) in the Cuntz-Krieger uniqueness Theorem 1.1. Theorem 2.1. Theorem 1.1 still holds if one replaces the required condition (C) by the weaker condition (C ). Proof. Assume that the system (A, F, I, H) satisfies the conditions (A), (B) and (C ). Introduce the formal alphabet P := {p (1) } { p (1,c2,...,c n) {1, 2} {1,...,n} n 2, c i {1, 2} }, and define F to be the free nonunital -algebra generated by the alphabet P. Let I be the two-sided self-adjoint ideal in F which is generated by pp p, p p, pq qp, p (1) q q and (2) p (1,c2,...,c n) p (1,c2,...,c n,1) p (1,c2,...,c n,2) for all p, q P, n 1 and c 1,..., c n {1, 2}. In this proof we will regard P as a subset of F /I. Note that P is a commuting set of projections which forms a kind of binary tree: p 1 is a unit, p 1 = p 11 +p 12, p 11 = p p 112, p 12 = p p 122, and so on. Trivially, the system (P, F, I, {e}) (where e denotes the neutral element of the group T P ) satisfies the properties (A), (B) and (C ), as bal(p) = 1 for all p P, in other words, all words of F /I are zero-balanced, whence F /I =

4 4 B. BURGSTALLER A (P,F,I,{e}). One also has F /I = Alg (P) = lin(p) since the elements in P are either orthogonal or comparable. Write C = F /I for the unique C -norm closure of F /I. Write S = (A P, G, J, H {e}) for the tensor product system of (A, F, I, H) and (P, F, I, {e}), that is, we define G to be the free nonuntal -algebra generated by A P, and J to be the kernel of the -epimorphism ϕ : G F/I F /I determined by ϕ((a, p)) = a p for all a A, p P. We will identify G/J with the image of ϕ. Note that every element x G/J allows a representation (3) x = y 1 q y n q n, for some elements y i F/I, and mutually orthogonal projections q i P (indeed, think of choosing the projections q i on a sufficiently large kth level on the tree of P, that is, q 1 = p 1,1,1,...,1, q 2 = p 1,1,1,...,2,..., q n = p 1,2,2,...,2 ). That the system S satisfies property (A) may be realized by the fact that when t λ is the induced gauge action on F/I for λ H, and id is the identity gauge action on F /I, then t λ id defines a gauge action on G/J, showing that J is invariant under t λ id. The system S satisfies property (B), as A (S) = A (A,F,I,H) F /I. Now assume that we are given two - homomorphisms π k : F/I A k mapping into C -algebras A k, and the π k s are faithful on A (A,F,I,H) and have dense images (k = 1, 2). Define homomorphisms π k id : G/J A k C by (π k id)(a p) = π k (a) p (k = 1, 2). By representation (3), for any nonzero projection x G/J there are nonzero projections y F/I and q P such that y q x. We are going to show that the system S satisfies condition (C). To this end, suppose we are given three nonzero projections e 1, e 2, e 3 P and a nonzero-balanced word x W \W e. We have to show that there are smaller projections p i e i in P such that p 1 xp 1 = p 2 xp 3 = 0. At first we choose nonzero smaller projections f i q i e i for some f i P (A,F,I,H) and q i P (1 i 3). We may write x = y q for some word y in the letters of A, and some projection q P. By construction of P, there exist smaller projections a i q i in P (2 i 3) such that a 2 a 3 = 0, and consequently p 2 xp 3 = f 2 xf 3 a 2 qa 3 = 0 for p i = f i a i e i. Assuming condition (C ) for the system (A, F, I, H), we may choose a

