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Graph-Based Models for Kirchberg Algebras

We give a construction of Kirchberg algebras from graphs. By using product graphs in the construction we are able to provide models for general (UCT) Kirchberg algebras while maintaining the explicit generators and relations of the underlying graphs.

theorems -graph C * -algebras have free K 1 -groups, and hence cannot be used to model general Kirchberg algebras. More recently, Kumjian and Pask have introduced a notion of higher-rank graphs, or k-graphs, and their C * -algebras. These have many features in common with ordinary (or 1-) graphs, and allow for more general K 1 -groups. However they are much less flexible than 1-graphs, and the theory has not yet been developed as extensively. In particular, the case of higher rank graphs that are not row-finite, which is crucial for our methods, has not yet been treated.
In this paper we present examples of hybrid objects mixing elements of k-graphs of different ranks. While a general treatment seems beyond current technique (and perhaps not worth the considerable effort), the special situation developed here allows us to model arbitrary Kirchberg algebras with the same flexibility exhibited by ordinary graph algebras. In particular, we use this construction in [15] and [16] to prove interesting properties of general (UCT) Kirchberg algebras: (1) any prime-order automorphism of the K-theory of a UCT Kirchberg algebra is induced from an automorphism of the algebra having the same order (generalizing work of [2]); (2) a UCT Kirchberg algebra is weakly semiprojective if and only if its K-groups are direct sums of cyclic groups.
We hope that the interest of these applications will justify the work required to establish these models. The rest of the paper is organized as follows. We conclude the introduction by recalling the basic notions of graph C * -algebras in a form convenient for our purposes. In part two we construct the hybrid object underlying our algebras, and use it to define an r-discrete groupoid whose unit space is an appropriate set of paths. We remark that because the underlying object is not "row-finite," the paths we use may be finite, infinite, or semi-infinite. In part three we give generators and relations for the C * -algebra of this groupoid, prove the gauge-invariant uniqueness theorem, and show that the C * -algebra we have constructed is a UCT Kirchberg algebra. Finally in part four we compute the K-theory of the algebra, showing that it is equal to the direct sum of the K-theory of tensor products of the ordinary graph algebras used in the construction of the underlying object. It is here that the flexibility inherent in graph algebras may be used to construct our models of Kirchberg algebras with arbitrary K-theory.
We now briefly recall the main facts about (ordinary) graph algebras (see [13]). A directed graph E consists of two sets, E 0 (the vertices) and E 1 (the edges), together with two maps o, t : E 1 → E 0 (origin and terminus). A path of length n in E is a string e 1 e 2 · · · e n of edges with t(e i ) = o(e i+1 ), 1 ≤ i < n. We let E n denote the set of paths of length n, and E * the set of all finite paths; the origin and terminus maps extend to E * in the obvious way. For a vertex a ∈ E 0 we use the notation E n (a), respectively E * (a), for the set of paths in E of length n, respectively of arbitrary length, with origin a. We let O(E) denote the C * -algebra of E. It is the universal C * -algebra defined by generators P a a ∈ E 0 and S e e ∈ E 1 with the Cuntz-Krieger relations: • P a a ∈ E 0 are pairwise orthogonal projections.
• S * e S e = P t(e) , for e ∈ E 1 .
(These are a variant of the relations given in [13], Theorem 2.21.) The relationship between the C * -algebras of a graph and a subgraph are crucial to our methods. We refer to [13]. The results are as follows. Let E be a graph and let F be a subgraph of E. We let S = S(F ) be the set of vertices in F 0 that do not emit more edges in E than in F . We let T O(F, S) denote the relative Toeplitz Cuntz-Krieger algebra of F in E. It is the universal C * -algebra defined by generators P a a ∈ F 0 and S e e ∈ F 1 with the relations (as above) for O(F ), modified by requiring the fourth relation only if a ∈ S. Then T O(F, S) is the C * -subalgebra of O(E) generated by the projections and partial isometries associated to the vertices and edges of F ( [13], Theorem 2.35).
