New York Journal of Mathematics

New York J. Math. 6 (2000) 1–20.

Higher Rank Graph C^{∗}-Algebras

Alex Kumjian and David Pask

Abstract. Building on recent work of Robertson and Steger, we associate a C^{∗}–algebra to a combinatorial object which may be thought of as a higher rank graph. This C^{∗}–algebra is shown to be isomorphic to that of the associated path groupoid. Various results in this paper give sufficient conditions on the higher rank graph for the associated C^{∗}–algebra to be: simple, purely infinite and AF. Results concerning the structure of crossed products by certain natural actions of discrete groups are obtained; a technique for constructing rank 2 graphs from “commuting” rank 1 graphs is given.

Contents

1. Higher rank graph C^{∗}–algebras 3 2. The path groupoid 6 3. The gauge invariant uniqueness theorem 9 4. Aperiodicity and its consequences 12 5. Skew products and group actions 14 6. 2-graphs 17 References 19

In this paper we shall introduce the notion of a higher rank graph and associate a C^{∗}–algebra to it in such a way as to generalise the construction of the C^{∗}–algebra of a directed graph as studied in [CK, KPRR, KPR] (amongst others). Graph C^{∗}– algebras include up to strong Morita equivalence Cuntz–Krieger algebras and AF algebras. The motivation for the form of our generalisation comes from the recent work of Robertson and Steger [RS1, RS2, RS3]. In [RS1] the authors study crossed product C^{∗}–algebras arising from certain group actions on Ã_{2}-buildings and show that they are generated by two families of partial isometries which satisfy certain relations amongst which are Cuntz–Krieger type relations [RS1, Equations (2), (5)] as well as more intriguing commutation relations [RS1, Equation (7)]. In [RS2] they give a more general framework for studying such algebras involving certain families

1^{ISSN 1076-9803/00}

Received November 8, 1999.

Mathematics Subject Classification. Primary 46L05; Secondary 46L55.

Key words and phrases. Graphs as categories, Graph algebra, Path groupoid, C^{∗}–algebra.

Research of the first author partially supported by NSF grant DMS-9706982.

Research of the second author supported by University of Newcastle RMC project grant.

of commuting 0 − 1 matrices. In particular the associated C^{∗}–algebras are simple, purely infinite and generated by a family of Cuntz–Krieger algebras associated to these matrices. It is this framework which we seek to cast in graphical terms to include a wider class of examples (including graph C^{∗}–algebras).

What follows is a brief outline of the paper. In the first section we introduce the notion of a higher rank graph as a purely combinatorial object: a small category Λ gifted with a degree map d : Λ → N^{k} (called shape in [RS2]) playing the role of the length function. No detailed knowledge of category theory is required to read this paper. The associated C^{∗}–algebra C^{∗}(Λ) is defined as the universal C^{∗}–algebra generated by a family of partial isometries {s_{λ} : λ ∈ Λ} satisfying relations similar to those of [KPR]. (Our standing assumption is that our higher rank graphs satisfy conditions analogous to a directed graph being row–finite and having no sinks.) We then describe some basic examples and indicate the relationship between our formalism and that of [RS2].

In the second section we introduce the path groupoid G_{Λ} associated to a higher rank graph Λ (cf. [R, D, KPRR]). Once the infinite path space Λ^{∞} is formed (and a few elementary facts are obtained) the construction is fairly routine. It follows from the gauge-invariant uniqueness theorem (Theorem 3.4) that C^{∗}(Λ) ≅ C^{∗}(G_{Λ}). By the universal property C^{∗}(Λ) carries a canonical action of T^{k} defined by

α_{t}(s_{λ}) = t^{d(λ)}s_{λ} (1)

called the gauge action. In the third section we prove the gauge–invariant uniqueness theorem, which is the key result for analysing C^{∗}(Λ) (cf. [BPRS, aHR], see also [CK, RS2] where similar techniques are used to prove simplicity). It gives conditions under which a homomorphism with domain C^{∗}(Λ) is faithful: roughly speaking, if the homomorphism is equivariant for the gauge action and nonzero on the generators then it is faithful. This theorem has a number of interesting consequences, amongst which are the isomorphism mentioned above and the fact that the higher rank Cuntz–Krieger algebras of [RS2] are isomorphic to C^{∗}–algebras associated to suitably chosen higher rank graphs.

In the fourth section we characterise, in terms of an aperiodicity condition on Λ, the circumstances under which the groupoid G_{Λ} is essentially free. This aperiodicity condition allows us to prove a second uniqueness theorem analogous to the original theorem of [CK]. In 4.8 and 4.9 we obtain conditions under which C^{∗}(Λ) is simple and purely infinite respectively which are similar to those in [KPR] but with the aperiodicity condition replacing condition (L).

In the next section we show that, given a functor c : Λ → G where G is a discrete group, then as in [KP] one may construct a skew product G ×_{c} Λ which is also a higher rank graph. If G is abelian then there is a natural action α^{c} : Ĝ → Aut C^{∗}(Λ) such that

α_{χ}^{c}(s_{λ}) = ⟨χ, c(λ)⟩s_{λ}; (2)

Higher Rank Graph C^{∗}-Algebras 3

moreover C^{∗}(Λ) ⋊_{α^{c}} Ĝ ≅ C^{∗}(G ×_{c} Λ). Comparing (1) and (2) we see that the gauge action α is of the form α^{d} and as a consequence we may show that the crossed product of C^{∗}(Λ) by the gauge action is isomorphic to C^{∗}(Z^{k} ×_{d} Λ); this C^{∗}–algebra is then shown to be AF. By Takai duality C^{∗}(Λ) is strongly Morita equivalent to a crossed product of this AF algebra by the dual action of Z^{k}. Hence C^{∗}(Λ) belongs to the bootstrap class N of C^{∗}–algebras for which the UCT applies (see [RSc]) and is consequently nuclear. If a discrete group G acts freely on a kgraph Λ, then the quotient object Λ/G inherits the structure of a k–graph; moreover (as a generalisation of [GT, Theorem 2.2.2]) there is a functor c : Λ/G → G such that Λ ≅ G ×_{c} (Λ/G) in an equivariant way. This fact allows us to prove that

C^{∗}(Λ) ⋊ G ≅ C^{∗}(Λ/G) ⊗ K (ℓ^{2}(G))

where the action of G on C^{∗}(Λ) is induced from that on Λ. Finally in Section 6, a technique for constructing a 2-graph from “commuting” 1-graphs A, B with the same vertex set is given. The construction depends on the choice of a certain bijection between pairs of composable edges: θ : (a, b) ↦ (b′, a′) where a, a′ ∈ A^{1} and b, b′ ∈ B^{1}; the resulting 2-graph is denoted A ∗_{θ} B. It is not hard to show that every 2-graph is of this form.

Throughout this paper we let N = {0, 1, … } denote the monoid of natural numbers under addition. For k ≥ 1, regard N^{k} as an abelian monoid under addition with identity 0 (it will sometimes be useful to regard N^{k} as a small category with one object) and canonical generators e_{i} for i = 1, …, k. We shall also regard N^{k} as the positive cone of Z^{k} under the usual coordinatewise partial order: thus m ≤ n if and only if m_{i} ≤ n_{i} for all i, where m = (m_{1}, …, m_{k}), and n = (n_{1}, …, n_{k}). (This makes N^{k} a lattice.)

We wish to thank Guyan Robertson and Tim Steger for providing us with an early version of their paper [RS2]; the first author would also like to thank them for a number of stimulating conversations and the staff of the Mathematics Department at Newcastle University for their hospitality during a recent visit.

1. Higher rank graph C^{∗}–algebras

In this section we first introduce what we shall call a higher rank graph as a purely combinatorial object. (We do not know whether this concept has been studied before.) Our definition of a higher rank graph is modelled on the path category of a directed graph (see [H], [Mu], [MacL, §II.7] and Example 1.3). Thus a higher rank graph will be defined to be a small category gifted with a degree map (called shape in [RS2]) satisfying a certain factorisation property. We then introduce the associated C^{∗}–algebra whose definition is modelled on that of the C^{∗}–algebra of a graph as well as the definition of [RS2].

Definitions 1.1. A k-graph (rank k graph or higher rank graph) (Λ, d) consists of a countable small category Λ (with range and source maps r and s respectively) together with a functor d : Λ → N^{k} satisfying the factorisation property: for every λ ∈ Λ and m, n ∈ N^{k} with d(λ) = m + n, there are unique elements μ, ν ∈ Λ such that λ = μν and d(μ) = m, d(ν) = n. For n ∈ N^{k} we write Λ^{n} := d^{−1}(n). A morphism between k-graphs (Λ_{1}, d_{1}) and (Λ_{2}, d_{2}) is a functor f : Λ_{1} → Λ_{2} compatible with the degree maps.

Remarks 1.2. The factorisation property of 1.1 allows us to identify Obj(Λ), the objects of Λ with Λ^{0}. Suppose λα = μα in Λ then by the the factorisation property λ = μ; left cancellation follows similarly. We shall write the objects of Λ as u, v, w, … and the morphisms as greek letters λ, μ, ν . . .. We shall frequently refer to Λ as a k-graph without mentioning d explicitly.