5 CUNTZ KRIEGER UNIQUENESS THEOREMS 5 projection g 1 P (A,F,I,H) such that g 1 f 1 and g 1 yg 1 q 1 = 0, and thus we may set p 1 = g 1 q 1 e 1. By Theorem 1.1 there is a homomorphism σ : A 1 C A 2 C such that σ(π 1 id) = π 2 id. The restriction σ = σ A1 C is the desired map σ : A 1 A 2 satisfying σπ 1 = π 2. Definition 2.2. If a system (A, F, I, H) satisfies the conditions (A), (B) and (C ), and there exists an A-faithful representation π : F/I A into a C -algebra A, then we call O A,F,I,H := π(f/i) (norm closure) the Cuntz-Krieger type algebra associated with (A, F, I, H). (By the last uniqueness theorem, O A,F,I,H is uniquely determined up to - isomorphism.) Given projections p, q A, we write p q in A if there exists a partial isometry s A (i.e. ss s = s) such that p = ss and s s q. We introduce the following condition: (C ) There exist subsets W F/I and P 2 P such that (i) W \W 0 lin(w ), (ii) for all p P there exists p 2 P 2 such that p 2 p in A, and (iii) for all x W and e P 2 there exists p P 2 such that p e and pxp = 0. Notice that W = W \W 0 is a valid candidate satisfying (i), and P 2 = P is a valid candidate satisfying (ii). Hence (C ) trivially implies (C ). But also the inverse implication holds true: Lemma 2.3. Assume that (A) holds. Then (C ) and (C ) are equivalent. Proof. Fix x W \W 0 and e P 2. By (C )(i) we may write x as x = n i=1 λ ix i for some λ i C, x i W. By condition (C )(iii) we may choose a decreasing sequence of projections e p 1... p n, p i P 2, such that p 1 x 1 p 1 = p 2 x 2 p 2 =... = p n x n p n = 0. Hence p n xp n = 0 as required in condition (C ). For the general case, when e P, we refer the reader to Step 2 of the proof of Lemma 2.4 in [3]. 3. Condition (D) In most cases of which we are aware a generalized Cuntz-Krieger algebra is generated by partial isometries with commuting range and

6 6 B. BURGSTALLER source projections. The aim of this section is to assume such a situation and to analyze the structure of A to obtain a condition (D) which implies (C ) and which may be easier to check sometimes. As proved in [4], the higher rank Exel-Laca algebras [4] satisfy (A), (B) and (D). Moreover, we will use condition (D) in the next section for C -algebras of labelled graphs. Let (A, F, I, H) be a system satisfying (A). By [3, Lemma 3.1] we have a balance function as described in the introduction. Define = { xx F/I x W }, A 0 = Alg ( ), and denote by P 0 the set of nonzero projections of A 0. Observe that A 0 A and P 0 P. Put Rg(s) = ss and Supp(s) = s s for a partial isometry s (i.e. ss s = s). We introduce the following properties. (D)(a) The alphabet A consists of partial isometries (i.e. aa a = a in F/I) and is a commuting set. (D)(b) If x W 0, p P 0 and Supp(pxp) = Rg(pxp) then pxp is a projection. (D)(c) If x W \W 0 and e P 0 then there exists p P 0 such that p e and pxp = 0. All three properties (D)(a)-(c) together are denoted by (D). Lemma 3.1. The following points hold when (D)(a) is satisfied. (1) Each word x W is a partial isometry,, and A 0 = Alg ( ) = lin( ) is a commutative -algebra. (2) If x W and p, q P 0 then px, xq and pxq are partial isometries with range and source projections in A 0. (3) The following two conditions are equivalent for x W and p P 0. First, px = xp and px x = pxx = p. Second, Supp(pxp) = Rg(pxp) = p. (4) Suppose that each x W 0 satisfying xx 0 is a projection. Then property (D)(b) holds. (5) Assume that there does not exist e P 0 and x W \W 0 such that e (x n x n )(x n x n ) for all n 1. Then property (D)(c) holds. (6) Under (D), the set of words W forms an inverse semigroup under multiplication.

