New York Journal of Mathematics

New York J. Math. 14 (2008) 1–31.

On the K-theory of higher rank graph C^{∗}-algebras

D. Gwion Evans

Abstract. Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C^{∗}-algebra, C^{∗}(Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C^{∗}(Λ). The K-groups of C^{∗}(Λ) for k > 2 can be calculated under certain circumstances and we consider the case k = 3. We prove that for arbitrary k, the torsion-free rank of K_{0}(C^{∗}(Λ)) and K_{1}(C^{∗}(Λ)) are equal when C^{∗}(Λ) is unital, and for k = 2 we determine the position of the class of the unit of C^{∗}(Λ) in K_{0}(C^{∗}(Λ)).

Contents

1. Introduction 1 2. Preliminaries 3 3. The K-groups of k-graph C^{∗}-algebras 6 4. The K-groups of unital k-graph C^{∗}-algebras 21 5. Applications and examples 24 References 30

1. Introduction

In [31] Spielberg realised that a crossed product algebra C(Ω) ⋊ Γ, where Ω is the boundary of a certain tree and Γ is a free group, is isomorphic to a Cuntz–Krieger algebra [6, 5]. Noticing that such a tree may be regarded as an affine building of type Ã_{1}, Robertson and Steger studied the situation when a group Γ acts simply transitively on the vertices of an affine building of type Ã_{2} with boundary Ω [23]. They found that the corresponding crossed

Received May 12, 2004. Revised December 18, 2007. Mathematics Subject Classification. Primary 46L80; Secondary 46L35. Key words and phrases. K-theory, C^{∗}-algebra, k-graphs, graph algebra. Research supported by the European Union Research Training Network in Quantum

Spaces — Noncommutative Geometry.

1^{ISSN 1076-9803/08}

product algebra C(Ω)⋊Γ is generated by two Cuntz–Krieger algebras. This led them to define a C^{∗}-algebra A via a finite sequence of finite 0-1 matrices (i.e., matrices with entries in {0, 1}) M_{1}, …, M_{r} satisfying certain conditions (H0)–(H3), such that A is generated by r Cuntz–Krieger algebras, one for each M_{1}, …, M_{r}. Accordingly they named their algebras higher rank Cuntz–Krieger algebras, the rank being r.

Kumjian and Pask [13] noticed that Robertson and Steger had constructed their algebras from a set, W of (higher rank) words in a finite alphabet A — the common index set of the 0-1 matrices — and realised that W could be thought of as a special case of a generalised directed graph — a higher rank graph or k-graph, where k is the rank. Subsequently, Kumjian and Pask associated a C^{∗}-algebra, C^{∗}(Λ) to a higher rank graph Λ and showed that A ≅ C^{∗}(W) [13, Corollary 3.5 (ii)]. Moreover, they derived a number of results elucidating the structure of higher rank graph C^{∗}-algebras. They show in [13, Theorem 5.5] that a simple, purely infinite k-graph C^{∗}-algebra C^{∗}(Λ) may be classified by its K-theory. This is a consequence of C^{∗}(Λ) satsifying the hypotheses of the Kirchberg–Phillips classification theorem ([12, 18]). Moreover, criteria on the underlying k-graph Λ were found that decided when C^{∗}(Λ) was simple and purely infinite (see [13, Proposition 4.8, Proposition 4.9] and [28, Proposition 8.8]). Thus a step towards the classification of k-graph C^{∗}-algebras is the computation of their K-groups.

In [25, Proposition 4.1] Robertson and Steger proved that the K-groups of a rank 2 Cuntz–Krieger algebra is given in terms of the homology of a certain chain complex, whoose differentials are defined in terms of M_{1} and M_{2}. Their proof relied on the fact that a rank 2 Cuntz–Krieger algebra is stably isomorphic to a crossed product of an AF-algebra by ℤ^{2}. We will generalise their method to provide explicit formulae for the K-groups of 2-graph C^{∗}-algebras and to gain information on the K-groups of k-graph C^{∗}-algebras for k > 2.

The rest of this paper is organised as follows. We begin in §2 by recalling the fundamental definitions relating to higher rank graphs and their C^{∗}- algebras we will need from [13].

On the K-theory of higher rank graph C^{∗}-algebras 3

In §3 we use the fact that the C^{∗}-algebra of a row-finite k-graph Λ with no sources is stably isomorphic to a crossed product of an AF algebra, B, by ℤ^{k} ([13, Theorem 5.5]) to apply a theorem of Kasparov [11, 6.10 Theorem] to deduce that there is a homological spectral sequence ([34, Chapter 5]) converging to K_{∗}(C^{∗}(Λ)) with initial term given by E_{p,q}^{2} ≅ H_{p}(ℤ^{k}, K_{q}(B)) (see [3] for the definition of the homology of a group G with coefficients in a left G-module M, denoted by H_{∗}(G, M)). We will see that it suffices to compute H_{∗}(ℤ^{k}, K_{0}(B)). It transpires that H_{∗}(ℤ^{k}, K_{0}(B)) is given by the so-called vertex matrices of Λ. These are matrices with nonnegative integer entries that encode the structure of the category Λ. Next we assemble the results of §3 and state them in our main theorem, Theorem 3.15. We then specialise to the cases k = 2 and k = 3. For k = 2 a complete description of the K-groups in terms of the vertex matrices can be given. For k = 3 we illustrate how Theorem 3.15 can be used to give a description of the K-groups of 3-graph C^{∗}-algebras under stronger hypotheses.

In section §4 we consider the K-theory of unital k-graph C^{∗}-algebras. We show that the torsion-free rank of K_{0}(C^{∗}(Λ)) is equal to that of K_{1}(C^{∗}(Λ)) when C^{∗}(Λ) is unital and give formulae for the torsion-free rank and torsion parts of the K-groups of 2-graph C^{∗}-algebras.

We conclude with §5, in which we consider some immediate applications to the classification of k-graph C^{∗}-algebras by means of the Kirchberg–Phillips classification theorem. We also consider some simple examples of K-group calculations using the results derived in the previous sections.

Acknowledgements. This paper was written while the author was an European Union Network in Quantum Spaces — Noncommutative Geometry funded post-doc at the University of Copenhagen. The paper develops a part of the author’s Ph.D. thesis, which was written under the supervision of David E. Evans at Cardiff University. We would like to take this opportunity to thank David for his guidance and support, and Johannes Kellendonk and Ryszard Nest for enlightening discussions on homological algebra. We would also like to express our gratitude to the members of the operator algebras groups in both universities, for maintaining stimulating environments for research. Finally, we thank the referee for their careful reading, comments and suggestions, which helped to clarify the exposition of the paper.

2. Preliminaries

By the usual slight abuse of notation we shall let the set of morphisms of a small category Λ be denoted by Λ and identify an object of Λ with its corresponding identity morphism. Also note that a monoid M (and hence a group) can be considered as a category with one object and morphism set equal to M, with composition given by multiplication in the monoid. For convenience of notation, we shall denote a monoid and its associated category by the same symbol.

The following notation will be used throughout this paper. We let ℕ denote the abelian monoid of nonnegative integers and we let ℤ be the group of integers. For a positive integer k, we let ℕ^{k} be the product monoid viewed as a category. Similarly, we let ℤ^{k} be the product group viewed, where appropriate, as a category. Let {e_{i}}_{i=1}^{k} be the canonical generators of ℕ^{k} as a monoid and ℤ^{k} as a group. Moreover, we choose to endow ℕ^{k} and ℤ^{k} with the coordinatewise order induced by the usual order on ℕ and ℤ, i.e., for all m, n ∈ ℤ^{k} m ≤ n ⟺ m − n ∈ ℕ^{k}.

We will denote by 𝕂(H) the C^{∗}-algebra of compact operators on a Hilbert space H. Where the Hilbert space H is separable and of infinite dimension we write 𝕂 for 𝕂(H).

The concept of a higher rank graph or k-graph (k = 1, 2, … being the rank) was introduced by A. Kumjian and D. Pask in [13]. We recall their definition of a k-graph. Definition 2.1 ([13, Definitions 1.1]). A k-graph (rank k graph or higher rank graph) (Λ, d) consists of a countable small category Λ (with range and source maps r and s respectively) together with a functor d : Λ ⟶ ℕ^{k} satisfying the factorisation property: for every λ ∈ Λ and m, n ∈ ℕ^{k} with d(λ) = m + n, there are unique elements μ, ν ∈ Λ such that λ = μν and d(μ) = m, d(ν) = n. For n ∈ ℕ^{k} and v ∈ Λ^{0} we write Λ^{n} := d^{−1}(n), Λ(v) := r^{−1}(v) and Λ^{n}(v) := {λ ∈ Λ^{n} | r(λ) = v}. Definition 2.2 ([13, Definitions 1.4]). A k-graph Λ is row-finite if for each n ∈ ℕ^{k} and v ∈ Λ^{0} the set Λ^{n}(v) is finite. We say that Λ has no sources if Λ^{n}(v) ≠ ∅ for all v ∈ Λ^{0} and n ∈ ℕ^{k}.

Unless stated otherwise, we will assume that each higher rank graph in this paper is row-finite with no sources. Furthermore, we shall denote such a generic higher rank graph by (Λ, d) (or more succinctly Λ with the understanding that the degree functor will be denoted by d).

We refer to [15] as an appropriate reference on category theory. There is no need for a detailed knowledge of category theory as we will be interested in the combinatorial graph-like nature of higher rank graphs. As the name suggests a higher rank graph can be thought of as a higher rank analogue of a directed graph. Indeed, every 1-graph is isomorphic (in the natural sense) to the category of finite paths of a directed graph ([13, Example 1.3]). By [13, Remarks 1.2] Λ^{0} is the set of identity morphisms of Λ. Indeed it is fruitful to view Λ^{0} as a set of vertices and Λ as a set of (coloured) paths with composition in Λ being concatention of paths. This viewpoint is discussed further in [7, 20].

In the sequel we will use the following higher rank graph constructions devised by Kumjian and Pask. For further examples of k-graphs see, for example, [13, 20, 21, 29]. Examples 2.3. (1) Let Δ_{k} be the category with morphism set equal to

{(m, n) ∈ ℤ^{k} × ℤ^{k} | m ≤ n}, object set equal to {(m, m) | m ∈ ℤ^{k}}, structure maps defined by r(m, n) = m, s(m, n) = n and composition defined by (m, l)(l, n) = (m, n) for all m, l, n ∈ ℤ^{k} such that m ≤ l ≤ n. One may define a degree functor d : Δ_{k} ⟶ ℕ^{k} by d(m, n) = n−m so that (Δ_{k}, d) is a k-graph. Furthermore, it is straightforward to check that (Δ_{k}, d) is row-finite and has no sources.

(2) The product higher rank graph ([13, Proposition 1.8]): Let (Λ_{1}, d_{1}) and (Λ_{2}, d_{2}) be rank k_{1} and k_{2} graphs respectively. Then their product higher rank graph (Λ_{1}×Λ_{2}, d_{1}×d_{2}) is a (k_{1}+k_{2})-graph, where Λ_{1}×Λ_{2} is the product category and the degree functor d_{1} × d_{2} : Λ_{1} × Λ_{2} ⟶ ℕ^{k_{1}+k_{2}} is given by d_{1} × d_{2}(λ_{1}, λ_{2}) = (d_{1}(λ_{1}), d_{2}(λ_{2})) ∈ ℕ^{k_{1}} × ℕ^{k_{2}} for λ_{1} ∈ Λ_{1} and λ_{2} ∈ Λ_{2}.

(3) The skew-product higher rank graph ([13, Definition 5.1]): Given a countable group G, a k-graph Λ and a functor c : Λ ⟶ G, the skewproduct k-graph G×_{c}Λ consists of a category with object set identified with G × Λ^{0} and morphism set identified with G × Λ. The structure maps are given by: s(g, λ) = (gc(λ), s(λ)) and r(g, λ) = (g, r(λ)). If s(λ) = r(μ) then (g, λ) and (gc(λ), μ) are composable in G ×_{c} Λ and (g, λ)(gc(λ), μ) = (g, λμ). The degree map is given by d(g, λ) = d(λ). Furthermore, G acts freely on G ×_{c} Λ by g · (h, λ) ↦ (gh, λ) for all g, h ∈ G and λ ∈ Λ (see [13, Remark 5.6] and its preceding paragraph).

To each row-finite k-graph with no sources, Kumjian and Pask associated an unique C^{∗}-algebra in the following way.

Definition 2.4. ([13, Definitions 1.5]) Let Λ be a row-finite k-graph with no sources. Then C^{∗}(Λ) is defined to be the universal C^{∗}-algebra generated by a family {s_{λ} | λ ∈ Λ} of partial isometries satisfying:

(i) {s_{v} | v ∈ Λ^{0}} is a family of mutually orthogonal projections. (ii) s_{λμ} = s_{λ}s_{μ} for all λ, μ ∈ Λ such that s(λ) = r(μ). (iii) s_{λ}^{∗}s_{λ} = s_{s(λ)} for all λ ∈ Λ. (iv) for all v ∈ Λ^{0} and n ∈ ℕ^{k} we have s_{v} = ∑_{λ∈Λ^{n}(v)} s_{λ}s_{λ}^{∗}.