It might be interesting to replace N^{k} in Definition 1.1 above by a monoid or perhaps the positive cone of an ordered abelian group.

Recall that λ, μ ∈ Λ are composable if and only if r(μ) = s(λ), and then λμ ∈ Λ; on the other hand two finite paths λ, μ in a directed graph may be composed to give the path λμ provided that r(λ) = s(μ); so in 1.3 below we will need to switch the range and source maps.

Example 1.3. Given a 1-graph Λ, define E^{0} = Λ^{0} and E^{1} = Λ^{1}. If we define s_{E}(λ) = r(λ) and r_{E}(λ) = s(λ) then the quadruple (E^{0}, E^{1}, r_{E}, s_{E}) is a directed graph in the sense of [KPR, KP]. On the other hand, given a directed graph E = (E^{0}, E^{1}, r_{E}, s_{E}), then E^{∗} = ∪_{n≥0}E^{n}, the collection of finite paths, may be viewed as small category with range and source maps given by s(λ) = r_{E}(λ) and r(λ) = s_{E}(λ). If we let d : E^{∗} → N be the length function (i.e., d(λ) = n iff λ ∈ E^{n}) then (E^{∗}, d) is a 1-graph.

We shall associate a C^{∗}–algebra to a k-graph in such a way that for k = 1 the associated C^{∗}–algebra is the same as that of the directed graph. We shall consider other examples later.

Definitions 1.4. The k-graph Λ is row finite if for each m ∈ N^{k} and v ∈ Λ^{0} the set Λ^{m}(v) := {λ ∈ Λ^{m} : r(λ) = v} is finite. Similarly Λ has no sources if Λ^{m}(v) ≠ ∅ for all v ∈ Λ^{0} and m ∈ N^{k}.

Clearly if E is a directed graph then E is row finite (resp. has no sinks) if and only if E^{∗} is row finite (resp. has no sources). Throughout this paper we will assume (unless otherwise stated) that any k-graph Λ is row finite and has no sources, that is

(3) 0 < #Λ^{n}(v) < ∞ for every v ∈ Λ^{0} and n ∈ N^{k}.

The Cuntz–Krieger relations [CK, p.253] and the relations given in [KPR, §1] may be interpreted as providing a representation of a certain directed graph by partial isometries and orthogonal projections. This view motivates the definition of C^{∗}(Λ).

Definitions 1.5. Let Λ be a k-graph (which satisfies the standing hypothesis (3)). Then C^{∗}(Λ) is defined to be the universal C^{∗}–algebra generated by a family {s_{λ} : λ ∈ Λ} of partial isometries satisfying:

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

For λ ∈ Λ, define p_{λ} = s_{λ}s_{λ}^{∗} (note that p_{v} = s_{v} for all v ∈ Λ^{0}). A family of partial isometries satisfying (i)–(iv) above is called a ∗–representation of Λ.

Remarks 1.6. (i) If {t_{λ} : λ ∈ Λ} is a ∗–representation of Λ then the map

s_{λ} ↦ t_{λ} defines a ∗–homomorphism from C^{∗}(Λ) to C^{∗}({t_{λ} : λ ∈ Λ}). (ii) If E^{∗} is the 1-graph associated to the directed graph E (see 1.3), then by

restricting a ∗–representation to E^{0} and E^{1} one obtains a Cuntz–Krieger family for E in the sense of [KPR, §1]. Conversely every Cuntz–Krieger family for E extends uniquely to a ∗–representation of E^{∗}.

(iii) In fact we only need the relation (iv) above to be satisfied for n = e_{i} ∈ N^{k} for i = 1, …, k, the relations for all n will then follow (cf. [RS2, Lemma 3.2]). Note that the definition of C^{∗}(Λ) given in 1.5 may be extended to the case where there are sources by only requiring that relation (iv) hold for n = e_{i} and then only if Λ^{e_{i}}(v) ≠ ∅ (cf. [KPR, Equation (2)]).

(iv) For λ, μ ∈ Λ if s(λ) ≠ s(μ) then s_{λ}s_{μ}^{∗} = 0. The converse follows from 2.11.

(v) Increasing finite sums of p_{v}’s form an approximate identity for C^{∗}(Λ) (if Λ^{0} is finite then ∑_{v∈Λ^{0}} p_{v} is the unit for C^{∗}(Λ)). It follows from relations (i) and (iv) above that for any n ∈ N^{k}, {p_{λ} : d(λ) = n} forms a collection of orthogonal projections (cf. [RS2, 3.3]); likewise increasing finite sums of these form an approximate identity for C^{∗}(Λ) (see 2.5).

(vi) The above definition is not stated most efficiently. Any family of operators {s_{λ} : λ ∈ Λ} satisfying the above conditions must consist of partial isometries. The first two axioms could also be replaced by:

s_{λ}s_{μ} = \begin{cases}s_{λμ} & if s(λ) = r(μ) \\ 0 & otherwise. \end{cases}

Examples 1.7. (i) If E is a directed graph, then by 1.6 (i) and (ii) we have

C^{∗}(E^{∗}) ≅ C^{∗}(E) (see 1.3). (ii) For k ≥ 1 let Ω = Ω_{k} be the small category with objects Obj (Ω) = N^{k},

and morphisms Ω = {(m, n) ∈ N^{k} × N^{k} : m ≤ n}; the range and source maps are given by r(m, n) = m, s(m, n) = n. Let d : Ω → N^{k} be defined by d(m, n) = n − m. It is then straightforward to show that Ω_{k} is a k-graph and C^{∗}(Ω_{k}) ≅ K (ℓ^{2}(N^{k})).

(iii) Let T = T_{k} be the semigroup N^{k} viewed as a small category, then if d : T → N^{k} is the identity map then (T, d) is a k-graph. It is not hard to show that C^{∗}(T) ≅ C(T^{k}), where s_{e_{i}} for 1 ≤ i ≤ k are the canonical unitary generators.

(iv) Let {M_{1}, …, M_{k}} be square {0, 1} matrices satisfying conditions (H0)–(H3) of [RS2] and let A be the associated C^{∗}-algebra. For m ∈ N^{k} let W_{m} be the collection of undecorated words in the finite alphabet A of shape m as defined in [RS2] then let

W = ⋃_{m∈N^{k}} W_{m}.

Together with range and source maps r(λ) = o(λ), s(λ) = t(λ) and product defined in [RS2, Definition 0.1] W is a small category. If we define d : W → N^{k} by d(λ) = σ(λ), then one checks that d satisfies the factorisation property, and then from the second part of (H2) we see that (W, d) is an irreducible k-graph in the sense that for all u, v ∈ W_{0} there is λ ∈ W such that s(λ) = u and r(λ) = v.

We claim that the map s_{λ} ↦ s_{λ,s(λ)} for λ ∈ W extends to a *-homomorphism C^{∗}(W) → A for which s_{λ}s_{μ}^{∗} ↦ s_{λ,μ} (since these generate A this will show that the map is onto). It suffices to verify that {s_{λ,s(λ)} : λ ∈ W } constitutes a ∗–representation of W. Conditions (i) and (iii) are easy to check, (iv) follows from [RS2, 0.1c,3.2] with u = v ∈ W^{0}. We check condition (ii): if s(λ) = r(μ) apply [RS2, 3.2]

s_{λ,s(λ)}s_{μ,s(μ)} = ∑_{W^{d(μ)}(s(λ))} s_{λν,ν}s_{μ,s(μ)} = s_{λμ,μ}s_{μ,s(μ)} = s_{λμ,s(λμ)}

where the sum simplifies using [RS2, 3.1, 3.3] . We shall show below that C^{∗}(W) ≅ A.

We may combine higher rank graphs using the following fact, whose proof is straightforward.

Proposition 1.8. Let (Λ_{1}, d_{1}) and (Λ_{2}, d_{2}) be rank k_{1}, k_{2} graphs respectively, then (Λ_{1} × Λ_{2}, d_{1} × d_{2}) is a rank k_{1} + k_{2} graph where Λ_{1} × Λ_{2} is the product category and d_{1}×d_{2} : Λ_{1}×Λ_{2} → N^{k_{1}+k_{2}} is given by d_{1}×d_{2}(λ_{1}, λ_{2}) = (d_{1}(λ_{1}), d_{2}(λ_{2})) ∈ N^{k_{1}}×N^{k_{2}} for λ_{1} ∈ Λ_{1} and λ_{2} ∈ Λ_{2}.

An example of this construction is discussed in [RS2, Remark 3.11]. It is clear that Ω_{k+ℓ} ≅ Ω_{k} × Ω_{ℓ} where k, ℓ > 0.

Definition 1.9. Let f : N^{ℓ} → N^{k} be a monoid morphism, then if (Λ, d) is a kgraph we may form the ℓ-graph f^{∗}(Λ) as follows: (the objects of f^{∗}(Λ) may be identified with those of Λ and) f^{∗}(Λ) = {(λ, n) : d(λ) = f(n)} with d(λ, n) = n, s(λ, n) = s(λ) and r(λ, n) = r(λ).

