We consider the higher-rank graphs introduced by Kumjian and Pask as models for higherrank Cuntz-Krieger algebras. We describe a variant of the Cuntz-Krieger relations which applies to graphs with sources, and describe a local convexity condition which characterizes the higher-rank graphs that admit a non-trivial Cuntz-Krieger family. We then prove versions of the uniqueness theorems and classifications of ideals for the C * -algebras generated by Cuntz-Krieger families.

We generalise the theory of Cuntz-Krieger families and graph algebras to the class of finitely aligned k-graphs. This class contains in particular all row-finite k-graphs. The CuntzKrieger relations for non-row-finite k-graphs look significantly different from the usual ones, and this substantially complicates the analysis of the graph algebra. We prove a gaugeinvariant uniqueness theorem and a Cuntz-Krieger uniqueness theorem for the C Ã -algebras of finitely aligned k-graphs. r 2004 Elsevier Inc. All rights reserved. MSC: primary 46L05 convex or row-finite, and we do allow them to have sources. When k ¼ 1 or the k-graph is row-finite and locally convex, our new Cuntz-Krieger relations are equivalent to the usual ones. We show that for every finitely aligned k-graph L; there is a family of nonzero partial isometries which satisfies the new relations, and we define C Ã ðLÞ to be the universal C Ã -algebra generated by such a family. We then prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for C Ã ðLÞ: Our analysis is elementary in the sense that we do not use groupoids, partial actions or Hilbert bimodules, though we cheerfully acknowledge that we have gained insight from the models these theories provide.The results in this paper extend the existing theory of graph algebras in several directions. Since 1-graphs are always finitely aligned, and our new relations are then equivalent to the usual ones (Proposition B.1), our approach provides the first elementary analysis of the C Ã -algebra of an arbitrary directed graph. Our results are also new for finitely aligned k-graphs without sources; those interested primarily in this situation may mentally replace all the symbols L pn by L n ; and thereby avoid several technical complications. Even for row-finite k-graphs we make significant improvements on the existing theory: for non-locally-convex row-finite k-graphs, our Cuntz-Krieger families may have every vertex projection nonzero, unlike those in [13] (see Example A.1).In Section 2, we describe our new Cuntz-Krieger relations for a finitely aligned kgraph L; define C Ã ðLÞ to be the universal C Ã -algebra generated by a Cuntz-Krieger family, and investigate some of its basic properties. We discuss a notion of boundary paths which we use to construct a Cuntz-Krieger family in which every vertex projection is nonzero.The core in C Ã ðLÞ is the fixed-point algebra C Ã ðLÞ g for the gauge action g of T k : In Section 3, we show that the core is AF, and deduce that a homomorphism p of C Ã ðLÞ which is nonzero at each vertex projection is injective on the core.Our proof that C Ã ðLÞ g is AF is quite different from the argument which we gave for row-finite k-graphs in [13] in that we do not describe C Ã ðLÞ g as a direct limit over N k : Instead, we describe C Ã ðLÞ g as the increasing union of finite-dimensional algebras indexed by finite sets of paths, and produce families of matrix units which span these algebras. In addition to showing that C Ã ðLÞ is AF, this formu...

Abstract. 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. IntroductionA 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.Date: March 10, 2018. 1991 Mathematics Subject Classification. Primary 46L05; Secondary 22A22; 20M18. Key words and phrases. graph algebra; Cuntz-Krieger algebra; higher-rank graph; groupoid; inverse semigroup.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 semi...

We provide groupoid models for Toeplitz and Cuntz-Krieger algebras of topological higher-rank graphs. Extending the groupoid models used in the theory of graph algebras and topological dynamical systems to our setting, we prove results on essential freeness and amenability of the groupoids which capture the existing theory, and extend results involving group crossed products of graph algebras.Notation 2.2. For m ∈ N k , we write m i for the ith coordinate of m. We use the partial ordering ≤ on N k defined by m ≤ n ⇐⇒ m i ≤ n i for all i ∈ {1, . . . , k}, so least upper bounds and greatest lower bounds are given by (m ∨ n) i = max{m i , n i } and (m ∧ n) i = min{m i , n i }, respectively. For m ∈ N k , define Λ m to be the setq for the set of minimal common extensions of paths from U and V . For λ, µ ∈ Λ, we write Λ min (λ, µ) := {(α, β)|λα = µβ, d(λα) = d(λ) ∨ d(µ)} for the set of pairs which give minimal common extensions of λ and µ; that is, Λ min (λ, µ) = {(α, β)|λα = µβ ∈ {λ} ∨ {µ}}. Definition 2.3. A topological k-graph (Λ, d) is compactly aligned if for all p, q ∈ N k and compact U ⊂ Λ p and V ⊂ Λ q , the set U ∨ V is compact.

Abstract. We consider the boundary-path groupoids of topological higher-rank graphs. We give a direct limit decomposition of the algebra of continuous functions vanishing at infinity on the unit space, and show that the groupoid is amenable. We use these results to prove versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the associated C * -algebras.