For λ ∈ Λ, define p_{λ} := s_{λ}s_{λ}^{∗}. A family of partial isometries satisfying (i)–(iv) above is a called a ∗-representation of Λ.

We consider the following C^{∗}-algebras associated with the constructions noted in Examples 2.3, which will be useful in the sequel.

Examples 2.5. (1) Let Δ_{k} be the row-finite k-graph with no sources defined in Examples 2.3.1. Let e_{m,n} := s_{(m,q)}s_{(n,q)}^{∗}, m, n ∈ ℤ^{k}, q := sup{m, n}. Since {e_{m,n} | m, n ∈ ℤ^{k}} is a complete system of matrix units, we have C^{∗}(Δ_{k}) ≅ 𝕂(ℓ^{2}(ℤ^{k})) (cf. [13, Examples 1.7 (ii)]).

(2) Let (Λ_{i}, d_{i}) be a row-finite k_{i}-graph with no sources for i = 1, 2. Then

C^{∗}(Λ_{1} × Λ_{2}) ≅ C^{∗}(Λ_{1}) ⊗ C^{∗}(Λ_{2})

by [13, Corollary 3.5 (iv)].^{1}

(3) Let G be a countable group, Λ a row-finite k-graph with no sources and c : Λ ⟶ G a functor. Then the action of G on G×_{c}Λ described in Examples 2.3.3 induces an action β : G ⟶ Aut(C^{∗}(G×_{c}Λ)) such that β_{g}(s_{(h,λ)}) = s_{(gh,λ)}. Furthermore C^{∗}(G×_{c}Λ)⋊_{β}G ≅ C^{∗}(Λ)⊗𝕂(ℓ^{2}(G)) [13, Theorem 5.7].

1The C^{∗}-algebra of a row-finite higher rank graph with no sources is nuclear [13, Theorem 5.5].

3. The K-groups of k-graph C^{∗}-algebras

For the remainder of this paper we shall denote by B_{Λ} (or simply B when there is no ambiguity) the C^{∗}-algebra of the skew-product of a row-finite k-graph (Λ, d), with no sources, by ℤ^{k} via the degree functor regarded as a functor into ℤ^{k}, i.e., B := C^{∗}(ℤ^{k} ×_{d} Λ), and by β the action of ℤ^{k} on B as described in Examples 2.5.3. Note that by [13, Corollary 5.3 and Theorem 5.5] and Takesaki–Takai duality [32] (or [13, Theorem 5.7], cf. Examples 2.5.3), C^{∗}(Λ) is stably isomorphic to the crossed product of an AF-algebra, B, by ℤ^{k}, i.e., B ⋊_{β} ℤ^{k} ≅ C^{∗}(Λ) ⊗ 𝕂. Therefore

K_{0}(C^{∗}(Λ)) ≅ K_{0}(B ⋊_{β} ℤ^{k}).

It will be useful for us in the sequel to have an explicit description of how an isomorphism K_{0}(C^{∗}(Λ)) ⟶ K_{0}(B ⋊_{β} ℤ^{k}) acts on the K_{0}-class of a canonical projection p_{v}, v ∈ Λ^{0} in C^{∗}(Λ). To this end, we prefer to investigate how the isomorphism acts by using an alternative approach to that outlined above. This we do below by using a standard technique in kgraph C^{∗}-algebra theory, namely by utilising the gauge-invariant uniqueness theorem for k-graph C^{∗}-algebras [13, Theorem 3.4].

Theorem 3.1. Let Λ be a row-finite k-graph with no sources. Then there exists a group isomorphism Φ_{0} : K_{0}(B ⋊_{β} ℤ^{k}) ⟶ K_{0}(C^{∗}(Λ)) such that Φ_{0}([i_{B}(p_{(0,v)})]) = [p_{v}] for all v ∈ Λ^{0} (where we adopt the notation used in [19] for crossed product C^{∗}-algebras).

Proof. Let (B⋊_{β}ℤ^{k}, i_{B}, i_{ℤ^{k}}) be a crossed product for the dynamical system (B, ℤ^{k}, β) in the sense of [19]. One checks that

{t_{(λ,(m,n))} | (λ, (m, n)) ∈ Λ × Δ}

is a ∗-representation of Λ × Δ, where Δ := Δ_{k} (see Examples 2.3.1), and for (λ, (m, n)) ∈ Λ × Δ we let

t_{(λ,(m,n))} := i_{B}(s_{(m,λ)})i_{ℤ^{k}}(m + d(λ) − n).

Moreover, C^{∗}(t_{ξ} | ξ ∈ Λ × Δ) = B ⋊_{β} ℤ^{k}. Thus by the universal property of C^{∗}(Λ × Δ), there exists a ∗-homomorphism π : C^{∗}(Λ × Δ) ⟶ B ⋊_{β} ℤ^{k} such that π(s_{ξ}) = t_{ξ} for all ξ ∈ Λ × Δ. Let α : 𝕋^{k} ⟶ Aut(B) denote the canonical gauge action on B (see [13, §3]) and let β̂ : 𝕋^{k} ⟶ Aut(B ⋊_{β} ℤ^{k}) denote the dual action of β. There exists an action α̃ of 𝕋^{k} on B ⋊_{β} ℤ^{k} such that i_{B}α_{z} = α̃_{z}i_{B} for all z ∈ 𝕋^{k}. It is clear that setting γ_{(z_{1},z_{2})} := α̃_{z_{1}z_{2}} β̂_{z_{2}^{−1}} for all (z_{1}, z_{2}) ∈ 𝕋^{k} × 𝕋^{k} defines an action γ of 𝕋^{2k} on B ⋊_{β} ℤ^{k}. Moreover, it satisfies πα_{z}^{×} = γ_{z}π for all z ∈ 𝕋^{2k} where α^{×} is the canonical gauge action on C^{∗}(Λ×Δ). Clearly π(p_{v}) = 0 for all v ∈ Λ×Δ, hence by the gauge-invariant uniqueness theorem [13, Theorem 3.4] we see that π : C^{∗}(Λ×Δ) ⟶ B⋊_{β} ℤ^{k} is a ∗-isomorphism.

For the zero element 0 ∈ ℤ^{k}, we see that p_{(0,0)} is a minimal projection in C^{∗}(Δ) ≅ 𝕂 (cf. Examples 2.5.1). Therefore, the homomorphism given by x ↦ x ⊗ p_{(0,0)} induces an isomorphism K_{0}(C^{∗}(Λ)) ≅ K_{0}(C^{∗}(Λ) ⊗ C^{∗}(Δ)). The latter is isomorphic to K_{0}(C^{∗}(Λ×Δ)) (see 2.5.2) and K_{0}(B ⋊_{β} ℤ^{k}). Let Ψ : K_{0}(C^{∗}(Λ)) ⟶ K_{0}(B ⋊_{β} ℤ^{k}) be the composition of the preceding group isomorphisms. Then it follows easily that Ψ([p_{v}]) = [i_{B}(p_{(0,v)})]. Setting Φ_{0} := Ψ^{−1} completes the proof. □

Therefore we may apply [11, 6.10 Theorem] to describe the K-groups of C^{∗}(Λ) by means of a homological spectral sequence with initial term given by H_{p}(ℤ^{k}, K_{q}(B)), i.e., the homology of the group ℤ^{k} with coefficients in the left ℤ^{k}-module K_{q}(B) [3], where the ℤ^{k}-action is given by e_{i}·m = K_{0}(β_{e_{i}})(m) for i = 1, …, k, (cf. the proof of [25, Proposition 4.1]). First, we recall the definition of a homology spectral sequence from [34, §5] and the notion of convergence (see also [14]).^{2}

Definition 3.2. A homology spectral sequence (starting at E^{a}) consists of the following data:

(1) a family {E_{p,q}^{r}} of modules defined for all integers p, q and r ≥ a,

(2) maps d_{pq}^{r} : E_{p,q}^{r} ⟶ E_{p−r,q+r−1}^{r} that are differentials in the sense that d_{p−r,q+r−1}^{r}d_{pq} = 0,

(3) isomorphisms E_{pq}^{r+1} ⟶ ker(d_{pq}^{r})/ im(d_{p+r,q−r+1}^{r}). We will denote the above data by {(E^{r}, d^{r})}. The total degree of the term E_{pq}^{r} is n = p + q. The homology spectral sequence is said to be bounded if for each n there are only finitely many nonzero terms of total degree n in {E_{pq}^{r}}, in which case, for each p and q there is an r_{0} such that E_{pq}^{r} ≅ E_{pq}^{r+1} for all r ≥ r_{0}. We write E_{pq}^{∞} for this stable value of E_{pq}^{r}.

We say that a bounded spectral sequence converges to K_{∗} if we are given a family of modules {K_{n}}, each having a finite filtration

0 = F_{s}(K_{n}) ⊆ ⋯ ⊆ F_{p−1}(K_{n}) ⊆ F_{p}(K_{n}) ⊆ F_{p+1}(K_{n}) ⊆ ⋯ ⊆ F_{t}(K_{n}) = K_{n},

and we are given isomorphisms E_{pq}^{∞} ⟶ F_{p}(K_{p+q})/F_{p−1}(K_{p+q}).

Lemma 3.3 (cf. [25, Proposition 4.1]). There exists a spectral sequence {(E^{r}, d^{r})} converging to K_{∗}(C^{∗}(Λ)) := {K_{n}}_{n∈ℤ} where

K_{n} := \begin{cases}K_{0}(C^{∗}(Λ)) & if n is even, \\ K_{1}(C^{∗}(Λ)) & if n is odd. \end{cases}

Moreover, for p, q ∈ ℤ,

E_{p,q}^{2} ≅ \begin{cases}H_{p}(ℤ^{k}, K_{0}(B)) & if p ∈ \{0, 1, …, k\} and q is even, \\ 0 & otherwise, \end{cases}

2The reader will notice that the definition is presented in a less general form than in [34, §5], but is adequate for our purposes.

E_{p,q}^{∞} ≅ E_{p,q}^{k+1} and E_{p,q}^{k+1} = 0 if p ∈ ℤ\{0, 1, …, k} or q is odd.

Proof. The first assertion follows from an application of [11, 6.10 Theorem] to B ⋊_{β} ℤ^{k}, which is ∗-isomorphic to C^{∗}(Λ)⊗𝕂 by the proof of Theorem 3.1, after noting that K_{∗}(B ⋊_{β} ℤ^{k}) coincides with its “γ-part” since the Baum– Connes Conjecture with coefficients in an arbitrary C^{∗}-algebra is true for the amenable group ℤ^{k} for all k ≥ 1.

By the proof of [11, 6.10 Theorem], K_{∗}(B ⋊_{β} ℤ^{k}) ≅ K_{∗}(D) for some C^{∗}- algebra D which has a finite filtration by ideals: 0 ⊂ D_{0} ⊂ D_{1} ⊂ ⋯ ⊂ D_{k} = D since the dimension of the universal covering space of the classifying space of ℤ^{k} is k.

Here, we are considering the spectral sequence {(E^{r}, d^{r})} in homology K_{∗} associated with the finite filtration 0 ⊂ D_{0} ⊂ D_{1} ⊂ ⋯ ⊂ D_{k} = D of D ([27, §6]) which has E_{p,q}^{1} = K_{(p+q mod 2)}(D_{p}/D_{p−1}) where D_{n} = 0 for n < 0 and D_{n} = D for n ≥ k. It follows easily that E_{p,q}^{r} = 0 for p ∈ ℤ\{0, 1, …, k}, for all q ∈ ℤ and for all r ≥ 1 and E_{p,q}^{∞} ≅ E_{p,q}^{k+1} (see also [27, Theorem 2.1]). This combined with Kasparov’s calculation in the proof of [11, 6.10 Theorem], giving E_{p,q}^{2} ≅ H_{p}(ℤ^{k}, K_{q}(B)), along with the observation that K_{q}(B) = 0 for odd q, since B is an AF-algebra, proves the second assertion. □

Now we will compute H_{∗}(ℤ^{k}, K_{0}(B)) in terms of the combinatorial data encoded in Λ. First, let us examine the structure of B, and hence K_{0}(B), in a little more detail.

Lemma 3.4. Let Λ be a row-finite k-graph with no sources. Then

B = \overline{⋃_{n∈ℤ^{k}} B_{n}},

where

B_{n} = \overline{span}\{s_{λ}s_{μ}^{∗} | λ, μ ∈ ℤ^{k} ×_{d} Λ, s(λ)=s(μ)=(n, v) for some v ∈ Λ^{0}\}

≅ ⨁_{v∈Λ^{0}} B_{n}(v),

where

B_{n}(v) := \overline{span}\{s_{λ}s_{μ}^{∗} | λ, μ ∈ ℤ^{k} ×_{d} Λ, s(λ) = s(μ) = (n, v)\}
≅ 𝕂(ℓ^{2}(s^{−1}(v)))

for all v ∈ Λ^{0} and n ∈ ℤ^{k}.