Examples 1.10. (i) Let Λ be a k–graph and put ℓ = 1, then if we define the morphism f_{i}(n) = ne_{i} for 1 ≤ i ≤ k, we obtain the coordinate graphs Λ_{i} := f_{i}^{∗}(Λ) of Λ (these are 1–graphs).

(ii) Suppose E is a directed graph and define f : N^{2} → N by (m_{1}, m_{2}) ↦ m_{1}+m_{2}; then the two coordinate graphs of f^{∗}(E^{∗}) are isomorphic to E^{∗}. We will show below that C^{∗}(f^{∗}(E^{∗})) ≅ C^{∗}(E^{∗}) ⊗ C(T).

(iii) Suppose E and F are directed graphs and define f : N → N^{2} by f(m) = (m, m) then f^{∗}(E^{∗} × F^{∗}) = (E × F)^{∗} where E × F denotes the cartesian product graph (see [KP, Def. 2.1]).

Proposition 1.11. Let Λ be a k-graph and f : N^{ℓ} → N^{k} a monoid morphism, then there is a ∗–homomorphism π_{f} : C^{∗}(f^{∗}(Λ)) → C^{∗}(Λ) such that s_{(λ,n)} ↦ s_{λ}; moreover if f is surjective, then π_{f} is too.

Proof. By 1.6(i) it suffices to show that this is a ∗–representation of f^{∗}(Λ). Properties (i)–(iii) are straightforward to verify and property (iv) follows by observing that for fixed n ∈ N^{ℓ} and v ∈ Λ^{0} the map f^{∗}(Λ)^{n}(v) → Λ^{f(n)}(v) given by (λ, n) ↦ λ is a bijection. If f is surjective, then it is clear that every generator s_{λ} of C^{∗}(Λ) is in the range of π_{f}. □

Later in 3.5 we will also show that π_{f} is injective if f is injective.

2. The path groupoid

In this section we construct the path groupoid G_{Λ} associated to a higher rank graph (Λ, d) along the lines of [KPRR, §2]. Because some of the details are not quite the same as those in [KPRR, §2] we feel it is useful to sketch the construction. First we introduce the following analog of an infinite path in a higher rank graph:

Definitions 2.1. Let Λ be a k-graph, then

Λ^{∞} = {x : Ω_{k} → Λ : x is a k-graph morphism},

is the infinite path space of Λ. For v ∈ Λ^{0} let Λ^{∞}(v) = {x ∈ Λ^{∞} : x(0) = v}. For each p ∈ N^{k} define σ^{p} : Λ^{∞} → Λ^{∞} by σ^{p}(x)(m, n) = x(m + p, n + p) for x ∈ Λ^{∞} and (m, n) ∈ Ω. (Note that σ^{p+q} = σ^{p} ∘ σ^{q}).

By our standing assumption (3) one can show that for every v ∈ Λ^{0} we have Λ^{∞}(v) ≠ ∅. Our definition of Λ^{∞} is related to the definition of W_{∞}, the space of infinite words, given in the proof of [RS2, Lemma 3.8]. If E^{∗} is the 1-graph associated to the directed graph E then (E^{∗})^{∞} may be identified with E^{∞}.

Remarks 2.2. By the factorisation property the values of x(0, m) for m ∈ N^{k} completely determine x ∈ Λ^{∞}. To see this, suppose that x(0, m) is given for all m ∈ N^{k} then for (m, n) ∈ Ω, x(m, n) is the unique element λ ∈ Λ such that x(0, n) = x(0, m)λ.

More generally, let {n_{j} : j ≥ 0} be an increasing cofinal sequence in N^{k} with n_{0} = 0 (for example one could take n_{j} = jp where p = (1, …, 1) ∈ N^{k}); then x ∈ Λ^{∞} is completely determined by the values of x(0, n_{j}). Moreover, given a sequence {λ_{j} : j ≥ 1} in Λ such that s(λ_{j}) = r(λ_{j+1}) and d(λ_{j}) = n_{j} − n_{j−1} there is a unique x ∈ Λ^{∞} such that x(n_{j−1}, n_{j}) = λ_{j}. For (m, n) ∈ Ω we define x(m, n) by the factorisation property as follows: let j be the smallest index such that n ≤ n_{j}. Then x(m, n) is the unique element of degree n−m such that λ_{1} ⋯ λ_{j} = μx(m, n)ν where d(μ) = m and d(ν) = n_{j} − n. It is straightforward to show that x has the desired properties.

We now establish a factorisation property for Λ^{∞} which is an easy consequence of the above remarks.

Proposition 2.3. Let Λ be a rank k graph. For all λ ∈ Λ and x ∈ Λ^{∞} with x(0) = s(λ), there is a unique y ∈ Λ^{∞} such that x = σ^{d(λ)}y and λ = y(0, d(λ)); we write y = λx. Note that for every x ∈ Λ^{∞} and p ∈ N^{k} we have x = x(0, p)σ^{p}x.

Proof. Fix λ ∈ Λ and x ∈ Λ^{∞} with x(0) = s(λ). The sequence {n_{j} : j ≥ 0} defined by n_{0} = 0 and n_{j} = (j − 1)p + d(λ) for j ≥ 1 is cofinal. Set λ_{1} = λ and λ_{j} = x((j − 2)p, (j − 1)p) for j ≥ 2 and let y ∈ Λ^{∞} be defined by the method given in 2.2. Then y has the desired properties. □

Next we construct a basis of compact open sets for the topology on Λ^{∞} indexed by Λ.

Definitions 2.4. Let Λ be a rank k graph. For λ ∈ Λ define

Z(λ) = {λx ∈ Λ^{∞} : s(λ) = x(0)} = {x : x(0, d(λ)) = λ}.

Remarks 2.5. Note that Z(v) = Λ^{∞}(v) for all v ∈ Λ^{0}. For fixed n ∈ N^{k} the sets {Z(λ) : d(λ) = n} form a partition of Λ^{∞} (see 1.6(v)); moreover for every λ ∈ Λ we have

Z(λ) = ⋃_{\begin{matrix}d(μ)=n \\ r(μ)=s(λ) \end{matrix}} Z(λμ). (4)

We endow Λ^{∞} with the topology generated by the collection {Z(λ) : λ ∈ Λ}. Note that the map given by λx ↦ x induces a homeomorphism between Z(λ) and Z(s(λ)) for all λ ∈ Λ. Hence, for every p ∈ N^{k} the map σ^{p} : Λ^{∞} → Λ^{∞} is a local homeomorphism.

Lemma 2.6. For each λ ∈ Λ, Z(λ) is compact.

Proof. By 2.5 it suffices to show that Z(v) is compact for all v ∈ Λ^{0}. Fix v ∈ Λ^{0} and let {x_{n}}_{n≥1} be a sequence in Z(v). For every m, x_{n}(0, m) may take only finitely many values (by (3)). Hence there is a λ ∈ Λ^{m} such that x_{n}(0, m) = λ for infinitely many n. We may therefore inductively construct a sequence {λ_{j} : j ≥ 1} in Λ^{p} such that s(λ_{j}) = r(λ_{j+1}) and x_{n}(0, jp) = λ_{1} ⋯ λ_{j} for infinitely many n (recall p = (1, …, 1) ∈ N^{k}). Choose a subsequence {x_{n_{j}}} such that x_{n_{j}}(0, jp) = λ_{1} ⋯ λ_{j}. Since {jp} is cofinal, there is a unique y ∈ Λ^{∞}(v) such that y((j − 1)p, jp) = λ_{j} for j ≥ 1; then x_{n_{j}} → y and hence Z(v) is compact. □

Note that Λ^{∞} is compact if and only if Λ^{0} is finite.

Definition 2.7. If Λ is k-graph then let

G_{Λ} = {(x, n, y) ∈ Λ^{∞} × Z^{k} × Λ^{∞} : σ^{ℓ}x = σ^{m}y, n = ℓ − m}.

Define range and source maps r, s : G_{Λ} → Λ^{∞} by r(x, n, y) = x, s(x, n, y) = y. For (x, n, y), (y, ℓ, z) ∈ G_{Λ} set (x, n, y)(y, ℓ, z) = (x, n+ℓ, z), and (x, n, y)^{−1} = (y, −n, x); G_{Λ} is called the path groupoid of Λ (cf. [R, D, KPRR]).

One may check that G_{Λ} is a groupoid with Λ^{∞} = G_{Λ}^{0} under the identification x ↦ (x, 0, x). For λ, μ ∈ Λ such that s(λ) = s(μ) define

Z(λ, μ) = {(λz, d(λ) − d(μ), μz) : z ∈ Λ^{∞}(s(λ))}.

We collect certain standard facts about G_{Λ} in the following result.

