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Higher-rank graph C*-algebras: an inverse semigroup and groupoid approach

We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.


Introduction
A higher-rank graph is a countable category Λ endowed with a degree functor d : Λ → N k satisfying the unique factorization property: For all λ ∈ Λ and m, n ∈ N k with d(λ) = m + n, there are unique elements µ, ν ∈ Λ such that d(µ) = m, d(ν) = n and λ = µν. The rank of Λ is k and for this reason, (Λ, d) is also called a k-graph. A 1-graph is simply the finite-path category generated freely by an ordinary directed graph. 1 In [2], Kumjian and Pask introduced the notion of a higher-rank graph in order to capture the essential features of the C * -algebras that Robertson and Steger associated to buildings [14,15,16,17] and to provide links between these and higher order shift dynamical systems. (See [3] also.) The C * -algebras associated to higher-rank graphs are generalizations of ordinary graph C * -algebras in that they are generated by families of partial isometries {s λ } λ∈Λ that satisfy certain relations that have received a lot of attention in recent years. Kumjian and Pask defined and studied the C * -algebra of (Λ, d), C * (Λ), in terms of a certain type of groupoid that encodes the graph. They were motivated by, and generalized, the theory in [5] which has been the source of considerable inspiration in our subject. However, just as in the setting of ordinary graphs, where the groupoid techniques of [5] require hypotheses that rule out many interesting examples, the work of Kumjian and Pask requires hypotheses that place significant limitations on the nature of the k-graphs that may be analyzed. Our first objective in this paper, then, is to overcome the limitations that Kumjian and Pask place on their k-graphs and to show how to build a groupoid that gives the C * -algebra of an arbitrary k-graph subject only to the condition that it is "finitely aligned" (see Definition 3.4). This condition seems to lie at the natural "boundary" of the subject. That is, with or without the use of groupoids, little can be said about k-graphs that are not finitely aligned.
The research of the first two authors was supported in part by a grant from the National Science Foundation, DMS-0070405. 1 For the purpose of motivation, we discuss the theory of 1-graphs at some length in the next section. For a very readable account of graph C * -algebras, including the rudiments of the C *algebras of k-graphs, we recommend the CBMS lectures by Iain Raeburn [9].
We were motivated in part by the important contribution of Paterson [8] in which he successfully circumvented the limitations of [5] that involve finiteness hypotheses on the graphs under consideration by first introducing an inverse semigroup that is naturally attached to the graph. Once this inverse semigroup is identified, he constructed a groupoid from it using technology that he and others have developed and which he exposed thoroughly in [7]. Paterson's success inspired our approach here. Given a k-graph (Λ, d), we first build a natural inverse semigroup from Λ, S Λ . However, in contrast to the rank-1 setting of [8], the groupoid we want is fairly far removed from the universal groupoid of S Λ . Rather, it is obtained directly from a natural action of S Λ on a certain "infinite-path space" and is realized in terms of the sheaf of germs of the action (cf. [1]).
One extra benefit of our analysis is that we obtain a groupoid presentation of the Toeplitz algebra of (Λ, d), T C * (Λ). Another is that we overcome the limitation of [8] whereby the graphs considered are free of sources. That is, our theory gives an extension of Paterson's analysis even when restricted to the context of ordinary graphs.
We note, too, that under the hypotheses invoked by Kumjian and Pask [2], our analysis is different from theirs. They build a groupoid directly from the k-graph; we obtain a groupoid by first considering an inverse semigroup. Their groupoid and ours are the same, however, under their hypotheses. (See Remarks 6.11.) The paper is arranged as follows. In the next section, we discuss some of the features of 1-graphs that inspired our analysis. The discussion here is informal and incomplete. Detailed work begins in Section 3, where some of the basic facts about k-graphs are exposed, notation is set up, and basic facts about the C * -algebras we will study are presented. Also, two propositions regarding the structure of k-graphs are presented to be used in Section 5. As will be seen in due course, they lie at the heart of our analysis.