Proof. Follows immediately from the proofs of [13, Lemma 5.4, Theorem 5.5] and the observation that for all n ∈ ℤ^{k} and v ∈ Λ^{0}, s^{−1}((n, v)) ⊂ ℤ^{k} ×_{d} Λ may be identified with s^{−1}(v) ⊂ Λ via (n − d(λ), λ) ↦ λ for all λ ∈ s^{−1}(v). □

Definition 3.5. Let ℤΛ^{0} be the group of all maps from Λ^{0} into ℤ that have finite support under pointwise addition. For each u ∈ Λ^{0}, we denote by δ_{u} the element of ℤΛ^{0} defined by δ_{u}(v) = δ_{u,v} (the Kronecker delta) for all v ∈ Λ^{0}. Note that ℤΛ^{0} is a free abelian group with free set of generators {δ_{u} | u ∈ Λ^{0}}.

Definition 3.6 (cf. [13, §6]). Define the vertex matrices of Λ, M_{i}, by the following. For u, v ∈ Λ^{0} and i = 1, 2, …, k,

M_{i}(u, v) := |{λ ∈ Λ^{e_{i}} | r(λ) = u, s(λ) = v}|.

Remarks 3.7. By the factorisation property, the vertex matrices of a kgraph pairwise commute [13, §6].

Convention 3.8. Given a matrix M with integer entries and index set Λ^{0}, by slight abuse of notation we shall, on occasion, regard M as the group endomorphism ℤΛ^{0} ⟶ ℤΛ^{0}, defined in the natural way as

(Mf)(u) = ∑_{v∈Λ^{0}} M(u, v)f(v) for all u ∈ Λ^{0}, f ∈ ℤΛ^{0}.

Lemma 3.9 (cf. [25] Lemma 4.5 and [17] Proposition 4.1.2). For all n, m ∈ ℤ^{k} such that m ≤ n, let A_{m} := ℤΛ^{0}. Moreover, define homomorphisms ȷ_{nm} : A_{m} ⟶ A_{n} by ȷ_{mm}(f) = f, ȷ_{m+e_{i},m} := M_{i}^{t} for all f ∈ A_{m}, u ∈ Λ^{0}, i ∈ {1, …, k}, and ȷ_{m+e_{i}+e_{j},m} := ȷ_{m+e_{i}+e_{j},m+e_{i}}ȷ_{m+e_{i},m} for all j ∈ {1, …, k}. Then (A_{m}; ȷ_{nm}) is a direct system of groups and

K_{0}(B) ≅ A := \underrightarrow{lim}(A_{m}; ȷ_{nm}).

Proof. It follows from Remarks 3.7 that the connecting homomorphisms are well-defined and that (A_{m}; ȷ_{nm}) is a direct system. From Lemma 3.4 we deduce that

K_{0}(B) ≅ \underrightarrow{lim}(K_{0}(B_{n}); K_{0}(ι_{n,m})),

where, for m, n ∈ ℤ^{k} with m ≤ n, ι_{n,m} : B_{m} ⟶ B_{n} are the inclusion maps [33, Proposition 6.2.9]. We also deduce that for n ∈ ℤ^{k} and v ∈ Λ^{0}, K_{0}(B_{n}(v)) is isomorphic to ℤ and is generated by the equivalence class comprising all minimal projections in B_{n}(v), of which p_{ξ} is a member for any ξ ∈ s^{−1}(n, v). Therefore, K_{0}(B_{n}) ≅ ⨁_{v∈Λ^{0}} K_{0}(B_{n}(v)) is generated by {[p_{(n,v)}]_{n} | v ∈ Λ^{0}}, where [ ·]_{n} denotes the equivalence classes of K_{0}(B_{n}) for all n ∈ ℤ^{k}. Thus the map ψ_{n} : A_{n} ⟶ K_{0}(B_{n}) given by f ↦ ∑_{u∈Λ^{0}} f(u)[p_{(n,u)}]_{n} is a group isomorphism for all n ∈ ℤ^{k}.

The embedding ι_{n,m} : B_{m} ⟶ B_{n} is given by

ι_{n,m}(s_{(m−d(λ),λ)}s_{(m−d(μ),μ)}^{∗}) = ∑_{α∈Λ^{n−m}(s(λ))} s_{(m−d(λ),λα)}s_{(m−d(μ),μα)}^{∗}

for all λ, μ ∈ Λ. Therefore,

K_{0}(ι_{n+e_{i},n}) ([p_{(n,v)}]_{n}) = [ι_{n+e_{i},n} (p_{(n,v)})]_{n+e_{i}} = [ ∑_{α∈Λ^{e_{i}}(v)} p_{(n,α)}]_{n+e_{i}}

= ∑_{u∈Λ^{0}} M_{i}(v, u)[p_{(n+e_{i},u)}]_{n+e_{i}}

and

K_{0}(ι_{n+e_{i},n}) ( ∑_{v∈Λ^{0}} f(v)[p_{(n,v)}]_{n}) = ∑_{u∈Λ^{0}} ( ∑_{v∈Λ^{0}} M_{i}(v, u)f(v)) [p_{(n+e_{i},u)}]_{n+e_{i}}

= ∑_{u∈Λ^{0}} ȷ_{n+e_{i},n}(f)(u)[p_{(n+e_{i},u)}]_{n+e_{i}}.

Thus the following square commutes for all i ∈ {1, 2, …, k}:

A_{n}  ⟶^{ȷ_{n+e_{i},n}}  A_{n+e_{i}}
ψ_{n}↓  ↓^{ψ_{n+e_{i}}}
K_{0}(B_{n}) ⟶^{K_{0}(ι_{n+e_{i},n})} K_{0}(B_{n+e_{i}}).

The result follows. □

Henceforth, we shall follow the notation introduced in Lemma 3.9 and its proof.

Now we begin to examine the action of ℤ^{k} on K_{0}(B) in terms of the description of K_{0}(B) provided by Lemma 3.9.

Lemma 3.10 (cf. [25] Lemma 4.10). Fix i ∈ {1, …, k} and define a homomorphism ϕ_{i,n} : A_{n} ⟶ A_{n} by ϕ_{i,n} := M_{i}^{t} for all n ∈ ℤ^{k}. Let ϕ_{i} : A ⟶ A be the homomorphism induced by the system of homomorphisms {ϕ_{i,n} | n ∈ ℤ^{k}}. Then ψϕ_{i} = K_{0}(β_{e_{i}})ψ, where ψ : A ⟶ K_{0}(B) is the isomorphism constructed in Lemma 3.9.

Proof. It follows from Remarks 3.7 that ϕ_{i,n}ȷ_{nm} = ȷ_{nm}ϕ_{i,m} for all m, n ∈ ℤ^{k} so that ϕ_{i} is well-defined for all i ∈ {1, …, k}.

Now, we let ψ̃: A ⟶ \underrightarrow{lim}(K_{0}(B_{m}); K_{0}(ι_{n,m})) be the unique isomorphism such that K_{0}(ι_{n}) ∘ ψ_{n} = ψ̃ ∘ ȷ_{n} for all n ∈ ℤ^{k} where

ψ_{n} : A_{n} ⟶^{≅} K_{0}(B_{n})
f ↦ ∑_{u∈Λ^{0}} f(u)[p_{(n,u)}]_{n}

(proof of Lemma 3.9). Then ψ : A ⟶ K_{0}(B) is the composition of ψ̃ with the canonical isomorphism \underrightarrow{lim}(K_{0}(B_{n}); K_{0}(ι_{n,m})) ≅ K_{0}(B).

We will show that

K_{0}(B_{n}) ⟶^{K_{0}(ι_{n})} K_{0}(B)
ϕ̃_{i,n}↓  ↓^{K_{0}(β_{e_{i}})}
K_{0}(B_{n}) ⟶^{K_{0}(ι_{n})} K_{0}(B)

commutes for all i = 1, 2, …, k and n ∈ ℤ^{k} where ι_{n} : B_{n} ⟶ B is the inclusion map for all n ∈ ℤ^{k} and ϕ̃_{i,n} = ψ_{n} ∘ ϕ_{i,n} ∘ ψ_{n}^{−1}. The lemma then follows from the universal properties of direct limits.

Fix i ∈ {1, …, k} and n ∈ ℤ^{k}. Since K_{0}(B_{n}) is generated by

{[p_{(n,v)}]_{n} | v ∈ Λ^{0}},

it suffices to show that K_{0}(β_{e_{i}}) ∘ K_{0}(ι_{n})([p_{(n,v)}]_{n}) = K_{0}(ι_{n}) ∘ ϕ̃_{i,n}([p_{(n,v)}]_{n}) for all v ∈ Λ^{0}. To see that this holds let v ∈ Λ^{0}. Then

K_{0}(β_{e_{i}}) ∘ K_{0}(ι_{n})([p_{(n,v)}]_{n}) = K_{0}(β_{e_{i}})([p_{(n,v)}]) = [p_{(n+e_{i},v)}],

while

\begin{aligned}K_{0}(ι_{n}) ∘ ϕ̃_{i,n}([p_{(n,v)}]_{n}) & = ∑_{u∈Λ^{0}} M_{i}(v, u)[p_{(n,u)}] \\ & = ∑_{α∈Λ^{e_{i}}(v)}[p_{(n+e_{i},α)}] = [p_{(n+e_{i},v)}]. \end{aligned}  □

Having established a description of K_{0}(B) as a left ℤ^{k}-module in terms of the structure of Λ, we are almost in a position to describe H_{∗}(ℤ^{k}, K_{0}(B)). First we recall some relevant notions from homological algebra.

It will be convenient to use multiplicative notation for the free abelian group ℤ^{k}, generated by k generators. Thus we set

G := ⟨s_{i} | s_{i}s_{j} = s_{j}s_{i} for all i, j ∈ {1, …, k}⟩

and R := ℤG, the group ring of G [3]. We shall compute H_{∗}(ℤ^{k}, K_{0}(B)) by means of a Koszul resolution K(x) for an appropriate regular sequence x on R, [34, Corollary 4.5.5].

By a regular sequence on R we mean a sequence x = {x_{i}}_{i=1}^{n} of elements of R such that:

(a) (x_{1}, …, x_{n}) ≠ R.

(b) For i = 1, …, n, x_{i} ∉ Z(R/(x_{1}, …, x_{i−1})). Here, we regard R as an R-module, write (x_{1}, …, x_{j}) for the ideal of R generated by {x_{i}}_{i=1}^{j}, and write Z(M) for the set of zero-divisors on a Rmodule M, i.e., Z(M) := {r ∈ R | r · m = 0 for some nonzero m ∈ M} (see [10, §3.1] for more details).

It is straightforward to check that for any finite set {t_{1}, …, t_{k}} of generators of G, the subset x = {x_{i}}_{i=1}^{k}, where x_{i} := 1 − t_{i} for i = 1, …, k, is a regular sequence on R.

Following [34, §4.5] we will describe the Koszul complex, K(x), in terms of the exterior algebra of a free R-module [2]. It will be convenient for us to describe the terms of the exterior algebra as follows.

Definition 3.11. For any nonnegative integer l, let E^{l}(R^{k}) denote the l^{th} term of the exterior algebra of the free R-module R^{k} := ⨁_{i=1}^{k} R_{i}, over R, where R_{i} = R for i = 1, …, k. Moreover, for any negative integer l, let E^{l}(R^{k}) = {0}. For l ∈ ℤ let

N_{l} := \begin{cases}\{(μ_{1}, …, μ_{l}) ∈ \{1, …, k\}^{l} | μ_{1} < ⋯ < μ_{l}\} & if l ∈ \{1, …, k\}, \\ \{⋆\} & if l = 0, \\ ∅ & otherwise. \end{cases}

For l ∈ {1, …, k} and μ = (μ_{1}, …, μ_{l}) ∈ N_{l}, i = 1, …, l we let

μ^{i} := \begin{cases}(μ_{1}, μ_{2} …, μ_{i−1}, μ_{i+1}, μ_{i+2} …, μ_{l}) ∈ N_{l−1} & if l ≠ 1, \\ ⋆ & if l = 1. \end{cases}

For n = 1, 2, … and r ∈ ℤ, let

\binom{n}{r} := \begin{cases}\frac{n!}{(n−r)!r!} & if 0 ≤ r ≤ n, \\ 0 & if r < 0 or r > n. \end{cases}

Using the above notation we may describe the l^{th}-term (l ∈ ℤ) of the exterior algebra over R^{k}, E^{l}(R^{k}), as being generated by the set N_{l} as a free R-module and having rank \binom{k}{l}.

