C^{∗}-Algebras Associated to Coverings of k-Graphs

Alex Kumjian, David Pask, Aidan Sims^{∗}

Received: March 15, 2007 Revised: May 24, 2008

Communicated by Joachim Cuntz

Abstract. A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C^{∗}-algebras. We show how to build a (k + 1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k + 1)-graph. Our main focus is on computing the K-theory of the (k +1)-graph algebra from that of the component k-graph algebras.

Examples of our construction include a realisation of the Kirchberg algebra P_{n} whose K-theory is opposite to that of O_{n}, and a class of AT-algebras that can naturally be regarded as higher-rank Bunce-Deddens algebras.

2000 Mathematics Subject Classification: Primary 46L05 Keywords and Phrases: Graph algebra; k-graph; covering; K-theory; C^{∗}-algebra

1. Introduction

C^{∗}-Algebras Associated to Coverings of k-Graphs 163

A directed graph E consists of a countable collection E^{0} of vertices, a countable collection E^{1} of edges, and maps r, s : E^{1} → E^{0} which give the edges their direction; the edge e points from s(e) to r(e). Following the convention established in [30], the associated graph algebra C^{∗}(E) is the universal C^{∗}- algebra generated by partial isometries {s_{e} : e ∈ E^{1}} together with mutually orthogonal projections {p_{v} : v ∈ E^{0}} such that p_{s(e)} = s_{e}^{∗}s_{e} for all e ∈ E^{1}, and p_{v} ≥ ∑_{e∈F} s_{e}s_{e}^{∗} for all v ∈ E^{0} and finite F ⊂ r^{−1}(v), with equality when F = r^{−1}(v) is finite and nonempty. Graph algebras, introduced in [13, 23], have been studied intensively in recent years because much of the structure of C^{∗}(E) can be deduced from elementary features of E. In particular, graph C^{∗}-algebras are an excellent class of models for Kirchberg algebras, because it is easy to tell from the graph E whether C^{∗}(E) is simple and purely infinite [22]. Indeed, a Kirchberg algebra can be realised up to Morita equivalence as a graph C^{∗}-algebra if and only if its K_{1}- group is torsion-free [39]. It is also true that every AF algebra can be realised up to Morita equivalence as a graph algebra; the desired graph is a Bratteli diagram for the AF algebra in question (see [11] or [40]). However, this is the full extent to which graph algebras model simple classifiable C^{∗}-algebras due to the following dichotomy: if E is a directed graph and C^{∗}(E) is simple, then C^{∗}(E) is either AF or purely infinite (see [22, Corollary 3.10], [2, Remark 5.6]). Higher-rank graphs, or k-graphs, and their C^{∗}-algebras were originally developed by the first two authors [20] to provide a graphical framework for the higher-rank Cuntz-Krieger algebras of Robertson and Steger [35]. A k-graph Λ is a kind of k-dimensional graph, which one can think of as consisting of vertices Λ^{0} together with k collections of edges Λ^{e_{1}}, …, Λ^{e_{k}} which we think of as lying in k different dimensions. As an aid to visualisation, we often distinguish the different types of edges using k different colours. Higher-rank graphs and their C^{∗}-algebras are generalisations of directed graphs and their algebras. Given a directed graph E, its path category E^{∗} is a 1-graph, and the 1-graph C^{∗}-algebra C^{∗}(E^{∗}) as defined in [20] is canonically isomorphic to the graph algebra C^{∗}(E) as defined in [23]. Furthermore, every 1-graph arises this way, so the class of graph algebras and the class of 1-graph algebras are one and the same. For k ≥ 2, there are many k-graph algebras which do not arise as graph algebras. For example, the original work of Robertson and Steger on higher-rank Cuntz-Krieger algebras describes numerous 2-graphs Λ for which C^{∗}(Λ) is a Kirchberg algebra and K_{1}(C^{∗}(Λ)) contains torsion. Recent work of Pask, Raeburn, Rørdam and Sims has shown that one can also realise a substantial class of AT-algebras as 2-graph algebras, and that one can tell from the 2-graph whether or not the resulting C^{∗}-algebra is simple and has real-rank zero [27]. The basic idea of the construction in [27] is as follows. One takes a Bratteli diagram in which the edges are coloured red, and replaces each vertex with a blue simple cycle (there are technical restrictions on the relationship between the lengths of the blue cycles and the distribution of the red edges joining them, but this is the gist of the construction). The resulting 2-graph is called a rank-2 Bratteli diagram. The associated C^{∗}-algebra is AT because the C^{∗}-algebra of a simple cycle of length n is isomorphic to M_{n}(C(T)) [17]. The results of [27] show how to read off from a rank-2 Bratteli diagram the K-theory, simplicity or otherwise, and real-rank of the resulting AT algebra. The construction explored in the current paper is motivated by the following example of a rank-2 Bratteli diagram. For each n ∈ N, let L_{2^{n}} be the simple directed loop graph with 2^{n} vertices labelled 0, …, 2^{n} − 1 and 2^{n} edges f_{0}, …, f_{2^{n}−1}, where f_{i} is directed from the vertex labelled i + 1 (mod 2^{n}) to the vertex labelled i. We specify a rank-2 Bratteli diagram Λ(2^{∞}) as follows. The n^{th} level of Λ(2^{∞}) consists of a single blue copy of L_{2^{n−1}} (n = 1, 2, ⋯). For 0 ≤ i ≤ 2^{n} − 1, there is a single red edge from the vertex labelled i at the (n+1)^{st} level to the vertex labelled i (mod 2^{n}) at the n^{th} level. The C^{∗}-algebra of the resulting 2-graph is Morita equivalent to the Bunce-Deddens algebra of type 2^{∞}, and this was one of the first examples of a 2-graph algebra which is simple but neither purely infinite nor AF (see [27, Example 6.7]).

∗This research was supported by the Australian Research Council.

The purpose of this paper is to explore the observation that the growing blue cycles in Λ(2^{∞}) can be thought of as a tower of coverings of 1-graphs (roughly speaking, a covering is a locally bijective surjection — see Definition 2.1), where the red edges connecting levels indicate the covering maps. In Section 2, we describe how to construct (k + 1)-graphs from coverings. In its simplest form, our construction takes k-graphs Λ and Γ and a covering map p : Γ → Λ, and produces a (k + 1)-graph Λ↽^{p}Γ in which each edge in the (k + 1)^{st} dimension points from a vertex v of Γ to the vertex p(v) of Λ which it covers^{†}. Building on this construction, we show how to take an infinite tower of coverings p_{n} : Λ_{n+1} → Λ_{n}, n = 1, 2, … and construct from it an infinite (k + 1)-graph \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}, p_{n}) with a natural inductive structure (Corollary 2.11). The next step, achieved in Section 3, is to determine how the universal C^{∗}- algebra of Λ↽^{p}Γ relates to those of Λ and Γ. We show that C^{∗}(Λ↽^{p}Γ) is Morita equivalent to C^{∗}(Γ) and contains an isomorphic copy of C^{∗}(Λ) (Proposition 3.2). We then show that given a system of coverings p_{n} : Λ_{n+1} → Λ_{n}, the C^{∗}-algebra C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}, p_{n})) is Morita equivalent to a direct limit of the C^{∗}(Λ_{n}) (Theorem 3.8). In Section 4, we use results of [34] to characterise simplicity of C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}, p_{n})), and we also give a sufficient condition for this C^{∗}-algebra to be purely infinite. In Section 5, we show how various existing methods of computing the K-theory of the C^{∗}(Λ_{n}) can be used to compute the K-theory of C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}, p_{n})). Our results boil down to checking that each of the existing K-theory computations for the C^{∗}(Λ_{n}) is natural in the appropriate sense. Given that K-theory for higher-rank graph C^{∗}-algebras has proven quite difficult to compute in general (see [14]), our K-theory computations are an important outcome of the paper. We conclude in Section 6 by exploring some detailed examples which illustrate the covering-system construction, and show how to apply our K-theory calculations to the resulting higher-rank graph C^{∗}-algebras. For integers 3 ≤ n < ∞, we obtain a 3-graph algebra realisation of Kirchberg algebra P_{n} whose K-theory is opposite to that of O_{n} (see Section 6.3). We also obtain, using 3-graphs, a class of simple AT-algebras with real-rank zero which cannot be obtained from the rank-2 Bratteli diagram construction of [27] (see Section 6.4), and which we can describe in a natural fashion as higher-rank analogues of the Bunce-Deddens algebras. These are, to our knowledge, the first explicit computations of K-theory for infinite classes of 3-graph algebras.

†In its full generality, our construction is more complicated (see Proposition 2.14), enabling us to recover the important example of the irrational rotation algebras discussed in [27]. To keep technical detail in this introduction to a minimum, we discuss only the basic construction here.

Acknowledgements. We thank Iain Raeburn for his generosity with his time and for a number of very helpful conversations about this work. We also thank Toke Carlsen for helpful discussions and for his comments on the manuscript.

2. Covering systems of k-graphs

For k-graphs we adopt the conventions of [20, 25, 31]; briefly, a k-graph is a countable small category Λ equipped with a functor d : Λ → N^{k} satisfying the factorisation property: for all λ ∈ Λ and m, n ∈ N^{k} such that d(λ) = m + n there exist unique μ, ν ∈ Λ such that d(μ) = m, d(ν) = n, and λ = μν. When d(λ) = n we say λ has degree n. By abuse of notation, we will use d to denote the degree functor in every k-graph in this paper; the domain of d is always clear from context. The standard generators of N^{k} are denoted e_{1}, …, e_{k}, and for n ∈ N^{k} and 1 ≤ i ≤ k we write n_{i} for the i^{th} coordinate of n. If Λ is a k-graph, the vertices are the morphisms of degree 0. The factorisation property implies that these are precisely the identity morphisms, and so can be identified with the objects. For α ∈ Λ, the source s(α) is the domain of α, and the range r(α) is the codomain of α (strictly speaking, s(α) and r(α) are the identity morphisms associated to the domain and codomain of α). For n ∈ N^{k}, we write Λ^{n} for d^{−1}(n). In particular, Λ^{0} is the vertex set. For u, v ∈ Λ^{0} and E ⊂ Λ, we write uE := E ∩ r^{−1}(u) and Ev := E ∩ s^{−1}(v). For n ∈ N^{k}, we write

Λ^{≤n} := {λ ∈ Λ : d(λ) ≤ n, s(λ)Λ^{e_{i}} = ∅ whenever d(λ) + e_{i} ≤ n}.

We say that Λ is connected if the equivalence relation on Λ^{0} generated by {(v, w) ∈ Λ^{0} × Λ^{0} : vΛw ≠ ∅} is the whole of Λ^{0} × Λ^{0}. A morphism between k-graphs is a degree-preserving functor. We say that Λ is row-finite if vΛ^{n} is finite for all v ∈ Λ^{0} and n ∈ N^{k}. We say that Λ is locally convex if whenever 1 ≤ i < j ≤ k, e ∈ Λ^{e_{i}}, f ∈ Λ^{e_{j}} and r(e) = r(f), we can extend both e and f to paths ee′ and ff′ in Λ^{e_{i}+e_{j}}. We next introduce the notion of a covering of one k-graph by another. For a more detailed treatment of coverings of k-graphs, see [25].

Definition 2.1. A covering of a k-graph Λ is a surjective k-graph morphism p : Γ → Λ such that for all v ∈ Γ^{0}, p maps Γv 1-1 onto Λp(v) and vΓ 1-1 onto p(v)Λ. A covering p : Γ → Λ is connected if Γ, and hence also Λ, is connected. A covering p : Γ → Λ is finite if p^{−1}(v) is finite for all v ∈ Λ^{0}.

Remarks 2.2. (1) A covering p : Γ → Λ has the unique path lifting property: for every λ ∈ Λ and v ∈ Γ^{0} with p(v) = s(λ) there exists a unique γ such that p(γ) = λ and s(γ) = v; likewise, if p(v) = r(λ) there is a unique ζ such that p(ζ) = λ and r(ζ) = v.

(2) If Λ is connected then surjectivity of p is implied by the unique path-lifting property.

(3) If there is a fixed integer n such that |p^{−1}(v)| = n for all v ∈ Λ^{0}, p is said to be an n-fold covering. If Γ is connected, then p is automatically an n-fold covering for some n.

Notation 2.3. For m ∈ N \ {0}, we write S_{m} for the group of permutations of the set {1, …, m}. We denote both composition of permutations in S_{m}, and the action of a permutation in S_{m} on an element of {1, …, m} by juxtaposition; so for ϕ, ψ ∈ S_{m}, ϕψ ∈ S_{m} is the permutation ϕ ∘ ψ, and for ϕ ∈ S_{m} and j ∈ {1, …, m}, ϕj ∈ {1, …, m} is the image of j under ϕ. When convenient, we regard S_{m} as (the morphisms of) a category with a single object.

Definition 2.4. Fix k, m ∈ N \ {0}, and let Λ be a k-graph. A cocycle 𝔰 : Λ → S_{m} is a functor λ ↦ 𝔰(λ) from the category Λ to the category S_{m}. That is, whenever α, β ∈ Λ satisfy s(α) = r(β) we have 𝔰(α)𝔰(β) = 𝔰(αβ).

We are now ready to describe the data needed for our construction.

Definition 2.5. A covering system of k-graphs is a quintuple (Λ, Γ, p, m, 𝔰) where Λ and Γ are k-graphs, p : Λ → Γ is a covering, m is a nonzero positive integer, and 𝔰 : Γ → S_{m} is a cocycle. We say that the covering system is row finite if the covering map p is finite and both Λ and Γ are row finite. When m = 1 and 𝔰 is the identity cocycle, we drop references to m and 𝔰 altogether, and say that (Λ, Γ, p) is a covering system of k-graphs.

Given a covering system (Λ, Γ, p, m, 𝔰) of k-graphs, we will define a (k + 1)- graph Λ↽^{p,𝔰}Γ which encodes the covering map. Before the formal statement of this construction, we give an intuitive description of Λ↽^{p,𝔰}Γ. The idea is that Λ↽^{p,𝔰}Γ is a (k + 1)-graph containing disjoint copies ı(Λ) and ȷ(Γ) of the k-graphs Λ and Γ in the first k dimensions. The image ȷ(v) of a vertex v ∈ Γ is connected to the image ı(p(v)) of the vertex it covers in Λ by m parallel edges e(v, 1), …, e(v, m) of degree e_{k+1}. Factorisations of paths in Λ↽^{p,𝔰}Γ involving edges e(v, l) of degree e_{k+1} are determined by the unique path-lifting property and the cocycle 𝔰. It may be helpful on the first reading to consider the case where m = 1 so that 𝔰 is necessarily trivial. To state the result formally, we first establish some notation.

Notation 2.6. Fix k > 0. For n ∈ N^{k} we denote by (n, 0_{1}) the element ∑_{i=1}^{k} n_{i}e_{i} ∈ N^{k+1} and for m ∈ N, we denote by (0_{k}, m) the element me_{k+1} ∈ N^{k+1}. We write (N^{k}, 0_{1}) for {(n, 0_{1}) : n ∈ N^{k}} and (0_{k}, N) for {(0_{k}, m) : m ∈ N}. Given a (k + 1)-graph Ξ, we write Ξ^{(0_{k},N)} for {ξ ∈ Ξ : d(ξ) ∈ (0_{k}, N)}, and we write Ξ^{(N^{k},0_{1})} for {ξ ∈ Ξ : d(ξ) ∈ (N^{k}, 0_{1})}. When convenient, we regard Ξ^{(0_{k},N)} as a 1-graph and Ξ^{(N^{k},0_{1})} as a k-graph, ignoring the distinctions between N and (0_{k}, N) and between N^{k} and (N^{k}, 0_{1}).

Proposition 2.7. Let (Λ, Γ, p, m, 𝔰) be a covering system of k-graphs. There is a unique (k + 1)-graph Λ↽^{p,𝔰}Γ such that:

(1) there are injective functors ı : Λ → Λ↽^{p,𝔰}Γ and ȷ : Γ → Λ↽^{p,𝔰}Γ such that d(ı(α)) = (d(α), 0_{1}) and d(ȷ(β)) = (d(β), 0_{1}) for all α ∈ Λ and β ∈ Γ;

(2) ı(Λ) ∩ ȷ(Γ) = ∅ and ı(Λ) ∪ ȷ(Γ) = {τ ∈ Λ↽^{p,𝔰}Γ : d(τ)_{k+1} = 0};

(3) there is a bijection e : Γ^{0} × {1, …, m} → (Λ↽^{p,𝔰}Γ)^{e_{k+1}};

(4) s(e(v, l)) = ȷ(v) and r(e(v, l)) = ı(p(v)) for all v ∈ Γ^{0} and 1 ≤ l ≤ m; and (5) e(r(λ), l)ȷ(λ) = ı(p(λ))e(s(λ), 𝔰(λ)^{−1}l) for all λ ∈ Γ and 1 ≤ l ≤ m.

If the covering system (Λ, Γ, p, m, 𝔰) is row finite, then Λ↽^{p,𝔰}Γ is row finite. Moreover, Λ is locally convex if and only if Γ is locally convex, and in this case Λ↽^{p,𝔰}Γ is also locally convex.

Notation 2.8. If m = 1 so that 𝔰 is necessarily trivial, we drop all reference to 𝔰. We denote Λ↽^{p,𝔰}Γ by Λ↽^{p}Γ, and write (Λ↽^{p}Γ)^{e_{k+1}} = {e(v) : v ∈ Γ^{0}}. In this case, the factorisation property is determined by the unique path-lifting property alone.

The main ingredient in the proof of Proposition 2.7 is the following fact from [15, Remark 2.3] (see also [31, Section 2]).

Lemma 2.9. Let E_{1}, …, E_{k} be 1-graphs with the same vertex set E^{0}. For distinct i, j ∈ {1, …, k}, let E_{i,j} := {(e, f) ∈ E_{i}^{1} × E_{j}^{1} : s(e) = r(f)}, and write r((e, f)) = r(e) and s((e, f)) = s(f). For distinct h, i, j ∈ {1, …, k}, let E_{h,i,j} := {(e, f, g) ∈ E_{h}^{1} × E_{i}^{1} × E_{j}^{1} : (e, f) ∈ E_{h,i}, (f, g) ∈ E_{i,j}}. Suppose we have bijections θ_{i,j} : E_{i,j} → E_{j,i} such that r ∘ θ_{i,j} = r, s ∘ θ_{i,j} = s and θ_{i,j} ∘ θ_{j,i} = id, and such that

(2.1)  (θ_{i,j} × id)(id ×θ_{h,j})(θ_{h,i} × id) = (id ×θ_{h,i})(θ_{h,j} × id)(id ×θ_{i,j})

as bijections from E_{h,i,j} to E_{j,i,h}. Then there is a unique k-graph Λ such that Λ^{0} = E^{0}, Λ^{e_{i}} = E_{i}^{1} for 1 ≤ i ≤ k, and for distinct i, j ∈ {1, …, k} and (e, f) ∈ E_{i,j}, the pair (f′, e′) ∈ E_{j,i} such that (f′, e′) = θ_{i,j}(e, f) satisfies ef = f′e′ as morphisms in Λ.

Remark 2.10. Every k-graph arises in this way: Given a k-graph Λ, let E^{0} := Λ^{0}, and E_{i}^{1} := Λ^{e_{i}} for 1 ≤ i ≤ k, and define r, s : E_{i}^{1} → E^{0} by restriction of the range and source maps in Λ. Define bijections θ_{i,j} : E_{i,j} → E_{j,i} via the factorisation property: θ_{i,j}(e, f) is equal to the unique pair (f′, e′) ∈ E_{j,i} such that ef = f′e′ in Λ. Then condition (2.1) holds by the associativity of the category Λ, and the uniqueness assertion of Lemma 2.9 implies that Λ is isomorphic to the k-graph obtained from the E_{i} and the θ_{i,j} using Lemma 2.9.

Lemma 2.9 tells us how to describe a k-graph pictorially. As in [31, 27], the skeleton of a k-graph Λ is the directed graph E_{Λ} with vertices E_{Λ}^{0} = Λ^{0}, edges E_{Λ}^{1} = ⋃_{i=1}^{k} Λ^{e_{i}}, range and source maps inherited from Λ, and edges of different degrees in Λ distinguished using k different colours in E_{Λ}: in this paper, we will often refer to edges of degree e_{1} as “blue” and edges of degree e_{2} as “red.” Lemma 2.9 implies that the skeleton E_{Λ} together with the factorisation rules fg = g′f′ where f, f′ ∈ Λ^{e_{i}} and g, g′ ∈ Λ^{e_{j}} completely specify Λ. In practise, we draw E_{Λ} using solid, dashed and dotted edges to distinguish the different colours, and list the factorisation rules separately.

Proof of Proposition 2.7. The idea is to apply Lemma 2.9 to obtain the (k+1)- graph Λ↽^{p,𝔰}Γ. We first define sets E^{0} and E_{i}^{1} for 1 ≤ i ≤ k + 1. As a set, E^{0} is a copy of the disjoint union Λ^{0} ⊔ Γ^{0}. We denote the copy of Λ^{0} in E^{0} by {ı(v) : v ∈ Λ^{0}} and the copy of Γ^{0} in E^{0} by {ȷ(w) : w ∈ Γ^{0}} where as yet the ı(v) and ȷ(w) are purely formal symbols. So

E^{0} = {ı(v) : v ∈ Λ^{0}} ⊔ {ȷ(w) : w ∈ Γ^{0}}.