Proposition 2.8. Let Λ be a k–graph. The sets {Z(λ, μ) : λ, μ ∈ Λ, s(λ) = s(μ)} form a basis for a locally compact Hausdorff topology on G_{Λ}. With this topology G_{Λ} is a second countable, r–discrete locally compact groupoid in which each Z(λ, μ) is a compact open bisection. The topology on Λ^{∞} agrees with the relative topology under the identification of Λ^{∞} with the subset G_{Λ}^{0} of G_{Λ}.

Proof. One may check that the sets Z(λ, μ) form a basis for a topology on G_{Λ}. To see that multiplication is continuous, suppose that (x, n, y)(y, ℓ, z) = (x, n + ℓ, z) ∈ Z(γ, δ). Since (x, n, y), (y, ℓ, z) are composable in G_{Λ} there are κ, ν ∈ Λ and t ∈ Λ^{∞} such that x = γκt, y = νt and z = δκt. Hence (x, k, y) ∈ Z(γκ, ν) and (y, ℓ, z) ∈ Z(ν, δκ) and the product maps the open set G_{Λ}^{2} ∩ (Z(γκ, ν) × Z(ν, δκ)) into Z(γ, δ). The remaining parts of the proof are similar to those given in [KPRR, Proposition 2.6]. □

Note that Z(λ, μ) ≅ Z(s(λ)), via the map (λz, d(λ) − d(μ), μz) ↦ z. Again we note that in the case k = 1 we have Λ = E^{∗} for some directed graph E and the groupoid G_{E^{∗}} ≅ G_{E}, the graph groupoid of E which is described in detail in [KPRR, §2].

Proposition 2.9. Let Λ be a k-graph and let f : N^{ℓ} → N^{k} be a morphism. The map x ↦ f^{∗}(x) given by f^{∗}(x)(m, n) = (x(f(m), f(n)), n−m) defines a continuous surjective map f^{∗} : Λ^{∞} → f^{∗}(Λ)^{∞}. Moreover, if the image of f is cofinal (equivalently f(p) is strictly positive in the sense that all of its coordinates are nonzero) then f^{∗} is a homeomorphism.

Proof. Given x ∈ f^{∗}(Λ)^{∞} choose a sequence {m_{i}} such that n_{j} = ∑_{i=1}^{j} m_{i} is cofinal in N^{ℓ}. Set n_{0} = 0 and let λ_{j} ∈ Λ^{f(m_{j})} be defined by the condition that x(n_{j−1}, n_{j}) = (λ_{j}, m_{j}). We must show that there is an x′ ∈ Λ^{∞} such that x′(f(n_{j−1}), f(n_{j})) = λ_{j}. It suffices to show that the the intersection ∩_{j}Z(λ_{1} ⋯ λ_{j}) ≠ ∅. But this follows by the finite intersection property. One checks that x = f^{∗}(x′). Furthermore the inverse image of Z(λ, n) is Z(λ) and hence f^{∗} is continuous.

Now suppose that the image of f is cofinal, then the procedure defined above gives a continuous inverse for f^{∗}. Given x ∈ f^{∗}(Λ)^{∞}, then since f(n_{j}) is cofinal, the intersection ∩_{j}Z(λ_{1} ⋯ λ_{j}) contains a single point x′. Note that x′ depends on x continuously. □

For higher rank graphs of the form f^{∗}(Λ) with f surjective (see 1.9), the associated groupoid G_{f^{∗}(Λ)} decomposes as a direct product as follows.

Proposition 2.10. Let Λ be a k-graph and let f : N^{ℓ} → N^{k} be a surjective morphism. Then

G_{f^{∗}(Λ)} ≅ G_{Λ} × Z^{ℓ−k}.

Proof. Since f is surjective, the map f^{∗} : Λ^{∞} → f^{∗}(Λ)^{∞} is a homeomorphism (see 2.9). The map f extends to a surjective morphism f : Z^{ℓ} → Z^{k}. Let j : Z^{k} → Z^{ℓ} be a section for f and let i : Z^{ℓ−k} → Z^{ℓ} be an identification of Z^{ℓ−k} with ker f. Then we get a groupoid isomorphism by the map

((x, n, y), m) ↦ (f^{∗}x, i(m) + j(n), f^{∗}y),

where ((x, n, y), m) ∈ G_{Λ} × Z^{ℓ−k}. □

Finally, as in [RS2, Lemma 3.8] we demonstrate that there is a nontrivial ∗–representation of (Λ, d).

Proposition 2.11. Let (Λ, d) be a k-graph. Then there exists a representation {S_{λ} : λ ∈ Λ} of Λ on a Hilbert space with all partial isometries S_{λ} nonzero.

Proof. Let H = ℓ^{2}(Λ^{∞}), then for λ ∈ Λ define S_{λ} ∈ B(H) by

S_{λ}e_{y} = \{ \begin{matrix}e_{λy} & if s(λ) = y(0), \\ 0 & otherwise, \end{matrix}

where {e_{y} : y ∈ Λ^{∞}} is the canonical basis for H. Notice that S_{λ} is nonzero since Λ^{∞}(s(λ)) ≠ ∅; one then checks that the family {S_{λ} : λ ∈ Λ} satisfies conditions 1.5(i)–(iv). □

3. The gauge invariant uniqueness theorem

By the universal property of C^{∗}(Λ) there is a canonical action of the k-torus T^{k}, called the gauge action: α : T^{k} → Aut C^{∗}(Λ) defined for t = (t_{1}, …, t_{k}) ∈ T^{k} and s_{λ} ∈ C^{∗}(Λ) by

α_{t}(s_{λ}) = t^{d(λ)}s_{λ} (5)

where t^{m} = t_{1}^{m_{1}} ⋯ t_{k}^{m_{k}} for m = (m_{1}, …, m_{k}) ∈ N^{k}. It is straightforward to show that α is strongly continuous. As in [CK, Lemma 2.2] and [RS2, Lemma 3.6] we shall need the following.

Lemma 3.1. Let Λ be a k-graph. Then for λ, μ ∈ Λ and q ∈ N^{k} with d(λ), d(μ) ≤ q we have

s_{λ}^{∗}s_{μ} = ∑_{\begin{matrix}λα=μβ \\ d(λα)=q \end{matrix}} s_{α}s_{β}^{∗}. (6)

Hence every nonzero word in s_{λ}, s_{μ}^{∗} may be written as a finite sum of partial isometries of the form s_{α}s_{β}^{∗} where s(α) = s(β); their linear span then forms a dense ∗–subalgebra of C^{∗}(Λ).

Proof. Applying 1.5(iv) to s(λ) with n = q − d(λ), to s(μ) with n = q − d(μ) and using 1.5 (ii) we get

s_{λ}^{∗}s_{μ} = p_{s(λ)}s_{λ}^{∗}s_{μ}p_{s(μ)} = ( ∑_{Λ^{q−d(λ)}(s(λ))} s_{α}s_{α}^{∗}) s_{λ}^{∗}s_{μ} ( ∑_{Λ^{q−d(μ)}(s(μ))} s_{β}s_{β}^{∗})

(7)  = ( ∑_{Λ^{q−d(λ)}(s(λ))} s_{α}s_{λα}^{∗}) ( ∑_{Λ^{q−d(μ)}(s(μ))} s_{μβ}s_{β}^{∗}) .

By 1.6(iv) if d(λα) = d(μβ) but λα ≠ μβ, then the range projections p_{λα}, p_{μβ} are orthogonal and hence one has s_{λα}^{∗}s_{μβ} = 0. If λα = μβ then s_{λα}^{∗}s_{μβ} = p_{v} where v = s(α) and so s_{α}s_{λα}^{∗}s_{μβ}s_{β}^{∗} = s_{α}p_{v}s_{β}^{∗} = s_{α}s_{β}^{∗}; formula (6) then follows from formula (7). The rest of the proof is now routine. □

Following [RS2, §4]: for m ∈ N^{k} let F_{m} denote the C^{∗}–subalgebra of C^{∗}(Λ) generated by the elements s_{λ}s_{μ}^{∗} for λ, μ ∈ Λ^{m} where s(λ) = s(μ), and for v ∈ Λ^{0} denote F_{m}(v) the C^{∗}–subalgebra generated by s_{λ}s_{μ}^{∗} where s(λ) = v.

Lemma 3.2. For m ∈ N^{k}, v ∈ Λ^{0} there exist isomorphisms

F_{m}(v) ≅ K (ℓ^{2}({λ ∈ Λ^{m} : s(λ) = v}))

and F_{m} ≅ ⨁_{v∈Λ^{0}} F_{m}(v). Moreover, the C^{∗}–algebras F_{m}, m ∈ N^{k}, form a directed system under inclusion, and F_{Λ} = \overline{∪F_{m}} is an AF C^{∗}–algebra.