Now let K(x) be the chain complex

0 ⟵ E^{0}(R^{k}) ⟵ E^{1}(R^{k}) ⟵ ⋯ ⟵ E^{k}(R^{k}) ⟵ 0

where the differentials E^{l}(R^{k}) ⟶ E^{l−1}(R^{k}) are given by mapping

μ ↦ ∑_{j=1}^{l}(−1)^{j+1}x_{μ_{j}}μ^{j} for all μ = (μ_{1}, …, μ_{l}) ∈ N_{l},

if l ∈ {1, …, k} and the zero map otherwise. By [34, Corollary 4.5.5] K(x) is a free resolution of R/I over R where I is the ideal of R generated by x. It is well-known (see, e.g., [34, Chapter 6],[3, §I.2]) that I = ker ϵ where ϵ : R ⟶ ℤ : g ↦ 1 is the augmentation map of the group ring R = ℤG. Thus we have a free (and hence projective) resolution of ℤ over ℤG, which we may use to compute H_{∗}(G, K_{0}(B)) (see [3, Chapter III]).

Lemma 3.12. Following the above notation, we have H_{∗}(G, K_{0}(B)) isomorphic to the homology of the chain complex

B : 0 ⟵ K_{0}(B) ⟵ ⋯ ⟵ ⨁_{N_{l}} K_{0}(B) ⟵ ⋯ ⟵ K_{0}(B) ⟵ 0,

where the differentials ˜∂_{l} : ⨁_{N_{l}} K_{0}(B) ⟶ ⨁_{N_{l−1}} K_{0}(B) (l ∈ {1, …, k}) are defined by

⨁_{μ∈N_{l}} m_{μ} ↦ ⨁_{λ∈N_{l−1}} ∑_{μ∈N_{l}} ∑_{i=1}^{l}(−1)^{i+1}δ_{λ,μ^{i}}(m_{μ} − K_{0}(β_{μ_{i}})(m_{μ})).

(Recall that the G-action on K_{0}(B) is given by s_{i} · m = K_{0}(β_{e_{i}})(m) for all m ∈ K_{0}(B), i = 1, …, k.)

Proof. By definition H_{∗}(G, K_{0}(B)) ≅ H_{∗}(K(x)⊗_{G}K_{0}(B)), where the latter chain complex is obtained by applying the functor − ⊗_{G} K_{0}(B) termwise to the chain complex K(x). The lemma follows via the canonical isomorphism between E^{l}(R^{k})⊗_{G} K_{0}(B) and ⨁_{N_{l}} K_{0}(B) (l ∈ {0, …, k}), setting t_{i} := s_{i}^{−1} (i = 1, …, k) as our generators of G to obtain x as described above. □

For m, n ∈ ℤ^{k} with m ≤ n, let A^{(n)} be the chain complex

0 ⟵ A_{n} ⟵ ⋯ ⟵ ⨁_{N_{l}} A_{n} ⟵ ⋯ ⟵ A_{n} ⟵ 0,

where l = 0, …, k, A_{n} = ℤΛ^{0} and for l > 0 the differentials

∂_{l}^{(n)} : ⨁_{N_{l}} A_{n} ⟶ ⨁_{N_{l−1}} A_{n}

are defined by

⨁_{μ∈N_{l}} m_{μ} ↦ ⨁_{λ∈N_{l−1}} ∑_{μ∈N_{l}} ∑_{i=1}^{l}(−1)^{i+1}δ_{λ,μ^{i}}(m_{μ} − ϕ_{μ_{i},n}(m_{μ})),

where for i = 1, …, k and n ∈ ℤ^{k}, ϕ_{i,n} is the homomorphism defined in Lemma 3.10. Furthermore, let (τ_{m}^{n})_{p} : A_{p}^{(m)} ⟶ A_{p}^{(n)} be the homomorphism defined by

(τ_{m}^{n})_{p} ( ⨁_{μ∈N_{p}} m_{μ}) = ⨁_{μ∈N_{p}} ȷ_{nm}(m_{μ})

for all p ∈ {0, 1, …, k} and the trivial map for p ∈ ℤ\{0, 1, …, k} (see Lemma 3.9).

Following [30, Chapter 4, §1], by a chain map τ : C ⟶ C′ we mean a collection {τ_{p} : C_{p} ⟶ C′_{p}} of homomorphisms that commute with the differentials in the sense that commutativity holds in each square:

C_{p} ⟶ C_{p−1}
τ_{p}↓  ↓^{τ_{p−1}}
C′_{p} ⟶ C′_{p−1}

Recall that there is a category of chain complexes whose objects are chain complexes and whose morphisms are chain maps. Moreover, the category of chain complexes admits direct limits.

Lemma 3.13. Following the above notation, for each m, n ∈ ℤ^{k} the system of homomorphisms {(τ_{m}^{n})_{p} | p ∈ ℤ} defines a chain map τ_{m}^{n} : A^{(m)} ⟶ A^{(n)} such that (A^{(n)}; τ_{m}^{n}) is a direct system in the category of chain complexes. Furthermore, (B; γ_{n}) is a direct limit for (A^{(n)}; τ_{m}^{n}), where γ_{n} : A^{(n)} ⟶ B is given by (γ_{n})_{p}(⨁_{μ∈N_{p}} m_{μ}) = ⨁_{μ∈N_{p}} ψȷ_{n}(m_{μ}) for all p ∈ {0, 1, …, k} and the trivial map otherwise.

Proof. That τ_{m}^{n} : A^{(m)} ⟶ A^{(n)} is a chain map for all m, n ∈ ℤ^{k}, follows immediately from the fact that ϕ_{i,n}ȷ_{nm} = ȷ_{nm}ϕ_{i,m} for all i ∈ {1, …, k} and m, n ∈ ℤ^{k} (cf. proof of Lemma 3.10). That (A^{(n)}; τ_{m}^{n}) is a direct system of chain complexes follows immediately from the fact that (A_{m}; ȷ_{nm}) is a direct system of groups (by Lemma 3.9).

Note that since K_{0}(β_{e_{i}})ψ = ψϕ_{i} for all i ∈ {1, …, k} by Lemma 3.10, a direct calculation shows that ˜∂_{p}(γ_{n})_{p} = (γ_{n})_{p−1}∂_{p}^{(n)} for all p ∈ ℤ^{k}, thus γ_{n} : A^{(n)} ⟶ B is a chain map for all n ∈ ℤ^{k}. The fact that γ_{m} = γ_{n}τ_{nm} for all m, n ∈ ℤ^{k} follows immediately by construction of the maps.

Now suppose that (A; τ_{n}) is a direct limit for (A^{(n)}; τ_{m}^{n}). Then by the above and the universal property of direct limits, there exists a morphism γ : A ⟶ B such that γτ_{m}^{n} = γ_{n} for all m, n ∈ ℤ^{k}. In order to show that γ is an isomorphism it suffices to show that each (γ)_{p} : A_{p} ⟶ B_{p} is an isomorphism for all p ∈ ℤ. This follows immediately from the fact that ψ : A ⟶ K_{0}(B) is an isomorphism (Lemma 3.9) and that direct limits commute with (finite) direct sums in the category of abelian groups in the obvious way. We have shown, therefore, that (B; γ_{n}) is a direct limit for (A^{(n)}; τ_{m}^{n}) in the category of chain complexes. □

Note that each chain complex A^{(n)} does not actually depend on n ∈ ℤ^{k}, thus for ease of notation we let D denote this common chain complex with differentials ∂_{p} := ∂_{p}^{(n)} for all p ∈ ℤ.

Theorem 3.14. Using the above notation, the homology of ℤ^{k} with coefficients in the left ℤ^{k}-module K_{0}(B) is given by the homology of the chain complex D, i.e., we have H_{∗}(G, K_{0}(B)) ≅ H_{∗}(D).

Proof. The homology functor commutes with direct limits ([30, Chapter 4, §1, Theorem 7]), therefore it follows that

H_{∗}(G, K_{0}(B)) ≅ \underrightarrow{lim}(H_{∗}(A^{(n)}); H_{∗}(τ_{m}^{n})).

Thus, it suffices to prove that H_{p}(τ_{m}^{m+e_{j}}) is the identity map for all p ∈ ℤ, m ∈ ℤ^{k}, j ∈ {1, …, k}. To prove that this is true, we fix j ∈ {1, …, k} and show that

(3.1) y − (τ_{m}^{m+e_{j}})_{p}y ∈ im ∂_{p+1} for y ∈ ker ∂_{p} and p ∈ {0, 1, …, k}.

Here, we recall that if y = ⨁_{μ∈N_{p}} y_{μ}, then

(τ_{m}^{m+e_{j}})_{p}y = ⨁_{μ∈N_{p}} M_{j}^{t}y_{μ},

so for a given p ∈ {0, 1, …, k}, (3.1) says

(3.2)  ⨁_{μ∈N_{p}}(1 − M_{j}^{t})y_{μ} ∈ im ∂_{p+1} for y = ⨁_{μ∈N_{p}} y_{μ} ∈ ker ∂_{p}.

The case p = 0 is obvious, so we concentrate on the case p ∈ {1, …, k}.

For μ ∈ N_{p}, let κ(μ) := ∑_{i=1}^{p} δ_{j,μ_{i}}i, i.e., if j is a component of μ, then κ(μ) denotes the unique i ∈ {1, …, p} such that μ_{i} = j; otherwise κ(μ) = 0. Now fix a μ′ = (μ′_{1}, …, μ′_{p}) ∈ N_{p} and let y = ⨁_{μ∈N_{p}} y_{μ} be in ker ∂_{p}.

First, suppose that κ(μ′) > 0 and let η = (μ′)^{κ(μ′)}. Then

0 = ∂_{p}(y)_{η} = ∑_{μ∈N_{p}} ∑_{i=1}^{p}(−1)^{i+1}δ_{η,μ^{i}}(1 − M_{μ_{i}}^{t})y_{μ}

= ∑_{i=1}^{p}(−1)^{i+1}δ_{η,(μ′)^{i}}(1 − M_{μ_{i}}^{t})y_{μ′} + ∑_{\begin{matrix}μ∈N_{p} \\ μ≠μ′ \end{matrix}} ∑_{i=1}^{p}(−1)^{i+1}δ_{η,μ^{i}}(1 − M_{μ_{i}}^{t})y_{μ}

= (−1)^{κ(μ′)+1}(1 − M_{j}^{t})y_{μ′} + ∑_{\begin{matrix}μ∈N_{p} \\ μ≠μ′ \end{matrix}} ∑_{i=1}^{p}(−1)^{i+1}δ_{η,μ^{i}}(1 − M_{μ_{i}}^{t})y_{μ},

so that

(1 − M_{j}^{t})y_{μ′} = ∑_{\begin{matrix}μ∈N_{p} \\ μ≠μ′ \end{matrix}} ∑_{i=1}^{p}(−1)^{i+κ(μ′)+1}δ_{η,μ^{i}}(1 − M_{μ_{i}}^{t})y_{μ}.

Note that the case p = k (where κ(μ) is necessarily greater than zero) follows immediately from the above. Therefore, we may assume that p ∈ {1, …, k − 1}, in which case we claim that

⨁_{μ∈N_{p}}(1 − M_{j}^{t})y_{μ} = ∂_{p+1} ( ⨁_{λ∈N_{p+1}} z_{λ})

where z_{λ} = ∑_{i=1}^{p+1}(−1)^{i+1}δ_{λ_{i},j}y_{λ^{i}} for all λ ∈ N_{p+1}.

Now

∂_{p+1} ( ⨁_{λ∈N_{p+1}} z_{λ})_{μ′} = ∑_{λ∈N_{p+1}} ∑_{i,r=1}^{p+1}(−1)^{i+r+2}δ_{μ′,λ^{i}}δ_{j,λ_{r}}(1 − M_{λ_{i}}^{t})y_{λ^{r}}

= ∑_{\begin{matrix}λ∈N_{p+1} \\ κ(λ)>0 \end{matrix}} ∑_{i=1}^{p+1}(−1)^{i+κ(λ)}δ_{μ′,λ^{i}}(1 − M_{λ_{i}}^{t})y_{λ^{κ(λ)}}

= ∑_{\begin{matrix}λ∈N_{p+1} \\ κ(λ)>0 \end{matrix}} \{ ∑_{i=1}^{κ(λ)−1}(−1)^{i+κ(μ′)+1}δ_{μ′,λ^{i}}(1 − M_{λ_{i}}^{t})y_{λ^{κ(λ)}}

+ ∑_{i=κ(λ)+1}^{p+1}(−1)^{i+κ(μ′)}δ_{μ′,λ^{i}}(1 − M_{λ_{i}}^{t})y_{λ^{κ(λ)}} }

= ∑_{\begin{matrix}μ∈N_{p} \\ κ(μ)=0 \end{matrix}} ∑_{i=1}^{p}(−1)^{i+κ(μ′)+1}δ_{η,μ^{i}}(1 − M_{μ_{i}}^{t})y_{μ}

since for every λ ∈ N_{p+1} such that κ(λ) > 0 and for every i ∈ {1, …, p + 1}\{κ(λ)} we have:

(1) δ_{μ′,λ^{κ(λ)}} = 0,

(2) κ(λ) = \begin{cases}κ(μ′) + 1 & if μ′ = λ^{i} with i < κ(λ), \\ κ(μ′) & if μ′ = λ^{i} with κ(λ) < i, \end{cases}

(3) μ′ = λ^{i} ⟺ η = \begin{cases}(λ^{κ(λ)})^{i} & if i < κ(λ), \\ (λ^{κ(λ)})^{i−1} & if κ(λ) < i, \end{cases}

and if μ ∈ N_{p} then κ(μ) = 0, η = μ^{i} for some i ∈ {1, …, p} ⟺ μ ≠ μ′, η = μ^{i} for some i ∈ {1, …, p}.