For 1 ≤ i ≤ k, we define, in a similar fashion,

E_{i}^{1} := {ı(f) : f ∈ Λ^{e_{i}}} ⊔ {ȷ(g) : g ∈ Γ^{e_{i}}}

to be a copy of the disjoint union Λ^{e_{i}} ⊔ Γ^{e_{i}}. We define E_{k+1}^{1} to be a copy of Γ^{0} × {1, …, m} which is disjoint from E^{0} and each of the other E_{i}^{1}, and use formal symbols {e(v, l) : v ∈ Γ^{0}, 1 ≤ l ≤ m} to denote its elements. For 1 ≤ i ≤ k, define range and source maps r, s : E_{i}^{1} → E^{0} by r(ı(f)) := ı(r(f)), s(ı(f)) := ı(s(f)), r(ȷ(g)) := ȷ(r(g)) and s(ȷ(g)) := ȷ(s(g)). Define r, s : E_{k+1}^{1} → E^{0} as in Proposition 2.7(4). For distinct i, j ∈ {1, …, k + 1}, define E_{i,j} as in Lemma 2.9. Define bijections θ_{i,j} : E_{i,j} → E_{j,i} as follows:

• For 1 ≤ i, j ≤ k and (e, f) ∈ E_{i,j}, we must have either e = ı(a) and f = ı(b) for some composable pair (a, b) ∈ Λ^{e_{i}} ×_{Λ^{0}} Λ^{e_{j}}, or else e = ȷ(a) and f = ȷ(b) for some composable pair (a, b) ∈ Γ^{e_{i}} ×_{Γ^{0}} 3Γ^{e_{j}}. If e = ı(a) and f = ı(b), the factorisation property in Λ yields a unique pair b′ ∈ Λ^{e_{j}}, a′ ∈ Λ^{e_{i}} such that ab = b′a′, and we then define θ_{i,j}(e, f) = (ı(b′), ı(a′)). If e = ȷ(a) and f = ȷ(b), we define θ_{i,j}(e, f) similarly using the factorisation property in Γ.

• For 1 ≤ i ≤ k, and (e, f) ∈ E_{k+1,i}, we have f = ȷ(b) and e = e(r(b), l) for some b ∈ Γ^{e_{i}} and 1 ≤ l ≤ m. Define θ_{k+1,i}(e, f) := (ı(p(b)), e(s(f), 𝔰(f)^{−1}l)).

• For 1 ≤ i ≤ k, to define θ_{i,k+1}, first note that if (f′, e′) = θ_{k+1,i}(e, f), then e′ = e(w, l) for some w ∈ Γ^{0} and l ∈ {1, …, m} such that p(w) = s(f′), f is the unique lift of f′ such that s(f) = ȷ(w), and e = e(r(f), 𝔰(f)l). It follows that θ_{k+1,i} is a bijection and we may define θ_{i,k+1} := θ_{k+1,i}^{−1}.

Since Λ and Γ are k-graphs, the maps θ_{i,j}, 1 ≤ i, j ≤ k are bijections with θ_{j,i} = θ_{i,j}^{−1}, and we have θ_{i,k+1} = θ_{k+1,i}^{−1} by definition, so to invoke Lemma 2.9, we just need to establish equation (2.1). Equation (2.1) holds when h, i, j ≤ k because Λ and Γ are both k-graphs. Suppose one of h, i, j = k + 1. Fix edges f_{h} ∈ E_{h}^{1}, f_{i} ∈ E_{i}^{1} and f_{j} ∈ E_{j}^{1}. First suppose that h = k + 1; so f_{h} = e(r(f_{i}), l) for some l, and f_{i} and f_{j} both belong to ȷ(Γ). Apply the factorisation property for Γ to obtain f′_{j} and f′_{i} such that f′_{i} ∈ E_{i}^{1}, f′_{j} ∈ E_{j}^{1} and f′_{j}f′_{i} = f_{i}f_{j}. We then have θ_{i,j}(f_{i}, f_{j}) = (f′_{j}, f′_{i}). If we write p̃ for the map from {ȷ(f) : f ∈ ⋃_{i=1}^{k} Γ^{e_{i}}} to {ı(f) : f ∈ ⋃_{i=1}^{k} Λ^{e_{i}}} given by p̃(ȷ(λ)) := ı(p(λ)), then the properties of the covering map imply that θ_{i,j}(p̃(f_{i}), p̃(f_{j})) = (p̃(f′_{j}), p̃(f_{i})′). Now

(θ_{i,j} × id)(id ×θ_{h,j})(θ_{h,i} × id)(f_{h}, f_{i}, f_{j})
= (θ_{i,j} × id)(id ×θ_{h,j})(p̃(f_{i}), e(s(f_{i}), 𝔰(f_{i})^{−1}l), f_{j})
= (θ_{i,j} × id)(p̃(f_{i}), p̃(f_{j}), e(s(f_{j}), 𝔰(f_{j})^{−1}(𝔰(f_{i})^{−1})l))

= (p̃(f′_{j}), p̃(f′_{i}), e(s(f_{j}), 𝔰(f_{i}f_{j})^{−1}l), (2.2)

where, in the last equality, 𝔰(f_{j})^{−1}𝔰(f_{i})^{−1} = 𝔰(f_{i}f_{j})^{−1} by the cocycle property. On the other hand,

(id ×θ_{h,i})(θ_{h,j} × id)(id ×θ_{i,j})(f_{h}, f_{i}, f_{j})
= (id ×θ_{h,i})(θ_{h,j} × id)(f_{h}, f′_{j}, f′_{i})
= (id ×θ_{h,i})(p̃(f′_{j}), e(s(f_{j}), 𝔰(f′_{j})^{−1}l), f′_{i})
= (p̃(f′_{j}), p̃(f′_{i}), e(s(f_{i}), 𝔰(f′_{i})^{−1}(𝔰(f′_{j})^{−1}l)))
= (p̃(f′_{j}), p̃(f′_{i}), e(s(f_{i}), 𝔰(f′_{j}f′_{i})^{−1}l)).

Since f′_{j}f′_{i} = f_{i}f_{j}, this establishes (2.1) when h = k + 1 and 1 ≤ i, j ≤ k. Similar calculations establish (2.1) when i = k + 1 and when j = k + 1. By Lemma 2.9, there is a unique (k + 1)-graph Λ↽^{p,𝔰}Γ with (Λ↽^{p,𝔰}Γ)^{0} = E^{0}, (Λ↽^{p,𝔰}Γ)^{e_{i}} = E_{i}^{1} for all i and with commuting squares determined by the θ_{i,j}. Since the θ_{i,j}, 1 ≤ i, j ≤ k agree with the factorisation properties in Γ and Λ, the uniqueness assertion of Lemma 2.9 applied to paths consisting of edges in E_{1}^{1} ∪ ⋯ ∪ E_{k}^{1} shows that ı and ȷ extend uniquely to injective functors from Λ and Γ to (Λ↽^{p,𝔰}Γ)^{(N^{k},0_{1})} which satisfy Proposition 2.7(2). Assertions (3) and (4) of Proposition 2.7 follow from the definition of E_{k+1}^{1}, and the last assertion (5) is established by factorising λ into edges from the E_{i}^{1}, 1 ≤ i ≤ k and then performing calculations like (2.2). Now suppose that p is finite. Then Γ is row-finite if and only if Λ is, and in this case, Λ↽^{p,𝔰}Γ is also row-finite because p is locally bijective and m < ∞. That p is locally bijective shows that Λ is locally convex if and only if Γ is. Suppose that Γ is locally convex. Fix 1 ≤ i < j ≤ k + 1, a ∈ (Λ↽^{p,𝔰}Γ)^{e_{i}} and b ∈ (Λ↽^{p,𝔰}Γ)^{e_{j}} with r(a) = r(b). If j < k + 1 then a and b can be extended to paths of degree e_{i} + e_{j} because Λ and Γ are locally convex. If j = k + 1, then b = e(v, l) for some v ∈ Γ^{0} and 1 ≤ l ≤ m. Let a′ be the lift of a such that r(a′) = s(v), then ae(s(a′), l) and ba′ extend a and b to paths of degree e_{i} + e_{j}. It follows that Λ↽^{p}Γ is locally convex. □

Corollary 2.11. Fix N ≥ 2 in N ∪ {∞}. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{N−1} be a sequence of covering systems of k-graphs. Then there is a unique (k+1)-graph Λ

such that Λ^{e_{i}} = ⨆_{n=1}^{N} Λ_{n}^{e_{i}} for 1 ≤ i ≤ k, Λ^{e_{k+1}} = ⨆_{n=1}^{N−1}(Λ_{n} ↽^{p_{n},𝔰_{n}}Λ_{n+1})^{e_{k+1}}, and

such that range, source and composition are all inherited from the Λ_{n} ↽^{p_{n},𝔰_{n}}Λ_{n+1}. If each (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n}) is row-finite then Λ is row-finite. If each Λ_{n} is locally convex, so is Λ, and if each Λ_{n} is connected, so is Λ.

Proof. For the first part we just apply Lemma 2.9; the hypotheses follow automatically from the observation that if h, i, j are distinct elements of {1, …, k + 1} then each path of degree e_{h} + e_{i} + e_{j} lies in some Λ_{n} ↽^{p_{n},𝔰_{n}}Λ_{n+1}, and these are all (k + 1)-graphs by Proposition 2.7. The arguments for row-finiteness, local convexity and connectedness are the same as those in Proposition 2.7. □

Notation 2.12. When N is finite, the (k+1)-graph Λ of the previous lemma will henceforth be denoted Λ_{1} ↽^{p_{1},𝔰_{1}} ⋯ ↽^{p_{N−1},𝔰_{N−1}}Λ_{N}. If N = ∞, we instead denote

Λ by \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}).

2.1. Matrices of covering systems. In this subsection, we generalise our construction to allow for a different covering system (Λ_{j}, Γ_{i}, p_{i,j}, m_{i,j}, 𝔰_{i,j}) for each pair of connected components Λ_{j} ⊂ Λ and Γ_{i} ⊂ Γ. The objective is to recover the example of the irrational rotation algebras [27, Example 6.5].

Definition 2.13. Fix nonnegative integers c_{Λ}, c_{Γ} ∈ N \ {0}. A matrix of covering systems (Λ_{j}, Γ_{i}, m_{i,j}, p_{i,j}, 𝔰_{i,j})_{i,j=1}^{c_{Γ},c_{Λ}} consists of:

(1) k-graphs Λ and Γ which decompose into connected components Λ = ⨆_{j=1,...,c_{Λ}} Λ_{j} and Γ = ⨆_{i=1,...,c_{Γ}} Γ_{i};

(2) a matrix (m_{i,j})_{i,j=1}^{c_{Γ},c_{Λ}} ∈ M_{c_{Γ},c_{Λ}}(N) with no zero rows or columns; and

(3) for each i, j such that m_{i,j} ≠ 0, a covering system (Λ_{i}, Γ_{j}, p_{i,j}, m_{i,j}, 𝔰_{i,j}) of k-graphs.

Proposition 2.14. Fix nonnegative integers c_{Λ}, c_{Γ} ∈ N \ {0} and a matrix of covering systems (Λ_{j}, Γ_{i}, m_{i,j}, p_{i,j}, 𝔰_{i,j})_{i,j=1}^{c_{Γ},c_{Λ}}. Then there is a unique (k + 1)- graph

( ⨆ Λ_{j})↽^{p,𝔰}( ⨆ Γ_{i})

such that

(( ⨆ Λ_{j})↽^{p,𝔰}( ⨆ Γ_{i}))^{e_{k+1}} = ⨆_{i,j}(Λ_{j} ↽^{p_{i,j},𝔰_{i,j}}Γ_{i})^{e_{k+1}},

each (( ⨆ Λ_{j})↽^{p,𝔰}( ⨆ Γ_{i}))^{e_{l}} for 1 ≤ l ≤ k is equal to Λ^{e_{l}} ⊔Γ^{e_{l}} and the commuting squares are inherited from the Λ_{j} ↽^{p_{i,j},𝔰_{i,j}}Γ_{i}. If each (Λ_{i}, Γ_{j}, p_{i,j}, m_{i,j}, 𝔰_{i,j}) is row finite then ( ⨆ Λ_{j})↽^{p,𝔰}( ⨆ Γ_{i}) is row finite. If Λ and Γ are locally convex, so is ( ⨆ Λ_{j})↽^{p,𝔰}( ⨆ Γ_{i}).

Proof. We apply Lemma 2.9; since the commuting squares are inherited from the Λ_{j} ↽^{p_{i,j},𝔰_{i,j}}Γ_{i}, they satisfy the associativity condition (2.1) because each Λ_{j} ↽^{p_{i,j},𝔰_{i,j}}Γ_{i} is a (k + 1)-graph. □

Corollary 2.15. Fix N ≥ 2 in N∪{∞}. Let (c_{n})_{n=1}^{N} ⊂ N\{0} be a sequence of positive integers. For 1 ≤ n < N, let (Λ_{n,j}, Λ_{n+1,i}, p_{i,j}^{n}, m_{i,j}^{n}, 𝔰_{i,j}^{n})_{i,j=1}^{c_{n+1},c_{n}} be a matrix of covering systems. Then there exists a unique (k + 1)-graph Λ such 170 Alex Kumjian, David Pask, Aidan Sims

that

Λ^{e_{i}} = ⋃_{n=1}^{N} ⋃_{j=1}^{c_{n}} Λ_{n,j}^{e_{i}} for 1 ≤ i ≤ k,
Λ^{e_{k+1}} = ⋃_{n=1}^{N−1} (( ⨆_{j=1}^{c_{n}} Λ_{n,j}) ↽^{p^{n},𝔰^{n}}( ⨆_{i=1}^{c_{n+1}} Λ_{n+1,i}))^{e_{k+1}},

and the range, source and composition functions are all inherited from the

(k + 1)-graphs ( ⨆_{j=1}^{c_{n}} Λ_{n,j}) ↽^{p^{n},𝔰^{n}}( ⨆_{i=1}^{c_{n+1}} Λ_{n+1,i}).

If each (Λ_{n,j}, Λ_{n+1,i}, p_{i,j}^{n}, m_{i,j}^{n}, 𝔰_{i,j}^{n}) is row finite, then Λ is row finite. If each Λ_{n} is locally convex, so is Λ.

Example 2.16 (The Irrational Rotation algebras). Fix θ ∈ [0, 1] \ Q. Let [a_{1}, a_{2}, …] be the simple continued fraction expansion of θ. For each n, let c_{n} = 2, let ϕ_{n} := ( \begin{matrix}a_{n} & 1 \\ 1 & 0 \end{matrix} ), and let m^{n} := (m_{i,j}^{n})_{i,j=1}^{2} be the matrix product ϕ_{T (n+1)} ⋯ ϕ_{T (n)+1} where T (n) := n(n + 1)/2 is the n^{th} triangular number. Of all the integers m_{i,j}^{n} obtained this way, only m_{1,2}^{1} is equal to zero, so the matrices m^{n} have no zero rows or columns. Whenever m_{i,j}^{n} ≠ 0, let 𝔰_{i,j}^{n} be the permutation of the set {1, …, m_{i,j}^{n}} given by 𝔰_{i,j}^{n}l = l + 1 if 1 ≤ l < m_{i,j}^{n}, and 𝔰_{i,j}^{n}m_{i,j}^{n} = 1. Let Λ_{n,i}, n ∈ N \ {0}, i = 1, 2 be mutually disjoint copies of the 1graph T_{1} whose skeleton consists of a single vertex and a single directed edge. For each n, let Λ_{n} be the 1-graph Λ_{n,1} ⊔ Λ_{n,2} so that for each n, (Λ_{n,j}, Λ_{n+1,i}, p_{i,j}^{n}, m_{i,j}^{n}, 𝔰_{i,j}^{n})_{i,j=1}^{2} is a matrix of covering systems.

• ........... ...................... ................................. ............ ........... •^{........... ...................... ................................. ............ ...........}  • ........... ...................... ................................. ............ ........... •^{........... ...................... ................................. ............ ...........}  • ........... ...................... ................................. ............ ........... •^{........... ...................... ................................. ............ ...........}  • ........... ...................... ................................. ............ ........... •^{........... ...................... ................................. ............ ...........} . . . . . . ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................ ..............  ................ ..............  ................ .............. ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................ ..............  ................ ..............  ................ .............. .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ............................................................................................................................................................................................................................................................................... ................ .............. a_{1} 1 1  ................ .............. ................ .............. m_{1,1}^{1} m_{2,2}^{1} m_{1,2}^{1} m_{2,1}^{1}  ................ .............. ................ .............. m_{1,1}^{2} m_{2,2}^{2} m_{1,2}^{2} m_{2,1}^{2}

Figure 1. A tower of coverings with multiplicities

Modulo relabelling the generators of N^{2}, the 2-graph \begin{matrix}lim \\ ↽− \end{matrix} (⨆_{j=1}^{c_{n}} Λ_{n,j}; p_{i,j}^{n}, 𝔰_{i,j}^{n}) obtained from this data as in Corollary 2.15 is precisely the rank-2 Bratteli diagram of [27, Example 6.5] whose C^{∗}-algebra is Morita equivalent to the irrational rotation algebra A_{θ}. Figure 1 is an illustration of its skeleton (parallel edges drawn as a single edge with a label indicating the multiplicity). The factorisation rules are all of the form fg = σ(g)f′ where f and f′ are the dashed loops at either end of a solid edge in the diagram, and σ is a transitive permutation of the set of edges with the same range and source as g. More generally, Section 7 of [27] considers in some detail the structure of the C^{∗}-algebras associated to rank-2 Bratteli diagrams with length-1 cycles. All such rank-2 Bratteli diagrams can be recovered as above from Corollary 2.15.

3. C^{∗}-algebras associated to covering systems of k-graphs

In this section, we describe how a covering system (Λ, Γ, p, m, 𝔰) induces an inclusion of C^{∗}-algebras C^{∗}(Λ) ↪ M_{m}(C^{∗}(Γ)) and hence a homomorphism of K-groups K_{∗}(C^{∗}(Λ)) → K_{∗}(C^{∗}(Γ)). The main result of the section is Theorem 3.8 which shows how to use these maps to compute the K-theory of C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n})) from the data in a sequence (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} of covering systems. The following definition of the Cuntz-Krieger algebra of a row-finite locally convex k-graph Λ is taken from [31, Definition 3.3]. Given a row-finite, locally convex k-graph (Λ, d), a Cuntz-Krieger Λ-family is a collection {t_{λ} : λ ∈ Λ} of partial isometries satisfying (CK1) {t_{v} : v ∈ Λ^{0}} is a collection of mutually orthogonal projections; (CK2) t_{λ}t_{μ} = t_{λμ} whenever s(λ) = r(μ); (CK3) t_{λ}^{∗}t_{λ} = t_{s(λ)} for all λ ∈ Λ; and (CK4) t_{v} = ∑_{λ∈vΛ^{≤n}} t_{λ}t_{λ}^{∗} for all v ∈ Λ^{0} and n ∈ N^{k}. The Cuntz-Krieger algebra C^{∗}(Λ) is the C^{∗}-algebra generated by a Cuntz-Krieger Λ-family {s_{λ} : λ ∈ Λ} which is universal in the sense that for every Cuntz-Krieger Λ-family {t_{λ} : λ ∈ Λ} there is a unique homomorphism π_{t} of C^{∗}(Λ) satisfying π_{t}(s_{λ}) = t_{λ} for all λ ∈ Λ.

Remarks 3.1. If Λ has no sources (that is vΛ^{n} ≠ ∅ for all v ∈ Λ^{0} and n ∈ N^{k}), then Λ is automatically locally convex, and the definition of C^{∗}(Λ) given above reduces to the original definition [20, Definition 1.5]. By [31, Theorem 3.15] there is a Cuntz-Krieger Λ-family {t_{λ} : λ ∈ Λ} such that t_{λ} ≠ 0 for all λ ∈ Λ. The universal property of C^{∗}(Λ) therefore implies that the generating partial isometries {s_{λ} : λ ∈ Λ} ⊂ C^{∗}(Λ) are all nonzero.

Let Ξ be a k-graph. The universal property of C^{∗}(Ξ) gives rise to an action γ of T^{k} on C^{∗}(Ξ), called the gauge-action (see, for example [31, §4.1]), such that γ_{z}(s_{ξ}) = z^{d(ξ)}s_{ξ} for all z ∈ T^{k} and ξ ∈ Ξ.

Proposition 3.2. Let (Λ, Γ, p, m, 𝔰) be a row-finite covering system of locally convex k-graphs. Let γ_{Λ} and γ_{Γ} denote the gauge actions of T^{k} on C^{∗}(Λ) and C^{∗}(Γ), and let γ denote the gauge action of T^{k+1} on C^{∗}(Λ↽^{p,𝔰}Γ).

(1) The inclusions ı : Λ → Λ↽^{p,𝔰}Γ and ȷ : Γ → Λ↽^{p,𝔰}Γ induce embeddings of C^{∗}(Λ) and C^{∗}(Γ) in C^{∗}(Λ↽^{p,𝔰}Γ) characterised by

ı_{∗}(s_{α}) = s_{ı(α)} and ȷ_{∗}(s_{β}) = s_{ȷ(β)} for α ∈ Λ and β ∈ Γ.

(2) The sum ∑_{v∈ȷ(Γ^{0})} s_{v} converges in the strict topology to a full projection

Q ∈ M(C^{∗}(Λ↽^{p,𝔰}Γ)), and the range of ȷ_{∗} is QC^{∗}(Λ↽^{p,𝔰}Γ)Q.