Proof. Fix v ∈ Λ^{0} and let λ, μ, α, β ∈ Λ^{m} be such that s(λ) = s(μ) and s(α) = s(β), then by 1.6(v) we have

(s_{λ}s_{μ}^{∗}) (s_{α}s_{β}^{∗}) = δ_{μ,α}s_{λ}s_{β}^{∗}, (8)

so that the map which sends s_{λ}s_{μ}^{∗} ∈ F_{m}(v) to the matrix unit

e_{λ,μ}^{v} ∈ K (ℓ^{2}({λ ∈ Λ^{m} : s(λ) = v}))

for all λ, μ ∈ Λ^{m} with s(λ) = s(μ) = v extends to an isomorphism. The second isomorphism also follows from (8) (since s(μ) ≠ s(α) implies μ ≠ α). We claim that F_{m} is contained in F_{n} whenever m ≤ n. To see this we apply 1.5(iv) to give

s_{λ}s_{μ}^{∗} = s_{λ}p_{s(λ)}s_{μ}^{∗} = ∑_{Λ^{ℓ}(s(λ))} s_{λ}s_{γ}s_{γ}^{∗}s_{μ}^{∗} = ∑_{Λ^{ℓ}(s(λ))} s_{λγ}s_{μγ}^{∗} (9)

where ℓ = n − m. Hence the C^{∗}–algebras F_{m}, m ∈ N^{k}, form a directed system as required. □

Note that F_{Λ} may also be expressed as the closure of ∪_{j=1}^{∞}F_{jp} where p = (1, …, 1) ∈ N^{k}.

Clearly for t ∈ T^{k} the gauge automorphism α_{t} defined in (5) fixes those elements s_{λ}s_{μ}^{∗} ∈ C^{∗}(Λ) with d(λ) = d(μ) (since α_{t}(s_{λ}s_{μ}^{∗}) = t^{d(λ)−d(μ)}s_{λ}s_{μ}^{∗}) and hence F_{Λ} is contained in the fixed point algebra C^{∗}(Λ)^{α}. Consider the linear map on C^{∗}(Λ) defined by

Φ(x) = ∫_{T^{k}} α_{t}(x) dt

where dt denotes normalised Haar measure on T^{k} and note that Φ(x) ∈ C^{∗}(Λ)^{α} for all x ∈ C^{∗}(Λ). As the proof of the following result is now standard, we omit it (see [CK, Proposition 2.11], [RS2, Lemma 3.3], [BPRS, Lemma 2.2]).

Lemma 3.3. Let Φ, F_{Λ} be as described above.

(i) The map Φ is a faithful conditional expectation from C^{∗}(Λ) onto C^{∗}(Λ)^{α}. (ii) F_{Λ} = C^{∗}(Λ)^{α}.

Hence the fixed point algebra C^{∗}(Λ)^{α} is an AF algebra. This fact is key to the proof of the gauge–invariant uniqueness theorem for C^{∗}(Λ) (see [BPRS, Theorem 2.1], [aHR, Theorem 2.3], see also [CK, RS2] where a similar technique is used in the proof of simplicity).

Theorem 3.4. Let B be a C^{∗}–algebra, π : C^{∗}(Λ) → B be a homomorphism and let β : T^{k} → Aut (B) be an action such that π ∘ α_{t} = β_{t} ∘ π for all t ∈ T^{k}. Then π is faithful if and only if π(p_{v}) ≠ 0 for all v ∈ Λ^{0}.

Proof. If π(p_{v}) = 0 for some v ∈ Λ^{0} then clearly π is not faithful. Conversely, suppose that π is equivariant and that π(p_{v}) ≠ 0 for all v ∈ Λ^{0}. We first show that π is faithful on C^{∗}(Λ)^{α} = \overline{⋃_{j≥0} F_{jp}}. For any ideal I in C^{∗}(Λ)^{α}, we have I = \overline{⋃_{j≥0}(I ∩ F_{jp})} (see [B, Lemma 3.1], [ALNR, Lemma 1.3]). Thus it is enough to prove that π is faithful on each F_{n}. But by 3.2 it suffices to show that it is faithful on F_{n}(v), for all v ∈ Λ^{0}. Fix v ∈ Λ^{0} and λ, μ ∈ Λ^{n} with s(λ) = s(μ) = v we need only show that π(s_{λ}s_{μ}^{∗}) ≠ 0. Since π(p_{v}) ≠ 0 we have

0 ≠ π(p_{v}^{2}) = π(s_{λ}^{∗}s_{λ}s_{μ}^{∗}s_{μ}) = π(s_{λ}^{∗})π(s_{λ}s_{μ}^{∗})π(s_{μ}).

Hence π(s_{λ}s_{μ}^{∗}) ≠ 0 and π is faithful on C^{∗}(Λ)^{α}. Let a ∈ C^{∗}(Λ) be a nonzero positive element; then since Φ is faithful Φ(a) ≠ 0 and as π is faithful on C^{∗}(Λ^{α}) we have

0 ≠ π(Φ(a)) = π (∫_{T^{k}} α_{t}(a) dt) = ∫_{T^{k}} β_{t}(π(a)) dt,

hence π(a) ≠ 0 and π is faithful on C^{∗}(Λ) as required. □

Corollary 3.5.

(i) Let (Λ, d) be a k-graph and let G_{Λ} be its associated groupoid. Then there is an isomorphism C^{∗}(Λ) ≅ C^{∗}(G_{Λ}) such that s_{λ} ↦ 1_{Z(λ,s(λ))} for λ ∈ Λ. Moreover, the canonical map C^{∗}(G_{Λ}) → C_{r}^{∗}(G_{Λ}) is an isomorphism.

(ii) Let {M_{1}, …, M_{k}} be a collection of matrices satisfying (H0)–(H3) of [RS2] and W the k-graph defined in 1.7(iv). Then C^{∗}(W) ≅ A, via the map s_{λ} ↦ s_{λ,s(λ)} for λ ∈ W.

(iii) If Λ is a k-graph and f : N^{ℓ} → N^{k} is injective, then the ∗-homomorphism π_{f} : C^{∗}(f^{∗}(Λ)) → C^{∗}(Λ) (see 1.11) is injective. In particular the C^{∗}–algebras of the coordinate graphs Λ_{i} for 1 ≤ i ≤ k form a generating family of subalgebras of C^{∗}(Λ). Moreover, if f is surjective then C^{∗}(f^{∗}(Λ)) ≅ C^{∗}(Λ) ⊗ C(T^{ℓ−k}).

(iv) Let (Λ_{i}, d_{i}) be k_{i}-graphs for i = 1, 2, then C^{∗}(Λ_{1} × Λ_{2}) ≅ C^{∗}(Λ_{1}) ⊗ C^{∗}(Λ_{2}) via the map s_{(λ_{1},λ_{2})} ↦ s_{λ_{1}} ⊗ s_{λ_{2}} for (λ_{1}, λ_{2}) ∈ Λ_{1} × Λ_{2}.

Proof. For (i) we note that s_{λ} ↦ 1_{Z(λ,s(λ))} for λ ∈ Λ is a ∗-representation of Λ; hence there is a ∗-homomorphism π : C^{∗}(Λ) → C^{∗}(G_{Λ}) such that π(s_{λ}) = 1_{Z(λ,s(λ))} for λ ∈ Λ (see 1.6(i)). Let β denote the T^{k}-action on C^{∗}(G_{Λ}) induced by the Z^{k}- valued 1–cocycle defined on G_{Λ} by (x, k, y) ↦ k (see [R, II.5.1]); one checks that π∘α_{t} = β_{t}∘π for all t ∈ T^{k}. Clearly for v ∈ Λ^{0} we have 1_{Z(v,v)} ≠ 0, since Λ^{∞}(v) ≠ ∅ and π is injective. Surjectivity follows from the fact that π(s_{λ}s_{μ}^{∗}) = 1_{Z(λ,μ)} together with the observation that C^{∗}(G_{Λ}) = \overline{span}\{1_{Z(λ,μ)}\}. The same argument shows that C_{r}^{∗}(G_{Λ}) ≅ C^{∗}(Λ) and so C_{r}^{∗}(G_{Λ}) ≅ C^{∗}(G_{Λ})^{1} .

For (ii) we note that there is a surjective ∗-homomorphism π : C^{∗}(W) → A such that π(s_{λ}) = s_{λ,s(λ)} for λ ∈ W (see 1.7(iv)) which is clearly equivariant for the respective T^{k}–actions. Moreover by [RS2, Lemma 2.9] we have s_{v,v} ≠ 0 for all v ∈ W_{0} = A and so the result follows.

For (iii) note that the injection f : N^{ℓ} → N^{k} extends naturally to a homomorphism f : Z^{ℓ} → Z^{k} which in turn induces a map f̂ : T^{k} → T^{ℓ} characterised by f̂(t)^{p} = t^{f(p)} for p ∈ N^{ℓ}. Let B be the fixed point algebra of the gauge action of T^{k} on C^{∗}(Λ) restricted to the kernel of f̂. The gauge action restricted to B descends to an action of T^{ℓ} = T^{k}/Ker f̂ on B which we denote \overline{α}. Observe that for t ∈ T^{k} and (λ, n) ∈ f^{∗}(Λ) we have

α_{t}(π_{f}(s_{(λ,n)})) = t^{f(n)}s_{λ} = f̂(t)^{n}s_{λ};

hence Im π_{f} ⊆ B (if t ∈ Ker f̂, then f̂(t)^{n} = 1). By the same formula we see that π_{f} ∘ α = \overline{α} ∘ π_{f} and the result now follows by 3.4. The last assertion follows from part (i) together with the fact that G_{f^{∗}(Λ)} ≅ G_{Λ} × Z^{ℓ−k} (see 2.10).