Hence,

∂_{p+1} ( ⨁_{λ∈N_{p+1}} z_{λ})_{μ′} = ∑_{\begin{matrix}μ∈N_{p} \\ μ≠μ′ \end{matrix}} ∑_{i=1}^{p}(−1)^{i+κ(μ′)+1}δ_{η,μ^{i}}(1 − M_{μ_{i}}^{t})y_{μ}

= (1 − M_{j}^{t})y_{μ′}.

Now, if κ(μ′) = 0, p ∈ {1, …, k − 1} and ⨁_{λ∈N_{p+1}} z_{λ} is as before, then

∂_{p+1} ( ⨁_{λ∈N_{p+1}} z_{λ})_{μ′} = ∑_{λ∈N_{p+1}} ∑_{i,r=1}^{p+1}(−1)^{i+r+2}δ_{μ′,λ^{i}}δ_{j,λ_{r}}(1 − M_{λ_{i}}^{t})y_{λ^{r}}

= (−1)^{κ(ξ)+κ(ξ)+2}(1 − M_{ξ_{κ(ξ)}}^{t})y_{ξ^{κ(ξ)}}
= (1 − M_{j}^{t})y_{μ′},

where ξ is the unique element of N_{p+1} satisfying κ(ξ) > 0 and ξ^{κ(ξ)} = μ′. □

Combining the results of this section we get the following theorem.

Theorem 3.15. Let Λ be a row-finite k-graph with no sources. Then there exists a spectral sequence {(E^{r}, d^{r})} converging to K_{∗}(C^{∗}(Λ)) with E_{p,q}^{∞} ≅ E_{p,q}^{k+1} and

E_{p,q}^{2} ≅ \begin{cases}H_{p}(D) & if p ∈ \{0, 1, …, k\} and q is even, \\ 0 & otherwise, \end{cases}

where D is the chain complex with

D_{p} := \begin{cases}⨁_{μ∈N_{p}} ℤΛ^{0} & if p ∈ \{0, 1, …, k\}, \\ 0 & otherwise, \end{cases}

and differentials

∂_{p} : D_{p} ⟶ D_{p−1} : ⨁_{μ∈N_{p}} m_{μ} ↦ ⨁_{λ∈N_{p−1}} ∑_{μ∈N_{p}} ∑_{i=1}^{p}(−1)^{i+1}δ_{λ,μ^{i}}(1 − M_{μ_{i}}^{t})m_{μ}

for p ∈ {1, …, k}.

Specialising Theorem 3.15 to the case when k = 2 gives us explicit formulae to compute the K-groups of the C^{∗}-algebras of row-finite 2-graphs with no sources.

Proposition 3.16. Let Λ be a row-finite 2-graph with no sources and vertex matrices M_{1} and M_{2}. Then there is an isomorphism

Φ : coker(1 − M_{1}^{t}, 1 − M_{2}^{t}) ⊕ ker \binom{M_{2}^{t} − 1}{1 − M_{1}^{t}} ⟶ K_{0}(C^{∗}(Λ))

such that Φ((δ_{u} + im ∂_{1}) ⊕ 0) = [p_{u}] for all u ∈ Λ^{0} (see Definition 3.5) and where we regard

(1 − M_{1}^{t}, 1 − M_{2}^{t}) : ℤΛ^{0} ⊕ ℤΛ^{0} ⟶ ℤΛ^{0},

\binom{M_{2}^{t} − 1}{1 − M_{1}^{t}} : ℤΛ^{0} ⟶ ℤΛ^{0} ⊕ ℤΛ^{0},

as group homomorphisms defined in the natural way.

Moreover, we have

K_{1}(C^{∗}(Λ)) ≅ ker(1 − M_{1}^{t}, 1 − M_{2}^{t})/ im \binom{M_{2}^{t} − 1}{1 − M_{1}^{t}}.

Proof. The Kasparov spectral sequence converging to K_{∗}(C^{∗}(Λ)) of Theorem 3.15 has E_{p,q}^{∞} ≅ E_{p,q}^{3} for all p, q ∈ ℤ. However, it follows from E_{p,q}^{2} = 0 for odd q that the differential d^{2} is the zero map and E_{p,q}^{3} ≅ E_{p,q}^{2} ≅ H_{p}(D) for all p ∈ {0, 1, …, k} and even q, where D is the chain complex

0 ⟵ ℤΛ^{0} ⟵^{∂_{1}} ℤΛ^{0} ⊕ ℤΛ^{0} ⟵^{∂_{2}} ℤΛ^{0} ⟵ 0

with ∂_{1} = (1 − M_{1}^{t}, 1 − M_{2}^{t}) and ∂_{2} = \binom{M_{2}^{t} − 1}{1 − M_{1}^{t}} for a suitable choice of

bases.

Convergence of the spectral sequence to K_{∗}(C^{∗}(Λ)) and the above means that we have the following finite filtration of K_{1} = K_{1}(C^{∗}(Λ)):

0 = F_{0}(K_{1}) ⊆ F_{1}(K_{1}) = F_{2}(K_{1}) = K_{1},

with F_{1}(K_{1}) ≅ H_{1}(D). Hence, K_{1}(C^{∗}(Λ)) ≅ H_{1}(D) as required.

Now, we could proceed to obtain an isomorphism of K_{0}(C^{∗}(Λ)) by use of the spectral sequence; however we choose to use the Pimsner–Voiculescu sequence in order to deduce relatively easily the action of the isomorphism on basis elements.

By applying the Pimsner–Voiculescu sequence to B ⋊ ℤ and

B ⋊ ℤ^{2} ≅ (B ⋊ ℤ) ⋊ ℤ

in succession, we may deduce that K_{0}(i_{B}) : K_{0}(B) ⟶ K_{0}(B ⋊ ℤ^{2}) factors through an injection Φ_{1} : H_{0}(B) ⟶ K_{0}(B ⋊ ℤ^{2}), where i_{B} is the canonical injection i_{B} : B ⟶ B ⋊ ℤ^{2} and B is the chain complex defined in Lemma 3.12. We may also deduce that there is an exact sequence that constitutes the first row of the following commutative diagram:

0 ⟶ H_{0}(B) ⟶^{Φ_{1}} K_{0}(B ⋊ ℤ^{2}) ⟶ H_{2}(B) ⟶ 0
Φ_{2}↓  ↓^{Φ_{0}}  ↓
0 ⟶ H_{0}(D) ⟶ K_{0}(C^{∗}(Λ)) ⟶ H_{2}(D) ⟶ 0.

where all downward arrows represent isomorphisms. In particular,

Φ_{0} : K_{0}(C^{∗}(Λ)) ⟶ K_{0}(B ⋊ ℤ^{2})

is the isomorphism constructed in Theorem 3.1 and

Φ_{2} : H_{0}(B) ⟶ H_{0}(D)

is one of the isomorphisms in Theorem 3.14.

Now H_{2}(D) ⊆ ℤΛ^{0} is a free abelian group, thus the exact sequences split and we have an isomorphism Φ : H_{0}(D) ⊕ H_{2}(D) ⟶ C^{∗}(Λ) such that Φ(g ⊕ 0) = Φ_{0}Φ_{1}Φ_{2}^{−1}(g) for all g ∈ H_{0}(D). It is straightforward to check that Φ(δ_{u} +im ∂_{1} ⊕0) = [p_{u}] for all u ∈ Λ^{0}. Thus the theorem is proved. □

Evidently complications arise when k > 2, however it is worth noting that under some extra assumptions on the vertex matrices it is possible to determine a fair amount about the K-groups of higher rank graph C^{∗}- algebras. For example, the case k = 3 is considered below.

Proposition 3.17. Let Λ be a row-finite 3-graph with no sources. Consider the following group homomorphisms defined by block matrices:

∂_{1} = (1 − M_{1}^{t} 1 − M_{2}^{t} 1 − M_{3}^{t}) : ⨁_{i=1}^{3} ℤΛ^{0} ⟶ ℤΛ^{0},

∂_{2} = (\begin{matrix}M_{2}^{t} − 1 & M_{3}^{t} − 1 & 0 \\ 1 − M_{1}^{t} & 0 & M_{3}^{t} − 1 \\ 0 & 1 − M_{1}^{t} & 1 − M_{2}^{t} \end{matrix}) : ⨁_{i=1}^{3} ℤΛ^{0} ⟶ ⨁_{i=1}^{3} ℤΛ^{0},

∂_{3} = (\begin{matrix}1 − M_{3}^{t} \\ M_{2}^{t} − 1 \\ 1 − M_{1}^{t} \end{matrix}) : ℤΛ^{0} ⟶ ⨁_{i=1}^{3} ℤΛ^{0}.

There exists a short exact sequence:

0 ⟶ coker ∂_{1}/G_{0} ⟶ K_{0}(C^{∗}(Λ)) ⟶ ker ∂_{2}/ im ∂_{3} ⟶ 0,

and

K_{1}(C^{∗}(Λ)) ≅ ker ∂_{1}/ im ∂_{2} ⊕ G_{1},

where G_{0} is a subgroup of coker ∂_{1} and G_{1} is a subgroup of ker ∂_{3}.

Proof. By Theorem 3.15, there exist short exact sequences

0 ⟶ E_{0,0}^{4} ⟶ K_{0}(C^{∗}(Λ)) ⟶ E_{2,−2}^{4} ⟶ 0,
0 ⟶ E_{1,0}^{4} ⟶ K_{1}(C^{∗}(Λ)) ⟶ E_{3,−2}^{4} ⟶ 0.

However, since E_{p,q}^{4} = 0 if p ∈ ℤ\{0, 1, 2, 3}, the only nonzero components of the differential d^{3} are d_{3,q}^{3} : E_{3,q}^{3} ⟶ E_{0,q+2}^{3}, where q ∈ 2ℤ. Moreover, as in the proof of Proposition 3.16, the differential d^{2} is the zero map. Thus we have

\begin{matrix}E_{1,0}^{4} ≅ E_{1,0}^{3} ≅ E_{1,0}^{2} ≅ H_{1}(D), & E_{0,0}^{3} ≅ E_{0,0}^{2} ≅ H_{0}(D), \\ E_{2,−2}^{4} ≅ E_{2,−2}^{3} ≅ E_{2,−2}^{2} ≅ H_{2}(D), & E_{3,−2}^{3} ≅ E_{3,−2}^{2} ≅ H_{3}(D). \end{matrix}

Also note that E_{3,−2}^{4} is isomorphic to ker d_{3,−2}^{3} ⊆ E_{3,−2}^{3}, which is a subgroup of the free abelian group H_{3}(D) ≅ ker ∂_{3}. Thus E_{3,−2}^{4} is itself a free abelian group, from which we deduce that the exact sequence for K_{1}(C^{∗}(Λ)) splits. Hence the result follows by setting G_{0} to be the image of im d_{3,−2}^{3} ⊆ E_{0,0}^{3} under the isomorphism E_{0,0}^{3} ⟶ E_{0,0}^{2} ⟶ H_{0}(D), and G_{1} to be the image of ker d_{3,−2}^{3} ⊆ E_{3,−2}^{3} under the isomorphism E_{3,−2}^{3} ⟶ E_{3,−2}^{2} ⟶ H_{3}(D). □

Now we consider two cases for which we can describe the K-groups of the C^{∗}-algebra of a row finite 3-graph with no sources in terms of its vertex matrices by the immediate application of Proposition 3.17.

Corollary 3.18. In addition to the hypothesis of Proposition 3.17: (1) If ∂_{1} is surjective then

K_{0}(C^{∗}(Λ)) ≅ ker ∂_{2}/ im ∂_{3},
K_{1}(C^{∗}(Λ)) ≅ ker ∂_{1}/ im ∂_{2} ⊕ ker ∂_{3}.

(2) If ∩_{i=1}^{3} ker(1 − M_{i}^{t}) = 0 then

K_{1}(C^{∗}(Λ)) ≅ ker ∂_{1}/ im ∂_{2}

and there exists a short exact sequence

0 ⟶ coker ∂_{1} ⟶ K_{0}(C^{∗}(Λ)) ⟶ ker ∂_{2}/ im ∂_{3} ⟶ 0.