The figures in this paper were prepared with X Y -pic.

Groupoid Models for Kirchberg algebras
We will construct models for Kirchberg algebras by using a mixture of directed 1-graphs and k-graphs, as studied by Kumjian and Pask in [7]. Since neither the results of [7] nor of [13] directly apply in this situation, we will carry out the necessary constructions in detail. Our models will consist of a sequence of product k-graphs connected to each other by (ordinary) 1-graphs. In fact, all details of the argument are already present in the case of two product graphs of rank 2, and it is this case that we will treat. The argument in the general case is essentially identical to the one we will give. (We remark that in the general case one may attach other product k-graphs to u 0 in the same way that E 1 × F 1 is attached in what follows.) Figure 1).

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For i = 0, 1 let E i and F i be irreducible graphs. Let L i , respectively M i , denote the set of vertices in E 0 i , respectively F 0 i , emitting infinitely many edges. We assume that L i and M i are nonempty. We choose distinguished elements v i ∈ L i and w i ∈ M i . We attach the 2-graphs E i ×F i to the graph D by identifying The entire object comprising D, E 0 × F 0 , and E 1 × F 1 will be denoted Ω.
Definition 2.2. By a vertex we mean an element of where we identify u i and (v i , w i ). By an edge we mean an element of Definition 2.3. A finite path element of type D is a finite directed path in D of non-zero length. An infinite path element of type D is either an infinite directed path in D, or a finite path element of type D which terminates at u 0 or u 1 . A finite path element of type (E i , F i ) is an ordered pair (p, q) ∈ E * i × F * i such that p, q are not both of length zero. An infinite path element of type (E i , F i ) is an ordered pair (p, q), where p, respectively q, is either an infinite path or a finite path terminating in L i , respectively in M i , in E i , respectively F i , and p, q are not both of length zero.
For a path element (p, q) of type (E i , F i ) we define origin and terminus by If (p, q) and (p ′ , q ′ ) are path elements of type (E i , F i ) we say that (p, q) extends (p ′ , q ′ ) if p extends p ′ and q extends q ′ in the usual sense of paths in a directed graph.
A finite path is either a vertex, or a finite string µ 1 . . . µ k of finite path elements such that (i) t(µ i ) = o(µ i+1 ), and (ii) µ i and µ i+1 are of different types.
An infinite path is either a vertex in L i × M i , an infinite string of finite path elements satisfying the conditions (i) and (ii) above, or a finite sequence µ 1 . . . µ k+1 such that • µ 1 . . . µ k is a finite path • µ k+1 is an infinite path element • (i) and (ii) above hold. We let X denote the set of all infinite paths. We will use the notation µ ν to indicate that the path ν extends the path µ.
We wish to define a topology on X making it a locally compact metrizable space. First we will define the length function on paths.
(In the right hand side above we have used the symbol ℓ also for the usual length function on paths in a 1-graph. There should be no confusion resulting from this usage.) If µ = µ 1 . . . µ k is a finite path, we define We also will use the extension of ℓ to infinite paths, with values in Z + ∪ {∞} 2 , defined in the obvious way.
Lemma 2.5. ℓ is additive with respect to concatenation of paths.
Proof. The proof is left to the reader.
Definition 2.6. Let µ = µ 1 µ 2 · · · µ r be a finite path, decomposed as a string of finite path elements satisfying Definition 2.3(i) and (ii). We write |µ| = r, the number of finite path elements in µ.
We wish to define a groupoid having having X as its unit space. Since Ω is not a higher rank graph, the space of paths does not have the factorization property (Definition 1.1 of [7]). However we do have the following simple observation.
Lemma 2.7. Let µ and µ ′ be finite paths, and let x, x ′ be infinite paths, with Proof. The proof is accomplished by an easy induction on n = max{|µ|, |µ ′ |}.
We define certain subsets of Z(µ).
