We associate to each row-finite directed graph E a universal Cuntz-Krieger C - algebra C (E), and study how the distribution of loops in E affects the structure of C (E). We prove that C (E) is AF if and only if E has no loops. We describe an exit condition (L) on loops in E which allows us to prove an analogue of the Cuntz-Krieger uniqueness theorem and give a characterisation of when C (E) is purely infinite. If the graph E satisfies (L) and is cofinal, then we have a dichotomy: if E has no loops, then C (E) is AF; if E has a loop, then C (E) is purely infinite.

Abstract. We prove versions of the fundamentaltheorems about Cuntz-Krieger algebras for the C ∗-algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many edges. Special cases of these results have previously been obtained using various powerful machines; our main point is that direct methods yield sharper results more easily. Contents 1. The C∗-algebras of graphs 309 2. The gauge-invariant uniqueness theorem 311 3. The Cuntz-Krieger uniqueness theorem 313 4. Ideals in graph algebras 316

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.

Abstract. There is now a substantial literature on graph C ∗-algebras. Under a locally finite condition on a countable, directed graph, Kumjian, Pask, Raeburn, Renault showed that the C ∗-algebra of the graph can be realized as the C ∗-algebra of the path groupoid, i.e. the groupoid determined by the infinite paths in the graph. In the present paper, we remove the local finiteness requirement. The path groupoid in the general context is obtained through the universal groupoid of a certain inverse semigroup associated with the graph. This inverse semigroup is called the graph inverse semigroup, and graph representations turn out to be just representations of this inverse semigroup. A certain reduction of the universal groupoid gives the path groupoid of the graph, and its C ∗-algebra is isomorphic to the C ∗-algebra of the graph. The unit space of the path groupoid contains the infinite paths of the graph, but also contains some finite paths. We show that, as in the locally finite case, the path groupoid is always amenable, and we give a groupoid proof of a recent theorem of W. Szymanski, characterizing when a graph C ∗-algebra is simple. 1.

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which maybe 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 su cient conditions on the higher rank graph for the associated C*-algebra to be: simple, purely innite 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.

We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C*-algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many edges. Special cases of these results have previously been obtained using various powerful machines; our main point is that direct methods yield sharper results more easily.

Given a free action of a group G on a directed graph E we show that the crossed product of C*(E), the universal C*-algebra of E, by the induced action is strongly Morita equivalent to C*(E/G). Since every connected graph E may be expressed as the quotient of a tree T by an action of a free group G we may use our results to show that C*(E) is strongly Morita equivalent to the crossed product C0 (@T ) \Theta G, where @T is a certain 0--dimensional space canonically associated to the tree.

Abstract. We prove that if Λ is a row-finite k-graph with no sources, then the associated C ∗-algebra is simple if and only if Λ is cofinal and satisfies Kumjian and Pask’s Condition (A). We prove that Condition (A) is equivalent to a suitably modified version of Robertson and Steger’s original nonperiodicity condition (H3) which in particular involves only finite paths. We also characterise both cofinality and aperiodicity of Λ in terms of ideals in C ∗ (Λ). 1.

Abstract. Let γ = (γ1,..., γN), N ≥ 2, be a system of proper contractions on a complete metric space. Then there exists a unique self-similar non-empty compact subset K. We consider the union G = ∪ N i=1 {(x, y) ∈ K2; x = γi(y)} of the cographs of γi. Then X = C(G) is a Hilbert bimodule over A = C(K). We associate a C ∗-algebra Oγ(K) with them as a Cuntz-Pimsner algebra OX. We show that if a system of proper contractions satisfies the open set condition in K, then the C ∗-algebra Oγ(K) is simple and purely infinite, which is not isomorphic to a Cuntz algebra in general. 1.

Abstract. This paper explores the effect of various graphical constructions upon the associated graph C ∗-algebras. The graphical constructions in question arise naturally in the study of flow equivalence for topological Markov chains. We prove that outsplittings give rise to isomorphic graph algebras, and in-splittings give rise to strongly Morita equivalent C ∗-algebras. We generalise the notion of a delay as defined in [D] to form in-delays and out-delays. We prove that these constructions give rise to Morita equivalent graph C ∗-algebras. We provide examples which suggest that our results are the most general possible in the setting of the C ∗-algebras of arbitrary directed graphs. 1.