7 CUNTZ KRIEGER UNIQUENESS THEOREMS 7 Proof. Points (1) and (2) are standard. Point (3) is playing around with partial isometries and projections (we skip the tedious proof). (4): We have to check (D)(b). Let x W 0 and p P 0. Assume that 0 Supp(pxp) = Rg(pxp). Then Supp(pxp)Rg(pxp) 0. Then xpx 0, and so x 2 0. By assumption x is a projection, and hence pxp is a projection. (5): We have to check (D)(c). Let e P 0 and x W \W 0. Let y = exe. If Rg(y) < e, and notice that necessarily Rg(y) e, then we put p = e Rg(y), and we get pxp = 0 as required in condition (D)(c). A similar argument works when Supp(y) < e. So assume the last possible case that Rg(y) = Supp(y) = e. Then ex = xe and ex x = exx = e by Lemma 3.1.(3). So we get e = exx = xex. Hence we obtain e = xex = xxex x =... = x n ex n x n x n and similarly e x n x n for all n 1. Consequently, e x n x n x n x n, which contradicts the assumption. (6): By Lemma 3.1.(1) we have inverses. By a well known characterization for inverse semigroups, it is enough to show that idempotent elements in W commute. Let x W and assume that xx = x. Then either x = 0, or bal(x) = 1. Applying Lemma 3.1.(3) to x = x and p = xx we get x = xx. Lemma 3.2. Suppose (A) and (B). Then A is the inductively ordered union of a family (M) M Ω of finite dimensional C -subalgebras M of A, where each M is the linear span of finitely many words in W. Proof. Given words x 1,..., x n W 0 define M to be the finite dimensional C -algebra generated by x 1,..., x n. Then M = lin(y ), where Y denotes the set of all finite products generated by {x i } {x j}. By choosing a finite linear basis in lin(y ), we obtain finitely many words Y 0 Y such that lin(y 0 ) = lin(y ) = M. Then we set Ω = {M = M x1,...,x n n 1, x i W 0 }. Proposition 3.3. Assume that (A), (B), (D)(a) and (D)(b) hold. (i) Then A is the inductively ordered union of a family (M) M Ω of finite dimensional C -subalgebras M A such that M A 0 is a maximal abelian subalgebra of M. (ii) In particular, each M Ω allows a representation M 1... M N of simple factors M t M (denote the generating matrix units of M t by {e t i,j} 1 i,j nt ) such that M A 0 = lin{ e t ii 1 t N, 1 i n t }.

8 8 B. BURGSTALLER (iii) In particular, for all p P there exists p 0 P 0 such that p 0 p in A. Proof. Step 1. Consider the family (M) M Ω of Lemma 3.2. Fix a finite dimensional C -algebra M Ω, and chose words x 1,..., x n W such that M = lin{x 1,..., x n }. Let C = M A 0 and notice that C is an abelian self-adjoint subalgebra of M. Enlarge C to a maximal abelian self-adjoint subalgebra Ĉ M. We want to show that C = Ĉ. It is sufficient if we can show that each minimal projection z of Ĉ is contained in C. Step 2. Let z Ĉ be a minimal projection of Ĉ. For each projection p C we either have zp = 0 or zp = z. We choose a representation n (4) z = α i p i x i q i i=1 where α i C and p i, q i C (for example put p i = x i x i and q i = x i x i ). By Lemma 3.1.(2) each p i x i q i is a partial isometry with range projection r i A 0 and source projection s i A 0. Hence r i, s i C. Let I C be the identity of C, and let p = I p C for p C. By representation (4) we have zi = z. Hence (5) z = IzI = (r 1 + r 1 )... (r n + r n )z(s 1 + s 1 )... (s n + s n ) = r 1 r 2... r n zs 1 s 2...s n r 1 r 2...r n zs 1 s 2... s n. Since z is minimal in Ĉ, each summand of (5) is either 0 or z, and consequently only one summand of (5) does not vanish. Hence for certain ɛ i, ν i {1, } we have (by commutativity) (6) z = r ɛ rn ɛn zs ν s νn n = r ɛ r ɛn n z = α i p ix i q i, i=1 n s ν s νn n zr ɛ r ɛn n s ν s νn where p i := r ɛ rn ɛn s ν s νn n p i C and q i := q i r ɛ rn ɛn s ν s νn n C. Notice that p i p i and q i q i. Now we replace representation (4) by representation (6), in other words we replace p i by p i and q i by q i in (4), and repeat the last procedure with this new representation (6) of z. We will then obtain new p i, q i C such that z = i p i x i q i. Then we apply the procedure again, and so on. The procedure is repeated until it no longer provides n,