Proof. To prove (1), note that we have 0 = coker ∂_{1} thus the exact sequence for K_{0}(C^{∗}(Λ)) collapses to give the result for K_{0}(C^{∗}(Λ)). Also note that 0 = coker ∂_{1} = H_{0}(D) ≅ E_{0,0}^{3}. Therefore d_{3,−2}^{3} : E_{3,−2}^{3} ⟶ E_{3,0}^{3} is the zero map and ker d_{3,−2}^{3} = E_{3,−2}^{3} ≅ E_{3,−2}^{2} ≅ H_{3}(D) = ker ∂_{3}. Therefore, G_{1} in Proposition 3.17 is ker ∂_{3} and (1) is proved.

To prove (2), if ⋂_{i=1}^{3} ker(1 − M_{i}^{t}) = 0 then ker ∂_{3} = 0, which implies that G_{1} in Proposition 3.17 is the trivial group. It also follows that E_{3,−2}^{3} = 0 so that im d_{3,−2}^{3} = 0 and G_{0} in Proposition 3.17 is the trivial group. Whence (2) follows immediately from Proposition 3.17. □

Remarks 3.19. (i) One may recover [16, Theorem 3.1] from Theorem

3.15 by setting k equal to 1. (ii) By [13, Corollary 3.5 (ii)] a rank k Cuntz–Krieger algebra ([24, 25]) is

isomorphic to a k-graph C^{∗}-algebra. Thus, Propsition 3.16 generalises [25, Proposition 4.1], the proof of which inspired the methods used throughout this paper.

(iii) By showing that the C^{∗}-algebra of a row-finite 2-graph, Λ, with no sources and finite vertex set, satisfying some further conditions, is isomorphic to a rank 2 Cuntz–Krieger algebra, Allen, Pask and Sims used Robertson and Steger’s [25, Proposition 4.1] result to calculate the K-groups of C^{∗}(Λ) [1, Theorem 4.1]. Moreover, in [1, Remark 4.7. (1)] they note that their formulae for the K-groups holds for more general 2-graph C^{∗}-algebras, namely the C^{∗}-algebras of row-finite 2graphs, Λ, with no sinks (i.e., s^{−1}(v) ∩ Λ^{n} ≠ ∅ for all n ∈ ℕ^{k}, v ∈ Λ^{0}) nor sources and finite vertex set.

(iv) The notion of associating a C^{∗}-algebra, C^{∗}(Λ), to a k-graph Λ was generalised by Raeburn, Sims and Yeend [20] to include the case where Λ is finitely-aligned; a property identified by them to enable an appropriate C^{∗}-algebra to be constructed. The family of finitely-aligned kgraphs and their associated C^{∗}-algebras admit k-graphs with sources and those that are not row-finite. In [8], Farthing devised a method of constructing, from an arbitrary finitely-aligned k-graph Λ with sources, a row-finite k-graph with no sources, Λ̄, which contains Λ as a subgraph. If, in addition, Λ is row-finite then Farthing showed that C^{∗}(Λ̄) is strong Morita equivalent to C^{∗}(Λ) and thus has isomorphic K-groups to those of C^{∗}(Λ). Therefore, in principle, the results in this paper could be extended to the case where Λ is row-finite but with sources.

4. The K-groups of unital k-graph C^{∗}-algebras

Recall that if Λ is a row-finite higher rank graph with no sources then Λ^{0} finite is equivalent to C^{∗}(Λ) being unital ([13, Remarks 1.6 (v)]). Thus in this section we specialise in the case where the vertex set of our higher rank graph, hence each vertex matrix, is finite. We will continue to denote the Kasparov spectral sequence converging to K_{∗}(C^{∗}(Λ)) of the previous section by {(E^{r}, d^{r})} and we shall denote the torsion-free rank of an abelian group G by r_{0}(G) (see, e.g., [9]).

Proposition 4.1. If Λ is a row-finite higher rank graph with no sources and Λ^{0} finite then K_{0}(C^{∗}(Λ)) and K_{1}(C^{∗}(Λ)) have equal torsion-free rank.

Proof. Let the rank of the given higher rank graph Λ be k and let |Λ^{0}| = n.

Since E_{p,q}^{∞} ≅ E_{p,q}^{k+1} for all p, q ∈ ℤ and E_{p,q}^{k+1} = 0 if p ∈ ℤ\{0, 1, …, k} or q odd by Lemma 3.3, it follows from the definition of convergence of {(E^{r}, d^{r})} (Definition 3.2) that there exist finite filtrations,

0 = F_{−1}(K_{0}) ⊆ E_{0,0}^{k+1} ≅ F_{0}(K_{0}) ⊆ F_{1}(K_{0}) ⊆⋯⊆ F_{k−1}(K_{0}) ⊆ F_{k}(K_{0}) = K_{0},

and

0 = F_{0}(K_{1}) ⊆ E_{1,0}^{k+1} ≅ F_{1}(K_{1}) ⊆ F_{2}(K_{1}) ⊆ ⋯ ⊆ F_{k−1}(K_{1}) ⊆ F_{k}(K_{1}) = K_{1}

of K_{0} = K_{0}(C^{∗}(Λ)) and K_{1} = K_{1}(C^{∗}(Λ)) respectively, such that

E_{p,q}^{k+1} ≅ F_{p}(K_{p+q})/F_{p−1}(K_{p+q}).

Thus,

r_{0}(K_{0}(C^{∗}(Λ))) = r_{0}(F_{k}(K_{0})) = r_{0}(F_{k−1}(K_{0})) + r_{0}(E_{k,−k}^{k+1}) = ⋯

= r_{0}(F_{0}(K_{0})) + ∑_{s≥1} r_{0}(E_{s,−s}^{k+1}) = ∑_{s∈ℤ}r_{0}(E_{s,−s}^{k+1}),

and

r_{0}(K_{1}(C^{∗}(Λ))) = r_{0}(F_{k}(K_{1})) = r_{0}(F_{k−1}(K_{1})) + r_{0}(E_{k,−k+1}^{k+1}) = ⋯

= r_{0}(F_{1}(K_{1})) + ∑_{s≥2} r_{0}(E_{s,−s+1}^{k+1}) = ∑_{s∈ℤ}r_{0}(E_{s,−s+1}^{k+1}).

Now we claim that

∑_{s∈ℤ}r_{0}(E_{s,−s}^{k+1}) − r_{0}(E_{s,−s+1}^{k+1}) = ∑_{s∈ℤ}r_{0}(E_{s,−s}^{2}) − r_{0}(E_{s,−s+1}^{2}).

To see that this holds it is sufficient to prove that for all r ≥ 2 we have

∑_{s∈ℤ}r_{0}(E_{s,−s}^{r+1}) − r_{0}(E_{s,−s+1}^{r+1}) = ∑_{s∈ℤ}r_{0}(E_{s,−s}^{r}) − r_{0}(E_{s,−s+1}^{r}).

Recall that for all r ≥ 1, p, q ∈ ℤ, E_{p,q}^{r+1} ≅ Z(E^{r})_{p,q}/B(E^{r})_{p,q} where Z(E^{r})_{p,q} = ker d_{p,q}^{r} and B(E^{r})_{p,q} = im d_{p+r,q−r+1}^{r}. Thus

r_{0}(E_{p,q}^{r+1}) = r_{0}(Z(E^{r})_{p,q}) − r_{0}(B(E^{r})_{p,q})
= r_{0}(Z(E^{r})_{p,q}) − r_{0}(E_{p+r,q−r+1}^{r}) + r_{0}(Z(E^{r})_{p+r,q−r+1})

for all r ≥ 1, p, q ∈ ℤ. Moreover, it follows from the definition of the Kasparov spectral sequence that given any r ≥ 1 and p, q, q′ ∈ ℤ with q = q′ mod 2 there exist isomorphisms

ρ : E_{p,q}^{r} ⟶ E_{p,q′}^{r},
σ : E_{p−r,q+r−1}^{r} ⟶ E_{p−r,q′+r−1}^{r},

such that d_{p,q′}^{r} ∘ ρ = σ ∘ d_{p,q}^{r}. Therefore,

∑_{s∈ℤ}r_{0}(E_{s,−s}^{r+1}) − r_{0}(E_{s,−s+1}^{r+1}) = ∑_{s∈ℤ}{r_{0}(Z(E^{r})_{s,−s}) − r_{0}(Z(E^{r})_{s+r,−s−r+2})

− r_{0}(Z(E^{r})_{s,−s+1}) + r_{0}(Z(E^{r})_{s+r,−s−r+1})

+ r_{0}(E_{s+r,−s−r+2}^{r}) − r_{0}(E_{s+r,−s−r+1}^{r})}

= ∑_{s∈ℤ}r_{0}(E_{s,−s}^{r}) − r_{0}(E_{s,−s+1}^{r})

for all r ≥ 1. Combining the above gives

r_{0}(K_{0}(C^{∗}(Λ))) − r_{0}(K_{1}(C^{∗}(Λ))) = ∑_{s∈ℤ}r_{0}(E_{s,−s}^{2}) − r_{0}(E_{s,−s+1}^{2}).

Now, recall that for all p ∈ ℤ and q ∈ 2ℤ,

E_{p,q}^{2} ≅ H_{p}(ℤ^{k}, K_{0}(B)) ≅ ker ∂_{p}/ im ∂_{p+1}

by Theorem 3.14. Therefore,

r_{0}(K_{0}(C^{∗}(Λ))) − r_{0}(K_{1}(C^{∗}(Λ))) = ∑_{s∈ℤ}r_{0}(E_{2s,−2s}^{2}) − r_{0}(E_{2s+1,−2s}^{2})

= ∑_{s∈ℤ}r_{0}(ker ∂_{2s}) − r_{0}(im ∂_{2s+1}) − r_{0}(ker ∂_{2s+1}) + r_{0}(im ∂_{2s+2})

= ∑_{s∈ℤ}r_{0}(ker ∂_{2s}) − r_{0} (( ⨁_{N_{2s+1}} ℤΛ^{0}) / ker ∂_{2s+1}) − r_{0}(ker ∂_{2s+1})

+ r_{0} (( ⨁_{N_{2s+2}} ℤΛ^{0}) / ker ∂_{2s+2})

= ∑_{s∈ℤ}r_{0}(ker ∂_{2s}) − \binom{k}{2s + 1}n + r_{0}(ker ∂_{2s+1}) − r_{0}(ker ∂_{2s+1})

+ \binom{k}{2s + 2}n − r_{0}(ker ∂_{2s+2})

= ∑_{s∈ℤ}\{\binom{k}{2s} − \binom{k}{2s − 1}\} n

= ∑_{s∈ℤ}\{\binom{k − 1}{2s} + \binom{k − 1}{2s − 1} − \binom{k − 1}{2s − 1} − \binom{k − 1}{2s − 2}\} n = 0. □

Corollary 4.2. If Λ is a row-finite higher rank graph with no sources and Λ^{0} is finite then there exists a nonnegative integer r such that for i = 0, 1,

K_{i}(C^{∗}(Λ)) ≅ ℤ^{r} ⊕ T_{i}

for some finite group T_{i}, where ℤ^{0} := {0}.

Proof. It is well-known that if B is a finitely generated subgroup of an abelian group A such that A/B is also finitely generated then A must be finitely generated too [9]. Now, for all p, q ∈ ℤ, E_{p,q}^{k+1} is isomorphic to a subquotient of the finitely generated abelian group E_{p,q}^{2} ≅ H_{p}(ℤ^{k}, K_{q}(B)), so E_{p,−p}^{k+1} is also finitely generated. Moreover, E_{0,i}^{k+1} ≅ F_{0}(K_{i}(C^{∗}(Λ))) and for p ∈ {1, 2, …, k}, E_{p,−p+i}^{k+1} ≅ F_{p}(K_{i}(C^{∗}(Λ)))/F_{p−1}(K_{i}(C^{∗}(Λ))), which implies that K_{i}(C^{∗}(Λ)) = F_{k}(K_{i}(C^{∗}(Λ))) is finitely generated. The result follows from Proposition 4.1 by noting that every finitely generated abelian group A is isomorphic to the direct sum of a finite group with ℤ^{r}, where r = r_{0}(A) (see, e.g., [9, Theorem 15.5]). □

Remarks 4.3. Note that it is well-known that when k = 1 we always have T_{1} = 0 in the above, i.e., K_{1}(C^{∗}(Λ)) is torsion-free. However, for k > 1, K_{1}(C^{∗}(Λ)) may contain torsion elements.

Formulae for the torsion-free rank and torsion parts of the K-groups of unital C^{∗}-algebras of row-finite 2-graphs with no sources can be given in terms of the vertex matrices (cf. [25, Proposition 4.13]). This we do in Proposition 4.4 below.

Proposition 4.4 (cf. [25, Proposition 4.13]). Let Λ be a row-finite 2-graph with no sources and finite vertex set. Then

r_{0}(K_{0}(C^{∗}(Λ))) = r_{0}(K_{1}(C^{∗}(Λ)))

= r_{0}(coker(1 − M_{1}^{t}, 1 − M_{2}^{t})) + r_{0}(coker(1 − M_{1}, 1 − M_{2})),

tor(K_{0}(C^{∗}(Λ))) ≅ tor(coker(1 − M_{1}^{t}, 1 − M_{2}^{t})),
tor(K_{1}(C^{∗}(Λ))) ≅ tor(coker(1 − M_{1}, 1 − M_{2})).