(3) For 1 ≤ i ≤ m, the sum ∑_{v∈Γ^{0}} s_{e(v,i)} converges strictly to a partial isometry V_{i} ∈ M(C^{∗}(Λ↽^{p,𝔰}Γ)). The sum ∑_{v∈ı(Λ^{0})} s_{v}, converges strictly to the full

projection P := ∑_{i=1}^{m} V_{i}V_{i}^{∗} ∈ M(C^{∗}(Λ↽^{p,𝔰}Γ)). Moreover, ı_{∗} is a nondegenerate homomorphism into P C^{∗}(Λ↽^{p,𝔰}Γ)P.

(4) There is an isomorphism ϕ : M_{m}(C^{∗}(Γ)) → P C^{∗}(Λ↽^{p,𝔰}Γ)P such that

ϕ((a_{i,j})_{i,j=1}^{m}) = ∑_{i,j=1}^{m} V_{i}ȷ_{∗}(a_{i,j})V_{j}^{∗}.

(5) There is an embedding ι_{p,𝔰} : C^{∗}(Λ) → M_{m}(C^{∗}(Γ)) such that ϕ ∘ ι_{p,𝔰} = ı_{∗}. The embedding ι_{p,𝔰} is equivariant in γ_{Λ} and the action id_{m} ⊗γ_{Γ} of T^{k} on M_{m}(C^{∗}(Γ)) by coordinate-wise application of γ_{Γ}.

(6) If we identify K_{∗}(C^{∗}(Γ)) with K_{∗}(M_{m}(C^{∗}(Γ))), then the induced homomorphism (ι_{p,𝔰})_{∗} may be viewed as a map from K_{∗}(C^{∗}(Λ)) → K_{∗}(C^{∗}(Γ)). When applied to K_{0}-classes of vertex projections, this map satisfies

(ι_{p,𝔰})_{∗}([s_{v}]) = ∑_{p(u)=v} m · [s_{u}] ∈ K_{0}(C^{∗}(Γ)).

The proofs of the last three statements require the following general Lemma. This is surely well-known but we include it for completeness.

Lemma 3.3. Let A be a C^{∗}-algebra, let q ∈ M(A) be a projection, and suppose that v_{1}, …, v_{n} ∈ M(A) satisfy v_{i}^{∗}v_{j} = δ_{i,j}q for 1 ≤ i, j ≤ n. Then p = ∑_{i=1}^{n} v_{i}v_{i}^{∗} is a projection and pAp ≅ M_{n}(qAq).

Proof. That v_{i}^{∗}v_{j} = δ_{i,j}q implies that the v_{i} are partial isometries with mutually orthogonal range projections v_{i}v_{i}^{∗}. Hence p is a projection in M(A). Define a map ϕ from pAp to M_{n}(qAq) as follows: for a ∈ pAp and 1 ≤ i, j ≤ n, let a_{i,j} := v_{i}^{∗}av_{j}, and define ϕ(a) to be the matrix ϕ(a) = (a_{i,j})_{i,j=1}^{n}. It is straightforward to check using the properties of the v_{i} that ϕ is a C^{∗}-homomorphism. It is an isomorphism because the homomorphism ψ : M_{n}(qAq) → pAp defined by

ψ((a_{i,j})_{i,j=1}^{n}) := ∑_{i,j=1}^{n} v_{i}a_{ij}v_{j}^{∗} ∈ qAq

is an inverse for ϕ. □

Proof of Proposition 3.2. (1) The collection {s_{ı(λ)} : λ ∈ Λ} forms a Cuntz-Krieger Λ-family in C^{∗}(Λ↽^{p,𝔰}Γ), and so by the universal property of C^{∗}(Λ) induces a homomorphism ı_{∗} : C^{∗}(Λ) → C^{∗}(Λ↽^{p,𝔰}Γ). For z ∈ T^{k}, write (z, 1) for the element (z_{1}, …, z_{k}, 1) ∈ T^{k+1}. Recall that γ denotes the gauge action of T^{k+1} on C^{∗}(Λ↽^{p,𝔰}Γ). Then the action z ↦ γ_{(z,1)} of T^{k} on C^{∗}(Λ↽^{p,𝔰}Γ) satisfies

ı_{∗}((γ_{Λ})_{z}(a)) = γ_{(z,1)}(ı_{∗}(a))

for all a ∈ C^{∗}(Λ) and z ∈ T^{k}. Since ı_{∗}(s_{v}) = s_{ı(v)} ≠ 0 for all v ∈ Λ^{0} it follows from the gauge-invariant uniqueness theorem [20, Theorem 2.1] that ı_{∗} is injective. A similar argument applies to ȷ_{∗}.

(2) As the projections s_{v}, v ∈ ȷ(Γ^{0}) are mutually orthogonal, a standard argument shows that the sum ∑_{v∈ȷ(Γ^{0})} s_{v} converges to a projection Q in the mul-

tiplier algebra (see [30, Lemma 2.1]). The range of ȷ_{∗} is equal to QC^{∗}(Λ↽^{p,𝔰}Γ)Q because ȷ(Γ^{0})(Λ↽^{p,𝔰}Γ)ȷ(Γ^{0}) = ȷ(Γ). To see that Q is full, it suffices to show that every generator of C^{∗}(Λ↽^{p,𝔰}Γ) belongs to the ideal I(Q) generated by Q. So let α ∈ Λ↽^{p,𝔰}Γ. Either s(α) ∈ ȷ(Γ^{0}) or s(α) ∈ ı(Λ^{0}). If s(α) ∈ ȷ(Γ^{0}), then s_{α} = s_{α}Q ∈ I(Q). On the other hand, if s(α) ∈ ı(Λ^{0}), the Cuntz-Krieger relation ensures that

s_{α} = ∑_{p(w)=s(α)} ∑_{i=1}^{m} s_{α}s_{e(w,i)}Qs_{e(w,i)}^{∗},

which also belongs to I(Q).

(3) For fixed i, the partial isometries s_{e(v,i)} have mutually orthogonal range projections and mutually orthogonal source projections. Hence an argument similar that of [30, Lemma 2.1] shows that ∑_{v∈Γ^{0}} s_{e(v,i)} converges strictly to a multiplier V_{i} ∈ M(C^{∗}(Λ↽^{p,𝔰}Γ)). A simple calculation shows that V_{i}^{∗}V_{j} = δ_{i,j}Q for all i, j. Hence each V_{i} is a partial isometry, and P is full because Q is full. The homomorphism ı_{∗} is nondegenerate because the net

(ı_{∗}( ∑_{v∈F} s_{v}))_{F ⊂Λ^{0} finite}

converges strictly to P ∈ M(C^{∗}(Λ↽^{p,𝔰}Γ)).

(4) This follows directly from Part (3) and Lemma 3.3.

(5) We define ι_{p,𝔰} := ϕ^{−1}∘ı_{∗}. For the gauge-equivariance, recall that ı_{∗} (respectively ȷ_{∗}) are equivariant in γ|_{(T^{k},1)} and γ_{Λ} (respectively γ_{Γ}). By definition, ϕ is equivariant in (id ⊗γ) and γ_{(T^{k},1)} ∘ ȷ_{∗}. The equivariance of ι_{p,𝔰} follows.

(6) By (CK4), for v ∈ Λ^{0} we have s_{ı(v)} = ∑_{f∈v(Λ↽^{p,𝔰}Γ)^{e_{k+1}}} s_{f}s_{f}^{∗}, so the K_{0}-

class [s_{ı(v)}] is equal to ∑_{f∈v(Λ↽^{p,𝔰}Γ)^{e_{k+1}}} [s_{f}s_{f}^{∗}]. We can write v(Λ↽^{p,𝔰}Γ)^{e_{k+1}} as the disjoint union

v(Λ↽^{p,𝔰}Γ)^{e_{k+1}} = ⨆_{p(u)=v}{e(u, i) : 1 ≤ i ≤ m}.

In K_{0}(C^{∗}(Λ↽^{p,𝔰}Γ)), we have [s_{e(u,i)}s_{e(u,i)}^{∗}] = [s_{e(u,i)}^{∗}s_{e(u,i)}] = [s_{ȷ(u)}], and the

result follows. □

Notation 3.4. As in Notation 2.8, when m = 1 so that 𝔰 is trivial, we continue to drop references to 𝔰 at the level of C^{∗}-algebras. So Proposition 3.2(5) gives an inclusion ι_{p} : C^{∗}(Λ) → C^{∗}(Γ) and the induced homomorphism of K-groups obtained from Proposition 3.2(6) is denoted (ι_{p})_{∗} : K_{∗}(C^{∗}(Λ)) → K_{∗}(C^{∗}(Γ)). This homomorphism satisfies

(ι_{p})_{∗}([s_{v}]) = ∑_{p(u)=v}[s_{u}].

When no confusion is likely to occur, we will suppress the maps ı, ȷ, ı_{∗} and ȷ_{∗} and regard Λ and Γ as subsets of Λ↽^{p,𝔰}Γ and C^{∗}(Λ) and C^{∗}(Γ) as C^{∗}-subalgebras

of C^{∗}(Λ↽^{p,𝔰}Γ).

Remark 3.5. (1) The isomorphism ϕ of Proposition 3.2(4) extends to an isomorphism ϕ̃ : M_{m+1}(C^{∗}(Γ)) → C^{∗}(Λ↽^{p,𝔰}Γ) which takes the block diagonal matrix ( \begin{matrix}0_{m×m} & 0_{m×1} \\ 0_{1×m} & a \end{matrix} ) to ȷ_{∗}(a). To see this, let V_{,} …, V_{m} be as in Proposition 2.7(3), let V_{m+1} = Q, and apply Lemma 3.3.

(2) If m = 1 then ϕ is an isomorphism of C^{∗}(Γ) onto P C^{∗}(Λ↽^{p}Γ)P, and ι_{p} : C^{∗}(Λ) ↪ C^{∗}(Γ) satisfies

ι_{p}(s_{λ}) = ∑_{p(λ̃)=λ} sλ̃.

Fix N ≥ 2 in N. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{N−1} be a sequence of row-finite covering systems of locally convex k-graphs. Recall that in Corollary 2.11 we obtained from such data a (k + 1)-graph Λ_{1} ↽^{p_{1},𝔰_{1}} ⋯ ↽^{p_{N−1},𝔰_{N−1}}Λ_{N}, which for convenience we will denote Λ_{N} (the subscript is unnecessary here, but will be needed in Proposition 3.7). We now examine the structure of C^{∗}(Λ_{N}) using Proposition 3.2.

Proposition 3.6. Continue with the notation established in the previous paragraph. For each v ∈ Λ_{N}^{0}, list Λ_{N}^{Ne_{k+1}}v as {α(v, i) : 1 ≤ i ≤ M} where M = m_{1}m_{2} ⋯ m_{N−1}.

(1) For 1 ≤ n ≤ N, the sum ∑_{v∈Λ_{n}^{0}} s_{v} converges strictly to a full projection P_{n} ∈ M(C^{∗}(Λ_{N})).

(2) For 1 ≤ i ≤ M, the sum ∑_{v∈Λ_{N}^{0}} s_{α(v,i)} converges strictly to a partial isometry V_{i} ∈ M(C^{∗}(Λ_{N})) such that V_{i}^{∗}V_{i} = P_{N}.

(3) We have ∑_{i=1}^{M} V_{i}V_{i}^{∗} = P_{1}, and there is an isomorphism

ϕ : M_{M}(C^{∗}(Λ_{N})) → P_{1}C^{∗}(Λ_{N})P_{1}

such that ϕ((a_{i,j})_{i,j=1}^{M}) = ∑_{i,j=1}^{M} V_{i}a_{i,j}V_{j}^{∗}.

Proof. Calculations like those in parts (2) and (3) of Proposition 3.2 show that the sums defining the P_{n} and the V_{i} converge in the multiplier algebra of C^{∗}(Λ_{N}) and that each P_{n} is full. Since distinct paths in Λ_{N}^{Ne_{k+1}} have orthogonal range projections and since paths in Λ_{N}^{Ne_{k+1}} with distinct sources have orthogonal source projections, each V_{i}^{∗}V_{i} = P_{N}, and ∑_{i=1}^{M} V_{i}V_{i}^{∗} = P_{1}. One checks as in Proposition 3.2(1) that the inclusions ı_{n} : Λ_{n} ↪ Λ_{N} induce inclusions (ı_{n})_{∗} : C^{∗}(Λ_{n}) ↪ P_{n}C^{∗}(Λ_{N})P_{n}, and in particular that (ı_{N})_{∗} : C^{∗}(Λ_{N}) → P_{N}C^{∗}(Λ_{N})P_{N} is an isomorphism. The final statement follows from Lemma 3.3. □

We now describe the inclusions of the corners determined by P_{1} as N increases. To do this, we first need some notation. Given a C^{∗}-algebra A, and positive integers m, n, we denote by π_{m,n} ⊗id_{A} : M_{m}(M_{n}(A)) → M_{mn}(A) the canonical isomorphism which takes the matrix a = ((a_{i,j,j′,i′})_{j,j′=1}^{n})_{i,i′=1}^{m} to the matrix π(a) satisfying

π(a)_{j+n(i−1),j′+n(i′−1)} = a_{i,j,j′,i′} for 1 ≤ i, i′ ≤ m, 1 ≤ j, j′ ≤ n.

Given C^{∗}-algebras A and B, a positive integer m, and a C^{∗}-homomorphism ψ : A → B, we write id_{m} ⊗ψ : M_{m}(A) → M_{m}(B) for the C^{∗}-homomorphism

(id_{m} ⊗ψ)((a_{i,j})_{i,j=1}^{m}) = (ψ(a_{i,j}))_{i,j=1}^{m}.

Finally, given a matrix algebra M_{m}(A) over a C^{∗}-algebra A, and given 1 ≤ i, i′ ≤ m and a ∈ A, we write θ_{i,i′}a for the matrix

(θ_{i,i′}a)_{j,j′} = \begin{cases}a & if j = i and j′ = i′ \\ 0 & otherwise. \end{cases}

Proposition 3.7. Fix N ≥ 2 in N. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{N} be a sequence of row-finite covering systems of locally convex k-graphs. We view the (k + 1)-graph Λ_{N} := Λ_{1} ↽^{p_{1},𝔰_{1}} ⋯ ↽^{p_{N−1},𝔰_{N−1}}Λ_{N} as a subcategory of Λ_{N+1} := Λ_{1} ↽^{p_{1},𝔰_{1}} ⋯ ↽^{p_{N} ,𝔰_{N}}Λ_{N+1} and likewise regard C^{∗}(Λ_{N}) as a C^{∗}-subalgebra of C^{∗}(Λ_{N+1}). In particular, we view P_{1} = ∑_{v∈Λ_{1}^{0}} s_{v} as a projection in both M(C^{∗}(Λ_{N})) and M(C^{∗}(Λ_{N+1})). Let M := m_{1}m_{2} … m_{N−1}, and let ϕ_{N} : M_{M}(C^{∗}(Λ_{N})) → P_{1}(C^{∗}(Λ_{N})P_{1} and ϕ_{N+1} : M_{Mm_{N}} (C^{∗}(Λ_{N+1})) → P_{1}C^{∗}(Λ_{N+1})P_{1} be the isomorphisms obtained from Proposition 3.6. Then the following diagram commutes.

P_{1}C^{∗}(Λ_{N})P_{1}  P_{1}C^{∗}(Λ_{N+1})P_{1} M_{M}(C^{∗}(Λ_{N}))  M_{Mm_{N}} (C^{∗}(Λ_{N+1}))^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ϕ_{N}  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ϕ_{N+1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 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⊆_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} (π_{M,m_{N}} ⊗ id_{C^{∗}(Λ_{N+1})}) ∘ (id_{M} ⊗ι_{p_{N} ,𝔰_{N}})

Proof. As in Proposition 3.6, write Λ_{N}^{Ne_{k+1}} = {α(v, i) : v ∈ Λ_{N}^{0}, i ∈ {1, ⋯, M}}. For i = 1, …, M, let V_{i} := ∑_{v∈Λ_{N}^{0}} s_{α(v,i)}. For j = 1, …, m_{N}, let

W_{j} := ∑_{w∈Λ_{N+1}^{0}} ∑_{i=1}^{M} s_{α(p_{N} (w),i)}s_{e(w,j)}.

For (i, j) in the cartesian product {1, …, M}×{1, …, m_{N}}, let U_{j+m_{N} (i−1)} := ∑_{u∈Λ_{N+1}^{0}} s_{α(p_{N} (u),i)e(u,j)}. In what follows, we suppress canonical inclusion maps, and regard C^{∗}(Λ_{N}) as a subalgebra of C^{∗}(Λ_{N}), and both C^{∗}(Λ_{N}) and C^{∗}(Λ_{N+1}) as subalgebras of C^{∗}(Λ_{N+1}). The corner P_{1}C^{∗}(Λ_{N})P_{1} is equal to the closed span of elements of the form V_{i}aV_{i′}^{∗} where a ∈ C^{∗}(Λ_{N}) and i, i′ ∈ {1, …, M}, and P_{1}C^{∗}(Λ_{N+1})P_{1} is equal to the closed span of elements of the form U_{l}bU_{l′}^{∗} where b ∈ C^{∗}(Λ_{N+1}), l, l′ ∈ {1, …, Mm_{N}}. We have ϕ_{N}((a_{i,i′})_{i,i′=1}^{M}) = ∑_{i,i′=1}^{M} V_{i}a_{i,i′}V_{i′}^{∗} by definition. The isomorphism ϕ_{N+1} from M_{Mm_{N}} (C^{∗}(Λ_{N+1})) to P_{1}C^{∗}(Λ_{N+1})P_{1} described in Proposition 3.6 satisfies

ϕ_{N+1}( ∑_{l,l′=1}^{Mm_{N}} U_{l}b_{l,l′}U_{l′}^{∗}) = (b_{l,l′})_{l,l′=1}^{Mm_{N}} .

The Cuntz-Krieger relations show that

V_{i}V_{i′}^{∗}W_{j}W_{j′}^{∗} = U_{j+m_{N} (i−1)}U_{j′+m_{N} (i′−1)}^{∗} = W_{j}W_{j′}^{∗}V_{i}V_{i′}^{∗}

for 1 ≤ i, i′ ≤ M, 1 ≤ j, j′ ≤ m_{N}, and this decomposition of the matrix units U_{l}U_{l′}^{∗} implements π_{M,m_{N}}. Hence ϕ_{N+1} ∘ (π_{M,m_{N}} ⊗ id_{C^{∗}(Λ_{N+1})}) satisfies

\begin{matrix}ϕ_{N+1} ∘ (π_{M,m_{N}} ⊗ id_{C^{∗}(Λ_{N+1})}) & (((b_{i,j,j′,i′})_{j,j′=1}^{m_{N}})_{i,i′=1}^{M}) \\ = ∑_{i,i′=1}^{M} ∑_{j,j′=1}^{m_{N}} U_{j+m_{N} (i−1)}b_{i,j,j′,i′}U_{j′+m_{N} (i′−1)}^{∗}. \end{matrix} ; (3.1)

The Cuntz-Krieger relations also show that V_{i} = ∑_{j=1}^{m_{N}} W_{j}W_{j}^{∗}V_{i} for all i, and hence V_{i}aV_{i′}^{∗} = ∑_{j} U_{j+m_{N} (i−1)}W_{j}^{∗}aW_{j}U_{j+m_{N} (i′−1)}^{∗} for all a ∈ P_{1}C^{∗}(Λ_{N})P_{1}. One now checks that for λ ∈ Λ_{N}, we have

W_{j}^{∗}s_{λ}W_{j} = ∑_{p_{N} (λ′)=λ} s_{e(r(λ′),j)}^{∗}s_{e(r(λ),𝔰_{N} (λ′)j)}s_{λ′},

and hence that V_{i}s_{λ}V_{i′}^{∗} = ∑_{j} ∑_{p_{N} (λ′)=λ} U_{𝔰_{N} (λ′)j+m_{N} (i−1)}s_{λ′}U_{j+m_{N} (i′−1)}^{∗}. Recall that θ_{i,i′}s_{λ} ∈ M_{M}(C^{∗}(Λ_{N})) denotes the matrix

(θ_{i,i′}s_{λ})_{j,j′} = \begin{cases}s_{λ} & if j = i and j′ = i′ \\ 0 & otherwise. \end{cases}

Then V_{i}s_{λ}V_{i′}^{∗} = ϕ_{N}(θ_{i,i′}s_{λ}) by definition of ϕ_{N}, so

ϕ_{N}(θ_{i,i′}s_{λ}) = ∑_{j} ∑_{p_{N} (λ′)=λ} U_{𝔰_{N} (λ′)j+m_{N} (i−1)}s_{λ′}U_{j+m_{N} (i′−1)}^{∗}.

Since (id_{M} ⊗ι_{p_{N} ,𝔰_{N}})(θ_{i,i′}s_{λ}) = θ_{i,i′} ∑_{p_{N} (λ′)=λ} s_{λ′}, we may therefore apply (3.1) to see that

ϕ_{N}(θ_{i,i′}s_{λ}) = ϕ_{N+1} ∘ (π_{M,m_{N}} ⊗ id_{C^{∗}(Λ_{N+1})}) ∘ (id_{M} ⊗ι_{p_{N} ,𝔰_{N}})(θ_{i,i′}s_{λ}).

Since elements of the form θ_{i,i′}s_{λ} generate M_{M}(C^{∗}(Λ_{N})) this proves the result. □

Theorem 3.8. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} be a sequence of row-finite coverings of locally convex k-graphs. For each n, let Λ_{n} := Λ_{1} ↽^{p_{1},𝔰_{1}} ⋯ ↽^{p_{n−1},𝔰_{n−1}}Λ_{n}, identify Λ_{n} with the corresponding subset of \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}), and likewise identify C^{∗}(Λ_{n}) with the corresponding C^{∗}-subalgebra of C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n})). Then

(3.2)  C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n})) = \overline{⋃_{n=1}^{∞} C^{∗}(Λ_{n})}.