For (iv), define a map π : C^{∗}(Λ_{1} × Λ_{2}) → C^{∗}(Λ_{1}) ⊗ C^{∗}(Λ_{2}) given by s_{(λ_{1},λ_{2})} ↦ s_{λ_{1}} ⊗ s_{λ_{2}}; this is surjective as these elements generate C^{∗}(Λ_{1}) ⊗ C^{∗}(Λ_{2}). We note that C^{∗}(Λ_{1}) ⊗ C^{∗}(Λ_{2}) carries a T^{k_{1}+k_{2}} action β defined for (t_{1}, t_{2}) ∈ T^{k_{1}+k_{2}} and (λ_{0}, λ_{1}) ∈ Λ_{1} × Λ_{2} by β_{(t_{1},t_{2})}(s_{λ_{1}} ⊗ s_{λ_{2}}) = α_{t_{1}}s_{λ_{1}} ⊗ α_{t_{2}}s_{λ_{2}}. Injectivity then follows by 3.4, since π is equivariant and for (v, w) ∈ (Λ_{1} × Λ_{2})^{0} we have p_{v} ⊗ p_{w} ≠ 0. □

Henceforth we shall tacitly identify C^{∗}(Λ) with C^{∗}(G_{Λ}).

Remark 3.6. Let Λ be a k-graph and suppose that f : N^{ℓ} → N^{k} is an injective morphism for which H, the image of f, is cofinal. Then π_{f} induces an isomorphism of C^{∗}(f^{∗}(Λ)) with its range, the fixed point algebra of the restriction of the gauge action to H^{⊥}.

4. Aperiodicity and its consequences

The aperiodicity condition we study in this section is an analog of condition (L) used in [KPR]. We first define what it means for an infinite path to be periodic or aperiodic.

1This can be also deduced from the amenability of G_{Λ} (see 5.5).

Definitions 4.1. For x ∈ Λ^{∞} and p ∈ Z^{k} we say that p is a period of x if for every (m, n) ∈ Ω with m + p ≥ 0 we have x(m + p, n + p) = x(m, n). We say that x is periodic if it has a nonzero period. We say that x is eventually periodic if σ^{n}x is periodic for some n ∈ N^{k}, otherwise x is said to be aperiodic.

Remarks 4.2. For x ∈ Λ^{∞} and p ∈ Z^{k}, p is a period of x if and only if σ^{m}x = σ^{n}x for all m, n ∈ N^{k} such that p = m − n. Similarly x is eventually periodic, with eventual period p ≠ 0 if and only if σ^{m}x = σ^{n}x for some m, n ∈ N^{k} such that p = m − n.

Definition 4.3. The k-graph Λ is said to satisfy the aperiodicity condition (A) if for every v ∈ Λ^{0} there is an aperiodic path x ∈ Λ^{∞}(v).

Remark 4.4. Let E be a directed graph which is row finite and has no sinks. Then the associated 1-graph E^{∗} satisfies the aperiodicity condition if and only if every loop in E has an exit (i.e., satisfies condition (L) of [KPR]). However, if we consider the 2-graph f^{∗}(E^{∗}) where f : N^{2} → N is given by f(m_{1}, m_{2}) = m_{1} + m_{2} then p = (1, −1) is a period for every point in f^{∗}(E^{∗})^{∞} (even if E has no loops).

Proposition 4.5. The groupoid G_{Λ} is essentially free (i.e., the points with trivial isotropy are dense in G_{Λ}^{0}) if and only if Λ satisfies the aperiodicity condition.

Proof. Observe that if x ∈ Λ^{∞} is aperiodic then σ^{m}x = σ^{n}x implies that m = n and hence x ∈ Λ^{∞} = G_{Λ}^{0} has trivial isotropy, and conversely. Hence G_{Λ} is essentially free if and only if aperiodic points are dense in Λ^{∞}. If aperiodic points are dense in Λ^{∞} then Λ clearly satisfies the aperiodicity condition, for Z(v) = Λ^{∞}(v) must then contain aperiodic points for every v ∈ Λ^{0}. Conversely, suppose that Λ satisfies the aperiodicity condition, then for every λ ∈ Λ there is x ∈ Λ^{∞}(s(λ)) which is aperiodic. Then λx ∈ Z(λ) is aperiodic. Hence the aperiodic points are dense in Λ^{∞}. □

The isotropy group of an element x ∈ Λ^{∞} is equal to the subgroup of its eventual periods (including 0).

Theorem 4.6. Let π : C^{∗}(Λ) → B be a ∗–homomorphism and suppose that Λ satisfies the aperiodicity condition. Then π is faithful if and only if π(p_{v}) ≠ 0 for all v ∈ Λ^{0}.

Proof. If π(p_{v}) = 0 for some v ∈ Λ^{0} then clearly π is not faithful. Conversely, suppose π(p_{v}) ≠ 0 for all v ∈ Λ^{0}; then by 3.5(i) we have C^{∗}(Λ) = C_{r}^{∗}(G_{Λ}) and hence from [KPR, Corollary 3.6] it suffices to show that π is faithful on C_{0}(G_{Λ}^{0}). If the kernel of the restriction of π to C_{0}(G_{Λ}^{0}) is nonzero, it must contain the characteristic function 1_{Z(λ)} for some λ ∈ Λ. It follows that π(s_{λ}s_{λ}^{∗}) = 0 and hence π(s_{λ}) = 0; in which case π(p_{s(λ)}) = π(s_{λ}^{∗}s_{λ}) = 0, a contradiction. □

Definition 4.7. We say that Λ is cofinal if for every x ∈ Λ^{∞} and v ∈ Λ^{0} there is λ ∈ Λ and n ∈ N^{k} such that s(λ) = x(n) and r(λ) = v.

Proposition 4.8. Suppose Λ satisfies the aperiodicity condition, then C^{∗}(Λ) is simple if and only if Λ is cofinal.

Proof. By 3.5(i) C^{∗}(Λ) = C_{r}^{∗}(G_{Λ}); since G_{Λ} is essentially free, C^{∗}(Λ) is simple if and only if G_{Λ} is minimal. Suppose that Λ is cofinal and fix x ∈ Λ^{∞} and λ ∈ Λ; then by cofinality there is a μ ∈ Λ and n ∈ N^{k} so that s(μ) = x(n) and r(μ) = s(λ). Then y = λμσ^{n}x ∈ Z(λ) and y is in the same orbit as x; hence all orbits are dense and G_{Λ} is minimal.

Conversely, suppose that G_{Λ} is minimal and that x ∈ Λ^{∞} and v ∈ Λ^{0}. Then there is y ∈ Z(v) such that x, y are in the same orbit. Hence there exist m, n ∈ N^{k} such that σ^{n}x = σ^{m}y; then it is easy to check that λ = y(0, m) and n have the desired properties. □

Notice that second hypothesis used in the following corollary is the analog of the condition that every vertex connects to a loop and it is equivalent to requiring that for every v ∈ Λ^{0}, there is an eventually periodic x ∈ Λ^{∞}(v) with positive eventual period (i.e., the eventual period lies in N^{k}\{0}). The proof follows the same lines as [KPR, Theorem 3.9].

Proposition 4.9. Let Λ satisfy the aperiodicity condition. Suppose that for every v ∈ Λ^{0} there are λ, μ ∈ Λ with d(μ) ≠ 0 such that r(λ) = v and s(λ) = r(μ) = s(μ). Then C^{∗}(Λ) is purely infinite in the sense that every hereditary subalgebra contains an infinite projection.

Proof. Arguing as in [KPR, Lemma 3.8] one shows that G_{Λ} is locally contracting. The aperiodicity condition guarantees that G_{Λ} is essentially free, hence by [A-D, Proposition 2.4] (see also [LS]) we have C^{∗}(Λ) = C_{r}^{∗}(G_{Λ}) is purely infinite. □

5. Skew products and group actions

Let G be a discrete group, Λ a k-graph and c : Λ → G a functor. We introduce an analog of the skew product graph considered in [KP, §2] (see also [GT]); the resulting object, which we denote G ×_{c} Λ, is also a k-graph. As in [KP] if G is abelian the associated C^{∗}–algebra is isomorphic to a crossed product of C^{∗}(Λ) by the natural action of Ĝ induced by c (more generally it is a crossed product by a coaction — see [Ma, KQR]). As a corollary we show that the crossed product of C^{∗}(Λ) by the gauge action, C^{∗}(Λ) ⋊_{α} T^{k}, is isomorphic to C^{∗}(Z^{k} ×_{d} Λ), the C^{∗}–algebra of the skew-product k-graph arising from the degree map. It will then follow that C^{∗}(Λ) ⋊_{α} T^{k} is AF and that G_{Λ} is amenable.