If t(µ) = (y, z) ∈ E 0 i × F 0 i , let B ⊆ E 1 (y) and C ⊆ E 1 (z) be finite subsets such that if y ∈ L i then B = ∅, and if z ∈ M i then C = ∅. Then we define: Remarks. We mention that the topology we intend to define on X requires that we be able to 'block' finitely many edges originating at a vertex of Ω that emits infinitely many edges. This explains our use of the notation V (µ; B, C). At a vertex emitting only finitely many edges, we can handle the situation by considering explicitly the finitely many possible extensions by an edge. Since we do not want to block all edges originating at such a vertex, we do not allow B and/or C to be nonempty at such a vertex. Notes 2.9. We note the following disjoint unions of sets.
(1) If t(µ) = u i for some i, then for B and C nonempty, (2) B is countable.
(3) A is a ring of sets.
(4) The intersection of any decreasing sequence of sets in B is nonempty.
Proof. A bit of thought will likely convince the reader that this is elementary. However we have found the details slightly messy, and we warn that the following is a bit of a slog. (1) and (2) are clear. For (3), we must show that if A, A ′ ∈ B then A ∩ A ′ , A \ A ′ ∈ A. We will first deal with intersections. Let A, A ′ ∈ B. First suppose that A = Z(µ) and A ′ = Z(µ ′ ). If µ µ ′ then A ∩ A ′ = A ′ . So suppose that µ, µ ′ are not comparable. We then have decomposed as a string of finite path elements satisfying Definition 2.3(i) and (ii), where µ k and µ ′ k are not comparable. We have Z(µ) ∩ Z(µ ′ ) = ∅, unless σ and σ ′ are absent, µ k and µ ′ k are of type (E, F ), and (2.11.2) µ k = µ(e 1 · · · e m , z) with m and n both nonzero. In this case Z(µ) ∩ Z(µ ′ ) = Z(µ ′′ ), where (2.11.3) µ ′′ = µ 1 · · · µ k−1 µ(e 1 · · · e m , f 1 · · · f n ).
, if σ 1 obeys the restrictions imposed by B and C ∅, otherwise.
Case (ii). Suppose that µ k µ ′ k . If µ k and µ ′ k are of type D then we have for the same reason as in case (i). If µ k and µ ′ k are of type (E, F ) we assume that µ obeys the restrictions imposed by B and C (since otherwise we would have A ∩ A ′ = ∅). If j > 0 or if ℓ( µ) ≥ (1, 1) then A ′ ⊆ A. Otherwise we have j = 0 and ℓ( µ) = (m, n) with exactly one of m, n equal to zero. For definiteness suppose n = 0. Then Now consider the situation where µ and µ ′ are not comparable. Write µ and µ ′ as in (2.11.1). Since A ⊆ Z(µ), we have A ∩ A ′ = ∅ unless we are in the situation of (2.11.2). In this case let µ ′′ be as in (2.11.3). We have . Again, we first consider the situation where µ and µ ′ are comparable. If µ = µ ′ then we have Suppose now that µ µ ′ Write µ and µ ′ as in (2.11.4). There are two cases. otherwise.