Let P_{1} := ∑_{v∈Λ_{1}^{0}} s_{v}, and for each n, let M_{n} := m_{1}m_{2} ⋯ m_{n−1}. Then P_{1} is a full projection in each M(C^{∗}(Λ_{n})), and we have

(3.3)  P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))P_{1} ≅ \underrightarrow{lim} (M_{M_{n}}(C^{∗}(Λ_{n})), id_{M_{n}} ⊗ι_{p_{n},𝔰_{n}}).

In particular,

K_{∗}(C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))) = K_{∗}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))P_{1})
≅ \underrightarrow{lim}(K_{∗}(C^{∗}(Λ_{n})), (ι_{p_{n},𝔰_{n}})_{∗}).

Proof. For the duration of the proof, let Λ := \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}). We have C^{∗}(Λ) = \overline{span}\{s_{μ}s_{ν}^{∗} : μ, ν ∈ Λ\}, so for the first statement, we need only show that

span{s_{μ}s_{ν}^{∗} : μ, ν ∈ Λ} ⊂ ⋃_{n=1}^{∞} C^{∗}(Λ_{n}).

To see this we simply note that for any finite F ⊂ Λ, the integer N := max{n ∈ N : s(F) ∩ Λ_{n}^{0} ≠ ∅} satisfies F ⊂ Λ_{N}. Since P_{1} is full in each C^{∗}(Λ_{n}) by Proposition 3.2(3), it is full in C^{∗}(Λ) by (3.2). Equation 3.3 follows from Proposition 3.7. The final statement then follows from continuity of the K-functor. □

Remark 3.9. Note that if we let γ denote the restriction of the gauge action to P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))P_{1} then γ_{(1,··· ,1,z)} is trivial for all z ∈ T. Indeed, if s_{μ}s_{ν}^{∗} is a nonzero element P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))P_{1}, then d(μ)_{n+1} = d(ν)_{n+1}. So γ may be regarded as an action by T^{k} rather than T^{k+1}.

We can extend Theorem 3.8 to the situation of matrices of covering systems as discussed in Section 2.1 as follows.

Proposition 3.10. Resume the notation of Corollary 2.15. Each C^{∗}(Λ_{n}) is canonically isomorphic to ⨁_{j=1}^{c_{n}} C^{∗}(Λ_{n,j}). There are homomorphisms (ι_{n})_{∗} : K_{∗}(C^{∗}(Λ_{n})) → K_{∗}(C^{∗}(Λ_{n+1})) such that the partial homomorphism which maps the j^{th} summand in K_{∗}(C^{∗}(Λ_{n})) to the i^{th} summand in K_{∗}(C^{∗}(Λ_{n+1})) is equal to 0 if m_{i,j}^{n} = 0, and is equal to (ι_{p_{i,j}^{n},𝔰_{i,j}^{n}})_{∗} otherwise. The sum ∑_{v∈Λ_{1}^{0}} s_{v} converges strictly to a full projection P_{1} ∈ M(C^{∗}(Λ)). Furthermore,

K_{∗}(P_{1}C^{∗}(Λ)P_{1}) ≅ \underrightarrow{lim} ( ⨁_{j=1}^{c_{n}} K_{∗}(C^{∗}(Λ_{n,j})), (ι_{n})_{∗}).

Proof. For each λ ∈ Λ_{n} = ⨆_{j=1}^{c_{n}} Λ_{n,j}, define a partial isometry t_{λ} ∈ ⨁_{j=1}^{c_{n}} C^{∗}(Λ_{n,j}) by t_{λ} := (0, …, 0, s_{λ}, 0, …, 0) (the nonzero term is in the j^{th} coordinate when λ ∈ Λ_{n,j}). These nonzero partial isometries form a Cuntz-Krieger Λ_{n}-family consisting of nonzero partial isometries. The universal property of C^{∗}(Λ_{n}) gives a homomorphism π_{t}^{n} : C^{∗}(Λ_{n}) → ⨁_{j=1}^{c_{n}} C^{∗}(Λ_{n,j}) which intertwines the direct sum of the gauge actions on the C^{∗}(Λ_{n,j}) and the gauge action on C^{∗}(Λ_{n}). The gauge-invariant uniqueness theorem [20, Theorem 3.4], and the observation that each generator of each summand in ⨁_{j=1}^{c_{n}} C^{∗}(Λ_{n,j}) is nonzero and belongs to the image of π_{t}^{n} therefore shows that π_{t}^{n} is an isomorphism. The individual covering systems (Λ_{n,j}, Λ_{n+1,i}, p^{n}, m^{n}, 𝔰^{n}) induce inclusions ι_{p_{i,j}^{n},𝔰_{i,j}^{n}} : C^{∗}(Λ_{n,j}) → M_{m_{i,j}^{n}}(C^{∗}(Λ_{n+1,i})) as in Proposition 3.2(5). We therefore obtain homomorphisms (ι_{p_{i,j}^{n},𝔰_{i,j}^{n}})_{∗} : K_{∗}(C^{∗}(Λ_{n,j})) → K_{∗}(C^{∗}(Λ_{n+1,i})). The statement about the partial homomorphisms of K-groups then follows from the properties of the isomorphism K_{∗}(⨁_{i} A_{i}) ≅ ⨁_{i} K_{∗}(A_{i}) for C^{∗}-algebras A_{i}. The final statement can then be deduced from arguments similar to those of Theorem 3.8. □

4. Simplicity and pure infiniteness

Theorem 3.1 of [34] gives a necessary and sufficient condition for simplicity of the C^{∗}-algebra of a row-finite k-graph with no sources. Specifically, C^{∗}(Λ) is simple if and only if Λ is cofinal and every vertex of Λ receives an aperiodic infinite path (see below for the definitions of cofinality and aperiodicity). In this section we present some means of deciding whether \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}) is cofinal (Lemma 4.7), and whether an infinite path in \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}) is aperiodic (Lemma 4.3). We also present a condition under which C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n})) is purely infinite (Proposition 4.8). We begin by recalling the notation and definitions required to make sense of the hypotheses of [34, Theorem 3.1]. For more detail, see Section 2 of [31].

Notation 4.1. We write Ω_{k} for the k-graph such that Ω_{k}^{q} := {(m, n) ∈ N^{k}×N^{k} : n − m = q} for each q ∈ N^{k}, with r(m, n) = (m, m) and s(m, n) = (n, n). We identify Ω_{k}^{0} = {(m, m) : m ∈ N^{k}} with N^{k}. An infinite path in a k-graph Ξ is a graph morphism x : Ω_{k} → Ξ, and we denote the image x(0) of the vertex 0 ∈ Ω_{k}^{0} by r(x). We write Ξ^{∞} for the collection of all infinite paths in Ξ, and for v ∈ Ξ^{0} we denote by vΞ^{∞} the collection {x ∈ Ξ^{∞} : r(x) = v}. For x ∈ Ξ^{∞} and q ∈ N^{k}, there is a unique infinite path σ^{q}(x) ∈ Ξ^{∞} such that σ^{q}(x)(m, n) = x(m + q, n + q) for all m ≤ n ∈ N^{k}.

Definition 4.2. We say that a row-finite k-graph Ξ with no sources is aperiodic if for each vertex v ∈ Ξ^{0} there is an infinite path x ∈ vΞ^{∞} such that σ^{q}(x) ≠ σ^{q′}(x) for all q ≠ q′ ∈ N^{k}. We say that Ξ is cofinal if for each v ∈ Ξ^{0} and x ∈ Ξ^{∞} there exists m ∈ N^{k} such that vΞx(m) ≠ ∅.

We continue to make use in the following of the notation established earlier (see Notation 2.6) for the embeddings of N^{k} and of N in N^{k+1}. If y is an infinite path in the (k+1)-graph Ξ, we write α_{y} for the infinite path in Ξ^{(0_{k},N)} defined by α_{y}(p, q) := y((0_{k}, p), (0_{k}, q)) for p ≤ q ∈ N, and we write x_{y} for the infinite path in Ξ^{(N^{k},0_{1})} defined by x_{y}(p, q) := y((p, 0_{1}), (q, 0_{1})) where p ≤ q ∈ N^{k}.

Proposition 4.3. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} be a sequence of row-finite covering systems of k-graphs with no sources. For a, b ∈ N^{k+1}, an infinite path y ∈ (\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))^{∞} satisfies σ^{a}(y) = σ^{b}(y) if and only if x_{σ^{a}(y)} = x_{σ^{b}(y)} and α_{σ^{a}(y)} = α_{σ^{b}(y)}.

Proof. The “only if” implication is trivial. For the “if” implication, note that the factorisation property implies that an infinite path z of \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}) is uniquely determined by x_{z} and the paths α_{σ^{(n,0_{1})}(z)}, n ∈ N^{k}. So it suffices to show that each α_{σ^{(n,0_{1})}(z)} is uniquely determined by x_{z}(0_{k}, n) and α_{z}. Fix n ∈ N^{k} and let λ := x_{z}(0_{k}, n) = z(0_{k+1}, (n, 0_{1})). Fix i ∈ N. We will show that α_{σ^{(n,0_{1})}(z)}(0_{1}, i) is uniquely determined by α_{z}(0_{1}, i) and λ. Let v = r(z), and let N ∈ N be the element such that v ∈ Λ_{N}^{0}. For 1 ≤ j ≤ i, let w_{j} = α_{z}(i) ∈ Λ_{N+j}^{0}, and let 1 ≤ l_{j} ≤ m_{N+j−1} be the integer such that α_{z}(j − 1, j) = e(w_{i}, l_{j}). We have p_{N}(w_{1}) = v, and p_{N+j−1}(w_{j}) = w_{j−1} for 2 ≤ j ≤ i. For each j, let λ_{j} be the unique lift of λ such that r(λ_{j}) = w_{j}. By definition of the (k + 1)-graph \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}), the path

λe(s(λ_{1}), 𝔰(λ_{1})^{−1}l_{1})e(s(λ_{2}), 𝔰(λ_{2})^{−1}l_{2}) … e(s(λ_{i}), 𝔰(λ_{i})^{−1}l_{i}) = α_{z}(0_{1}, i)λ_{i}

is the unique minimal common extension of λ and α_{z}(0_{1}, i) in \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}). Hence

α_{σ^{(n,0_{1})}(z)}(0_{1}, i) = e(s(λ_{1}), 𝔰(λ_{1})^{−1}l_{1})e(s(λ_{2}), 𝔰(λ_{2})^{−1}l_{2}) … e(s(λ_{i}), 𝔰(λ_{i})^{−1}l_{i})

which is uniquely determined by λ and α_{z}(0_{1}, i). □

Corollary 4.4. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} be a sequence of row-finite covering systems of k-graphs with no sources. Suppose that Λ_{n} is aperiodic for some n. Then so is \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}).

Proof. Since each vertex in Λ_{n} receives an aperiodic path in Λ_{n}, Proposition 4.3, guarantees that each vertex in Λ_{n} receives an aperiodic path in \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}). Since the p_{n} are coverings, it follows that every vertex of \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}) receives an infinite path of the form λy or of the form σ^{p}(y) where y is an aperiodic path with range in Λ_{n}. If y is aperiodic, then λy is aperiodic for any λ and σ^{a}(y) is aperiodic for any a and the result follows. □

Lemma 4.5. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} be a sequence of row-finite covering systems of k-graphs with no sources. Fix y ∈ (\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))^{∞}, with y(0) ∈ Λ_{n} and a, b ∈ N^{k+1}. Let ã and b̃ denote the elements of N^{k} determined by the first k coordinates of a and b. For each m ≥ n, let v_{m} and i_{m} be the unique pair such that α_{y}(m, m + 1) = e(v_{m}, i_{m}). For each m ≥ n, let μ_{m} and ν_{m} be the unique lifts of x_{y}(0, ã) and x_{y}(0, b̃) such that r(μ_{m}) = r(ν_{m}) = v_{m}. Then α_{σ^{a}(y)} = α_{σ^{b}(y)} if and only if the following three conditions hold:

(1) a_{k+1} = b_{k+1};

(2) s(μ_{m}) = s(ν_{m}) for all m ≥ n; and

(3) 𝔰_{m}(μ_{m})i_{m} = 𝔰_{m}(ν_{m})i_{m} for all m ≥ n.

Proof. We have α_{σ^{a}(y)}(m, m+1) = e(s(μ_{m+a_{k+1}}), 𝔰_{m}(μ_{m+a_{k+1}})i_{m+a_{k+1}}) for all m, and likewise for b and ν. □

Remark 4.6. Lemma 5.4 of [27] implies that an infinite path in a rank-2 Bratteli diagram Λ is aperiodic if and only if the factorisation permutations of its red coordinate-paths are of unbounded order. Lemma 4.5 is the analogue of this result for general systems of coverings. To see the analogy, note that in a rank-2 Bratteli diagram, every x_{y} is of the form λλλ … for some blue cycle Λ, so that condition (3) fails for all a ≠ b precisely when the orders of the permutations 𝔰_{m}(μ_{m}) grow arbitrarily large with m.

Lemma 4.7. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} be a sequence of row-finite coverings of k-graphs with no sources. If infinitely many of the Λ_{n} are cofinal, then \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}) is also cofinal.

Proof. Fix a vertex v and an infinite path z ∈ (\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))^{∞}. Let n_{1}, n_{2} ∈ N be the elements such that v ∈ Λ_{n_{1}}^{0} and r(z) ∈ Λ_{n_{2}}^{0}. Choose N ≥ n_{1}, n_{2} such that Λ_{N} is cofinal. Fix w ∈ Λ_{N}^{0} such that p_{n} ∘ p_{n+1} ∘ ⋯ ∘ p_{N−1}(w) = v; so v(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n}))w ≠ ∅. We have x_{σ^{(0_{k},N−n_{2})}(z)} ∈ Λ_{N}^{∞}, and since Λ_{N} is cofinal, it follows that wΛ_{N}x_{σ^{(0_{k},N−n_{2})}(z)}(q) ≠ ∅ for some q ∈ N^{k}. Since x_{σ^{(0_{k},N−n_{2})}(z)}(q) = z(q, N − n_{2}), this completes the proof. □

As in [38], we say that a path λ in a k-graph Λ is a cycle with an entrance if s(λ) = r(λ), and there exists μ ∈ r(λ)Λ with d(μ) ≤ d(λ) and λ(0, d(μ)) ≠ μ.

Proposition 4.8. Let (Λ_{n}, Λ_{n+1}, p_{n}, m_{n}, 𝔰_{n})_{n=1}^{∞} be a sequence of row-finite coverings of k-graphs with no sources. There exists n such that Λ_{n} contains a cycle with an entrance if and only if every Λ_{n} contains a cycle with an entrance. Moreover, if C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n})) is simple and Λ_{1} contains a cycle with an entrance, then C^{∗}(Λ) is purely infinite.

Proof. That the presence of a cycle with an entrance in Λ_{1} is equivalent to the presence of a cycle with an entrance in every Λ_{n} follows from the properties of covering maps. Now the result follows from [38, Proposition 8.8] □

5. K-Theory

In this section, we consider the K-theory of C^{∗}(Λ↽^{p,𝔰}Γ). Specifically, we show how the homomorphism from K_{∗}(C^{∗}(Λ)) to K_{∗}(C^{∗}(Γ)) obtained from Proposition 3.2 behaves with respect to existing calculations of K-theory for various classes of higher-rank graph C^{∗}-algebras. We will use these results later to compute the K-theory of C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{n}; p_{n}, 𝔰_{n})) for a number of sequences of covering systems. Throughout this section, given a k-graph Λ, we view the ring ZΛ^{0} as the collection of finitely supported functions f : Λ^{0} → Z. For v ∈ Λ^{0}, we denote the point-mass at v by δ_{v}. Given a finite covering p : Γ → Λ of row-finite k-graphs, we define p^{∗} : ZΛ^{0} → ZΓ^{0} by p^{∗}(δ_{v}) = ∑_{p(u)=v} δ_{w}; equivalently, p^{∗}(f)(w) = f(p(w)).

5.1. Coverings of 1-graphs and the Pimsner-Voiculescu exact sequence. It is shown in [26, 32] how to compute the K-theory of a graph C^{∗}-algebra using the Pimsner-Voiculescu exact sequence. In this subsection, we show how this calculation interacts with the inclusion of C^{∗}-algebras arising from a covering of 1-graphs. The K-theory computations for arbitrary graph C^{∗}-algebras [12, 1] are somewhat more complicated than for the C^{∗}-algebras of row-finite graphs with no sources. Moreover, every graph C^{∗}-algebra is Morita equivalent to the C^{∗}- algebra of a row-finite graph with no sources [12]. We therefore restrict out attention here to the simpler setting.

Theorem 5.1. Let (E^{∗}, F^{∗}, p, m, 𝔰) be a row-finite covering system of 1-graphs with no sources. Let A, B be the vertex connectivity matrices of the underlying graphs E and F respectively. Then the diagram

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commutes and the rows are exact.

The proof of this theorem occupies the remainder of Section 5.1. We fix, for the duration, a finite covering p : F^{∗} → E^{∗} of row-finite 1-graphs with no sources, a multiplicity m and a cocycle 𝔰 : F^{∗} → S_{m}. It is relatively straightforward to prove that the right-hand two squares of (5.1) commute and that the rows are exact.

Lemma 5.2. Resume the notation of Theorem 5.1. We have (1 − B^{t})p^{∗} = p^{∗}(1 − A^{t}), the right-hand two squares of (5.1) commute, and the rows are exact.

Proof. For the first statement, consider a generator δ_{v} ∈ ZE^{0}. We have

(p^{∗} ∘ (1 − A^{t}))(δ_{v}) = p^{∗}(δ_{v} − ∑_{e∈vE^{1}} δ_{s(e)}) = ∑_{p(u)=v} δ_{u} − ∑_{e∈vE^{1}} ∑_{p(f)=e} δ_{s(f)}.

On the other hand,

((1 − B^{t}) ∘ p^{∗})(δ_{v}) = (1 − B^{t}) ∑_{p(u)=v} δ_{u} = ∑_{p(u)=v} (δ_{u} − ∑_{f∈uF^{1}} δ_{s(f)}).

Since p is a covering the double-sums occurring in these two equations each contain exactly one term for each edge f ∈ F^{1} such that p(r(f)) = v, and it follows that the two are equal. Multiplying by m throughout the above calculation shows that the middle square of (5.1) commutes. The identification of K_{0}(C^{∗}(E^{∗})) with coker(1 − A^{t}) takes the class of the projection s_{v} ∈ C^{∗}(E^{∗}) to the class of the corresponding generator δ_{v} ∈ ZE^{0} (see [30]). That the right-hand square commutes then follows from Proposition 3.2(6). Exactness of the rows is precisely the computation of K-theory for 1-graph C^{∗}-algebras [8, 26, 32]. □

It remains to prove that the left-hand square of (5.1) commutes. The strategy is to assemble the eight-term commuting diagrams which describe the K-theory of each of C^{∗}(E^{∗}) and C^{∗}(F^{∗}) (see equation (5.3) below) into a sixteen-term diagram, one face of which is the left-hand square of (5.1). We then focus on the cube in the sixteen-term diagram which contains left-hand square of (5.1) as one of its faces, and show that the remaining five faces of this cube commute. A diagram-chase then establishes that the sixth face commutes as well. The majority of the work involved goes into defining the connecting maps needed to write down the sixteen-term diagram in the first place. The proof that the various squares in it commute is then relatively straightforward. To begin, we recall how one shows that the rows of (5.1) are exact. Let E^{∗} ×_{d} Z be the skew-product of E^{∗} by the length functor d (see [20, Section 5]). Let γ be the gauge action of T on C^{∗}(E^{∗}) satisfying γ_{z}(s_{e}) = zs_{e} for e ∈ E^{1} and z ∈ T. Let (i_{T}, i_{C^{∗}(E^{∗})}) be the universal covariant representation of (C^{∗}(E^{∗}), T, γ) in the crossed product C^{∗}(E^{∗})×_{γ} T. By [32, Lemma 3.1], there is an isomorphism

(5.2)  ψ_{E} : C^{∗}(E^{∗} ×_{d} Z) → C^{∗}(E^{∗}) ×_{γ} T

satisfying ψ_{E}(s_{(λ,n)}) = i_{T}(z)^{n}i_{C^{∗}(E^{∗})}(s_{λ}). The C^{∗}-algebra C^{∗}(E^{∗} ×_{d} Z) is AF with K_{0}-group \underrightarrow{lim}(ZE^{0}, A^{t}) (see [26, 32]). Hence one may apply the dual Pimsner-Voiculescu sequence [4, Section 10.6] to the crossed product algebra C^{∗}(E^{∗}) ×_{γ} T to show that the top row of (5.1) is exact (the bottom row is the same after replacing E with F). From the point of view of coverings, the skew-product graph E^{∗}×_{d}Z and its C^{∗}- algebra are more natural to work with than the crossed product C^{∗}(E^{∗}) ×_{γ} T. Before proving that the final square of (5.1) commutes, we therefore detail first how coverings p : F^{∗} → E^{∗} interact with the isomorphisms ψ_{E} : C^{∗}(E^{∗}×_{d}Z) → C^{∗}(E^{∗}) ×_{γ} T.

Lemma 5.3. With the above notation, let E^{∗} ×_{d} Z and F^{∗} ×_{d} Z be the skewproduct graphs by the length functors d, and let ψ_{E} and ψ_{F} be the isomorphisms described in (5.2). Let γ_{E} and γ_{F} denote the gauge actions of T on C^{∗}(E^{∗}) and C^{∗}(F^{∗}).