Definition 5.1. Let G be a discrete group, (Λ, d) a k-graph. Given c : Λ → G a functor then define the skew product G ×_{c} Λ as follows: the objects are identified with G×Λ^{0} and the morphisms are identified with G×Λ with the following structure maps

s(g, λ) = (gc(λ), s(λ)) and r(g, λ) = (g, r(λ)).

If s(λ) = r(μ) then (g, λ) and (gc(λ), μ) are composable in G ×_{c} Λ and

(g, λ)(gc(λ), μ) = (g, λμ).

The degree map is given by d(g, λ) = d(λ).

One must check that G ×_{c} Λ is a k-graph. If k = 1 then any function c : E^{1} → G extends to a unique functor c : E^{∗} → G (as in [KP, §2]). The skew product graph E(c) of [KP] is related to our skew product in a simple way: G ×_{c} E^{∗} = E(c)^{∗}. A key example of this construction arises by regarding the degree map d as a functor with values in Z^{k}.

The functor c induces a cocycle c̃: G_{Λ} → G as follows: given (x, ℓ − m, y) ∈ G_{Λ} so that σ^{ℓ}x = σ^{m}y then set

c̃(x, ℓ − m, y) = c(x(0, ℓ))c(y(0, m))^{−1}.

As in [KP] one checks that this is well-defined and that c̃ is a (continuous) cocycle; regarding the degree map d as a functor with values in Z^{k}, we have d̃(x, n, y) = n for (x, n, y) ∈ G_{Λ}. In the following we show that the skew product groupoid obtained from c̃ (as defined in [R]) is the same as the path groupoid of the skew product (cf. [KP, Theorem 2.4]).

Theorem 5.2. Let G be a discrete group, Λ a k-graph and c : Λ → G a functor. Then G_{G×_{c}Λ} ≅ G_{Λ}(c̃) where c̃: G_{Λ} → G is defined as above.

Proof. We first identify G × Λ^{∞} with (G ×_{c} Λ)^{∞} as follows: for (g, x) ∈ G × Λ^{∞} define (g, x) : Ω → G ×_{c} Λ by

(g, x)(m, n) = (gc(x(0, m)), x(m, n));

it is straightforward to check that this defines a degree–preserving functor and thus an element of (G×_{c} Λ)^{∞}. Under this identification σ^{n}(g, x) = (gc(x(0, n)), σ^{n}x) for all n ∈ N^{k}, (g, x) ∈ (G ×_{c} Λ)^{∞}. As in the proof of [KP, Theorem 2.4] define a map ϕ : G_{Λ}(c̃) → G_{G×_{c}Λ} as follows: for x, y ∈ Λ^{∞} with σ^{ℓ}x = σ^{m}y set ϕ([x, ℓ−m, y], g) = (x′, ℓ − m, y′) where x′ = (g, x) and y′ = (gc̃(x, ℓ − m, y), y). Note that

σ^{m}y′ = σ^{m}(gc̃(x, ℓ − m, y), y) = σ^{m}(gc(x(0, ℓ))c(y(0, m))^{−1}, y)
= (gc(x(0, ℓ)), σ^{m}y) = (gc(x(0, ℓ)), σ^{ℓ}x) = σ^{ℓ}(g, x)
= σ^{ℓ}x′,

and hence (x′, ℓ−m, y′) ∈ G_{G×_{c}Λ}. The rest of the proof proceeds as in [KP, Theorem 2.4] mutatis mutandis. □

Corollary 5.3. Let G be a discrete abelian group, Λ a k-graph and c : Λ → G a functor. There is an action α^{c} : Ĝ → Aut C^{∗}(Λ) such that for χ ∈ Ĝ and λ ∈ Λ

α_{χ}^{c}(s_{λ}) = ⟨χ, c(λ)⟩s_{λ}.

Moreover C^{∗}(Λ)⋊_{α^{c}} Ĝ ≅ C^{∗}(G×_{c} Λ). In particular the gauge action is of the form, α = α^{d}, and so C^{∗}(Λ) ⋊_{α} T^{k} ≅ C^{∗}(Z^{k} ×_{d} Λ).

Proof. Since C^{∗}(Λ) is defined to be the universal C^{∗}–algebra generated by the s_{λ}’s subject to the relations (1.5) and α^{c} preserves these relations it is clear that it defines an action of Ĝ on C^{∗}(Λ). The rest of the proof follows in the same manner as that of [KP, Corollary 2.5] (see [R, II.5.7]). □

In order to show that C^{∗}(Λ) ⋊_{α} T^{k} is AF, we need the following lemma.

Lemma 5.4. Let Λ be a k-graph and suppose there is a map b : Λ^{0} → Z^{k} such that d(λ) = b(s(λ)) − b(r(λ)) for all λ ∈ Λ, then C^{∗}(Λ) is AF.

Proof. For every n ∈ Z^{k} let A_{n} be the closed linear span of elements of the form s_{λ}s_{μ}^{∗} with b(s(λ)) = n. Fix λ, μ ∈ Λ with b(s(λ)) = b(s(μ)) = n. We claim that s_{λ}^{∗}s_{μ} = 0 if λ ≠ μ. If s_{λ}^{∗}s_{μ} ≠ 0 then by 3.1 there are α, β ∈ Λ with s(λ) = r(α) and s(μ) = r(β) such that λα = μβ; but then we have

d(α) + n = d(α) + b(s(λ)) = b(s(λα)) = b(s(μβ)) = d(β) + b(s(μ)) = d(β) + n.

Thus d(α) = d(β) and hence by the factorisation property α = β. Consequently λ = μ by cancellation and the claim is established. It follows that for each v with b(v) = n the elements s_{λ}s_{μ}^{∗} with s(λ) = s(μ) = v form a system of matrix units and two systems associated to distinct v’s are orthogonal (see 3.2). Hence we have

A_{n} ≅ ⨁_{b(v)=n} K (ℓ^{2}(s^{−1}(v)) .

By an argument similar to that in the proof of Lemma 3.2, if n ≤ m then A_{n} ⊆ A_{m} (see equation (9)); our conclusion now follows. □

Note that A_{n} in the above proof is the C^{∗}–algebra of a subgroupoid of G_{Λ} which is isomorphic to the disjoint union

⨆_{b(v)=n} R_{v} × Λ^{∞}(v)

where R_{v} is the transitive principal groupoid on s^{−1}(v). Since G_{Λ} is the increasing union of these elementary groupoids, it is an AF-groupoid and hence amenable (see [R, III.1.1]). The existence of such a function b : Λ^{0} → Z^{k} is not necessary for C^{∗}(Λ) to be AF since there are 1–graphs with no loops which do not have this property (see [KPR, Theorem 2.4]).

Theorem 5.5. Let Λ be a k-graph, then C^{∗}(Λ)⋊_{α} T^{k} is AF and the groupoid G_{Λ} is amenable. Moreover, C^{∗}(Λ) falls in the bootstrap class N of [RSc] and is therefore nuclear. Hence, if C^{∗}(Λ) is simple and purely infinite (see Proposition 4.9), then it may be classified by its K-theory.

Proof. Observe that the map b : (Z^{k} ×_{d} Λ)^{0} → Z^{k} given by b(n, v) = n satisfies

b(s(n, λ)) − b(r(n, λ)) = b(n + d(λ), λ) − b(n, r(λ)) = n + d(λ) − n = d(n, λ)

The first part of the result then follows from 5.4 and 5.3. To show that G_{Λ} is amenable we first observe that G_{Λ}(d̃) ≅ G_{Z^{k}×_{d}Λ} is amenable. Since Z^{k} is amenable, we may apply [R, Proposition II.3.8] to deduce that G_{Λ} is amenable. Since C^{∗}(Λ) is strongly Morita equivalent to the crossed product of an AF algebra by a Z^{k}– action, it falls in the bootstrap class N of [RSc]. The final assertion follows from the Kirchberg-Phillips classification theorem (see [K, P]). □

We now consider free actions of groups on k-graphs (cf. [KP, §3]). Let Λ be a kgraph and G a countable group, then G acts on Λ if there is a group homomorphism G → Aut Λ (automorphisms are compatible with all structure maps, including the degree): write (g, λ) ↦ gλ. The action of G on Λ is said to be free if it is free on Λ^{0}. By the universality of C^{∗}(Λ) an action of G on Λ induces an action β on C^{∗}(Λ) such that β_{g}s_{λ} = s_{gλ}.

Given a free action of a group G on a k-graph Λ one forms the quotient Λ/G by the equivalence relation λ ∼ μ if λ = gν for some g ∈ G. One checks that all structure maps are compatible with ∼ and so Λ/G is also a k-graph.

Remark 5.6. Let G be a countable group and c : Λ → G a functor, then G acts freely on G ×_{c} Λ by g(h, λ) = (gh, λ); furthermore (G ×_{c} Λ)/G ≅ Λ.