9 CUNTZ KRIEGER UNIQUENESS THEOREMS 9 a new representation of z. Clearly this loop will halt after some finite steps since the projections p i p i p i... and q i q i q i... form a decreasing sequence in the finite dimensional C -algebra C. Step 3. We assume the loop has stopped with a representation as in (4), i.e. z = i p ix i q i for some p i, q i C. We apply the above procedure once more and obtain new r i, s i, ɛ i, ν i and clearly p i = p i and q i = q i. Then we have (since p i = p i and q i = q i) p i = r ɛ r ɛn n s ν s νn n p i q i = r ɛ r ɛn n s ν s νn n q i. Thus r i = Rg(p i x i q i ) p i r ɛ i i. If ɛ i = then such an inequality can only hold if r i = 0. Hence we have ɛ i = 1 and r i = p i for all i Γ where Γ = { j {1,..., n} α j p j x j q j 0 }. By a similar argument we get ν i = 1 and s i = q i for all i Γ. Hence r i = p i s ν j j = s j = q j r ɛ i i = r i for all i, j Γ. Hence p := p i = q i = r i = s i for all i Γ. This yields p = s i = Supp(p i x i q i ) = Supp(px i p) = Rg(px i p) for all i Γ. By condition (D)(b) we get px i p = p and consequently z = n α i p i x i q i = α i px i p = ( α i p = α i )p. i Γ i Γ i=1 Hence z = p C. This shows that C = Ĉ. Step 4. We have proved that C = M A 0 = Ĉ is a maximal abelian subalgebra of M. This proves the claim (i). Thus C contains the center C(M) of M, and for each minimal projection p C(M), pc C is a maximal abelian subalgebra of the simple factor pm. It is well-known, see for example [12, 11.2], that one can construct a matrix representation M = lin{e ij } of a simple finite dimensional C -algebra M (in our case M = pm) such that lin{e ii } = S for a beforehand given maximal subalgebra S (in our case S = pc) of M. This proves the point (ii). Hence for each projection p M there exists a projection q C such that q p in the algebra M. This proves the point (iii). i Γ

10 10 B. BURGSTALLER Corollary 3.4. Assume that the system (A, F, I, H) satisfies (A), (B) and (D). Then it also satisfies (C ). Moreover, a representation π : F/I A is faithful on A if and only if it is faithful on P 0. Proof. By Proposition 3.3.(iii), condition (D)(c) and Lemma 2.3 (one sets W := W \W 0 and P 2 := P 0 in condition (C )) we get condition (C ). Using Proposition 3.3, it is easy to check that π is faithful on A if and only if π is faithful on all matrix diagonal entries {e t ii} P 0 for all M Ω. 4. C -algebras of labelled graphs Our next aim is to prove two Cuntz-Krieger uniqueness theorems for C -algebras associated to weakly left-resolving labelled spaces (E, L, B) introduced by Bates and Pask in [1]. We will adopt the notations introduced in [1] and briefly recall it. The system E = (E 0, E 1, r, s) denotes a directed graph with range and source maps r, s : E 1 E 0. A labelled space consists of a system (E, L, B) where E is a directed graph, L : E 1 A is an (arbitrary) labelling map into an alphabet A, and B 2 E0 is an accommodation for (E, L) (see [1]). Write E = n 1 En for the set of finite paths in E. We will assume that A = L(E 1 ), and write L (E) = n 1 L(E1 ) n for the set of all finite labelled paths L(x 1 )... L(x n ) L(E 1 ) n (x i E 1, x 1 x 2... x n E n ). Labelled paths can be concatenated (written as a product) whenever the concatenation is also in L (E). We introduce source and range maps s, r : L (E) 2 E0 by s(α) = {s(x) L(x) = α, x E } and r(α) = {r(x) L(x) = α, x E } for all α L (E). The relative range of α L (E) with respect to A 2 E0 is defined to be r(a, α) = {r(x) E 0 s(x) A, L(x) = α, x E }. The labelled space (E, L, B) is called weakly left-resolving if for every A, B B and every α L (E) we have r(a, α) r(b, α) = r(a B, α). For A B one sets L 1 A = {b L(E 1 ) s(b) A }, and (E, L, B) is called set-finite if L 1 A is finite for all A B. For the definition of a representation (s a, p A ) a L(E 1 ),A B of (E, L, B) we refer to Definition 4.1 of [1]. We define L (E) = L (E) { } (not used in

11 CUNTZ KRIEGER UNIQUENESS THEOREMS 11 [1]), with the convention that α = α = α for all α L (E) and r( ) = 2 E0. By Theorem 4.5 of [1], there exists a representation (S a, P A ) a L(E 1 ),A B of (E, L, B) which does not annihilate any of the generators S α, P A. Hence the generators p A, s a of the universal representation C (E, L, B) (also denoted by C (s a, p A )) of (E, L, B) are all nonzero. In the rest of this section we will assume that the graph E has no sink (that is, every vertex emits an edge). Lemma 4.1. If A, B B and B A then p A p B > 0 and P A P B > 0. Proof. By the proof of Theorem 4.5 of [1] there is a representation S a, P A of (E, L, B) such that P A is the orthogonal projection onto H A = H (b,e), b L 1 v s(b) A {e L A 1 (b) s(e)=v} where each H (b,e) is a copy of an infinite dimensional Hilbert space. By assumption there is a v A\B, which (by assumption) also emits an edge e E 1 with label b = L(e). Clearly, the vertex v distinguishes H A from H B. We give here a combinatorial (though rather technical) condition for a labelled space. Definition 4.2. A labelled space (E, L, B) is called cancellable if for all α, µ, ν L (E) with µ ν, and all A, B, X B with B A and r(α) A r(α) B, there exist β L (E) with αβ L (E) and Â, ˆB B with B ˆB Â A such that and U := r(αβ) r(â, β) V := r(αβ) r( ˆB, β), (p U p V )s αβs µ p X s νs αβ (p U p V ) = 0. It is not difficult to check that the usual aperiodicity condition (that is, condition (L) in [8]) implies cancelability in case that E is an ordinary directed graph (meaning that L : E 1 E 1 is the identity map): Indeed, if r(α) A r(α) B then r(α) A\B and so s(β) = r(α) B,