(1) the formulae p̃(λ, n) := (p(λ), n) and 𝔰̃(λ, n) := 𝔰(λ) determine a covering p̃: F^{∗} ×_{d} Z → E^{∗} ×_{d} Z and a cocycle 𝔰̃: F^{∗} ×_{d} Z → S_{m}.

(2) the inclusion ι_{p,𝔰} : C^{∗}(E^{∗}) → M_{m}(C^{∗}(F^{∗})) is equivariant in the actions γ_{E} and id_{m} ⊗γ_{F}, and induces an inclusion ι̃_{p,𝔰} : C^{∗}(E^{∗}) ×_{γ_{E}} T → M_{m}(C^{∗}(F^{∗})) ×_{id_{m} ⊗γ_{F}} T.

(3) The following diagram commutes.

C^{∗}(E^{∗} ×_{d} Z)  M_{m}(C^{∗}(F^{∗} ×_{d} Z)) C^{∗}(E^{∗}) ×_{γ_{E}} T  M_{m}(C^{∗}(F^{∗})) ×_{id_{m} ⊗γ_{F}} T^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{ψ_{E}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{id_{m} ⊗ψ_{F}}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{ι_{p̃,𝔰̃}} ι̃_{p,𝔰}

Proof. (1) It is straightforward to check that p̃ is a covering. To see that 𝔰̃ is a cocycle, note that (μ, m) and (ν, n) are composable in the skew-product precisely when μ and ν are composable, and n = m−d(ν). So for i ∈ {1, …, m} we may calculate

𝔰̃(μ, m)(𝔰̃(ν, m − d(ν))i) = 𝔰(μ)(𝔰(ν)i) = 𝔰(μν)i = 𝔰̃(μν, m − d(ν))i.

(2) That ι_{p,𝔰} is equivariant in γ_{E} and id_{m} ⊗γ_{F} follows from Proposition 3.2(5). That it induces the desired inclusion ι̃_{p,𝔰} of crossed-products follows from the universal properties of the crossed-product algebras.

(3) That the diagram commutes follows from a simple calculation using the definitions of the maps involved. □

Proof of Theorem 5.1. Lemma 5.2 establishes everything except that the lefthand square in the diagram (5.1) commutes. To establish this last claim, recall from [32, Theorem 3.2] (see also [26]) that there is a homomorphism ϕ_{E} : ZE^{0} → K_{0}(C^{∗}(E^{∗}) ×_{γ_{E}} T) satisfying ϕ_{E}(δ_{v}) = [i_{T}(1)i_{C^{∗}(E^{∗})}(s_{v})]. Moreover, the rows of the following commutative diagram are exact and the left- and right-most vertical maps are isomorphisms (see [30, Lemma 7.15], [26]).

(5.3) 0^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ker(1 − A^{t})^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ZE^{0}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{1−A^{t}}  ZE^{0}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} coker(1 − A^{t})^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 0 0_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} K_{1}(C^{∗}(E^{∗}))_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} K_{0}(C^{∗}(E^{∗})×_{γ_{E}}T)_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{1−γ̂_{∗}^{−1}}K_{0}(C^{∗}(E^{∗})×_{γ_{E}}T)_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} K_{0}(C^{∗}(E^{∗}))_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 0^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{≅}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{ϕ_{E}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{ϕ_{E}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{≅}

A similar commutative diagram holds for F^{∗}, and using the standard isomorphism of K_{∗}(M_{m}(C^{∗}(F^{∗}))) with K_{∗}(C^{∗}(F^{∗})), we may assemble these two diagrams can into a single three-dimensional diagram by connecting each term in the diagram for E^{∗} to the corresponding term in the diagram for F^{∗} using the appropriate maps induced from (p, 𝔰). The map connecting the K_{0}-groups of the skew-product graph algebras is induced from the connecting map in the bottom row of the commuting diagram in Lemma 5.3(3) by applying the K-functor and using the canonical isomorphisms

K_{∗}(M_{m}(C^{∗}(F^{∗}) ×_{γ_{F}} T)) ≅ K_{∗}(C^{∗}(F^{∗}) ×_{γ_{F}} T) and
M_{m}(C^{∗}(F^{∗}) ×_{γ_{F}} T) ≅ M_{m}(C^{∗}(F^{∗})) ×_{id_{m} ⊗γ_{F}} T.

Let η denote the unlabelled inclusion K_{1}(C^{∗}(F^{∗})) ↪ K_{0}(C^{∗}(F^{∗}) ×_{γ_{F}} T)) in the bottom row of the diagram of the form (5.3) for F^{∗}. Notice that injectivity of the map m·p^{∗} : ZE^{0} → ZF^{0} together with the first statement of Lemma 5.2 ensures that m · p^{∗} restricts to a map from ker(1 − A^{t}) to ker(1 − B^{t}); abusing notation, we denote this map m · p^{∗} too. With this notation the diagram (5.4) below is the left-hand cube of the three-dimensional diagram described in the previous paragraph.

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ZE^{0}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{m·p^{∗}} ZF^{0}  K_{0}(C^{∗}(E^{∗}) ×_{γ_{E}} T)_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{(ι̃_{p,𝔰})_{∗}} K_{0}(C^{∗}(F^{∗}) ×_{γ_{F}} T)  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{ϕ_{E}}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{ϕ_{E}}

We have shown the whole cube because we prove that the left-hand face — which is none other than the left-hand square of (5.1) — commutes by showing that the other five faces commute. To see why this suffices, suppose that the other five faces do indeed commute. Since η is an injection by the exactness of the rows of (5.3), we just need to show that the two maps from ker(1 − A^{t}) into K_{0}(C^{∗}(F^{∗}) ×_{γ} T)) obtained from the maps in the left-hand face of the cube followed by η agree. A diagram chase shows that this is the case. It therefore remains only to show that the top, bottom, front, back and righthand faces of (5.4) commute. The top square commutes by definition. The bottom square commutes by the naturality of the dual Pimsner-Voiculescu exact sequence (see the argument at the beginning of [32, Section 3]). The back and front faces commute because (5.3) commutes. To see that the right-hand face commutes, recall that C^{∗}(E^{∗} ×_{d} Z) is AF with K_{0}-group \underrightarrow{lim}(ZE^{0}, 1 − A^{t}). Hence there is an inclusion ε_{E} : ZE^{0} → K_{0}(C^{∗}(E^{∗} ×_{d} Z)) which takes δ_{v} to the K_{0}-class of the vertex projection s_{(v,0)}, and likewise for F. Consider the map ψ_{E} defined in (5.2) and the map ϕ_{E} appearing in (5.3). It is clear that ϕ_{E} = (ψ_{E})_{∗} ∘ ε_{E} and similarly for F. So it suffices to show that the following diagram commutes.

(5.5)  ZE^{0}  ZF^{0} K_{0}(C^{∗}(E^{∗} ×_{d} Z))  K_{0}(C^{∗}(F^{∗} ×_{d} Z)) K_{0}(C^{∗}(E^{∗}) ×_{γ_{E}} T)  K_{0}(C^{∗}(F^{∗}) ×_{γ_{F}} T)^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{ε_{E}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{ε_{F}}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{(ψ_{E})_{∗}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{(ψ_{F})_{∗}}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{m·p^{∗}}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{(ι_{p̃,𝔰̃})_{∗}}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{(ι̃_{p,𝔰})_{∗}}

If one applies the K-functor to all terms and maps in the diagram of Lemma 5.3(3), and then applies the natural isomorphism

K_{∗}(M_{m}(C^{∗}(E^{∗}) ×_{γ_{E}} T)) ≅ K_{∗}(C^{∗}(E^{∗}) ×_{γ_{E}} T)

to the terms on the right, one obtains precisely the bottom rectangle of (5.5). The bottom rectangle of (5.5) therefore commutes by naturality of the Kfunctor together with Lemma 5.3(3). To see that the top rectangle of (5.5) commutes, recall that ε_{E} takes the image of the point-mass δ_{v} in the direct limit \underrightarrow{lim}(ZE^{0}, A^{t}) to the class of the projection s_{(v,0)}. The image of s_{(v,0)} under the homomorphism ι_{p̃,𝔰̃} is the diagonal matrix in M_{m}(C^{∗}(F^{∗}×_{d}Z)) whose diagonal entries are all equal to ∑_{p(w)=v} s_{(w,0)}. Under the standard isomorphism K_{0}(M_{m}(C^{∗}(F^{∗} ×_{d} Z))) ≅ K_{0}(C^{∗}(F^{∗} ×_{d} Z)), we therefore obtain the following equality in K_{0}(C^{∗}(F^{∗} ×_{d} Z)):

[ι_{p̃,𝔰̃}(s_{(v,0)})] = ∑_{p(w)=v} m · [s_{(w,0)}] = m · ( ∑_{p(w)=v}[s_{(w,0)}]).

Using once again the characterisation of the maps ε_{E} and ε_{F}, we see that this is precisely the statement that the bottom rectangle of (5.5) commutes. □

5.2. Coverings of higher-rank graphs and Kasparov’s spectral sequence theorem. We turn to the case where k > 1. We invoke the K-theory computations of [14] which are based on Kasparov’s spectral sequence theorem for the computation of the K-theory of crossed products by groups for which the Baum-Connes conjecture holds (see [18, Theorem 6.10], [14, Lemma 3.3] and [35]). We are grateful to Gennadi Kasparov for pointing out that the spectral sequence is natural. The standard notation for spectral sequences is that a spectral sequence (E^{r}, d^{r}) has terms E_{p,q}^{r} and differentials d^{r} : E_{p,q}^{r} → E_{p−r,q+r−1}^{r} where r > 0 and p, q ∈ Z. This however is problematic in the current situation because p clashes with our notation for a covering map. To avoid this, we replace the indexing variables p, q in the spectral sequence with a, b. That is, our spectral sequences have terms E_{a,b}^{r} and differentials d^{r} : E_{a,b}^{r} → E_{a−r,b+r−1}^{r} where r > 0 and a, b ∈ Z. Since each higher rank graph C^{∗}-algebra C^{∗}(Λ) is Morita equivalent to a crossed product by Z^{k} [21, Theorem 5.6], Kasparov’s result applies to give a spectral sequence which converges to K_{∗}(C^{∗}(Λ)) with E^{2} terms given by the homology of Z^{k} with appropriately chosen coefficients. In [14] Evans computes these homology groups using a resolution related to the Koszul complex. It follows that the above spectral sequence may be extended so that the terms of the resolution become the terms E_{a,b}^{1} for b even. The main result of this subsection is to show that given a finite covering p : Γ → Λ of row-finite k-graphs with no sources, a multiplicity m and a cocycle 𝔰 : Γ → S_{m}, there is a natural morphism of spectral sequences defined on E^{1} terms using m · p^{∗} : ZΛ^{0} → ZΓ^{0} which is compatible (see [41, p. 126]) with (ι_{p,𝔰})_{∗} the induced map on K-theory. This result is specialised to the case k = 2 with a view to applications in Section 6. The following is an immediate Corollary of [18, Theorem 6.10] (see [14, Lemma 3.3] and [35]). For more detail on spectral sequences used in this context, see [35, 14].

Proposition 5.4. Let F be a C^{∗}-algebra and let α : Z^{k} → Aut F be an action of Z^{k} on F. Then there is a spectral sequence (E^{r}, d^{r}) with differentials d^{r} : E_{a,b}^{r} → E_{a−r,b+r−1}^{r} which converges to K_{∗}(F ×_{α} Z^{k}) with E_{a,b}^{2} = H_{a}(Z^{k}, K_{b}(F)). Moreover, the spectral sequence is natural with respect to equivariant maps of C^{∗}-algebras.

Proof. As noted in the proof of [14, Lemma 3.3] this follows immediately from [18, Theorem 6.10] since Z^{k} is amenable and the Baum-Connes conjecture is known to hold for amenable groups [16, Theorem 1.1], so the γ part of K_{∗}(F ×_{α} Z^{k}) exhausts. The naturality of the spectral sequence with respect to equivariant maps follows from the construction in the proof of [18, Theorem 6.10], since every step is functorial. □

Naturality means that given Z^{k} actions α_{i} on F_{i}, a Z^{k}-equivariant map φ : F_{1} → F_{2} induces a morphism of spectral sequences and this morphism is compatible with

φ̂_{∗} : K_{∗}(F_{1} ×_{α_{1}} Z^{k}) → K_{∗}(F_{2} ×_{α_{2}} Z^{k})

where φ̂: F_{1} ×_{α_{1}} Z^{k} → F_{2} ×_{α_{2}} Z^{k} is the natural map. Evans applies this when F = F_{Λ} is the crossed product C^{∗}(Λ) ×_{γ} T^{k} of C^{∗}(Λ) by the gauge action, and α is the dual action γ̂ of Z^{k}. Hence, by Takai duality we have K_{∗}(C^{∗}(Λ)) = K_{∗}(F_{Λ} ×_{α} Z^{k}). In this case we have more specific results (see [14, Lemma 3.3]):

E_{a,b}^{2} = \begin{cases}H_{a}(Z^{k}, K_{0}(F_{Λ})) & if 0 ≤ a ≤ k and b is even, \\ 0 & otherwise. \end{cases}

In [14, Theorem 3.14]), Evans shows that these homology groups may be computed as the homology of the complex D_{∗}^{Λ} = ⋀^{∗} Z^{k} ⊗ ZΛ^{0}. That is, D_{a}^{Λ} = ⋀^{a} Z^{k} ⊗ ZΛ^{0} for 0 ≤ a ≤ k and D_{a}^{Λ} = 0 for a > k. For 1 ≤ j ≤ k let M_{j} denote the vertex connectivity matrix of the coordinate graph (Λ^{0}, Λ^{e_{j}}, r, s). For 1 ≤ a ≤ k define the differential ∂_{a} : D_{a}^{Λ} → D_{a−1}^{Λ} by

∂_{a}(ϵ_{i_{1}} ∧ ⋯ ∧ ϵ_{i_{a}} ⊗ e_{v}) = ∑_{j=1}^{a}(−1)^{j+1}ϵ_{i_{1}} ∧ ⋯ ∧ ϵ̂_{i_{j}} ∧ ⋯ ∧ ϵ_{i_{a}} ⊗ (1 − M_{j}^{t})e_{v}

where ϵ_{1}, …, ϵ_{k} constitute the canonical basis for Z^{k}, 1 ≤ i_{1} < ⋯ < i_{a} ≤ k and v ∈ Λ^{0}. It is straightforward to verify that D_{∗}^{Λ} is a complex. The first part of the following theorem is a restatement of [14, Theorem 3.15]).

Theorem 5.5. Fix k > 1. Let Λ be a row-finite k-graph with no sources. With notation as above there is a spectral sequence (E^{r}, d^{r}) with differentials d^{r} : E_{a,b}^{r} → E_{a−r,b+r−1}^{r} which converges to K_{∗}(C^{∗}(Λ)) = K_{∗}(F_{Λ} ×_{α} Z^{k}) with

E_{a,b}^{1} = D_{a}^{Λ} := ⋀^{a} Z^{k} ⊗ ZΛ^{0},

if 0 ≤ a ≤ k and b is even, and 0 otherwise. The differential d^{1} : E_{a,b}^{1} → E_{a−1,b}^{1} is given by ∂_{a} if b is even. Let (Λ, Γ, p, m, 𝔰) be a row-finite covering system of k-graphs with no sources. There is a morphism f of spectral sequences which is compatible with (ι_{p,𝔰})_{∗} : K_{∗}(C^{∗}(Λ)) → K_{∗}(C^{∗}(Γ)) such that f^{1} : D_{a}^{Λ} → D_{a}^{Γ} is given by id ⊗(m · p^{∗}).

Proof. Evans computes the homology groups using a Koszul complex (see [41, §4.5]). Set G = Z^{k} = ⟨s_{1}, … s_{k}⟩, R = ZG and let I be the ideal in R generated by {1 − s_{a}^{−1} : 1 ≤ a ≤ k}. Let ϵ_{1}, …, ϵ_{k} constitute the canonical basis for R^{k}. For each a, define ∂_{a} : ⋀^{a} R^{k} → ⋀^{a−1} R^{k} as follows: for 1 ≤ i_{1} < ⋯ < i_{a} ≤ k so that ϵ_{i_{1}} ∧ ⋯ ∧ ϵ_{i_{a}} ∈ ⋀^{a} R^{k}, define

∂_{a}(ϵ_{i_{1}} ∧ ⋯ ∧ ϵ_{i_{a}}) = ∑_{j=1}^{a}(−1)^{j+1}(1 − s_{j}^{−1})ϵ_{i_{1}} ∧ ⋯ ∧ ϵ̂_{i_{j}} ∧ ⋯ ∧ ϵ_{i_{a}}

where the symbol “ ˆ · ” denotes deletion of an element (note that ∂_{1}(ϵ_{i}) = 1 − s_{i}^{−1}). Then R/I ≅ Z and the following sequence of R-modules is exact (see [41, Corollary 4.5.5])

0 → ⋀^{k} R^{k} → ⋯ → ⋀^{1} R^{k} → ⋀^{0} R^{k} → Z → 0.

Note that ⋀^{0} R^{k} = R and ⋀^{a} R^{k} is a free R-module with basis

{ϵ_{i_{1}} ∧ ⋯ ∧ ϵ_{i_{a}} : 1 ≤ i_{1} < ⋯ < i_{a} ≤ k}.

Hence, ⋀^{∗} R^{k} yields a projective resolution of Z. Thus, by [6, §III.1] we have

H_{∗}(G, K_{0}(F_{Λ})) ≅ H_{∗}(⋀^{∗} R^{k} ⊗_{G} K_{0}(F_{Λ})).

We follow Evans here but have adopted slightly different notation to make naturality more apparent (see [14, Definition 3.11] and following). Under the isomorphism ⋀^{a} R^{k} ⊗_{G} K_{0}(F_{Λ}) ≅ ⋀^{a} Z^{k} ⊗ K_{0}(F_{Λ}) (as abelian groups), the boundary map ∂_{a} : ⋀^{a} Z^{k} ⊗ K_{0}(F_{Λ}) → ⋀^{a−1} Z^{k} ⊗ K_{0}(F_{Λ}) is given by

∂_{a}(ϵ_{i_{1}} ∧ ⋯ ∧ ϵ_{i_{a}} ⊗ x) = ∑_{j=1}^{a}(−1)^{a+1}ϵ_{i_{1}} ∧ ⋯ ∧ ϵ̂_{i_{j}} ∧ ⋯ ∧ ϵ_{i_{a}} ⊗ (1 − s_{j})x

where 1 ≤ i_{1} < ⋯ < i_{a} ≤ k and x ∈ K_{0}(F_{Λ}). Let D_{a}^{Λ} be given as above. There is a natural map ε^{Λ} : C_{0}(Λ^{0}) ↪ F_{Λ} which induces a map ε_{∗}^{Λ} : ZΛ^{0} → K_{0}(F_{Λ}). Moreover (see [14, Theorem 3.14]) the natural map

id ⊗ε_{∗}^{Λ} : ⋀^{∗} Z^{k} ⊗ ZΛ^{0} → ⋀^{∗} Z^{k} ⊗ K_{0}(F_{Λ})

is a map of complexes which induces an isomorphism on homology and hence

H_{∗}(G, K_{0}(F_{Λ})) ≅ H_{∗}(⋀^{∗} Z^{k} ⊗ ZΛ^{0}).

Therefore, setting

E_{a,b}^{1} = \begin{cases}⋀^{a} Z^{k} ⊗ ZΛ^{0} & if 0 ≤ a ≤ k and b is even, \\ 0 & otherwise \end{cases}

and defining d^{1} : E_{a,b}^{1} → E_{a−1,b}^{1} to be ∂_{a} if b is even (and 0 otherwise), yields

E_{a,b}^{2} ≅ \begin{cases}H_{p}(G, K_{0}(F_{Λ})) & if 0 ≤ a ≤ k and b is even, \\ 0 & otherwise. \end{cases}

It follows by [14, Lemma 3.3] that the spectral sequence converges to K_{∗}(C^{∗}(Λ)) = K_{∗}(F_{Λ} ×_{α} Z^{k}) as required. For the second part of the theorem, fix (Λ, Γ, p, m, 𝔰). The embedding ι_{p,𝔰} : C^{∗}(Λ) → M_{m}(C^{∗}(Γ)) induces an embedding ι̃_{p,𝔰} : F_{Λ} → M_{m}(F_{Γ}). Functoriality yields a map of complexes

id ⊗(ι̃_{p,𝔰})_{∗} : ⋀^{∗} Z^{k} ⊗ K_{0}(F_{Λ}) → ⋀^{∗} Z^{k} ⊗ K_{0}(F_{Γ}).

Since group homology is a covariant functor of its coefficient module we obtain the functorial maps for each n = 0, 1, …, k

H_{n}((ι̃_{p,𝔰})_{∗}) : H_{n}(Z^{k}, K_{0}(F_{Λ})) → H_{n}(Z^{k}, K_{0}(F_{Γ})).