In Section 4, we restrict our attention to finitely aligned k-graphs (Λ, d) (Definition 3.4), and define our inverse semigroup S Λ from the structure of paths in Λ. We then define a second countable, locally compact, Hausdorff space X Λ and an inverse semigroup action θ of S Λ on X Λ . The space X Λ comprises all paths constructed on Λ -finite, infinite and partially infinite -and S Λ acts naturally by "removing and adding initial segments". Section 6 defines the groupoid of germs (cf. [1]) G Λ of the system (X Λ , S Λ , θ). With a topology naturally arising from the action and topology on X Λ , G Λ becomes an ample groupoid with unit space X Λ , and S Λ is identified with an inverse semigroup of ample subsets of G Λ . It is then shown that C * (G Λ ) is isomorphic to the Toeplitz algebra T C * (Λ). A closed invariant subset ∂Λ of the unit space G (0) Λ is then identified and it is shown that Finally, in section 7 we present a Cuntz-Krieger uniqueness theorem for finitely aligned k-graphs. A similar theorem was given in [12], and we compare the two results here.

Motivation
In this section we call attention to the salient features of graph C * -algebras that are the inspiration for the present work. Our intention is to provide an outline of the crucial points of our analysis, especially to help those unfamiliar with groupoid and inverse semigroup methods in operator algebra. Those familiar with the theory of graph C * -algebras, and with the associated theories of inverse semigroups and groupoids, may skip directly to the next section.
Let E = (E 0 , E 1 , r, s) be an ordinary directed graph. This means that E 0 and E 1 are sets, called respectively the set of vertices and the set of edges, and that r and s are functions from E 1 to E 0 , called respectively the range and source maps. We will assume here that our graphs have countable vertex and edge sets. From E we build the associated finite-path category E * . This is just the free category generated by E. It may and will be viewed as the collection of finite words (e 1 , e 2 , . . . , e n ) over E 1 , where s(e i ) = r(e i+1 ) for all i ≤ n − 1, together with the vertices v ∈ E 0 . The range and source maps extend in the obvious way to E * so that E * is also a graph. Composition in E * is defined through concatenation. Thus, paths (e 1 , e 2 , . . . , e n ) and (f 1 , f 2 , . . . , f m ) are composable if and only if s(e n ) = r(f 1 ), and in this event, their product is (e 1 , e 2 , . . . , e n , f 1 , f 2 , . . . , f m ); if v is a vertex and e is of the form (e 1 , e 2 , . . . , e n ), then v and e are composable if and only if v = r(e), and in this event, ve is just e.
The category E * is countable and is endowed with a degree functor d from E * to the semigroup, or small category, of non-negative integers, N. Namely, d describes the length of paths in E * , so d(v) = 0 for all vertices v, and d(e) = n for e of the form (e 1 , . . . , e n ). Further, the "freeness" of the construction of E * is expressed by the fact that if d(e) = n 1 + n 2 , n 1 , n 2 ∈ N, then there are unique paths e 1 and e 2 in E * such that d(e i ) = n i , i = 1, 2, and e = e 1 e 2 . Thus E * and d satisfy the unique factorization property that we mentioned at the outset, and (E * , d) is a 1-graph. The innocuous-seeming factorization property is essential for the role graphs play in operator algebra.