If j = 1 and ℓ(σ 1 ) = (m, n) where exactly one of m, n is nonzero, suppose without loss of generality that n = 0. Then Case (iv). (This case is nearly identical to the earlier case (ii).) Suppose that µ k µ ′ k . If µ k and µ ′ k are of type D then we have If µ k and µ ′ k are of type (E, F ) we assume that µ obeys the restrictions imposed by B and C (since otherwise we would have Otherwise we have j = 0 and ℓ( µ) = (m, n) with exactly one of m, n equal to zero. For definiteness suppose n = 0. Then Now consider the case where µ and µ ′ are not comparable. Write µ and µ ′ as in (2.11.1). As before, A ∩ A ′ = ∅ unless we have (2.11.2) with σ and σ ′ absent. In this case we have where µ ′′ is as in (2.11.3). This finishes the proof that A is closed under intersections. We now show that A \ A ′ ∈ A. We first suppose that A = Z(µ) and A ′ = Z(µ ′ ). If µ µ ′ let µ and µ ′ be as in (2.11.4). We have a disjoint union: Thus it suffices to consider Z(µσ) \ Z(µ). If σ = e 1 · · · e m is of type D, then and for e ∈ D 1 we have where in the second summation we let {e m+1 } and {f n+1 } denote the empty set. If µ and µ ′ are not comparable, an earlier part of the proof implies that A∩A ′ = ∅ unless µ and µ ′ are as in (2.11.1) and (2.11.2), with σ and σ ′ absent. Then and the differences in the above union are treated by Next suppose that A = V (µ; B, C) and A ′ = Z(µ ′ ). We have a disjoint union: We have already proved that the first piece is in A. We saw in Notes 2.9(1) that In the case A = V (µ; B, C) and A ′ = V (µ ′ ; B ′ , C ′ ), a similar argument together with the above case applies: This completes the proof of item (3) of the lemma. Finally we prove (4).
Proof. Since A is countable, we may list the elements or E as A 1 , A 2 , . . . . Thus our assumption on E takes the form Then the lemma will follow by Lemma 2.11 (4).
Let A 1 = k1 j=1 A 1j be written as a disjoint union of elements of B. We claim that there exists j 1 such that for all p ≥ 2, For if not, then for all j = 1, . . . , k 1 there exists p j such that A ′′ nj be written as a disjoint union of elements of B. We claim that there exists j n such that for all p ≥ n + 1, For suppose not. Then for all j = 1, . . . , k n there exists p j ≥ n + 1 such that Therefore j n exists as claimed, and we set A ′ n = A ′′ n,jn . Lemma 2.13. The collection, B, of all V (µ; B, C) is a base for a locally compact metrizable topology on X.
Proof. It follows from Lemma 2.11 (2) and (3) that B is a base for a second countable topology on X. Thus metrizability will follow from local compactness and the Hausdorff property. To establish local compactness we must show that if A ∈ B is covered by U ⊆ B, then A is finitely covered by U. Suppose not. Then for every finite subcollection F ⊆ U, To verify the Hausdorff property, let x = x ′ in X. If x and x ′ are comparable, say x x ′ . Then there exists a finite path element µ such that µ x and µ x ′ . Then Z(µ) and Z o(x) \ Z(µ) are disjoint neighborhoods of x and x ′ . If x and x ′ are not comparable, write x = νx 1 and x ′ = νx ′ 1 , where x 1 and x ′ 1 have no common nonzero initial subpath. Let ν 1 and ν ′ 1 be nonzero initial finite subpaths of x 1 and x ′ 1 . Then Z(νν 1 ) and Z(νν ′ 1 )\Z(νν 1 ) are disjoint neighborhoods of x and x ′ .
Lemma 2.17. G is topologically free, minimal, and locally contractive.
Proof. The proof is virtually identical to the (easy) proofs for graph algebras in [13].
is simple and purely infinite. Proof. This follows from [11] and [1] (see also [9]). We next define the gauge action of T 2 on C * (G). We will use the notation ζ m , m ∈ Z 2 , for the characters of T 2 : ζ m (z) = z m1 1 z m2 2 . Definition 2.19. The gauge action α : T 2 → Aut C * (G) is dual to the cocycle c. Thus for f ∈ C c (G), α z (f )(g) = ζ c(g) (z)f (g).

Generators and Relations
For a finite path µ we let s µ ∈ C c (G) denote the partial isometry s µ = χ U(µ,t(µ)) .
In the sequel we will let A denote C * (G). We wish to give a presentation of A by generators and relations. For this we recall the hybrid graph structure of Ω given in Definition 2.2.
for the set of all finite paths. For x ∈ Ω (0,0) we will write Ω (i,j) (x) = {µ ∈ Ω (i,j) : o(µ) = x}. We remark that this notation is patterned on that of [7]. Note that the "origin" of a path in a graph corresponds to its "range" when it is thought of as a morphism in a small category. Definition 3.3. We let S denote the set of symbols P x x is a vertex ∪ S y y is an edge .