Then arguing as in Lemma 5.2 with p^{∗} : ZΛ^{0} → ZΓ^{0} defined as above we see that

(1 − (M_{j}^{Γ})^{t})(m · p^{∗}) = (m · p^{∗})(1 − (M_{j}^{Λ})^{t})

for all j = 1, …, k. It follows that the natural map

id ⊗(m · p^{∗}) : ⋀^{∗} Z^{k} ⊗ ZΛ^{0} → ⋀^{∗} Z^{k} ⊗ ZΓ^{0} is a map of complexes. Arguing as in the proof of Theorem 5.1, we see that (ι̃_{p,𝔰})_{∗} ∘ ε_{∗}^{Λ} = ε_{∗}^{Γ} ∘ (m · p^{∗}), so the map on homology induced by id ⊗(m · p^{∗}) coincides with the functorial map above (under the identifications of the homology groups induced by id ⊗ε_{∗}^{Λ} and id ⊗ε_{∗}^{Γ}). This combined with the naturality of Proposition 5.4 yields a morphism f of spectral sequences compatible with the map

( ˆι̃_{p,𝔰})_{∗} : K_{∗}(F_{Λ} ×_{α} Z^{k}) → K_{∗}(F_{Γ} ×_{α} Z^{k})

such that f^{1} : D_{a}^{Λ} → D_{a}^{Γ} is given by id ⊗(m · p^{∗}). Under the identifications K_{∗}(C^{∗}(Λ)) = K_{∗}(F_{Λ}×_{α}Z^{k}) and K_{∗}(C^{∗}(Γ)) = K_{∗}(F_{Γ}×_{α}Z^{k}), we have ( ˆι̃_{p,𝔰})_{∗} = (ι_{p,𝔰})_{∗}. □

The following corollary is an immediate consequence of the above theorem restricted to the case k = 2; for the first assertion see [14, Proposition 3.16] and its proof (see also [35]). Given a 2-graph Λ, recall that M_{1} and M_{2} denote the vertex connectivity matrices of the coordinate graphs (Λ^{0}, Λ^{e_{1}}, r, s) and (Λ^{0}, Λ^{e_{2}}, r, s).

Corollary 5.6. Suppose that (Λ, Γ, p, m, 𝔰) is a row-finite covering system of 2-graphs with no sources. With the notation of Theorem 5.5, the complex D_{a}^{Λ} = ⋀^{a} Z^{2} ⊗ ZΛ^{0} may be written as follows:

(5.6)  0 ← ZΛ^{0} ⟵^{∂_{1}} ZΛ^{0} ⊕ ZΛ^{0} ⟵^{∂_{2}} ZΛ^{0} ← 0

where ∂_{1} = (1 − M_{1}^{t}, 1 − M_{2}^{t}) and ∂_{2} = (\begin{matrix}M_{2}^{t} − 1 \\ 1 − M_{1}^{t} \end{matrix}) . We have E_{a,b}^{2} = E_{a,b}^{∞}, and

K_{0}(C^{∗}(Λ)) ≅ coker ∂_{1} ⊕ ker ∂_{2}
K_{1}(C^{∗}(Λ)) ≅ ker ∂_{1}/ Im ∂_{2} ≅ H_{1}(Z^{k}, K_{0}(F_{Λ})).

(5.7)

Moreover, the following diagram commutes

(5.8)

0 ⟵ ZΛ^{0} ⟵^{∂_{1}^{Λ}} ZΛ^{0} ⊕ ZΛ^{0} ⟵^{∂_{2}^{Λ}} ZΛ^{0} ⟵ 0
↓^{m·p^{∗}}  ↓^{m·p^{∗}⊕m·p^{∗}}  ↓^{m·p^{∗}}
0 ⟵ ZΓ^{0} ⟵^{∂_{1}^{Γ}} ZΓ^{0} ⊕ ZΓ^{0} ⟵^{∂_{2}^{Γ}} ZΓ^{0} ⟵ 0

and by naturality induces (ι_{p,𝔰})_{∗} : K_{∗}(C^{∗}(Λ)) → K_{∗}(C^{∗}(Γ)).

The inclusion of coker ∂_{1} into K_{0}(C^{∗}(Λ)) obtained from (5.7) takes the equivalence class (in the quotient group coker ∂_{1} = ZΛ^{0}/ Im(∂_{1})) of the generator δ_{v} of ZΛ^{0} to the K_{0}-class of the vertex projection [s_{v}] in C^{∗}(Λ). The proof of this fact can be obtained from the proof of [14, Proposition 4.4]. We thank Gwion Evans for pointing this out to us.

5.3. Product coverings and the Künneth formula. In this section we consider covering systems (Λ_{n}, p_{n}) in which each k-graph Λ_{n} is a cartesian product of two lower-dimensional graphs, and the covering maps p_{n} respect the product decomposition. Recall from [20, Proposition 1.8] that given a k-graph (Λ, d) and a k′- graph (Λ′, d′), the cartesian-product category Λ × Λ′ becomes a (k + k′)- graph when we endow it with the degree functor d × d′ : (λ, λ′) ↦ (d(λ)_{1}, …, d(λ)_{k}, d′(λ′)_{1}, …, d′(λ′)_{k′}).

Proposition 5.7. Fix k, k′ ∈ N \ {0}. Let (Λ, Γ, p, m, 𝔰) and (Λ′, Γ′, p′, m′, 𝔰′) be row-finite covering systems of k- and k′-graphs with no sources. Then

p × p′ : Γ × Γ′ → Λ × Λ′

is a finite covering of row-finite (k + k′)-graphs with no sources. Let f : {1, …, m} × {1 …, m′} → {1, …, mm′} denote the bijection f(j, j′) := j + (j′ − 1)m. There is a cocycle 𝔰 × 𝔰′ : Γ × Γ′ → S_{mm′} determined by ((𝔰 × 𝔰′)(α, α′))f(j, j′) := f(𝔰(α)j, 𝔰′(α′)j′). Moreover, the following diagram commutes.

C^{∗}(Λ × Λ′)  ⟶^{≅}  C^{∗}(Λ) ⊗ C^{∗}(Λ′)
↓^{ι_{p×p′,𝔰×𝔰′}}  ↓^{ι_{p,𝔰}⊗ι_{p′,𝔰′}}
M_{mm′}(C^{∗}(Γ × Γ′)) ⟶^{≅} M_{m}(C^{∗}(Γ)) ⊗ M_{m′}(C^{∗}(Γ′))

Suppose that at least one of K_{∗}(C^{∗}(Λ)), K_{∗}(C^{∗}(Λ′)) and at least one of K_{∗}(C^{∗}(Γ)), K_{∗}(C^{∗}(Γ′)) are torsion-free. Then the following diagram commutes and the horizontal connecting maps are zero-graded isomorphisms:

K_{∗}(C^{∗}(Λ)) ⊗ K_{∗}(C^{∗}(Λ′)) ⟶^{≅} K_{∗}(C^{∗}(Λ × Λ′))
↓^{(ι_{p,𝔰})_{∗}⊗(ι_{p′,𝔰′})_{∗}}  ↓^{(ι_{p×p′,𝔰×𝔰′})_{∗}}
K_{∗}(C^{∗}(Γ)) ⊗ K_{∗}(C^{∗}(Γ′)) ⟶^{≅} K_{∗}(C^{∗}(Γ × Γ′))

If Γ^{0} and (Γ′)^{0} (and hence also Λ^{0} and (Λ′)^{0}) are finite then the C^{∗}-algebras are unital, and the horizontal isomorphisms take [1] ⊗ [1] to [1].

Proof. It is straightforward to check that p×p′ is a covering using the properties of the covering maps p and p′ and the definition of the cartesian-product graph. A simple calculation shows that 𝔰 × 𝔰′ defines a cocycle. Theorem 5.5 of [20] shows that C^{∗}(Λ), C^{∗}(Λ′), C^{∗}(Γ) and C^{∗}(Γ′) are nuclear, and so there is just one tensor-product C^{∗}-algebra C^{∗}(Λ) ⊗ C^{∗}(Λ′). Corollary 3.5(iv) of [20] shows that the map s_{(λ,μ)} ↦ s_{λ} ⊗ s_{μ} is an isomorphism of C^{∗}(Λ×Λ′) onto C^{∗}(Λ)⊗C^{∗}(Λ′), and similarly for C^{∗}(Γ) and C^{∗}(Γ′). It is easy to check using the formulae for the maps ι_{p,𝔰}, ι_{p′,𝔰′}, and ι_{p×p′,𝔰×𝔰′} and using the chain of isomorphisms

M_{mm′}(C^{∗}(Γ × Γ′)) ≅ M_{mm′}(C) ⊗ C^{∗}(Γ × Γ′)
≅ M_{m}(C) ⊗ C^{∗}(Γ) ⊗ M_{m′}(C) ⊗ C^{∗}(Γ′)
≅ M_{m}(C^{∗}(Γ)) ⊗ M_{m′}(C^{∗}(Γ′))

that the first diagram commutes. In the presence of the additional hypothesis concerning torsion-free K-groups, the Künneth Theorem of [37] (see also Theorem 23.1.3 of [4]) implies: (1) that

K_{∗}(C^{∗}(Λ)) ⊗ K_{∗}(C^{∗}(Λ′)) ≅ K_{∗}(C^{∗}(Λ) ⊗ C^{∗}(Λ′))

and similarly for Γ, Γ′; (2) that these isomorphisms are natural and are zerograded; and (3) that these isomorphisms take [1]⊗[1] to [1]. The result therefore follows from the naturality of the K-functor. □

Note that in general when no assumption is made about torsion, the Künneth Theorem of [37] gives a short exact sequence which is still natural. The analogue of Proposition 5.7 still holds and gives a (fairly complicated) commuting diagram in which the rows are short exact sequences.

6. Examples

In this section we discuss a number of examples. A recurring theme will be supernatural numbers and the associated dimension groups, so we pause here to establish some notation. We will think of a supernatural number as an infinite product α = ∏_{n=1}^{∞} α_{n} where each α_{n} is an integer greater than 1. Any two such expressions in which the same prime factors occur with the same cardinality correspond to the same supernatural number. Given supernatural numbers α, β, we will abuse notation and write αβ for the supernatural number ∏_{n=1}^{∞} α_{n}β_{n}. We write α[1, n] for the product ∏_{i=1}^{n} α_{i} of the first n terms in α. For z_{1}, …, z_{n} ∈ C, we write Z[z_{1}, …, z_{n}] for the ring obtained by adjoining z_{1}, …, z_{n} to Z; we regard Z[z_{1}, …, z_{n}] as a group under addition. Abusing notation, for a supernatural number α, we write Z[\frac{1}{α}] for the dimension group \underrightarrow{lim}(Z, ×α_{n}) which we identify with the group

⋃_{n=1}^{∞} Z[\frac{1}{α[1, n]}] ⊂ Q

consisting of all fractions p/q where p, q ∈ Z, and q is a divisor of some α[1, n].

6.1. Rank-2 Bratteli diagrams. A rank-2 Bratteli diagram is a 2-graph in which the blue edges form a Bratteli diagram and the red edges determine simple cycles so that every vertex lies on precisely one red cycle, and all vertices on a given red cycle are at the same level in the blue Bratteli diagram. The C^{∗}-algebras of these 2-graphs were studied in [27] and provided the initial motivation for the covering construction. A rank-2 Bratteli diagram Λ can be constructed using Proposition 2.14 and Corollary 2.15 precisely when the length of each red cycle at level n of Λ is divisible by the lengths of all the cycles at level n − 1 to which it connects. In particular, the 2-graphs whose C^{∗}-algebras are Morita equivalent to the Bunce-Deddens algebras [27, Example 6.7] and the irrational rotation algebras [27, Example 6.5] arise in this fashion.

6.2. Coverings of dihedral graphs D_{n}. For n ∈ N \ {0}, let D_{n} be the directed graph with n vertices {v_{0}, …, v_{n−1}} and edges {x_{i}, y_{i} : 0 ≤ i ≤ n − 1} where r(x_{i}) = v_{i} = s(y_{i}) and s(x_{i}) = v_{i+1} = r(y_{i}) (throughout this section, addition in the subscripts is understood to be evaluated modulo n). More descriptively, D_{n} is a ring of n vertices, each of which connects to both of its neighbours (see Figure 2). Let D_{n}^{∗} be the path-category of D_{n}, regarded as a 1-graph. Note that for n ∈ N \ {0}, the graph D_{2n} is the Cayley graph for the

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Figure 2. The 1-graph D_{n}

dihedral group with 2n elements.

Example 6.1. For n, m ≥ 1 there are m-fold covering maps p_{n,mn} : D_{mn}^{∗} → D_{n}^{∗} as follows: for 0 ≤ i ≤ mn − 1 let i′ = i mod n and define

p_{n,mn}(v_{i}) := v_{i′}, p_{n,mn}(x_{i}) := x_{i′} and p_{n,mn}(y_{i}) := y_{i′}.

Hence for each pair of positive integers n, m, we obtain a row-finite covering system (D_{n}^{∗}, D_{mn}^{∗}, p_{n,mn}) of 1-graphs with no sources (see Notation 2.8). Fix an infinite supernatural number α = ∏_{i=1}^{∞} α_{i}. Consider the sequence of covering systems (D_{6α[1,n]}^{∗}, D_{6α[1,n+1]}^{∗}, p_{6α[1,n],6α[1,n+1]})_{n=1}^{∞} as in Notation 2.8. Applying Corollary 2.11, we obtain a 2-graph

D := \begin{matrix}lim \\ ↽− \end{matrix}(D_{6α[1,n]}^{∗}, p_{6α[1,n],6α[1,n+1]}).

Proposition 6.2. Consider the situation discussed in Example 6.1. We have K_{0}(C^{∗}(D)) = Z[\frac{1}{α}] ⊕ Z[\frac{1}{α}] and K_{1}(C^{∗}(D)) = Z ⊕ Z. Let P_{1} := ∑_{v∈D_{6}^{0}} s_{v}. Then [P_{1}] is the 0 element of K_{0}(P_{1}C^{∗}(D)P_{1}). Moreover, C^{∗}(D) is simple and purely infinite.

Before proving the proposition, we describe the K-theory of C^{∗}(D_{n}^{∗}) in general.

Lemma 6.3. (1) K_{0}(C^{∗}(D_{n}^{∗})) is generated by [s_{v_{0}}] and [s_{v_{1}}], and for each i, we

have [s_{v_{i}}] = −[s_{v_{i+3}}] in K_{0}(C^{∗}(D_{n}^{∗})).

(2) K_{1}(C^{∗}(D_{n}^{∗})) ≅ {(a_{1}, …, a_{n}) ∈ Z^{n} : a_{i+2} = a_{i+1} − a_{i} for all i}.

(3) the following table describes the K-theory of each C^{∗}(D_{n}^{∗}).

n mod 6 K_{0}(C^{∗}(D_{n}^{∗})) K_{1}(C^{∗}(D_{n}^{∗}))
0  Z^{2}  Z^{2}
1  0  0
2  Z/3Z  0
3  Z/2Z ⊕ Z/2Z  0
4  Z/3Z  0
5  0  0

Proof. (1) The K_{0} group is generated by the classes [s_{v_{0}}], …, [s_{v_{n−1}}] subject to the relations [s_{v_{i}}] = [s_{v_{i+1}}]+[s_{v_{i−1}}]. This relation forces [s_{v_{i+2}}] = [s_{v_{i+1}}]−[s_{v_{i}}], from which we conclude first that K_{0} is generated by [s_{v_{0}}] and [s_{v_{1}}] and second that

[s_{v_{i+3}}] = [s_{v_{i+2}}] − [s_{v_{i+1}}] = ([s_{v_{i+1}}] − [s_{v_{i}}]) − [s_{v_{i+1}}] = −[s_{v_{i}}].

(2) Let A_{n} denote the vertex connectivity matrix of D_{n}; so A_{n}(i, j) = 1 when i = j ± 1 (mod n) and zero otherwise. As in Theorem 5.1, we have K_{1}(C^{∗}(D_{n}^{∗})) ≅ ker(1 − A_{n}^{t}). For m ∈ Z^{n}, ((1 − A_{n}^{t})m)_{i} = −m_{i−1} + m_{i} − m_{i+1} by definition of A_{n}, and this establishes (2).

(3) If E is a finite 1-graph with no sinks or sources, then C^{∗}(E) is isomorphic to the Cuntz-Krieger algebra of the adjacency matrix A_{E} of E [23]. In particular, K_{1}(C^{∗}(E)) is torsion-free and has the same rank as K_{0}(C^{∗}(E)) [9]. Hence it suffices to verify that the first column of the table is correct. To calculate K_{0}, we use (1) to check by hand that the cases n = 1, 2, … 6 are as claimed. If n > 6, then applying the relations we find that [s_{v_{i+6}}] = [s_{v_{i}}] for all i which accounts for all remaining cases. □

Proof of Proposition 6.2. Lemma 6.3(1) shows that K_{0}(C^{∗}(D_{6α[1,n]}^{∗}) is generated by [s_{v_{0}^{n}}] and [s_{v_{1}^{n}}] where the v_{i}^{n} are the vertices of D_{6α[1,n]}^{∗}. Fix i ∈ {1, 2}. We have

(6.1)  (ι_{p_{n}})_{∗}[s_{v_{i}^{n}}] = [s_{v_{i}^{n+1}}] + [s_{v_{i+6α[1,n]}^{n+1}}] + ⋯ + [s_{v_{i+6(α_{n+1}−1)α[1,n]}^{n+1}}].

By Lemma 6.3(1), each [s_{v_{i}^{n+1}+6k}] = [s_{v_{i}^{n+1}}] in K_{0}(C^{∗}(D_{6α[1,n+1]}^{∗})), so (6.1) implies (ι_{p_{n}})_{∗}[s_{v_{i}^{n}}] = α_{n} · [s_{v_{i}^{n+1}}]. Hence K_{0}(ι_{p_{n}}) : Z^{2} → Z^{2} is multiplication by α_{n}. Fix m ∈ N \ {0}. By Lemma 6.3(2), K_{1}(C^{∗}(D_{6m}^{∗})) is identified with the set of sequences (a_{1}, …, a_{6m}) which satisfy a_{i+2} = a_{i+1} − a_{i} for all i. By Lemma 6.3(2), this forces a_{i+2} = a_{i+1} − a_{i} for all i. Consequently, the map a = (a_{1}, …, a_{6m}) ↦ (a_{1}, a_{2}) yields an isomorphism ζ_{m} : K_{1}(C^{∗}(D_{6m}^{∗})) → Z^{2}. As ζ_{α[1,n+1]} ∘ K_{1}(ι_{p_{6α[1,n],6α[1,n+1]}}) = ζ_{α[1,n]}, it follows that K_{1}(ι_{p_{n}}) : Z^{2} → Z^{2} is the identity map. Recall that D denotes \begin{matrix}lim \\ ↽− \end{matrix}(D_{6α[1,n]}^{∗}, p_{6α[1,n],6α[1,n+1]}). By Theorem 5.1 the Kgroups of C^{∗}(D) are as claimed. To compute the class of the identity, let P_{1} ∈ C^{∗}(D) be the sum of the six vertex projections in the bottom level. The final statement of Lemma 6.3(1) shows that the classes of the vertex projections in K_{0}(C^{∗}(D_{6}^{∗})) cancel, so that the class of the identity in K_{0}(C^{∗}(D_{6}^{∗})) is the zero element. It follows that the class of the identity P_{1} in K_{0}(P_{1}C^{∗}(D)P_{1}) is also the zero element. Each D_{n}^{∗} is aperiodic and cofinal (see Definition 4.2), so we may conclude from Corollary 4.4 and Lemma 4.7 that D is aperiodic and cofinal. Hence Proposition 4.8 of [20] implies that C^{∗}(D) is simple. The path x_{1}y_{1} is a cycle with an entrance (namely y_{0}) in D_{1}^{∗}. Proposition 4.8 now shows that C^{∗}(D) is purely infinite. □

6.3. Direct limits of O_{n} ⊗ C(T).

Example 6.4. Fix n ≥ 3, and let B_{n} be the bouquet of n loops. For m ≥ 1, let L_{m} denote the loop with m vertices, and let Λ_{m} be the cartesian-product 2-graph Λ_{m} = L_{(n−1)^{m}}^{∗} × B_{n}^{∗} obtained from the path categories of L_{(n−1)^{m}} and B_{n}. For each m, Let p_{m} denote the obvious (n − 1)-fold covering of L_{(n−1)^{m}}^{∗} by L_{(n−1)^{m+1}}^{∗}, and let p′ be the identity covering of B_{n} by B_{n}.

Proposition 6.5. Consider the situation of Example 6.4. Let v be a vertex of Λ_{1}. Then s_{v}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Λ_{m}, p_{m} × p′))s_{v} is isomorphic to the Kirchberg algebra P_{n} (see [5]) whose K-theory is opposite to that of O_{n}.

Proof. Since C^{∗}(B_{n}) is generated by n isometries whose range projections sum to the identity, C^{∗}(B_{n}) is canonically isomorphic to O_{n} [7]. Hence

C^{∗}(Λ_{m}) ≅ C^{∗}(L_{(n−1)^{m}}^{∗}) ⊗ O_{n}

by [20, Corollary 3.5(iv)]. As in [17, Lemma 2.4], there is an isomorphism C^{∗}(L_{(n−1)^{m}}^{∗}) ≅ M_{(n−1)^{m}}(C(T)) for each m, and in particular we have K_{∗}(C^{∗}(L_{(n−1)^{m}}^{∗})) ≅ (Z, Z). Since K_{∗}(O_{n}) = (Z/(n − 1)Z, 0) [9], the Künneth theorem implies that K_{∗}(C^{∗}(Λ_{m})) ≅ (Z/(n − 1)Z, Z/(n − 1)Z). A special case of [27, Equation (4.7)] implies that the covering map p_{m} induces multiplication by n − 1 from K_{0}(C^{∗}(L_{(n−1)^{m}}^{∗})) to K_{0}(C^{∗}(L_{(n−1)^{m+1}}^{∗})), and the identity homomorphism from K_{1}(C^{∗}(L_{(n−1)^{m}}^{∗})) to K_{1}(C^{∗}(L_{(n−1)^{m+1}}^{∗})). Clearly p′ induces the identity map on K_{∗}(O_{n}). Let Λ = \begin{matrix}lim \\ ↽− \end{matrix}(Λ_{m}, p_{m} × p′). Theorem 3.8 and Proposition 5.7 combine to show that

K_{∗}(C^{∗}(Λ)) ≅ \underrightarrow{lim}((Z/(n − 1)Z, Z/(n − 1)Z), (×(n − 1), id)).