Given a graph E, one would like to build a C * -algebra C * (E) that is generated by a family {s e } e∈E * which, at the very least, consists of partial isometries (i.e., s * e s e is a projection for each e) satisfying s e s f = s ef for all composable e, f ∈ E * . That is, one would like to build a C * -algebra that codifies the representation theory of E * by partial isometries. Experimental investigation reveals that the crucial elements in C * (E) are or ought to be s e s * f , e, f ∈ E * , and that one would like the products s e s * f to behave like matrix units when e and f have the same degree. Further, in order for the matrix units s e s * f of one degree to be linked nicely with the matrix units of a different degree, one is led naturally to require the partial isometries s e satisfy the so-called Cuntz-Krieger condition: Note that since s v must be an idempotent partial isometry, it is a projection, and customarily, one writes p v instead of s v to highlight this. Note, too, that for equation (2.1) to make sense, one requires that v must be the range of at least one edge, so, as one says, v is not a source. Also, since an infinite sum of projections cannot converge in a C * -algebra, one requires also that the sum is finite; i.e., one requires that v is not an infinite receiver. Problems with sources and infinite receivers play an important role in the subject, and this paper contributes to their solution, but for this discussion, we will assume that our graph has no sources and that there are no infinite receivers. The history and theory of graph C * -algebras is fairly complex and involved, but to keep matters short, we jump to Paterson's wonderful insight [8] that it is very helpful to embed E * into a certain inverse semigroup S E , and then to use the theory he and others have been developing to realize the C * -algebra C * (E) as the C * -algebra universal for particular representations of S E (see [8,Theorem 2]). In fact, in [7] Paterson advocates that the C * -algebras universal for particular representations of an inverse semigroup may be effectively studied as the C * -algebras of groupoids naturally associated to the semigroup. This is the tack we take here. Any inverse semigroup acts by partially defined homeomorphisms on the semicharacter space of its idempotent subsemigroup, and the collection of germs of these maps forms a groupoid. In our setting, there is a closed subset of the semicharacter space which is invariant under the action, and so we consider the groupoid of germs associated to the restricted action.
In the context of our graph E, the inverse semigroup S E consists of a zero element z together with all pairs (α, β) ∈ E * × E * such that s(α) = s(β). Multiplication in S E is given by the formula and involution is given by z * = z and (e, f ) * = (f, e). The path category E * is embedded in S E via the formula e → (e, s(e)).
As we just mentioned, we want to think of S E acting by partially defined homeomorphisms on the semicharacter spaceÊ of the idempotent semigroup E of S E . Clearly, E consists of z together with all the pairs (e, e), e ∈ E * . On the other hand, by definition,Ê consists of all (nonzero) semigroup homomorphisms from E to the multiplicative semigroup {0, 1}. With respect to the topology of pointwise convergence,Ê is a locally compact Hausdorff space. The semicharacter spaceÊ may be identified as the disjoint union of {z}, E * and the infinite-path space E ∞ ; that is,Ê comprises z, the finite paths (e 1 , e 2 , . . . , e n ) of E * , and the infinite sequences (e 1 , e 2 , . . . ) satisfying s(e i ) = r(e i+1 ) for i ≥ 1. The infinite-path space is a closed subset ofÊ which is invariant under the action of S E . Under the identification ofÊ with {z} ∪ E * ∪ E ∞ , the restricted action of S E on E ∞ is given by removing and affixing initial segments of infinite paths according to the formula (α, β) · x = αy, where x = βy.
The groupoid G E formed from the germs of the action may be realized as all triples from E ∞ × Z × E ∞ of the form (αx, d(α) − d(β), βx), where α and β are finite paths and x is an infinite path such that s(α) = s(β) = r(x). The C * -algebra of E may then be realized as the C * -algebra of G E .
As we indicated at the outset, a k-graph is a countable category Λ, endowed with a degree functor d taking values in the semigroup N k , that satisfies the unique factorization property: for every path λ ∈ Λ and m, n ∈ N k with d(λ) = m + n, there exist unique paths µ, ν ∈ Λ such that λ = µν, d(µ) = m and d(ν) = n. We want to build an inverse semigroup and groupoid for each k-graph. If one studies the outline of how Paterson does this for 1-graphs, one realizes that there are two significant hurdles that are not anticipated by his work which must be overcome.
The first hurdle arises when one tries to put an inverse semigroup structure on the collection of pairs (λ, µ) ∈ Λ × Λ such that s(λ) = s(µ). More precisely, the multiplication structure described by equation (2.2) is not appropriate for general k-graphs with k ≥ 2. Our way around this problem is to look at finite sets of such pairs satisfying certain conditions to be described in Section 4. In a natural way, this collection of finite sets turns out to be an inverse semigroup S Λ containing a natural embedding of Λ. Indeed, the image of Λ generates S Λ as an inverse semigroup with a partially defined addition structure. Further, when Λ = E * is a 1-graph, S Λ contains S E as an inverse subsemigroup, and there is a one-to-one correspondence between representations of S E and additive representations 2 of S Λ .