We let R denote the following set of relations on S: (i) P x is a projection for every vertex x, S y is a partial isometry for every edge y. We let Θ = C * S, R denote the universal C * -algebra given by these generators and relations. For a finite path written as a product of edges: µ = y 1 y 2 · · · y k , we let S µ denote the product S y1 S y2 · · · S y k (it follows from the relation (v) that this is unambiguous). It is easily seen from Definition 2.8 and Definitions 2.16 that S µ → s µ determines a surjective * -homomorphism, π, of Θ onto A. We will show below (Corollary 3.19) that π is an isomorphism. First we need to study Θ more closely.
For the next lemma recall from Definition 2.6 the notation |µ| for the number of finite path elements in µ.
where µ n and ν n are of the same type. Moreover, if µ n and ν n are of type D then one extends the other, while if they are of type E i × F i then in each coordinate one extends the other. Proof.
It follows from Lemma 3.4 that Θ is spanned by elements of the form S µ S * ν for which t(µ) = t(ν). It also follows from the relations that there is an action, β, of T 2 on Θ, defined by β z (S µ S * ν ) = ζ ℓ(µ)−ℓ(ν) (z) S µ S * ν . We note that π : Θ → A is equivariant for β and α. We make some elementary computations in Θ × β T 2 . By means of the surjection π we see that the analogous results hold in A × α T 2 . The elements (in C(T 2 , Θ) ⊆ Θ × β T 2 ) of the form ζ m S µ S * ν , where µ, ν ∈ Ω * , make up a total subset of Θ × β T 2 . The fixed-point algebra Θ β sits inside Θ × β T 2 as the closed linear span of the constant functions S µ S * ν for which ℓ(µ) = ℓ(ν). We recall the formulas for multiplication and adjoint in C(T 2 , Θ) ⊆ Θ × β T 2 : Definitions 3.7. For i = 0, 1, let E i,1 ⊆ E i,2 ⊆ · · · and F i,1 ⊆ F i,2 ⊆ · · · be finite irreducible non-circuit subgraphs of E i and F i , with k E i,k = E i and k F i,k = F i . Let Ω k denote the subobject of Ω comprising D, E 0,k × F 0,k , and E 1,k × F 1,k . We further let Proposition 3.8. Θ β is an AF algebra, with {Θ β k : k = 1, 2, . . . } as an approximating system of finite dimensional C * -subalgebras.
Proof. Since π : Θ → A is equivariant, and Θ α is AF, then so is A α . For any r-discrete groupoid G with continuous cocycle c taking values in a discrete abelian group, and α the induced action of the dual group on C * (G), it is a fact that C * (G) α = C * c −1 (0) .
We now wish to show that π : Θ → A is an isomorphism. To accomplish this we need a detailed study of the finite dimensional approximating subalgebras of Θ β . (This analysis will be needed again in Lemma 4.2.) We begin with several definitions. We remark that in general, the structure of the finite dimensional subalgebras of the AF core of a Toeplitz graph algebra is made complicated by the fact that the Cuntz-Krieger relations are not satisfied at all vertices, and hence that there will be nonzero defect projections at some vertices. In Lemma 4.3 below we treat this situation, which is analogous to, but much easier, than the situation of Θ β . The case of Θ β is further complicated by the hybrid graph structure of Ω.
Remarks 3.14. Note that ω (0) k (u i ) is a projection by relation (iv) of Definition 3.3. We list some easy consequences of the definitions. Recall (from the end of section 1) that S(E i,k ) is the set of vertices of E i,k that do not emit edges in E i other than those emitted in E i,k .
We remark that µ ∈ X k is maximal if and only if one of the following occurs: (1) ℓ(µ) = (k, k).