Since multiplication by n − 1 is the 0 homomorphism from Z/(n − 1)Z to Z/(n − 1)Z, it follows that K_{∗}(C^{∗}(Λ)) ≅ (0, Z/(n − 1)Z). Lemma 4.7 proves that Λ is cofinal. For an infinite path y ∈ Λ^{∞}, Lemma 4.5 combined with the observation that the cycles in the L_{(n−1)^{m}}^{∗} grow with m shows that if a, b ∈ N^{3} and σ^{a}(y) = σ^{b}(y), then a and b differ only in their first coordinates. It follows from Proposition 4.3 that the aperiodicity of Λ is implied by the well-known aperiodicity of B_{n}. Hence C^{∗}(Λ) is simple by [20, Proposition 4.8]. Moreover, since every vertex of Λ hosts a cycle with an entrance, C^{∗}(Λ) is also purely infinite (see [20, Proposition 4.9], [38, Proposition 8.8]). The result therefore follows from the Kirchberg-Phillips classification theorem [28]. □

6.4. Higher-rank Bunce-Deddens algebras. In this subsection we describe a class of simple AT algebras with real-rank 0 which arise from sequences of covering systems of 2-graphs and which cannot in general be obtained from the construction of [27] (see Example 6.6 and Theorem 6.7). We indicate in Remark 6.12 why we think of these algebras as higher-rank analogues of the Bunce-Deddens algebras. For k ≥ 1, let Δ_{k} be the k-graph with vertices Z^{k}, morphisms {(m, n) ∈ Z^{k} ×Z^{k} : m ≤ n} where r(m, n) = m, s(m, n) = n and d(m, n) = n−m. There is a free action of Z^{k} on Δ_{k} given by translation; that is m·(p, q) = (p+m, q+m) for m ∈ Z^{k} and (p, q) ∈ Δ_{k}. Given a finite-index subgroup H of Z^{k}, we denote by Δ_{k}/H the quotient of Δ_{k} by the action of H. That is, for q ∈ N^{k}, (Δ_{k}/H)^{q} = {[g, g + q] : g ∈ Z^{k}}; in particular, (Δ_{k}/H)^{0} = {[g, g] : g ∈ Z^{k}}, and we henceforth identify (Δ_{k}/H)^{0} with Z^{k}/H via the map [g, g] ↦ [g] where [g] denotes the class g + H of g in Z^{k}/H. The range and source maps in Δ_{k}/H are then given by r([g, g+q]) = [g] and s([g, g + q]) = [g + q]. If H′ ⊂ H is a finite-index subgroup of H, then it also has finite index in Z^{k}, and there is a natural surjection p : Z^{k}/H′ → Z^{k}/H which induces a finite covering map, also denoted p of Δ_{k}/H by Δ_{k}/H′. Most of the remainder of this section is concerned with the following example of a sequence of covering systems.

Example 6.6. Let H_{1} ⊃ H_{2} ⊃ H_{3} ⊃ … be a chain of finite-index subgroups of Z^{2}. For each n, let p_{n} : Δ_{2}/H_{n+1} → Δ_{2}/H_{n} be the canonical covering induced by the quotient maps described above, let m_{n} = 1, and let 𝔰_{n} : Δ_{2}/H_{n+1} → S_{1} be the trivial cocycle. This data specifies a sequence (Δ_{2}/H_{n}, Δ_{2}/H_{n+1}, p_{n})_{n=1}^{∞} of row-finite covering systems of 2-graphs with no sources. Applying Corollary 2.11, we obtain a 3-graph \begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n})). As

always, P_{1} denotes ∑_{v∈(Δ_{2}/H_{1})^{0}} s_{v} ∈ C^{∗}(Δ_{2}/H_{1}) ⊂ C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n})).

Theorem 6.7. Consider the situation of Example 6.6.

(1) We have

K_{0}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) ≅ \underrightarrow{lim}(Z, ×[H_{n} : H_{n+1}]) ⊕ Z,

and this isomorphism takes [P_{1}] to (g, 0) where g is the image of [Z^{2} : H_{1}] in the direct limit \underrightarrow{lim}(Z, ×[H_{n} : H_{n+1}]).

(2) For each n the homomorphism from Z^{2} to Z^{2} determined by coordinatewise multiplication by the integer [H_{n} : H_{n+1}] restricts to a homomorphism m_{H_{n},H_{n+1}} : H_{n} → H_{n+1}. Moreover,

K_{1}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) ≅ \underrightarrow{lim}(H_{n}, m_{H_{n},H_{n+1}}).

(3) C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n})) is simple if and only if ⋂_{n=1}^{∞} H_{n} = \{0\}, and is an AT algebra with real-rank 0 when it is simple.

The proof of this result will occupy the bulk of this section. Before presenting it, we state a Corollary and use it to formulate some concrete examples.

Corollary 6.8. Consider the situation of Example 6.6. There are sequences (h_{1}^{n})_{n=1}^{∞} and (h_{2}^{n})_{n=1}^{∞} in Z^{2} such that: (1) for each n, the elements h_{1}^{n} and h_{2}^{n} generate H_{n}; and (2) the matrix M_{n} = ( \begin{matrix}m_{1,1}^{n} & m_{1,2}^{n} \\ m_{2,1}^{n} & m_{2,2}^{n} \end{matrix} ) satisfying h_{1}^{n+1} =

m_{1,1}^{n}h_{1}^{n} + m_{1,2}^{n}h_{2}^{n} and h_{2}^{n+1} = m_{2,1}^{n}h_{1}^{n} + m_{2,2}^{n}h_{2}^{n} has positive determinant for

all n. Moreover, if M_{n}^{ca} denotes the classical adjoint ( \begin{matrix}m_{2,2}^{n} & −m_{1,2}^{n} \\ −m_{2,1}^{n} & m_{1,1}^{n} \end{matrix} ) of M_{n} for

each n, and if we regard these matrices as homomorphisms of Z^{2}, then

(6.2)  K_{1}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) ≅ \underrightarrow{lim}(Z^{2}, M_{n}^{ca}).

Proof. That we can choose the h_{i}^{n} so that the matrices M_{n} all have positive determinant follows from an inductive argument based on the observation that replacing h_{i}^{n+1} with −h_{i}^{n+1} reverses the sign of det(M_{n}). For each n, let ψ_{n} be the isomorphism of Z^{2} onto H_{n} satisfying ψ_{n}(e_{i}) = h_{i}^{n}, and let m_{H_{n},H_{n+1}} : H_{n} → H_{n+1} be the homomorphism described in Theorem 6.7(2). We claim that ψ_{n+1} ∘ M_{n}^{ca} = m_{H_{n},H_{n+1}} ∘ ψ_{n}. To see this, observe that m_{H_{n},H_{n+1}} is multiplication by the determinant of M_{n}. Hence, as rational transformations, m_{H_{n},H_{n+1}}^{−1} ∘ M_{n}^{ca} = M_{n}^{−1}. Since m_{H_{n},H_{n}+1} commutes with ψ_{n+1}, the desired equality ψ_{n+1} ∘ M_{n}^{ca} = m_{H_{n},H_{n+1}} ∘ ψ_{n} is therefore equivalent to ψ_{n+1} = ψ_{n} ∘ M_{n}, which follows from the definitions of the maps involved. This establishes the claim. The claim guarantees that \underrightarrow{lim}(H_{n}, m_{H_{n},H_{n+1}}) ≅ \underrightarrow{lim}(Z^{2}, M_{n}^{ca}), and (6.2) then follows from Theorem 6.7(2). □

Examples 6.9. (1) Let α and β be supernatural numbers. For n ∈ N \ {0}, let ϕ_{n} be the homomorphism of Z^{2} determined by the diagonal matrix

M_{n} := ( \begin{matrix}α_{n} & 0 \\ 0 & β_{n} \end{matrix} ).

For each n, let

H_{n} := α[1, n]Z × β[1, n]Z = ϕ_{n}(Z^{2}) ⊂ Z^{2}.

We deduce from Theorem 6.7 that

K_{∗}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) = (Z[\frac{1}{αβ} ] ⊕ Z, Z[\frac{1}{α}] ⊕ Z[\frac{1}{β} ]),

that the position of the unit in K_{0} corresponds to the element (α_{1}, 0), and that P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1} is a simple AT algebra of real-rank 0.

We claim that this is an example of an AT algebra which cannot be realised using a rank-2 Bratteli diagram as in [27]. To see this, suppose otherwise. Then [27, Theorem 6.1] implies that there exists an injective homomorphism ϕ : Z[\frac{1}{α}] ⊕ Z[\frac{1}{β} ] → Z[\frac{1}{αβ} ] ⊕ Z such that each element of coker(ϕ) has finite order. Hence there exists (x, y) ∈ Z[\frac{1}{α}]⊕Z[\frac{1}{β} ] such that ϕ(x, y) = (z, m) with m ≠ 0. Since Z[\frac{1}{α}]⊕Z[\frac{1}{β} ] is generated by elements of the form (x, 0) and (0, y), we may in fact assume without loss of generality that there is an element x ∈ Z[\frac{1}{α}] such that ϕ(x, 0) = (z, m). Since α is infinite, there exist n > m and x′ ∈ Z[\frac{1}{α}] such that n · x′ = x, and this forces n · ϕ(x′, 0) = (z, m) which is impossible by our choice of n.

Since each Δ_{2}/H_{n} ≅ L_{α[1,n]}^{∗} × L_{β[1,n]}^{∗}, the K-theory calculations for this example can also be verified using the Künneth formula (Theorem 3.8 and Proposition 5.7).

(2) Let ϕ be the homomorphism of Z^{2} determined by the integer matrix M := ( \begin{matrix}a & b \\ c & d \end{matrix} ). Suppose that M is diagonalisable as a real 2 × 2 matrix, and that its eigenvalues are greater than 1 in modulus. Let D := ad − bc be the determinant of M. For n ≥ 1, let H_{n} := M^{n}Z^{2} and Λ_{n} := Δ_{2}/H_{n}. Our assumption regarding the eigenvalues of M ensures that ⋂_{n=1}^{∞} H_{n} = {0}, so Theorem 6.7 and Corollary 6.8 imply that C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n})) is a simple AT algebra of real rank zero with

K_{∗}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) ≅ (Z[\frac{1}{D} ] ⊕ Z, \underrightarrow{lim} (Z^{2}, ( \begin{matrix}d & −b \\ −c & a \end{matrix} )) ).

In particular, let M = ( \begin{matrix}a & −b \\ b & a \end{matrix} ) with a^{2} + b^{2} > 1. We may identify Z^{2} with the group of Gaussian integers Z[i] by (m, n) ↦ m+in, and then the group homomorphism of Z^{2} obtained from multiplication by M coincides with the group homomorphism of Z[i] obtained from multiplication by a + ib. Likewise M^{ca} implements multiplication by the conjugate a − ib. With D := a^{2} + b^{2} and ζ := \frac{1}{a−ib} = \frac{a+ib}{a^{2}+b^{2}} , we have

K_{∗}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) ≅ (Z[\frac{1}{D} ] ⊕ Z, Z[i, \frac{1}{ζ}]).

by Theorem 6.7.

(3) More generally, a sequence of Gaussian integers ζ_{j} := a_{j} + b_{j}i with |ζ_{j}| > 1 for all j gives rise to a natural notion of a Gaussian supernatural number ζ = ∏_{j=1}^{∞} ζ_{j}. Generalising the construction of the latter part of example (2) above, let H_{n} := (∏_{j=1}^{n} \overline{ζ_{j}})Z[i] for each n, and identify Z[i] with Z^{2} as a group to obtain a decreasing chain of subgroups of H_{n} of Z^{2} with trivial intersection.

Let α be the supernatural number α = ∏_{j=1}^{∞} |ζ_{j}|^{2}. Then

K_{∗}(P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n}))P_{1}) ≅ (Z[\frac{1}{α}] ⊕ Z, Z[i, \frac{1}{ζ}]).

by Theorem 6.7 and Corollary 6.8.

We now turn to the proof of Theorem 6.7; in particular, we adopt the notation and conventions of Example 6.6. Our first step is to describe explicitly the K-theory of C^{∗}(Δ_{2}/H_{n}) for a fixed n ∈ N \ {0}. We do this using the results of Section 5.2. For q ∈ Z^{k} we write q_{+} and q_{−} for the positive and negative parts of q. That is to say that q_{+} and q_{−} are the unique elements of N^{k} whose coordinate-wise minimum q_{+} ∧ q_{−} is equal to 0, and which satisfy q = q_{+} − q_{−}. For q ∈ Z^{k}, a cycle of degree q in a k-graph Λ is a pair (μ, ν) where μ ∈ Λ^{q_{+}} and ν ∈ Λ^{q_{−}} such that r(μ) = r(ν) and s(μ) = s(ν). When q ∈ N^{k}, q = q_{+} and q_{−} = 0, so ν is a vertex, and μ is a cycle in the usual sense: a path whose range and source coincide. Let H ⊂ Z^{2} be a finite-index subgroup of Z^{2}. Let G = Z^{2}/H. We view the ring ZG as the collection of functions f : G → Z. For X ⊆ G we denote the indicator function of X by 1_{X}. We denote the point-mass at g ∈ G by δ_{g}. Let Λ := Δ_{2}/H. Let E be the skeleton of Λ. That is E is the directed graph with the same vertices as Λ, and edges Λ^{e_{1}}∪Λ^{e_{2}}, with range and source inherited from Λ. The degree map from Λ restricts to a map from E^{1} to {e_{1}, e_{2}}. As in [31, 27] we call edges in E blue when they are of degree e_{1} in Λ, and red when they are of degree e_{2}. We often blur the distinction between concatenation of edges in E and the corresponding factorisation of a path in Λ. Recall that we are identifying Λ^{0} with G = Z^{2}/H. Hence, given a path α = a_{0}a_{1} ⋯ a_{n} in E, we define functions f_{α}^{b} and f_{α}^{r} in ZG by

f_{α}^{b}(g) = #{0 ≤ j ≤ n : r(a_{j}) = g, d(a_{j}) = e_{1}}
f_{α}^{r}(h) = #{0 ≤ k ≤ n : r(a_{k}) = h, d(a_{k}) = e_{2}}.

The idea is that f_{α}^{b}(g) counts the number of blue edges in α whose range is g, and f_{α}^{r}(g) does the same thing for red edges. We define f_{α} ∈ ZG ⊕ ZG by f_{α} = f_{α}^{b} ⊕ f_{α}^{r}. For a vertex g ∈ Λ^{0} = G, we define f_{g}^{b} and f_{g}^{r} to be the zero element of ZG, and f_{g} = f_{g}^{b} ⊕ f_{g}^{r} is then the zero element of ZG ⊕ ZG. As Λ = Δ_{2}/H, for each g ∈ Λ^{0} = G there is a unique path [g, g + (1, 1)] of degree (1, 1) with range g. Using the factorisation property, we can express this path as b_{g}r_{g+[e_{1}]} = r_{g}b_{g+[e_{2}]} where r_{g} and b_{g} denote the unique red and blue edges in E with range g (for n ∈ Z^{2}, [n] denotes the class of n in the quotient group G = Z^{2}/H). We write z_{g} for the function (δ_{g+[e_{2}]} − δ_{g}) ⊕ (δ_{g} − δ_{g+[e_{1}]}) in ZG ⊕ ZG. Given paths α = a_{0} ⋯ a_{m} and β = b_{0} ⋯ b_{n} in the skeleton E of Λ such that r(a_{0}) = r(b_{0}) and s(a_{m}) = s(b_{n}), let f_{α,β} := f_{α} − f_{β} ∈ ZG ⊕ ZG. Fix generators h_{1}, h_{2} for H; so [h_{i}] = [0] in G. By definition of Λ, there are unique paths μ_{1}^{+} ∈ Λ^{(h_{1})_{+}} and μ_{1}^{−} ∈ Λ^{(h_{1})_{−}} with r(μ_{1}^{±}) = 0. Fix factorisations α_{1}^{±} of μ_{1}^{±} into edges from the skeleton E. Since

s(μ_{1}^{+}) = [(h_{1})_{+}] = [(h_{1})_{−}] = s(μ_{1}^{−})

in G, the pair (μ_{1}^{+}, μ_{1}^{−}) is a cycle of degree h_{1} in Λ with range [0]. The same construction for h_{2} gives a cycle (μ_{2}^{+}, μ_{2}^{−}) of degree h_{2} with range [0] and fixed factorisations α_{2}^{±} of μ_{2}^{±} into edges from the skeleton E.

Lemma 6.10. With the notation established in the preceding paragraphs, the chain complex (5.6) can be described as follows:

(1) for each g ∈ G, ∂_{1}(δ_{g} ⊕ 0) = δ_{g} − δ_{g+[e_{1}]}, ∂_{1}(0 ⊕ δ_{g}) = δ_{g} − δ_{g+[e_{2}]}, and

∂_{2}(δ_{g}) = (δ_{g+[e_{2}]} − δ_{g}) ⊕ (δ_{g} − δ_{g+[e_{1}]}) = z_{g}.

(2) coker(∂_{1}) ≅ Z is generated by δ_{0} + Im(∂_{1});

(3) ker(∂_{2}) ≅ Z is generated by 1_{G}; (4) For each h ∈ G, the set {z_{g} : g ∈ G \ {h}} is a basis for Im(∂_{2}) ≅ Z^{|G|−1}.

(5) Fix any two factorisations α and β of a path μ in Λ into edges from E. Then f_{α} − f_{β} ∈ Im(∂_{2}), and ∂_{1}(f_{α}) = ∂_{1}(f_{β}) = δ_{r(α)} − δ_{s(α)}.

(6) ker(∂_{1}) is the subgroup of ZG ⊕ ZG generated by the elements f_{α,β} where α and β are paths in the skeleton E with r(α) = r(β) and s(α) = s(β).

(7) There is an isomorphism ψ of H onto ker(∂_{1})/ Im(∂_{2}) which takes d(μ) − d(ν) to f_{α,β} + Im(∂_{2}) for each cycle (μ, ν) in Λ and pair of factorisations α of μ and β of ν. In particular, for any basis B for Im(∂_{2}), the set B ∪ {f_{α_{1}^{+},α_{1}^{−}}, f_{α_{2}^{+},α_{2}^{−}}} is a basis for ker(∂_{1}) ≅ Z^{|G|+1} (where α_{i}^{±} are the fixed factorisations of the paths μ_{i}^{±} of degree (h_{i})_{±} described above).

In particular, K_{∗}(C^{∗}(Λ)) ≅ (Z^{2}, H) where the class of the identity in K_{0} is identified with the element (|G|, 0) of Z^{2}.

Proof. (1) The adjacency matrix M_{1} associated to (Λ^{0}, Λ^{e_{1}}, r, s) is the permutation matrix determined by translation by [e_{1}] in G and similarly for M_{2}. The first statement then follows from the formulae for ∂_{1} and ∂_{2} in terms of M_{1} and M_{2}.

(2) The formulae for ∂_{1}(δ_{g} ⊕0) and ∂_{1}(0⊕δ_{g}) show that δ_{g} +Im(∂_{1}) = δ_{g+[e_{i}]} + Im(∂_{1}) in coker(∂_{1}) for i = 1, 2 and g ∈ G. Since the action of Z^{2} on G by translation is transitive, this establishes (2).

(3) Using the formula for ∂_{2} established in (1), one can see that for f ∈ ZG, ∂_{2}(f) = f_{1} ⊕ f_{2} where

f_{1}(g) = −f(g) + f(g − [e_{1}])  and  f_{2}(g) = f(g) − f(g − [e_{2}])).

Hence f ∈ ker(∂_{2}) if and only if f(g) = f(g − [e_{1}]) = f(g − [e_{2}]) for all g ∈ G, and since the action of Z^{2} on G is transitive, this establishes (3).

(4) Part (1) establishes that Im(∂_{2}) is generated by {z_{g} : g ∈ G}. A simple calculation shows that ∑_{g∈G} z_{g} = 0 in ZG ⊕ ZG, and it follows that for any h ∈ G, the set {z_{g} : g ∈ G \ {h}} generates Im(∂_{2}) ≅ Z^{|G|−1}. Since ker(∂_{2}) has rank 1, the rank of its image is |G| − 1, establishing (4).

(5) By part (4), the image of ∂_{2} is generated by elements of the form f_{α} − f_{β} where α and β are the two possible factorisations of a path in Λ^{(1,1)}. Since f_{αβ} = f_{α} + f_{β} when α and β are paths in E which can be concatenated, this establishes the first claim. The second statement follows from a straightforward calculation using that

(6.3)  ∂_{1}(f^{b} ⊕ f^{r})(g) = f^{b}(g) − f^{b}(g − [e_{1}]) + f^{r}(g) − f^{r}(g − [e_{2}]).