The second hurdle concerns the semicharacter space of the subsemigroup E(S Λ ) of idempotents in S Λ : it appears to be much larger than the space on which we want S Λ to act. We pass instead directly to an analogue of the infinite-path space. More accurately, we consider two analogues. The first, denoted X Λ , is, in the directed graph setting, an analogue of E * ∪ E ∞ . We show that the inverse semigroup S Λ acts by partially defined homeomorphisms on X Λ . Further, we show in Section 6 that the groupoid of germs of this action, G Λ , parameterizes the Toeplitz algebra of Λ, T C * (Λ); this turns out to be new even in the 1-graph setting. The second analogue of E ∞ is the subset of X Λ called the space of boundary paths, denoted ∂Λ.
In the setting of a graph E with no sources and no infinite receivers, ∂Λ is E ∞ . In general, ∂Λ is a closed subset of X Λ that is invariant under the action of G Λ . The reduction of G Λ to ∂Λ, G Λ | ∂Λ , is our choice for the groupoid that parameterizes the Cuntz-Krieger algebra of Λ, C * (Λ).

3.1.
Higher-Rank Graphs and their C * -algebras. Throughout the remainder of this paper, Λ will denote a fixed k-graph and d : Λ → N k will be the associated degree functor. As we have spelled out above, the crucial property of d is the unique factorization property: If d(λ) = m + n in N k , then there are unique µ and ν in Λ such that d(µ) = m, d(ν) = n, and λ = µν. Ordinarily a category Λ is viewed as a system (Obj(Λ), Mor(Λ), r, s) where Obj(Λ) and Mor(Λ) are separate sets and r and s are maps from the second set to the first. However, it is convenient here to take the "arrows only" approach to categories [6]. So all elements of Λ are morphisms and Obj(Λ) is distinguished by virtue of being the idempotent morphisms. This perspective is especially appropriate because of our use of the degree functor. The unique factorization property allows us to identify Obj(Λ) with the elements of Λ that have degree zero. Because of our desire to generalize 1-graphs, we will also call the elements of Λ (finite) paths. For m ∈ N k , we define Λ m := d −1 ({m}), so as we just mentioned, Obj(Λ) = Λ 0 .
We follow [11,Section 2] for the basic facts about k-graphs that we shall use.
Here, and throughout, given m, n ∈ N k we write m ∨ n for the coordinatewise maximum of m and n. That is, the i th coordinate of m ∨ n is the maximum of the i th coordinates of m and n.
2 These are the representations of S Λ which preserve its partially defined addition structure; cf. [7, page 193].
Remark 3.5. If Λ is a 1-graph, then Λ is automatically finitely aligned: for λ, µ ∈ Λ, Λ min (λ, µ) is either empty or a singleton. Also every row-finite k-graph is finitely aligned, where a k-graph is called row-finite if vΛ n is finite for each v ∈ Λ 0 and n ∈ N k . Definition 3.6. Let Λ be a k-graph and let v ∈ Λ 0 . We say a subset E ⊆ vΛ is exhaustive if for every µ ∈ vΛ there exists a λ ∈ E such that Λ min (λ, µ) = ∅. We denote the set of all finite exhaustive subsets of Λ by F E(Λ), and for v ∈ Λ 0 , we Examples 3.7.
(2) Consider the k-graph Λ below: Dashed edges represent edges of degree (0, 1) and solid edges represent edges of degree (1, 0). The edges τ i where i ∈ N each have degree (1, 0). Any finite exhaustive subset of wΛ must contain w. The set {µ} is a finite exhaustive subset of vΛ, whereas {λ} is not because Λ min (λ, µβτ i ) = ∅ for any i ∈ N.