Before proving (3) we make some preliminary observations. Let x ∈ Ω (0,0) k . Using Definition 3.13 and Remarks 3.14 we find that by Remarks 3.14. Finally, if t(µ 1 ) = u i and µ 3 ∈ {α i , ǫ i } then by Definition 3.13 and Definition 3.3 (iv). Hence for µ 3 ∈ Ω An analogous result holds with µ 2 replacing µ 1 and ρ replacing λ. Now we prove (3). Let x ∈ Ω . Use the above observations to eliminate zero terms and to simplify, then repeat. This process must stop in a finite number of steps, giving be an equivariant * -homomorphism between C * -dynamical systems. If φ Θ β is injective then φ is injective.
Proof. We have that φ : Θ × β T 2 → C × γ T 2 is injective, since it is so on the ideals The result now follows from Takesaki-Takai duality.
Corollary 3.19. A = C * (G) is isomorphic to Θ, and is simple, purely infinite, nuclear and classifiable, i.e. a UCT Kirchberg algebra.
Proof. Let λ : C * (G) → C * r (G) be the left regular representation. Then λ • π : Θ → C * r (G). The action of T 2 on C * (G) clearly descends equivariantly to C * r (G), so that λ • π is equivariant. We note that λ • π Θ β is injective. This follows from the facts that Θ β is AF, and that the minimal projections θ i (µ) (Proposition 3.17) in the subalgebras Θ β k have nonzero image in C c (G) α ⊆ C * r (G) α . Thus λ • π is injective by Theorem 3.18. It follows that λ is injective (this could also be deduced from nuclearity of C * (G), proved below, and [1]). We have that C * (G) × α T 2 = π(Θ× β T 2 ) is AF, and hence A, which is strongly Morita equivalent to A× α T 2 × α Z 2 , is nuclear and classifiable. Simplicity and pure infiniteness follow from Corollary 2.18.
The following proposition is necessary for our application of the results of this paper in [15]. The proof is immediate from the description of Θ by generators and relations. (We remark that in that application, the graphs F i will be chosen to represent Kirchberg algebras having K-theory of the form (Z, 0) or (0, Z), while the graphs E i will be chosen to represent Kirchberg algebras with preassigned K 0 and trivial K 1 .) Proposition 3.20. Let Γ i be a subgroup of Aut (E i ) fixing the vertex v i . There is a homomorphism Γ 0 × Γ 1 → Aut (Θ) defined on generators by letting Γ 0 act on E 0 , Γ 1 on E 1 , and the trivial action on D, F 0 and F 1 . Moreover, if x 0 ∈ E 0 0 is fixed by Γ 0 , then the corner of Θ defined by the projection P (x0,w0) is invariant for the action of Γ 0 × Γ 1 .

The K-theory of A
We may now omit the use of the * -isomorphism π, and identify Θ with A. It follows from the fact that the ideals I n are AF that the inclusion of I n into A × α T 2 induces an injection in K 0 . We let φ denote the automorphism α (1,1) * of K 0 (A × α T 2 ). We have We let W = K 0 (A× α T 2 ) and W n = K 0 (I n ). Thus W n ⊇ W n+1 and W = ∪ n∈Z W n . Since W 0 ∼ = K 0 (A α ), W 0 is generated by elements of the form [S µ S * µ ], where µ is a finite path. If t(µ) = t(ν) and ℓ(µ) = ℓ(ν) then S µ S * ν is a partial isometry in A α implementing an equivalence between S µ S * µ and S ν S * ν , so that , and f f * = S σµ S * σµ .
We let B denote the subalgebra of A generated by the edges of Ω in ∪ i E i × F i . Thus B is isomorphic to the direct sum of the algebras C * (E i ) ⊗ C * (F i ). We note that B is invariant under α. The crossed product B × α T 2 is an AF subalgebra of A × α T 2 , and is isomorphic to a direct sum of tensor products of AF algebras: Before giving the proof, we give a preliminary lemma describing the finite dimensional approximants to the AF core of the relative Toeplitz algebra of an ordinary graph. We remark that this is an easier version of Proposition 3.17.