Proof. We have already seen in Proposition 4.1 that the torsion-free rank of the K_{0}-group and K_{1}-group of a k-graph are equal so it is sufficient to calculate the torsion-free rank of K_{0}(C^{∗}(Λ)). Let n := |Λ^{0}|. By Proposition 3.16 we have

r_{0}(K_{0}(C^{∗}(Λ))) = r_{0}(coker(1 − M_{1}^{t}, 1 − M_{2}^{t})) + r_{0} (ker \binom{1 − M_{1}^{t}}{1 − M_{2}^{t}})
= r_{0}(coker(1 − M_{1}^{t}, 1 − M_{2}^{t})) + n − r_{0}(im(1 − M_{1}, 1 − M_{2}))
= r_{0}(coker(1 − M_{1}^{t}, 1 − M_{2}^{t})) + r_{0}(coker(1 − M_{1}, 1 − M_{2})).

Furthermore, the assertion about the torsion part of K_{0}(C^{∗}(Λ)) is obvious. The torsion part of K_{1}(C^{∗}(Λ)) is given by

tor(K_{1}(C^{∗}(Λ))) ≅ tor (ker(1 − M_{1}^{t}, 1 − M_{2}^{t})/ im \binom{M_{2}^{t} − 1}{1 − M_{1}^{t}}) ,

which is clearly isomorphic to tor(coker \binom{M_{2}^{t}−1}{1−M_{1}^{t}}). However, by reduction to

Smith normal forms, coker \binom{M_{2}^{t}−1}{1−M_{1}^{t}} is isomorphic to coker(1−M_{1}, 1−M_{2}). □

Remarks 4.5. We note that, in the case where Λ is a row-finite 3-graph with no sources and finite vertex set, and with ∂_{1}, ∂_{2} defined as in Proposition 3.17, it is straightforward to show that if ∂_{1} is surjective then

K_{0}(C^{∗}(Λ)) ≅ K_{1}(C^{∗}(Λ)) ≅ ℤ^{m},

where m := r_{0}(ker ∂_{2}) − |Λ^{0}| = r_{0}(coker ∂_{2}) − |Λ^{0}| (with ℤ^{0} := 0).

5. Applications and examples

We begin this section with two corollaries to the results in the preceding section, which facilitate the classification of the C^{∗}-algebras of row-finite 2graphs with no sources. We then end the paper with some simple illustrative examples.

Corollary 5.1. Let Λ be a row-finite 2-graph with no sources, finite vertex set and vertex matrices M_{1} and M_{2}. Then there exists an isomorphism

Φ : coker(1 − M_{1}^{t}, 1 − M_{2}^{t}) ⊕ ker \binom{M_{2}^{t} − 1}{1 − M_{1}^{t}} ⟶ K_{0}(C^{∗}(Λ))

such that Φ(e + im ∂_{0}) = [1], where e(v) = 1 for all v ∈ Λ^{0}.

Proof. Follows from Proposition 3.16 and that ∑_{u∈Λ^{0}} p_{u} = 1. □

Remarks 5.2. We note that the C^{∗}-algebra of a row-finite k-graph Λ, with no sources, is separable, nuclear and satisfies the UCT [26]. If in addition the C^{∗}-algebra is simple and purely infinite we say that it is a Kirchberg algebra, and note that by the Kirchberg–Phillips classification theorem ([12, 18]) it is classifiable by its K-theory (see [13, Theorem 5.5]). We also note that conditions on the underlying k-graph have been identified, which determine whether the C^{∗}-algebra is simple ([13, Proposition 4.8], [22, Theorem 3.2]) and purely infinite ([28, Proposition 8.8]).

Corollary 5.3. Let Λ and Δ be two row-finite 2-graphs with no sources. Furthermore, suppose that C^{∗}(Λ) and C^{∗}(Δ) are both simple and purely infinite, and that Λ and Δ share the same vertex matrices. Then

C^{∗}(Λ) ≅ C^{∗}(Δ).

Proof. Let Λ and Δ be two 2-graphs satisfying the hypothesis. Then, by Proposition 3.16 their K-groups are isomorphic.

Suppose that the vertex set of Λ (and hence that of Δ) is infinite. Then C^{∗}(Λ) and C^{∗}(Δ) are both nonunital, and thus stable, Kirchberg algebras with isomorphic K-groups. Thus by the Kirchberg–Phillips classification theorem C^{∗}(Λ) ≅ C^{∗}(Δ).

In the case where the vertex set of Λ (and hence that of Δ) is finite, C^{∗}(Λ) and C^{∗}(Δ) are both unital Kirchberg algebras with isomorphic Kgroups. Furthermore, by Corollary 5.1 we see that the isomorphism of Kgroups maps the K_{0}-class of the unit of one of the C^{∗}-algebras onto that of the other. Therefore, by the Kirchberg–Phillips classification theorem we conclude that C^{∗}(Λ) ≅ C^{∗}(Δ) and the result is proved. □

Examples 5.4. (1) Let Λ be a row-finite 2-graph with no sources. Suppose that the vertex matrices of Λ are both equal to M, say. By Proposition 3.16 and [16, Theorem 3.1] we have:

K_{i}(C^{∗}(Λ)) ≅ ℤΛ^{0}/ im(1 − M^{t}) ⊕ ker(1 − M^{t})
≅ K_{0}(C^{∗}(E)) ⊕ K_{1}(C^{∗}(E)),

for i = 1, 2, where E is the 1-graph with vertex matrix M. The isomorphism for K_{0}(C^{∗}(Λ)) is immediately obvious. That for K_{1}(C^{∗}(Λ)) is given by

\binom{x}{y} + im \binom{M^{t} − 1}{1 − M^{t}} ↦ (x + im(1 − M^{t})) ⊕ (x + y)

for all \binom{x}{y} ∈ ker(1 − M^{t}, 1 − M^{t}).

(2) Fix nonzero n_{1}, n_{2} ∈ ℕ^{2} and let Λ be a 2-graph with one vertex and vertex matrices M_{1} = (n_{1}) and M_{2} = (n_{2}) respectively. By Proposition 3.16, we have K_{0}(C^{∗}(Λ)) ≅ K_{1}(C^{∗}(Λ)) ≅ ℤ/gℤ, where g is the greatest common divisor of n_{1} − 1 and n_{2} − 1.

Note that we recover the K-groups of tensor products of Cuntz algebras [4] by letting Λ be the product 2-graph of two 1-graphs each with one vertex and a finite number of edges (i.e., morphisms of degree 1). We also note that tensor products of two Cuntz algebras are not the only examples of C^{∗}-algebras of such 2-graphs Λ (cf. [13, §6]). However, by Corollary 5.3, they are, up to ∗-isomorphism, the only examples of Kirchberg algebras arising from row-finite 2-graphs with one vertex.

(3) For each positive integer n, let O_{n} be the 1-graph with 1 vertex, ⋆, and n edges, α_{1}, α_{2}, …, α_{n}. Let c : O_{3} × O_{n} ⟶ ℤ be the unique functor that satisfies c(α_{i}, ⋆) = δ_{i,1} (i = 1, 2, 3) and c(⋆, α_{i}) = 1 (i = 1, …, n). Define Λ to be the 2-graph ℤ ×_{c} (O_{3} × O_{n}). Let T_{i} := 1 − M_{i}^{t}, where M_{1} and M_{2} are the vertex matrices of Λ. Then

(5.1)  T_{1}δ_{u} = −δ_{u} − δ_{u+1} and
(5.2)  T_{2}δ_{u} = δ_{u} − nδ_{u+1},

where we identify Λ^{0} with ℤ.

Clearly, ker \binom{−T_{2}}{T_{1}} = 0. Now consider coker(T_{1}, T_{2}) and for each g ∈ ℤΛ^{0} let [g] be the image of g under the natural homomorphism ℤΛ^{0} ⟶ coker(T_{1}, T_{2}). By (5.1) and (5.2) we have (n + 1)[δ_{u}] = 0. Therefore, coker(T_{1}, T_{2}) is a cyclic group, generated by [δ_{0}] say, whose order divides n + 1. We claim that ρ[δ_{0}] ≠ 0 for each ρ = 1, …, n. Suppose the contrary, then we have

ρδ_{0} = T_{1}x + T_{2}y

for some x, y ∈ ℤΛ^{0} and ρ ∈ {1, …, n}. Thus, for each u ∈ Λ^{0} we have

ρδ_{0}(u) = −x(u) − x(u − 1) + y(u) − ny(u − 1).

Since x, y ∈ ℤΛ^{0}, there exists N such that x(u) = y(u) = 0 if |u| > N, which we assume, without loss of generality, to be greater than zero. It follows that

y(u) =

\begin{cases}x(u), & u = −N, \\ x(u) + (n + 1) ∑_{j=0}^{N−1+u} n^{N−1+u−j}x(−N + j), & −N + 1 ≤ u ≤ −1, \\ x(u) + (n + 1) ∑_{j=0}^{N−1+u} n^{N−1+u−j}x(−N + j)+ρn^{u}, & u ≥ 0. \end{cases}

Setting u = N + 1, we arrive at the contradiction (n + 1)|ρn^{N+1}. Therefore, by Proposition 3.16 K_{0}(C^{∗}(Λ)) ≅ ℤ/(n + 1)ℤ.

Now we turn our attention to ker(T_{1}, T_{2})/ im \binom{−T_{2}}{T_{1}}. Suppose that x ⊕ y ∈ ker(T_{1}, T_{2}), then there exists N such that x(u) = y(u) = 0 if |u| > N,

(5.3)  y(−N) = x(−N) and
(5.4)  y(u) = x(u) + (n + 1) ∑_{j=−N}^{u−1} n^{u−1−i}x(j) for all u ≥ −N + 1.

Let P : ℤΛ^{0} ⊕ ℤΛ^{0} ⟶ ℤΛ^{0} be the projection onto the second component, i.e., P (x ⊕ y) = y. From (5.3) and (5.4) we see that P is injective on ker(T_{1}, T_{2}) and thus induces an isomorphism

ker(T_{1}, T_{2})/ im \binom{−T_{2}}{T_{1}} ≅ P (ker(T_{1}, T_{2}))/ im T_{1}.

Moreover, P (ker(T_{1}, T_{2})) = {y ∈ ℤΛ^{0} | ∑_{j∈ℤ}(−1)^{j}y(j) = 0}. Now given y ∈ P (ker(T_{1}, T_{2})), define z : Λ^{0} ⟶ ℤ by

z(u) = 0 if u < −N,
z(u) = ∑_{j=−N}^{u}(−1)^{u−j+1}y(j) if u ≥ −N.

Then it is straightforward to show that z ∈ ℤΛ^{0} and T_{1}z = y, and thus ker(T_{1}, T_{2})/ im \binom{−T_{2}}{T_{1}} is the trivial group. Therefore, by Proposition 3.16 K_{1}(C^{∗}(Λ)) = 0.

Note that Λ satisfies the hypotheses of [13, Proposition 4.8] and [28, Proposition 8.8] and thus C^{∗}(Λ) is a (stable) Kirchberg algebra. It now follows from the Kirchberg–Phillips classification theorem that C^{∗}(Λ) is ∗-isomorphic to the stabilized Cuntz algebra O_{n+2} ⊗ 𝕂.

(4) Let c : O_{3} × O_{3} × O_{3} ⟶ ℤ_{2} be the unique functor that satisfies^{3}

\begin{aligned}c(α_{1}, ⋆, ⋆) & = & 0, & c(⋆, α_{1}, ⋆) & = & 0, & c(⋆, ⋆, α_{1}) & = & 1, \\ c(α_{2}, ⋆, ⋆) & = & 0, & c(⋆, α_{2}, ⋆) & = & 0, & c(⋆, ⋆, α_{2}) & = & 1, \\ c(α_{3}, ⋆, ⋆) & = & 1, & c(⋆, α_{3}, ⋆) & = & 1, & c(⋆, ⋆, α_{3}) & = & 1. \end{aligned}

Then the vertex matrices of Λ := ℤ_{2} ×_{c} (O_{3} × O_{3} × O_{3}) are

M_{1} = M_{2} = ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) , M_{3} = ( \begin{matrix}0 & 3 \\ 3 & 0 \end{matrix} ) .

3We extend the definition of the product of two higher rank graphs (Examples 2.3.2)

to the product higher rank graph of three higher rank graphs in the natural way. Note that if Λ_{i} is a k_{i}-graph for i = 1, 2, 3 then both (Λ_{1} × Λ_{2}) × Λ_{3} and Λ_{1} × (Λ_{2} × Λ_{3}) are clearly pairwise isomorphic as (k_{1} + k_{2} + k_{3})-graphs to Λ_{1} × Λ_{2} × Λ_{3} .