. Of course, the hypothesis that (Λ, d) is finitely aligned guarantees that the sums in Definition 3.8 are finite sums, and hence make sense in any C * -algebra. The following remark summarizes the ontological properties of these relations and the C * -algebras that codify them. Remark 3.9. Let (Λ, d) be a finitely aligned k-graph. Then there is a C * -algebra T C * (Λ), called the Toeplitz algebra of Λ, generated by a Toeplitz-Cuntz-Krieger Λfamily {s T λ : λ ∈ Λ}, which is universal in the sense that if {t λ : λ ∈ Λ} is a Toeplitz-Cuntz-Krieger Λ-family in a C * -algebra B, then there exists a C * -homomorphism π : T C * (Λ) → B such that π(s T λ ) = t λ for all λ ∈ Λ. The Cuntz-Krieger algebra of Λ is the C * -algebra C * (Λ), generated by a Cuntz-Krieger Λ-family {s λ : λ ∈ Λ}, which is universal for Cuntz-Krieger Λ-families. If we let I be the ideal in T C * (Λ) generated by products , then C * (Λ) can be identified with T C * (Λ)/I. Furthermore, [12,Proposition 2.12] says that the s T λ and s λ are nonzero for all λ ∈ Λ, implying that T C * (Λ) and C * (Λ) are nontrivial.
We want to call special attention to [12, Appendix A] for a thorough explanation of the Cuntz-Krieger relations of finitely aligned k-graphs, and to [12,Appendix B] for an account of how the theory encompasses the Cuntz-Krieger relations described in Equation 2.1.

3.2.
Extending paths. In the study of graph C * -algebras, i.e., in the study of C * -algebras defined by 1-graphs, problems arise if the graph has sources or sinks. Recall that a source in a graph is a vertex v that does not receive any edges, i.e., There is now a substantial literature on how to tackle these. In a k-graph Λ, a vertex v is a source if vΛ ei = ∅ for some i ∈ {1, . . . , k} where e i is the element in N k with 1 in the i th coordinate and 0 in every other coordinate. Therefore in the setting of higher-rank graphs, the situations can be considerably more complicated. A vertex may receive edges from some directions and not from others, while emitting edges in still other directions, but not in all. The highly ramified collection of possibilities creates numerous difficulties. One of the achievements of our approach to the analysis of C * -algebras associated to higher-rank graphs is to circumvent difficulties that sources and sinks can cause. We do not eliminate all problems, of course. Rather, we show that sources and sinks do not prevent one from defining and analyzing a groupoid associated to such k-graphs.
A key tool in our analysis are the two propositions of this subsection, Propositions 3.11 and 3.12.

Inverse Semigroups of Higher-Rank Graphs
For the remainder of the paper (Λ, d) will be a finitely aligned k-graph. To build the groupoids associated to Λ, we first construct an inverse semigroup S Λ associated to Λ that captures the salient features of Toeplitz-Cuntz-Krieger families. Recall that a semigroup S is an inverse semigroup if for all s ∈ S there exists a unique element s * ∈ S such that ss * s = s and s * ss * = s * . We follow [7] for the general theory and notation concerning inverse semigroups. In particular, we denote the semilattice of idempotents of S by E(S).
We claim that β = s(ξ). Since (ξ, η) ∈ G, (λ, µ) ∈ F , (β, s(λ)) ∈ Λ min (η, λ) and (s(λ), β) ∈ Λ min (µ, ξ), it follows from the definition of multiplication that (ξβ, ηβ) ∈ GF G. Since GF G = G, it then follows from Definition 4.1 that (ξβ, ηβ) = (ξ, η), which gives β = s(ξ). Therefore (4.5) gives λ = η and µ = ξ. Thus for every (λ, µ) ∈ F , we have (µ, λ) ∈ G, so F * ⊆ G. A similar argument gives G * ⊆ F which is equivalent to G ⊆ F * . Therefore G = F * , and S Λ is an inverse semigroup. Remark 4.5. As we noted in Section 2, the inverse semigroup Paterson used in [8] is Λ * s Λ ∪ {z}. The multiplication defined on Paterson's inverse semigroup corresponds to the multiplication of singleton sets in S Λ . If Λ is a k-graph with k ≥ 2 and λ, µ ∈ Λ, then Λ min (λ, µ) generally contains more than one element. Thus S Λ was chosen to consist of finite subsets of Λ * s Λ in order to obtain an inverse semigroup that is closed under a multiplication and which captures the properties of the k-graph. We note, too, that in [8] one has to add a zero element to Λ * s Λ, but this is not necessary in our setting, since the empty subset of Λ * s Λ is included in our S Λ and serves as a zero.