For y ∈ E 0 \ S let ξ y ∈ C k (E, S) be given by For 0 ≤ j < k and y ∈ E 0 \ S let Then  Proof. We first note that ξ y S e = 0 for y ∈ E 0 \ S, e ∈ E 1 , It follows easily that S p ξ t(p) S * p S q ξ t(q) S * q = 0 iff p = q and t(p), t(q) ∈ E 0 \ S; S p S * p S q S * q = 0 if ℓ(p) = ℓ(q) = k and p = q; S p ξ t(p) S * p S q S * q = 0 if ℓ(p) < ℓ(q) and t(p) ∈ E 0 \ S.
Therefore the projections of N (k) are pairwise orthogonal. We now consider the minimality and equivalence of projections in N (k) together. For this, fix paths r, s ∈ ∪ j≤k E j with ℓ(r) = ℓ(s). Let t(p), t(q) ∈ E 0 \ S with ℓ(p), ℓ(q) < k, and suppose that (4.3.5) S p ξ t(p) S * p S r S * s S q ξ t(q) S * q = 0. Then (4.3.3) and (4.3.4) imply that r p, s q, and S * p S r = S * q S s . Hence t(p) = t(q) and ℓ(p) = ℓ(q), as required by (4.3.2). Moreover if p = q then the product in (4.3.5) equals S p ξ t(p) S * p , proving that S p ξ t(p) S * p is minimal. If p, q ∈ E k and if (4.3.6) S p S * p S r S * s S q S * q = 0, then r p, s q, and S * r S p = S * s S q , so t(p) = t(q) as required by (4.3.2). Moreover if p = q then the product in (4.3.6) equals S p S * p , proving that S p S * p is minimal. Finally, let t(p) ∈ E 0 \ S with ℓ(p) < k, and ℓ(q) = k, and consider S p ξ t(p) S * p S r S * s S q S * q . It follows from (4.3.3) and (4.3.4) that r p and s q.
Since ℓ(p) < k, it follows from the same considerations that ξ t(p) S * p S q ′ = 0. Thus S p ξ t(p) S * p and S q S * q are inequivalent. For the reverse implication in (4.3.2), note that if ℓ(p) = ℓ(q) < k and y = t(p) = t(q) ∈ E 0 \ S then (S q ξ y S * p ) * (S q ξ y S * p ) = S p ξ y S * p (S q ξ y S * p )(S q ξ y S * p ) * = S q ξ y S * q , and if ℓ(p) = ℓ(q) = k and t(p) = t(q), then Finally, we show that N (k) = 1. For convenience we will let ξ y = 0 for y ∈ S. Then for any y ∈ E 0 we have The result follows by summing over y ∈ E 0 .
Proof of Lemma 4.2. By repeated application of φ it suffices to show that i * : We define finite dimensional approximating subalgebras to B α in a manner similar to the Θ β k . Namely, let X (Thus X (0) k is the set of paths µ in X k that do not contain edges from D.) Now let We will show that the inclusion B α k ⊆ A α k induces an injection in K 0 . This will prove the lemma. We note that Thus every minimal projection in B α k is a tensor product of minimal projections from C k E i,k , S(E i,k ) and C k F i,k , S(F i,k ) , and two such are equivalent in B α k if and only if they are separately equivalent in each factor. Note that if (y, z) ∈ Ω For µ ∈ X (0) k with ℓ(µ) ≤ (k − 1, k − 1) and t(µ) = (y, z) with y ∈ S(E i,k ), z ∈ S(F i,k ), we define θ (0) 4 (µ) = S µ ω (0) k t(µ) S * µ ∈ B α k , analogously to Definition 3.16 (4). Then the projections in B α k of the form θ 1 (µ), θ 2 (µ), θ 3 (µ) and θ (0) 4 (µ) form a complete family of pairwise orthogonal minimal projections. By Lemma 4.3 we see that they satisfy the conditions for equivalence given in Proposition 3.17 (2). From Definition 3.13 we see that where ℓ(τ 1 ), ℓ(τ 2 ), . . . > ℓ(µ). This observation has the following consequence. Choose bases for K 0 (A α k ) and K 0 (B α k ) consisting of classes of minimal projections as above. If the bases are ordered by increasing length of the underlying paths, then the matrix of the map K 0 (B α k ) → K 0 (A α k ) induced from inclusion is lower triangular, with 1's on the diagonal. Thus the map is injective.