Suppose now that G acts freely on Λ with quotient Λ/G; we claim that Λ is isomorphic, in an equivariant way, to a skew product of Λ/G for some suitably chosen c (see [GT, Theorem 2.2.2]). Let q denote the quotient map. For every v ∈ (Λ/G)^{0} choose v′ ∈ Λ^{0} with q(v′) = v and for every λ ∈ Λ/G let λ′ denote the unique element in Λ such that q(λ′) = λ and r(λ′) = r(λ)′. Now let c : Λ/G → G be defined by the formula

s(λ′) = c(λ)s(λ)′.

We claim that c(λμ) = c(λ)c(μ) for all λ, μ ∈ Λ with s(λ) = r(μ). Note that

r(c(λ)μ′) = c(λ)r(μ′) = c(λ)r(μ)′ = c(λ)s(λ)′ = s(λ′);

hence, we have (λμ)′ = λ′(c(λ)μ′) (since the image of both sides agree under q and r). Thus

c(λμ)s(μ)′ = c(λμ)s(λμ)′ = s[(λμ)′] = s(c(λ)μ′) = c(λ)s(μ′) = c(λ)c(μ)s(μ)′

which establishes the desired identity (since G acts freely on Λ). The map (g, λ) ↦ gλ′ defines an equivariant isomorphism between G ×_{c} (Λ/G) and Λ as required.

The following is a generalization of [KPR, 3.9, 3.10] and is proved similarly.

Theorem 5.7. Let Λ be a k-graph and suppose that the countable group G acts freely on Λ, then

C^{∗}(Λ) ⋊_{β} G ≅ C^{∗}(Λ/G) ⊗ K (ℓ^{2}(G)) .

Equivalently, if c : Λ′ → G is a functor, then

C^{∗}(G ×_{c} Λ′) ⋊_{β} G ≅ C^{∗}(Λ′) ⊗ K (ℓ^{2}(G))

where β, the action of G on C^{∗}(G×_{c}Λ′), is induced by the natural action on G×_{c}Λ′. If G is abelian this action is dual to α^{c} under the identification of 5.3.

Proof. The first statement follows from the second with Λ′ = Λ/G; indeed, by 5.6 there is a functor c : Λ/G → G such that Λ ≅ G ×_{c} (Λ/G) in an equivariant way. The second statement follows from applying [KP, Proposition 3.7] to the natural G-action on G_{G×_{c}Λ′} ≅ G_{Λ′}(c̃). The final statement follows from the identifications

C^{∗}(Λ) ⋊_{α^{c}} Ĝ ≅ C^{∗}(G ×_{c} Λ) ≅ C^{∗}(G_{Λ}(c̃))

and [R, II.2.7]. □

6. 2-graphs

Given a k-graph Λ one obtains for each n ∈ N^{k} a matrix

M_{Λ}^{n}(u, v) = #{λ ∈ Λ^{n} : r(λ) = u, s(λ) = v}.

By our standing assumption the entries are all finite and there are no zero rows. Note that for any m, n ∈ N^{k} we have M_{Λ}^{m+n} = M_{Λ}^{m}M_{Λ}^{n} (by the factorisation property); consequently, the matrices M_{Λ}^{m} and M_{Λ}^{n} commute for all m, n ∈ N^{k}. If W is the k-graph associated to the commuting matrices {M_{1}, …, M_{k}} satisfying conditions (H0)–(H3) of [RS2] which was considered in Example 1.7(iv), then one checks that M_{W}^{e_{i}} = M_{i}^{t}. Further, if Λ = E^{∗} is a 1-graph derived from the directed graph E, then M_{Λ}^{1} is the vertex matrix of E.

Now suppose that A and B are 1-graphs with A^{0} = B^{0} = V such the associated vertex matrices commute. Set A^{1} ∗ B^{1} = {(α, β) ∈ A^{1} × B^{1} : s(α) = r(β)} and B^{1} ∗ A^{1} = {(β, α) ∈ B^{1} × A^{1} : s(β) = r(α)}; since the associated vertex matrices commute there is a bijection θ : (α, β) ↦ (β′, α′) from A^{1} ∗ B^{1} to B^{1} ∗ A^{1} such that r(α) = r(β′) and s(β) = s(α′). We construct a 2-graph Λ from A, B and θ. This construction is very much in the spirit of [RS2]; roughly speaking an element in Λ of degree (m, n) ∈ N^{2} will consist of a rectangular grid of size (m, n) with edges of A horizontally, edges of B vertically and nodes in V arranged compatibly. First identify Λ^{0} = V. For (m, n) ∈ N^{2} set W (m, n) = {(i, j) ∈ N^{2} : (i, j) ≤ (m, n)}. An element in Λ^{(m,n)} is given by v(i, j) ∈ V for (i, j) ∈ W (m, n), α(i, j) ∈ A^{1} for (i, j) ∈ W (m − 1, n) and β(i, j) ∈ B^{1} for (i, j) ∈ W (m, n − 1) (set W (m, n) = ∅ if m or n is negative) satisfying the following compatibility conditions wherever they make sense:

i. r(α(i, j)) = v(i, j) and r(β(i, j)) = v(i, j)

ii. s(α(i, j)) = v(i + 1, j) and s(β(i, j)) = v(i, j + 1)

iii. θ(α(i, j), β(i + 1, j)) = (β(i, j), α(i, j + 1)); for brevity and with a slight abuse of notation we regard this element as a triple (v, α, β) (note that α disappears if m = 0 and β disappears if n = 0 and v is determined by α and/or β if mn ≠ 0). Set

Λ = ⋃_{(m,n)} Λ^{(m,n)}

and define s(v, α, β) = v(m, n) and r(v, α, β) = v(0, 0).

Note that if λ ∈ A^{m} and μ ∈ B^{n} with m, n > 0 such that s(λ) = r(μ) there is a unique element (v, α, β) ∈ Λ^{(m,n)} such that λ = α(0, 0)α(1, 0) ⋯ α(m − 1, 0) and μ = β(m, 0)β(m, 1) ⋯ β(m, n − 1); denote this element λμ. Further if λ ∈ A^{m} and μ ∈ B^{n} with m, n > 0 such that r(λ) = s(μ) there is a unique element (v, α, β) in Λ^{(m,n)} such that λ = α(0, n)α(1, n) ⋯ α(m−1, n) and μ = β(0, 0)β(0, 1) ⋯ β(0, n− 1); denote this element μλ. Using these two facts it is not difficult to verify that given elements (v, α, β) ∈ Λ^{(m,n)} and (v′, α′, β′) ∈ Λ^{(m′,n′)} with v(m, n) = v′(0, 0) there is a unique element (v′′, α′′, β′′) ∈ Λ^{(m+m′,n+n′)} such that v′′(i, j) = v(i, j), α′′(i, j) = α(i, j), β′′(i, j) = β(i, j), v′′(m + i, n + j) = v′(i, j), α′′(m + i, n + j) = α′(i, j) and β′′(m + i, n + j) = β′(i, j) wherever these formulas make sense. Write (v′′, α′′, β′′) = (v, α, β)(v′, α′, β′). This defines composition in Λ; note that associativity and the factorisation property are built into the construction (as in [RS2]). Finally, we write Λ = A ∗_{θ} B. It is straightforward to verify that up to isomorphism any 2-graph may be obtained from its constituent 1-graphs in this way.

If A = B, then we may take θ = ι the identity map. In that case one has A ∗_{ι} A ≅ f^{∗}(A) where f : N^{2} → N is given by f(m, n) = m + n. Hence, by Corollary 3.5(iii) we have C^{∗}(A ∗_{ι} A) ≅ C^{∗}(A) ⊗ C(T).

To further emphasise the dependence of the product A ∗_{θ} B on the bijection θ : A^{1} ∗ B^{1} → B^{1} ∗ A^{1} consider the following example.

Example 6.1. Let A = B be the 1-graph derived from the directed graph which consists of one vertex and two edges, say A^{1} = {e, f} (note C^{∗}(A) ≅ O_{2}). Then A^{1} ∗ A^{1} = {(e, e), (e, f), (f, e), (f, f)}, and we define the bijection θ to be the flip. It is easy to show that A ∗_{θ} A ≅ A × A; hence,

C^{∗}(A ∗_{θ} A) ≅ O_{2} ⊗ O_{2} ≅ O_{2}

where the first isomorphism follows from Corollary 3.5(iv) and the second from the Kirchberg-Phillips classification theorem (see [K, P]). But

C^{∗}(A ∗_{ι} A) ≅ O_{2} ⊗ C(T);

hence, A ∗_{θ} A ≅ A̸ ∗_{ι} A.

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20 Alex Kumjian and David Pask

Department of Mathematics (084), University of Nevada, Reno NV 89557-0045, USA. alex@unr.edu http://equinox.comnet.unr.edu/homepage/alex/

Department of Mathematics, University of Newcastle, NSW 2308, Australia davidp@maths.newcastle.edu.au http://maths.newcastle.edu.au/˜ davidp/

This paper is available via http://nyjm.albany.edu:8000/j/2000/6-1.html.