Following the notation in Proposition 3.17, for i = 1, 2, 3, we have

\begin{aligned}∂_{1} & = & ( \begin{matrix}−1 & −1 & −1 & −1 & 1 & −3 \\ −1 & −1 & −1 & −1 & −3 & 1 \end{matrix} ) , & ∂_{3} & = & ( \begin{matrix}1 & −3 \\ −3 & 1 \\ 1 & 1 \\ 1 & 1 \\ −1 & −1 \\ −1 & −1 \end{matrix} ) . \\ ∂_{2} & = & ( \begin{matrix}1 & 1 & −1 & 3 & 0 & 0 \\ 1 & 1 & 3 & −1 & 0 & 0 \\ −1 & −1 & 0 & 0 & −1 & 3 \\ −1 & −1 & 0 & 0 & 3 & −1 \\ 0 & 0 & −1 & −1 & −1 & −1 \\ 0 & 0 & −1 & −1 & −1 & −1 \end{matrix} ) , \end{aligned}

To compute the K-groups of C^{∗}(Λ) we reduce the relevant matrices to their Smith normal forms (for a (not necessarily square) matrix M we shall denote its Smith normal form by S(M)). In particular,

U_{1}∂_{1}V_{1} = S(∂_{1}) = S(∂_{3})^{t} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 & 0 \end{matrix} )

for some invertible matrices U_{1}, V_{1}. Thus, coker ∂_{1} ≅ ℤ/4ℤ, ker ∂_{3} = 0 and Corollary 3.18 can be applied to deduce that there exists a short exact sequence

0 ⟶ ℤ/4ℤ ⟶ K_{0}(C^{∗}(Λ)) ⟶ ker ∂_{2}/ im ∂_{3} ⟶ 0

and K_{1}(C^{∗}(Λ)) ≅ ker ∂_{1}/ im ∂_{2}. Now

U_{2}∂_{2}V_{2} = S(∂_{2}) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) ,

for some invertible matrices U_{2}, V_{2}, and we see that there exists an isomorphism (induced by V_{2}^{−1} or by U_{1}) of ker ∂_{2}/ im ∂_{3} onto ℤ/4ℤ. Furthermore, we can now see that ker ∂_{1}/ im ∂_{2} is isomorphic to ℤ/4ℤ ⊕ ℤ/4ℤ. Hence, K_{0}(C^{∗}(Λ)) is a group order 16 and K_{1}(C^{∗}(Λ)) ≅ ℤ/4ℤ ⊕ ℤ/4ℤ.^{4}

We note that it is, perhaps, slightly surprising that C^{∗}(Λ) shares, at least, one of its K-groups with that of O_{5} ⊗ O_{5} ⊗ O_{5}, given that this relationship is not obvious at the level of C^{∗}-algebras.

4It is well-known that there are, up to isomorphism, 5 abelian groups of order 16.

(5) Let c : O_{2} × O_{3} × O_{3} ⟶ ℤ_{2} be the unique functor that satisfies

c(α_{1}, ⋆, ⋆) = 0,  c(⋆, α_{1}, ⋆) = 0,  c(⋆, ⋆, α_{1}) = 1,
c(α_{2}, ⋆, ⋆) = 1,  c(⋆, α_{2}, ⋆) = 1,  c(⋆, ⋆, α_{2}) = 1,
c(⋆, α_{3}, ⋆) = 1,  c(⋆, ⋆, α_{3}) = 1.

Then the vertex matrices of Λ := ℤ_{2} ×_{c} (O_{3} × O_{3} × O_{3}) are

M_{1} = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) ,  M_{2} = ( \begin{matrix}1 & 2 \\ 2 & 1 \end{matrix} ) ,  M_{3} = ( \begin{matrix}0 & 3 \\ 3 & 0 \end{matrix} ) .

Now, for i = 1, 2, 3, we have

\begin{aligned}∂_{1} & = & ( \begin{matrix}0 & −1 & 0 & −2 & 1 & −3 \\ −1 & 0 & −2 & 0 & −3 & 1 \end{matrix} ) , & ∂_{3} & = & ( \begin{matrix}1 & −3 \\ −3 & 1 \\ 0 & 2 \\ 2 & 0 \\ 0 & −1 \\ −1 & 0 \end{matrix} ) . \\ ∂_{2} & = & ( \begin{matrix}0 & 2 & −1 & 3 & 0 & 0 \\ 2 & 0 & 3 & −1 & 0 & 0 \\ 0 & −1 & 0 & 0 & −1 & 3 \\ −1 & 0 & 0 & 0 & 3 & −1 \\ 0 & 0 & 0 & −1 & 0 & −2 \\ 0 & 0 & −1 & 0 & −2 & 0 \end{matrix} ) , \end{aligned}

As in the previous example we compute the Smith normal form of ∂_{1} (and hence that of ∂_{3}) first, to determine whether Corollary 3.18 is applicable. We find that

S(∂_{1}) = S(∂_{3})^{t} = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{matrix} ) ,

thus coker ∂_{1} ≅ ker ∂_{3} ≅ 0 and we may apply Corollary 3.18 to deduce that

K_{0}(C^{∗}(Λ)) = ker ∂_{2}/ im ∂_{3} and K_{1}(C^{∗}(Λ)) = ker ∂_{1}/ im ∂_{2}.

Now,

S(∂_{2}) = ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) .

It follows that both ker ∂_{2}/ im ∂_{3} and ker ∂_{1}/ im ∂_{2} are trivial. Thus, the K-groups of C^{∗}(Λ) are isomorphic to those of the Cuntz algebra O_{2}. Furthermore, It is clear that Λ satisfies the hypotheses of [13, Proposition 4.8] and [28, Proposition 8.8], and therefore C^{∗}(Λ) is an 30 D. Gwion Evans

unital Kirchberg algebra (cf. Remarks 5.2). Applying the Kirchberg– Phillips classification theorem, we conclude that C^{∗}(Λ) ≅ O_{2}.

References

[1] Allen, Stephen; Pask, David; Sims, Aidan. A dual graph construction for higherrank graphs, and K-theory for finite 2-graphs. Proc. Amer. Math. Soc. 134 (2006), no. 2, 455–464. MR2176014 (2006e:46078), Zbl pre02227186.

[2] Auslander, Maurice; Buchsbaum, David A. Groups, rings, modules. Harper’s Series in Modern Mathematics Harper & Row Publishers, New York, 1974. MR0366959 (51 #3205), Zbl 0325.13001.

[3] Brown, Kenneth S. Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York, 1994. MR1324339 (96a:20072), Zbl 0584.20036.

[4] Cuntz, Joachim. Simple C^{∗}-algebras generated by isometries. Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR0467330 (57 #7189), Zbl 0399.46045.

[5] Cuntz, Joachim. A class of C^{∗}-algebras and topological Markov chains. II. Reducible chains and the Ext-functor for C^{∗}-algebras. Invent. Math. 63 (1981), no. 1, 25–40. MR0608527 (82f:46073b), Zbl 0461.46047.

[6] Cuntz, Joachim; Krieger, Wolfgang. A class of C^{∗}-algebras and topological Markov chains. Invent. Math. 56 (1980), no. 3, 251–268. MR0561974 (82f:46073a), Zbl 0434.46045.

[7] Evans, D. Gwion. On higher rank graph C^{∗}-algebras. Ph.D. Thesis, Univ. Wales, 2002.

[8] Farthing, Cynthia. Removing sources from higher-rank graphs.

arXiv:math/0603037v2.

[9] Fuchs, László. Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York, 1970. MR0255673 (41 #333), Zbl 0209.05503.

[10] Kaplansky, Irving. Commutative rings. Revised ed. The University of Chicago Press, Chicago, Ill.-London, 1974. MR0345945 (49 #10674), Zbl 0296.13001.

[11] Kasparov, G. G. Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91 (1988), no. 1, 147–201. MR0918241 (88j:58123), Zbl 0647.46053.

[12] Kirchberg, Eberhard. The classification of purely infinite C^{∗}-algebras using Kasparov’s theory. Preprint.

[13] Kumjian, Alex; Pask, David. Higher rank graph C^{∗}-algebras. New York J. Math. 6 (2000) 1–20 (electronic). MR1745529 (2001b:46102), Zbl 0946.46044.

[14] Mac Lane, Saunders. Homology. Die Grundlehren der mathematischen Wissenschaften, Bd. 114. Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Gttingen-Heidelberg, 1963 x+422 pp. MR0156879 (28 #122), Zbl 0133.26502.

[15] Mac Lane, Saunders. Categories for the Working Mathematician. Second ed. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. MR1712872 (2001j:18001), Zbl 0906.18001.

[16] Pask, David. Cuntz–Krieger algebras associated to directed graphs. Operator algebras and quantum field theory (Rome, 1996), Internat. Press, Cambridge, MA, 1997, 85–92. MR1491109 (98m:46072), Zbl 0921.46050.

[17] Pask, David; Raeburn, Iain. On the K-theory of Cuntz–Krieger algebras. Publ. Res. Inst. Math. Sci. 32 (1996), no. 3, 415–443. MR1409796 (97m:46111), Zbl 0862.46043.

[18] Phillips, N. Christopher. A classification theorem for nuclear purely infinite simple C^{∗}-algebras. Doc. Math. 5 (2000) 49–114 (electronic). MR1745197 (2001d:46086b), Zbl 0943.46037.

[19] Raeburn, Iain. On crossed products and Takai duality. Proc. Edinburgh Math. Soc. (2) 31 (1988), no. 2, 321–330. MR0989764 (90d:46093), Zbl 0674.46038.

[20] Raeburn, Iain; Sims, Aidan; Yeend, Trent. Higher-rank graphs and their C^{∗}-algebras. Proc. Edinb. Math. Soc. (2) 46 (2003), no. 1, 99–115. MR1961175 (2004f:46068), Zbl pre01925883.

[21] Raeburn, Iain; Sims, Aidan; Yeend, Trent. The C^{∗}-algebras of finitely aligned higher-rank graphs. J. Funct. Anal. 213 (2004), no. 1, 206–240. MR2069786 (2005e:46103), Zbl 1063.46041.

[22] Robertson, David I.; Sims, Aidan. Simplicity of C^{∗}-algebras associated to higherrank graphs. Bull. Lond. Math. Soc. 39 (2007), no. 2, 337–344. MR2323468.

[23] Robertson, Guyan; Steger, Tim. C^{∗}-algebras arising from group actions on the boundary of a triangle building. Proc. London Math. Soc. (3) 72 (1996), no. 3, 613– 637. MR1376771 (98b:46088), Zbl 0869.46035.

[24] Robertson, Guyan; Steger, Tim. Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras. J. Reine Angew. Math. 513 (1999), 115–144. MR1713322 (2000j:46109), Zbl pre01333057.

[25] Robertson, Guyan; Steger, Tim. Asymptotic K-theory for groups acting on Ã_{2} buildings. Canad. J. Math. 53 (2001), no. 4, 809–833. MR1848508 (2002f:46141), Zbl 0993.46039.

[26] Rosenberg, Jonathan; Schochet, Claude. The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor. Duke Math. J. 55 (1987) 431–474. MR88i:46091, Zbl 644.46051.

[27] Schochet, Claude. Topological methods for C^{∗}-algebras. I. Spectral sequences. Pacific J. Math. 96 (1981), no. 1, 193–211. MR0634772 (84g:46105a), Zbl 0426.46057.

[28] Sims, Aidan. Gauge-invariant ideals in the C^{∗}-algebras of finitely aligned higher-rank graphs. Canad. J. Math. 58 (2006), no. 6, 1268–1290. MR2270926 (2007j:46095).

[29] Sims, A. Relative Cuntz–Krieger algebras of finitely aligned higher-rank graphs. Indiana Univ. Math. J. 55 (2006), no. 2, 849–868. MR2225454 (2007c:46053).

[30] Spanier, Edwin H. Algebraic topology. McGraw-Hill Book Co., New York, 1966. MR0210112 (35 #1007), Zbl 0145.43303.

[31] Spielberg, John. Free-product groups, Cuntz–Krieger algebras, and covariant maps. Internat. J. Math. 2 (1991), no. 4, 457–476. MR1113572 (92j:46120), Zbl 0769.46044.

[32] Takai, Hiroshi. On a duality for crossed products of C^{∗}-algebras. J. Funct. Anal. 19 (1975) 25–39. MR0365160 (51 #1413), Zbl 0295.46088.

[33] Wegge-Olsen, Niels Erik. K-theory and C^{∗}-algebras: A friendly approach. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1993. MR1222415 (95c:46116), Zbl 0780.46038.

[34] Weibel, Charles A. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994. MR1269324 (95f:18001), Zbl 0797.18001.

Institute of Mathematical and Physical Sciences, Aberystwyth University, Penglais Campus, Aberystwyth, Ceredigion, SY23 3BZ, Wales, UK dfe@aber.ac.uk

This paper is available via http://nyjm.albany.edu/j/2008/14-1.html.