(6) If α, β are paths in the skeleton with r(α) = r(β) and s(α) = s(β) then f_{α,β} belongs to ker(∂_{1}) by (5). We must show that every f ∈ ker(∂_{1}) can be written as a Z-linear combination of elements of the form f_{α,β}. First note that it suffices to treat the case where f takes only nonnegative values (this is because 1_{G} ⊕1_{G} can be so expressed). So suppose that f takes nonnegative values, and write f = f^{b} ⊕ f^{r}. Let E_{f} be the directed graph with vertices G and which contains f^{b}(g) parallel copies of the blue edge in E with range g and f^{r}(g) copies of the red edge in E with range g. If E_{f} contains a terminal vertex g which receives at least one edge but emits no edges at all, then f^{b}(g) + f^{r}(g) ≠ 0, but f^{b}(g − [e_{1}]) = f^{r}(g − [e_{2}]) = 0, and (6.3) shows that ∂_{1}(f)(g) ≠ 0. Hence E_{f} contains no such vertex, and therefore must either contain a cycle α or contain no edges at all. In the latter case, the claim is trivial, and in the former case, f ≥ f_{α}, and removing the cycle α from E_{f} produces the graph E_{f−f_{α}} for the function f − f_{α}. After finitely many such steps, we must obtain a forest with no terminal vertex. The only such forest is the empty graph which corresponds to the function 0 ⊕ 0. That is f − ∑_{α∈L} f_{α} = 0 ⊕ 0 for some collection L of cycles, and this proves (6).

(7) Suppose that (μ, ν) is a cycle in Λ. Then

s(μ) − [d(μ)] = r(μ) = r(ν) = s(ν) − [d(ν)] = s(μ) − [d(ν)]

in G = Λ^{0} = Z^{2}/H, so d(μ) − d(ν) ∈ H. It is clear from the definition of Λ that each element of H arises as d(μ) − d(ν) for some cycle (μ, ν) in Λ. To see that the assignment d(μ) − d(ν) ↦ f_{α,β} + Im(∂_{2}) is well defined, we must show two things. First that for distinct factorisations α and α′ of μ and distinct factorisations β and β′ of ν, the difference f_{α,β} −f_{α′,β′} lies in the image of ∂_{2}. This follows from (5). Second, we must show that if (μ, ν) and (μ′, ν′) are cycles in Λ with d(μ) − d(ν) = d(μ′) − d(ν′), then there exist factorisations α of μ, β of ν, α′ of μ′, and β′ of ν′ such that f_{α,β} − f_{α′,β′} is in Im(∂_{2}). To see this, first note that by factorising μ = μ′τ and ν = ν′τ where d(τ) = d(μ) ∧ d(ν), we can reduce to the case where d(μ) ∧ d(ν) = 0. Next we claim that it suffices to consider the case where r(μ) = r(ν) = r(μ′) = r(ν′) = [0]. To see this, fix η in [0]Λr(μ) and note that the cycle (ημ, ην) corresponds to the same class as (μ, ν) in ker(∂_{1})/ Im(∂_{2}). Factorise ημ = ξρ and ην = ωσ where d(ξ) = d(μ), d(ω) = d(ν) and d(ρ) = d(σ) = d(η). Since each gΛ^{n} is a singleton and since Z^{2} acts on Λ by translation, (ξ, ω) is a cycle with range [0], and ρ = σ. Hence the cycle (ξ, ω) corresponds to the same class in ker(∂_{1})/ Im(∂_{2}) as (μ, ν). After shifting (μ′, ν′) in a similar way we may assume that both cycles have range [0]. We now have cycles (μ, ν) and (μ′, ν′) with range [0] and such that d(μ) − d(ν) = d(μ′) − d(ν′) and d(μ) ∧ d(ν) = 0 = d(μ′) ∧ d(ν′). Since [0]Λ^{n} is a singleton for any n ∈ Z^{2}, this forces μ = μ′ and ν = ν′. This completes the proof that d(μ) − d(ν) ↦ f_{α,β} + Im(∂_{2}) is well defined. That f_{αβ} = f_{α} +f_{β} ensures that ψ(g+h) = ψ(g)+ψ(h), and that f_{β,α} = −f_{α,β} shows that ψ(−g) = −ψ(g). Hence ψ is a homomorphism. By part (6), to see that ψ is surjective, we just need to show that each f_{α,β} +Im(∂_{2}) is in the range of ψ. This is clear because f_{α,β} + Im(∂_{2}) is precisely ψ(d(μ) − d(ν)) where μ factorises as α and ν factorises as β. Finally, to see that ψ is injective, note that if f_{α,β} ∈ Im(∂_{2}), then d(μ) = d(ν) where μ factorises as α and ν factorises as β. This completes the proof that ψ : H → ker(∂_{1})/ Im(∂_{2}) is an isomorphism. The remaining statement follows from (4) and that (μ_{1}^{+}, μ_{1}^{−}) and (μ_{2}^{+}, μ_{2}^{−}) are cycles whose degrees form a basis for H. This proves (7). The final statement of the Lemma follows from (5.7). □

We now consider two consecutive graphs in the sequence of covering systems described in Example 6.6, and describe the homomorphism of K-invariants obtained from Proposition 3.2(6).

Theorem 6.11. Consider the situation described in Example 6.6, and fix n ∈ N \ {0}. For i = n, n + 1, let Λ_{i} := Δ_{2}/H_{i}, and consider the commuting diagram

0 ⟵ ZΛ_{n}^{0} ⟵^{∂_{1}^{Λn}} ZΛ_{n}^{0} ⊕ ZΛ_{n}^{0}  ⟵^{∂_{2}^{Λn}} ZΛ_{n}^{0} ⟵ 0
↓^{p_{n}^{∗}}  ↓^{p_{n}^{∗}⊕p_{n}^{∗}}  ↓^{p_{n}^{∗}}
0 ⟵ ZΛ_{n+1}^{0} ⟵^{∂_{1}^{Λ_{n+1}}} ZΛ_{n+1}^{0} ⊕ ZΛ_{n+1}^{0} ⟵^{∂_{2}^{Λ_{n+1}}} ZΛ_{n+1}^{0} ⟵ 0

(1) The right-hand vertical map p_{n}^{∗} : ZΛ_{n}^{0} → ZΛ_{n+1}^{0} restricts to a homomorphism p_{n}^{∗}|_{ker(∂_{2}^{Λn})} : ker(∂_{2}^{Λ_{n}}) → ker(∂_{2}^{Λ_{n+1}}) which is characterised by p_{n}^{∗}|_{ker(∂_{2}^{Λn})}(1_{G_{n}}) = 1_{G_{n+1}}.

(2) The left-hand vertical map p_{n}^{∗} : ZΛ_{n}^{0} → ZΛ_{n+1}^{0} induces a homomorphism p̃_{n}^{∗} : coker(∂_{1}^{Λ_{n}}) → coker(∂_{1}^{Λ_{n+1}}) characterised by

p̃_{n}^{∗}(δ_{0} + Im(∂_{1}^{Λ_{n}})) = [H_{n} : H_{n+1}] · δ_{0} + Im(∂_{1}^{Λ_{n+1}}).

(3) The middle vertical map p_{n}^{∗} ⊕ p_{n}^{∗} : ZΛ_{n}^{0} ⊕ ZΛ_{n}^{0} → ZΛ_{n+1}^{0} ⊕ ZΛ_{n+1}^{0} induces a homomorphism (p_{n}^{∗} ⊕ p_{n}^{∗})^{∼} : ker(∂_{1}^{Λ_{n}})/ Im(∂_{2}^{Λ_{n}}) → ker(∂_{1}^{Λ_{n+1}})/ Im(∂_{2}^{Λ_{n+1}}) such that the following diagram commutes.

H_{n} ⟶^{ψ_{n}} ker(∂_{1}^{Λ_{n}})/ Im(∂_{2}^{Λ_{n}})
↓^{m_{Hn,H_{n+1}}}  ↓^{(p_{n}^{∗}⊕p_{n}^{∗})^{∼}}
H_{n+1} ⟶^{ψ_{n+1}} ker(∂_{1}^{Λ_{n+1}})/ Im(∂_{2}^{Λ_{n+1}})

where ψ_{n} and ψ_{n+1} are the isomorphisms obtained from Lemma 6.10(7), and m_{H_{n},H_{n+1}} is as in Theorem 6.7(2).

Under the isomorphism

K_{∗}(C^{∗}(Λ_{i})) ≅ ( coker(∂_{1}^{Λ_{i}}) ⊕ ker(∂_{2}^{Λ_{i}}), ker(∂_{1}^{Λ_{i}})/ Im(∂_{2}^{Λ_{i}}))

obtained from Corollary 5.6, the maps described in (1), (2) and (3) determine the map (ι_{p_{n}})_{∗} : K_{∗}(C^{∗}(Λ_{n})) → K_{∗}(C^{∗}(Λ_{n+1})) obtained from Proposition 3.2(6).

Proof. Lemma 6.10(3) ensures that 1_{G_{i}} generates ker(∂_{2}^{Λ_{i}}) for i = n, n+1. The formula for p_{n}^{∗} shows that p_{n}^{∗}(1_{G_{n}}) = 1_{G_{n+1}}, which gives (1). Statement (2) follows from the formula for p_{n}^{∗} combined with the observation that for i = n, n + 1, the δ_{g}, g ∈ G_{i} are all equivalent modulo Im(∂_{1}^{Λ_{i}}). It remains only to prove (3). We first consider the case where H_{n} = Z^{2}, so G_{n} = {0} and Λ_{n} is a copy of the 2-graph T_{2} ≅ N^{2} (as a category) with one vertex and one morphism λ_{m} of each degree m ∈ N^{2}. In this case, ψ_{n} is just the identity map from Z^{2} to Z⊕Z. Let h_{1}, h_{2} be a pair of generators for H_{n+1}.

Since H_{n+1} has finite index in Z^{2}, the assignments (1, 0) ↦ h_{1} and (0, 1) ↦ h_{2} determine an endomorphism of H_{n} which is a rational isomorphism. Hence it suffices to show that (p_{n}^{∗}⊕p_{n}^{∗})^{∼}∘ψ_{n}(h_{i}) = ψ_{n+1}([Z^{2} : H_{n+1}]·h_{i}) for i = 1, 2. We just argue that this happens for i = 1 (the case i = 2 follows from a symmetric argument). Writing h_{1} = (x, y) where x, y ∈ Z, the formula for p_{n}^{∗} ensures that (p_{n}^{∗} ⊕ p_{n}^{∗})^{∼} takes ψ_{n}(h_{1}) to the class of x1_{G_{n+1}} ⊕ y1_{G_{n+1}}. To see that this is ψ_{n+1}([Z^{2} : H_{n+1}] · h_{1}), let f := f_{α_{1}^{+},α_{1}^{−}} = ψ_{n+1}(h_{1}) be the function in ZG_{n+1} ⊕ ZG_{n+1} obtained from Lemma 6.10(7). By definition of f, we have f = f_{b} ⊕ f_{r} where the entries of f_{b} sum to x and the entries of f_{r} sum to y. For g ∈ G_{n+1}, let g · f_{b} be the function determined by g · f_{b}(h) = f_{b}(h − g), and similarly for f_{r}. Since G_{n+1} acts freely and transitively on Λ_{n+1}^{0} = G_{n+1}, it follows that

(6.4)  ∑_{g∈G_{n+1}} g · f = x1_{G_{n+1}} ⊕ y1_{G_{n+1}} = (p_{n}^{∗} ⊕ p_{n}^{∗})^{∼} ∘ ψ_{n}(h_{1}).

The proof of statement (7) in Lemma 6.10 shows that each g · f := g · f_{b} ⊕ g · f_{r} represents the same class as f in ker(∂_{1}^{Λ_{n+1}})/ Im(∂_{2}^{Λ_{n+1}}). Hence the left-hand side of (6.4) has the same class in ker(∂_{1}^{Λ_{n+1}})/ Im(∂_{2}^{Λ_{n+1}}) as ψ_{n+1}(|G_{n+1}| · h_{1}) as required. For the general case, first note that we may assume without loss of generality that H_{1} = Z^{2} so that Λ_{1} = T_{2}. Let p_{[1,n]} := p_{1} ∘ ⋯ ∘ p_{n−1} and p_{[1,n+1]} := p_{1} ∘⋯∘p_{n} be the coverings of Λ_{1} = T_{2} by Λ_{n} and Λ_{n+1} obtained by composing the first n and n + 1 levels of the covering system; we may apply the argument of the previous paragraph to these coverings. Then p_{[1,n+1]} = p_{[1,n]} ∘ p_{n}, so p_{[1,n+1]}^{∗} ⊕ p_{[1,n+1]}^{∗} = (p_{[1,n]}^{∗} ⊕ p_{[1,n]}^{∗}) ∘ (p_{n}^{∗} ⊕ p_{n}^{∗}), and since these maps induce

homomorphisms between ker(∂_{1}^{T_{2}})/ Im(∂_{2}^{T_{2}}) and ker(∂_{1}^{Λ_{n+1}})/ Im(∂_{2}^{Λ_{n+1}}) which are rational isomorphisms, it follows that (p_{n}^{∗} ⊕ p_{n}^{∗})^{∼} behaves as claimed. The final statement follows from Corollary 5.6. □

We are now ready to prove Theorem 6.7.

Proof of Theorem 6.7. Proposition 3.2 shows that P_{1} is full so that compression by P_{1} induces an isomorphism on K-theory. The formulae for the K-groups in statements (1) and (2) follow from Lemma 6.10 and Theorem 6.11 and the continuity of the K-functor. Since v(Δ_{2}/H_{n})w ≠ ∅ for all n ∈ N \ {0}, and v, w ∈ Δ_{2}^{0}/H_{n}, the 3-graph \begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}, p_{n}) is cofinal. Moreover a given infinite path x in \begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}, p_{n}) is periodic with period m ∈ Z^{2} if and only if every infinite path in \begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}, p_{n}) is periodic with period m, which in turn is equivalent to the condition that m ∈ ⋂_{n=1}^{∞} H_{n}. It follows from Lemma 4.5 that \begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}, p_{n}) is simple if and only if ⋂ H_{n} = {0}; moreover, in this case, the argument of the second part of [27, Section 5] shows that C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}, p_{n})) has unique trace. We next claim that each C^{∗}(Δ_{2}/H_{n}) ≅ M_{[Z^{2}:H_{n}]}(C(T^{2})). To verify this, one first checks that h ↦ s_{[(0,h_{+})]}s_{[(0,h_{−})]}^{∗} is a group isomorphism H_{n} → U(s_{[0]}C^{∗}(Δ_{2}/H_{n})s_{[0]}) for each n. The standard argument used in [27, Lemma 3.9] shows that each s_{[(0,h_{+})]}s_{[(0,h_{−})]}^{∗} has full spectrum. One can then deduce that s_{[0]}C^{∗}(Δ_{2}/H_{n})s_{[0]} ≅ C^{∗}(H_{n}) ≅ C^{∗}(Z^{2}) ≅ C(T^{2}). For m ∈ Z^{2}/H_{n}, define V_{m} := s_{[0,m]}^{∗} ∈ C^{∗}(Δ_{2}/H_{n}). Applying Lemma 3.3 to these partial isometries with p = s_{[0]} and q = 1_{C^{∗}(Δ_{2}/H_{n})}proves that C^{∗}(Δ_{2}/H_{n}) ≅ M_{[Z^{2}:H_{n}]}(C(T^{2})). It now follows from [3, Theorem 1.3] that C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{2}/H_{n}; p_{n})) has real-rank 0. The classification of such algebras of Dădărlat-Elliott-Gong (see [36, Theorem 3.3.1]), and the K-theory calculations above complete the proof. □

Remark 6.12. Higher-rank Bunce-Deddens algebras and generalised odometer actions. We consider a slightly more general version of the situation described in Example 6.6. Let H_{1} ⊃ H_{2} ⊃ H_{3} ⊃ … be a chain of finite-index subgroups of Z^{k} such that ⋂_{n} H_{n} = {0}. For each n, let p_{n} : Δ_{k}/H_{n+1} → Δ_{k}/H_{n} be the canonical covering induced by the quotient maps described above, let m_{n} = 1, and let 𝔰_{n} : Δ_{k}/H_{n+1} → S_{1} be the trivial cocycle. This data specifies a sequence (Δ_{k}/H_{n}, Δ_{k}/H_{n+1}, p_{n})_{n=1}^{∞} of row-finite covering systems of k-graphs with no sources. Applying Corollary 2.11, we obtain a (k + 1)-graph

\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{k}/H_{n}; p_{n}).

We claim that the corner P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{k}/H_{n}; p_{n}))P_{1} can be thought of as a higher-rank Bunce-Deddens algebra. We justify this by giving a description of P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{k}/H_{n}; p_{n}))P_{1} as a crossed product by a generalised odometer action. We assume here that H_{1} = Z^{k} so that Δ_{k}/H_{1} is a copy of the k-graph T_{k} ≅ N^{k} (as a category) with one vertex and one morphism λ_{m} of each degree m ∈ N^{k}. One way to realise the Bunce-Deddens algebras is as crossed products of algebras of continuous functions on Cantor sets by generalised odometer actions. Given a supernatural number α = α_{1}α_{2} ⋯, let G_{n} := Z/α[1, n]Z for all n. Then for each n, since α[1, n+1]Z ⊃ α[1, n]Z, there is a natural surjective group homomorphism from G_{n+1} to G_{n}. Hence, we may form the projective limit group \underleftarrow{lim}(G_{n}, p_{n}). The automorphism τ(g_{1}, g_{2}, …) = (g_{1} + [1], g_{2} + [1], …) for (g_{1}, g_{2}, …) ∈ \underleftarrow{lim}(G_{n}, p_{n}) can then naturally be regarded as an odometer action on \underleftarrow{lim}(G_{n}, p_{n}). The Bunce-Deddens algebra of type α is the crossed product C(\underleftarrow{lim}(G_{n}, p_{n})) ⋊τ̃ Z where τ̃ is the automorphism of C(\underleftarrow{lim}(G_{n}, p_{n})) induced by τ (see [33, Examples 1(3)]). There is an analogous realisation of P_{1}C^{∗}(\begin{matrix}lim \\ ↽− \end{matrix}(Δ_{k}/H_{n}, p_{n}))P_{1} as follows. Let Λ := \begin{matrix}lim \\ ↽− \end{matrix}(Δ_{k}/H_{n}, p_{n}). Let F denote the fixed-point algebra of C^{∗}(Λ) for the gauge action γ of T^{k+1}. Note that by Remark 3.9, the restriction of the gauge action to P_{1}C^{∗}(Λ)P_{1} is trivial on the last coordinate of T^{k+1} and therefore becomes an action by T^{k} denoted γ̃. Recall that Λ^{∞} denotes the collection of infinite paths in Λ (see Notation 4.1). It is not hard to see that P_{1}F P_{1} is canonically isomorphic to C(vΛ^{∞}) where v is the unique vertex of Δ_{k}/H_{1} ≅ T_{k}. Let G_{n} := Z^{k}/H_{n} for each n, and let p_{n} : G_{n+1} → G_{n} be the induced map p_{n}(m + H_{n+1}) := m + H_{n}. Observe that G = \underleftarrow{lim}(G_{n}, p_{n}) is a compact abelian group. By functoriality of the projective limit the quotient maps Z^{k} → Z^{k}/H_{n} induce a homomorphism j : Z^{k} → G; injectivity of j follows from the fact that ⋂_{n} H_{n} = {0}. There is an action τ of Z^{k} on G given by τ_{m}(g_{1}, g_{2}, …) = (g_{1} +[m], g_{2} +[m], …), which generalises the odometer action discussed above. Since there is just one infinite path in T_{k}, the arguments of Section 4 show that vΛ^{∞} ≅ G as a topological space. Note that for every m ∈ N^{k}, the generator s_{λ_{m}} associated to the unique path λ_{m} ∈ T_{k}^{m} is a unitary in P_{1}C^{∗}(Λ)P_{1} and that under the identification of P_{1}F P_{1} with C(vΛ^{∞}) = C(G) conjugation by s_{λ_{m}} implements the automorphism induced by the homeomorphism τ_{m} of G. It follows that the reduction of the path groupoid (see [20, Section 2]) of Λ to vΛ^{∞} is isomorphic to the semidirect product groupoid G ⋊_{τ} Z^{k}. Therefore, standard arguments show that

P_{1}C^{∗}(Λ)P_{1} ≅ C(G) ⋊τ̃ Z^{k}

where τ̃ is the action induced by τ. Note that under this identification the restricted gauge action γ̃ coincides with the dual action of T^{k} = Ẑ^{k}. The action of G on C(G) induced by translation in G yields an action of G on C(G)⋊τ̃ Z^{k} which commutes with the dual action of T^{k} = Ẑ^{k}. Thus we obtain an action α by the compact abelian group G × T^{k} with fixed point algebra isomorphic to C. Hence, C(G) ⋊τ̃ Z^{k} (and thus P_{1}C^{∗}(Λ)P_{1}) admits an ergodic action of a compact abelian group. Such ergodic actions have been classified in [24, 4.5, 6.1]; the invariant is a symplectic bicharacter χ_{α} on Ĝ × Z^{k}, the dual of G × T^{k}. This gives rise to an alternative description of the C^{∗}-algebra as a twisted group C^{∗}-algebra with the group Ĝ × Z^{k} and a 2-cocycle associated to the bicharacter χ_{α} (only its cohomology class is determined by the bicharacter). It follows that

C(G) ⋊τ̃ Z^{k} ≅ C(T^{k}) ⋊ Ĝ

where the action of Ĝ on C(T^{k}) arises by translation from the embedding Ĝ → T^{k} dual to j : Z^{k} → G.

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Alex Kumjian Department of Mathematics (084) University of Nevada Reno NV 89557-0084 USA alex@unr.edu David Pask School of Mathematics and Applied Statistics University of Wollongong NSW 2522 AUSTRALIA dpask@uow.edu.au

Aidan Sims

School of Mathematics and Applied Statistics University of Wollongong NSW 2522 AUSTRALIA asims@uow.edu.au 206

Documenta Mathematica 13 (2008)