12 12 B. BURGSTALLER which shows that U V if we choose  = A and ˆB = B. Further, we choose an aperiodic continuation αβ of α such that αβ µ ν and s αβ s µs νs αβ = 0. This proves the claim. Theorem 4.3. The C -algebra of a cancellable set-finite weakly leftresolving labelled space (E, L, B) satisfies the Cuntz-Krieger uniqueness theorem: a C -algebra C (s a, p A ) generated by a representation (s a, p A ) of (E, L, B) is canonically isomorphic to the universal C - algebra C (s a, p A ) if p A p B 0 for all A, B B with B A. Proof. Consider the alphabet A = { p A, s a A B, a L(E 1 ) }. Let σ : F C (E, L, B) be the canonical map into the universal C - algebra. Write I for the kernel of σ. It is clear that the canonical map π 1 : F/I C (E, L, B) derived from σ is injective, all identities introduced in [1, Definition 4.1] hold in F/I, and there is a canonical representation π 2 : F/I C (s a, p A ). Recall that each word x W F/I allows a representation x = s α p A s β (α, β L (E), A B) (with the convention that s = 1). Then the system (A, F, I, H) satisfies property (A) for H = {(λ a ) T L(E1 ) B λ a = λ b if a, b L(E 1 ), λ c = 1 if c B}, as by [1], after Lemma 4.8, there is a gauge action on C (E, L, B). One computes that H = T, and bal(s α p A s β ) = α β under the identification Ĥ = Z. Moreover, the system (A, F, I, H) satisfies property (B) (see the proof of [1, Theorem 5.3]), and the conditions (D)(a)-(D)(b) (follows immediately from Lemma 3.1.(4)). We are going to check condition (C ). Given a p P, there is some v = k i=1 γ kx i x i (γ i C, x i W ) in P 0 such that v p by Proposition 3.3.(iii). By choosing a common refinement of the commuting x i x i s and (1 x i x i ) s (1 i k), we may choose a u v, u P 0, of the form (7) u = s a0 p A0 s a 0 (1 s a1 p A1 s a 1 )... (1 s an p An s a n ) for some a i L (E), A i B. Using set-finiteness and Definition 4.1.(iv) of [1], we get (8) p A = p A s a s a, A B, a L(E 1 )

13 CUNTZ KRIEGER UNIQUENESS THEOREMS 13 and the appearing sum becomes actually finite. Define N = a 0... a n 1, and rewrite each s ai p Ai s a i (1 i n) as a (finite) sum (9) s ai p Ai s a i = s α p Ai,α s α α L (E N ) for some A i,α B by successive application of identity (8), for instance, ( (10) s ai p Ai s a i = s ai p Ai s a s a )s ai = s ai s a p r(ai,a)s as a i. a L(E 1 ) a L(E 1 ) Entering the formulas (9) into (7) and then expanding (7) yields (11) u = s α (p A0,α p A0,α A α )s α α L (E N ) for some A α B by the lattice rules for the map A p A. Since there is at least one non-vanishing summand q in (11), we can pick a q P 2 such that q u v p, where P 2 = { s α (p A p B )s α α L (E), A, B B, B A }\{0}. So we checked condition (C )(ii). Note that s α (p A p B )s α = 0 if and only if p r(α) (p A p B )p r(α) = 0 if and only if r(α) A = r(α) B by Lemma 4.1. Condition (C )(ii) shows that the C -representations π 1 and π 2, which are faithful on { p A p B A, B B, B A } (and hence faithful on P 2 by Lemma 4.1), are automatically faithful on A. Set W = W \W 0 in condition (C )(i). Given x = s µ p X s ν W (where µ ν ) and e = s α (p A p B )s α P 2, by the cancelability condition we may choose a β L (E) and Â, ˆB B such that e p := s αβ (p r( Â,β) p r( ˆB,β) )s αβ P 2, r(αβ) r(â, β) r(αβ) r( ˆB, β)) (this inequality implies p 0 by Lemma 4.1), and pxp = 0. This verifies condition (C )(iii). The claim then follows by Lemma 2.3 and Theorem 2.1.