We let Y denote K 0 (B × α T 2 ). By Lemma 4.2 we may identify Y with i * (Y ) ⊆ W . We now give a key lemma, that is based on the fact that the (ordinary) graph D connecting the 2-graphs E i × F i is a "bit of O 2 ". Proof. Let x ∈ W . For n ∈ Z large enough we have φ n (x) ∈ W 0 . If (2φ−id)φ n (x) ∈ Y , then since Y is φ-invariant we get (2φ − id)(x) ∈ φ −n Y = Y . So we may assume that x ∈ W 0 . Since W 0 = K 0 (A α ) is generated by elements of the form [S µ S * µ ] for finite paths µ, we may assume that x = [S µ S * µ ]. Case (i). Suppose t(µ) ∈ E 0 i ×F 0 i for some i. Since the K 0 -class of the projection is unchanged if the path is replaced by a new path with the same length and terminus, we may assume that µ ∈ (E i × F i ) * , and so that x ∈ Y .
Case (ii). Suppose t(µ) ∈ {a 0 , a 1 }. For definiteness we suppose t(µ) = a 0 . For the rest of this argument we will omit the subscript on a 0 , β 0 , γ 0 and δ 0 . Then we may assume that µ = νβ m for some m ≥ 0 and some path ν with t(ν) = a. We note that Hence  Proof. Let x ∈ ker (id − φ). Then The preceding and following lemmas will allow us to show that the K-theory of A × α T 2 × φ Z is given by the subalgebra B. We let ψ = α (1,0) * , so that φ and ψ generate the action of Z 2 on W . We note that since B is invariant for α, Y is invariant for ψ as well as for φ. Proof. First we show that W = (id − φ)W + Y . Let x ∈ W . By Lemma 4.5 we have Applying φ −1 we see that x ∈ (id − φ)W + Y . Now we have We will show that Y ∩ (id − φ)W = (id − φ)Y , which will conclude the proof. The containment "⊇" is clear. For the containment "⊆", let y ∈ Y with y = (id − φ)(x) for some x ∈ W . Then φ(x) = y + (2φ − id)(x) ∈ Y , by Lemma 4.4. It follows that x ∈ Y , so that y ∈ (id − φ)Y . Proof. Lemma 4.5 and Lemma 4.6, and the Pimsner-Voiculescu exact sequence, show that K * (A × α T 2 × φ Z) and K * (B × α T 2 × φ Z) are isomorphic, equivariantly for ψ. Another application of Pimsner-Voiculescu, together with Takai-Takesaki duality, gives a commuting diagram of long exact sequences: It follows from the five lemma that K * (A) ∼ = K * (B).
Theorem 4.8. Let k ≥ 1 be given. For 0 ≤ i and 1 ≤ j ≤ k let E i,j be an irreducible directed graph with distinguished vertex v i,j emitting infinitely many edges. For i ≥ 0 let D i be a copy of the graph D in Definition 2.1 (with vertices u i−1 , u i , a i , a ′ i -see figure 2). Let Ω be the object obtained from the 1-graphs {D i } and the product k-graphs {E i1 × · · · × E i,k } by identifying the vertex u i with (v i,1 , . . . , v i,k ) as in Definition 2.1. Let A be the C * -algebra defined by the generators S and relations R as in Definition 3.3 (modified in the obvious way). Then A is the unique UCT Kirchberg algebra with K-theory equal to ∞ i=0 K * ⊗ k j=1 O(E i,j ) .