5.
The action of S Λ on the path space X Λ Definition 5.1. Let X Λ be the set of all graph morphisms x : Ω k,m → Λ where m ∈ (N∪{∞}) k (See Definition 3.3 and Example 3.2). We extend range and degree maps to all x : Ω k,m → Λ in X Λ by setting r(x) := x(0) and d(x) := m.
Without loss of generality assume that n + e i ≤ d(x). Then x ∈ D x(0,n+ei) and y ∈ D c x(0,n+ei) , as required.

the family of sets
{D λ ∩ D c λν1 ∩ · · · ∩ D c λν l : λ ∈ Λ and ν 1 , . . . , ν l ∈ s(λ)Λ} is a basis for the given topology on X Λ . Remark 5.6. An infinite sequence of paths in Λ is called wandering if for any finite set E ⊆ Λ, the sequence is eventually in Λ \ E. Convergence in X Λ is then given by: A sequence x i converges to x if and only if the following two conditions occur: (1) for all n ∈ N k such that n ≤ d(x), there exists I ∈ N such that i ≥ I implies x i (0, n) = x(0, n), and (2) if d(x) j < ∞, then for any n ∈ N k such that n ≤ d(x) and n j = d(x) j , either J(j, n) := {i ∈ N : d(x i ) ≥ n + e j } is finite, or x i (n, n + e j ) i∈J(j,n) is wandering.
Proposition 5.7. For each v ∈ Λ 0 , D v is compact, and so, in particular, X Λ is a locally compact Hausdorff space.
n=1 be a listing of all finite subsets of vΛ containing v. There is at least one λ ∈ E 1 such that x i (0, d(λ)) = λ for infinitely many i ∈ N (namely λ = v). Let λ 1 be such a λ of maximal degree.
Proof. For each λ ∈ Λ, D λ is a closed subset of D r(λ) , and hence is compact by Proposition 5.7. The result then follows since D F = (λ,µ)∈F D µ -a finite union.
Remark 5.9. In the next section we will construct two topological groupoids associated to a finitely aligned k-graph Λ. The unit space of the first groupoid G Λ is homeomorphic to X Λ , and C * (G Λ ) is isomorphic to the Toeplitz algebra T C * (Λ). The second groupoid is a reduction of G Λ to a closed invariant subset ∂Λ of (G Λ ) (0) ; we describe ∂Λ as a subspace of X Λ . The C * -algebra of G Λ | ∂Λ is isomorphic to the Cuntz-Krieger algebra C * (Λ).
Definition 5.10. An element x ∈ X Λ is called a boundary path if for all n ∈ N k with n ≤ d(x), and for all E ∈ x(n) F E(Λ), there exists λ ∈ E such that x(n, n + d(λ)) = λ. We write ∂Λ for the set of all boundary paths, and for v ∈ Λ 0 , write v(∂Λ) for {x ∈ ∂Λ : r(x) = v}.
(2) For the k-graph in Example 3.7 (2), any path starting with γ, η, ξ, ω or τ i for some i is a boundary path. Furthermore, since any finite exhaustive subset of wΛ must contain w, every element in Λw is also a boundary path.
Straightforward calculations give the following lemma.
Remark 5.14. Lemma 5.12 and Lemma 5.13 imply that ∂Λ is a locally compact Hausdorff space which is invariant under the action of S Λ . Following on from Remark 5.9, this fact will give us the required closed invariant subset of the unit space of G Λ . The next lemma ensures that the partial isometries inside C * (G Λ | ∂Λ ) giving a Cuntz-Krieger Λ-family are nonzero.