14 14 B. BURGSTALLER Lemma 4.4. C (S a, P A ) is represented on a Hilbert space such that, with respect to strong operator topology, (12) P A S a Sa A B. a L(E 1 ) Proof. One checks that the representation constructed in Theorem 4.5 of [1] satisfies the claim. Without assuming set-finiteness we can say the following: Theorem 4.5. Let (E, L, B) be a cancellable weakly left-resolving labelled space. Let (S a, P A ) be a representation of (E, L, B) acting on Hilbert space and satisfying P A a L(E 1 ) S as a in the strong operator topology for all A B, and assume that P A P B > 0 for all B A. Then C (S a, P A ) and C (S a, P A ) are canonically isomorphic. Proof. (Sketch.) The proof is quite similar to the proof of Theorem 4.3 with the following adaption. Assume (S a, P A ) and (S a, P A ) are represented on Hilbert spaces H and H, respectively. Redefine s a and p A, where a L(E 1 ) and A B, by s a := λ T λs a λs a and p A := λ T P A P A, that is, they are represented on the Hilbert space λ T H H. On C (s a, p A ) we can define a gauge action Γ µ (s a ) = µs a, Γ µ (p A ) = p A (µ T), as Γ µ acts only as a shifting operator along the direct sum λ T. Moreover, one has p A a L(E 1 ) s as a (strong operator topology sum) for all A B. We copy then the proof of Theorem 4.3 two times, once putting (s a, p A ) := (S a, P A ) and the second time putting (s a, p A ) := (S a, P A ). We also replace in the original proof the universal C -algebra C (E, L, B) by our redefined C -algebra C (s a, p A ). This shows that C (s a, p A ) = C (S a, P A ) = C (S a, P A ). What we still have not said is, how we circumvent the problem that (E, L, B) is not set-finite. We do this by simply replacing in the identities (8), (9), (10) and (11) the appearing finite sums by (possibly infinite) strong operator topology sums. We remark that Bates and Pask also proved a Cuntz Krieger uniqueness theorem for labelled graph C -algebras in [2] completely independently from the author (and vice versa) at the same time (a previous version of this paper appeared as a preprint in November 2007).

15 CUNTZ KRIEGER UNIQUENESS THEOREMS 15 References [1] Bates, T., Pask, D., C -algebras of labelled graphs. J. Operator Th. 57, (2007). [2] Bates, T., Pask, D., C -algebras of labelled graphs II - simplicity results. Math. Scand., to appear. [3] Burgstaller, B., The uniqueness of Cuntz-Krieger type algebras. J. reine angew. Math. 594, (2006). [4] Burgstaller, B., A class of higher rank Exel-Laca algebras. Acta. Sci. Math. 73, (2007). [5] Cuntz, J., Krieger, W., A class of C -Algebras and Topological Markov Chains. Invent. math. 56, (1980). [6] Exel, R., Laca, M., Cuntz-Krieger algebras for infinite matrices. J. reine angew. Math. 512, (1999). [7] Kumjian, A., Pask, D., Higher rank graph C -algebras. New York J. Math. 6, 1-20 (2000). [8] Kumjian, A., Pask, D., Raeburn, I., Cuntz-Krieger algebras of directed graphs. Pacific J. Math. 184, No. 1, (1998). [9] Kumjian, A., Pask, D., Raeburn, I., Renault, J., Graphs, Groupoids, and Cuntz-Krieger algebras. J. Funct. Anal. 144, (1997). [10] Raeburn, I., Graph algebras. CBMS Regional Conference Series in Mathematics, 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the AMS, Providence, RI, [11] Robertson, G., Steger, T., Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras. J. reine angew. Math. 513, (1999). [12] Takesaki, M., Theory of operator algebras. I. Springer-Verlag, New York- Heidelberg, Institute of Mathematics, University of Münster, Einsteinstrasse 62, Münster, Germany address: bernhardburgstaller@yahoo.de