6. The groupoid of the system (X Λ , S Λ , θ) We begin this section by recalling notions from groupoid theory. We follow Paterson's book [7]. Let G be a topological groupoid. The unit space of G is denoted G (0) and the space of composable pairs is denoted G (2) . We use r and s for the range and source maps r(g) = gg −1 and s(g) = g −1 g (for g ∈ G). A subset U ⊆ G (0) is invariant if r(s −1 (U )) is contained in U . We denote by G op the family of open Hausdorff subsets A of G such that r| A and s| A are homeomorphisms. A locally compact groupoid G is called r-discrete if G (0) is open in G. In fact, a locally compact groupoid G is r-discrete and admits a left Haar system if and only if G op is a basis for the topology of G, which happens if and only if r : G → G (0) is a local homeomorphism (see [13,Proposition 2.8]). We define G a := {A ∈ G op : A is compact}.
An r-discrete groupoid G is called ample if G a forms a basis for the topology of G.
We now construct the groupoid of germs of the system (X Λ , S Λ , θ). Define and define a relation ∼ on Ξ Λ by requiring that (F, x) ∼ (G, y) if and only if x = y and there exists P ∈ E(S Λ ) such that x ∈ D P and F P = GP .
Lemma 6.1. The relation ∼ is an equivalence relation on Ξ Λ .
The equivalence class of (F, x) will be denoted [F, x]. It is called the germ of F at x.
To see that π T is injective, it suffices, by [10,Theorem 8.1], to show that for all v ∈ Λ 0 and all finite E ⊆ (vΛ) \ {v}, where the product is convolution. For v ∈ Λ 0 and finite E ⊆ (vΛ) \ {v}, regarding v as an element of Thus the product in (6.6) is bounded below by since the left Haar system on G Λ is given by the counting measures.
Proposition 6.10. ∂Λ is a nonempty closed invariant subset of G Λ is homeomorphic to X Λ (see Remark 6.6) and since r([F, x]) = θ F (x), Remark 5.14 implies that ∂Λ is a closed invariant subset of G (0) Λ . It is nonempty by Lemma 5.15. Proposition 6.10 implies that G Λ | ∂Λ is a locally compact, Hausdorff and ample groupoid.

The Cuntz-Krieger Uniqueness Theorem
In this section we present a Cuntz-Krieger uniqueness theorem for finitely aligned k-graphs. The theorem generalizes [2, Theorem 4.6] but differs from the Cuntz-Krieger uniqueness theorem [12,Theorem 4.5] obtained by direct methods. These differences will be discussed in Remarks 7.3.
Theorem 7.1. Let (Λ, d) be a finitely aligned k-graph, and suppose that for each v ∈ Λ 0 there exists x ∈ v(∂Λ) such that σ m (x) = σ n (x) =⇒ m = n for all m, n ∈ N k , m, n ≤ d(x).

(A)
Suppose that π : C * (Λ) → B is a homomorphism such that π(s v ) is nonzero for all v ∈ Λ 0 . Then π is injective.
The proof of Theorem 7.1 relies on the following proposition. Proof. First, observe that x ∈ (G Λ | ∂Λ ) (0) has trivial isotropy if and only if (7.1) for all m, n ∈ N k with m, n ≤ d(x), σ m x = σ n x implies m = n.
If G Λ | ∂Λ is essentially free, then since each D v is nonempty by Lemma 5.15 it follows that each D v = v(∂Λ) contains a boundary path x satisfying (7.1). Therefore Λ satisfies Condition (A).
Although this property of G Λ | ∂Λ is quite different to essential freeness, it may still yield similar consequences to essential freeness. In particular, the conclusion [4, Lemma 3.5] may hold, the key result in the proof of [2, Theorem 4.6] and Theorem 7.1. If this were the case, and if condition (B) was not equivalent to (B ′ ), then we would obtain a generalization of [12,Theorem 4.5] using groupoid methods (since (B) asks for an element of v(∂Λ) for each v ∈ Λ 0 , whereas (B ′ ) asks for an element of the smaller set vΛ ≤∞ ).