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 classify the gauge-invariant ideals in the C ∗-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant primitive ideals in terms of the structural properties of the graph, and describe the K-theory of the C ∗-algebras of arbitrary infinite graphs. 1.

We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of one--sided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C --algebras are Morita equivalent; the groupoid C --algebra C (G) is the Cuntz--Krieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C --algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C --algebras using the general theory of Renault, and calculate their K-theory. 1 Introduction Over the past fifteen years many C -algebras and classes of C -algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...

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.

The notion of extension of a given C*-category C by a C*-algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging

Abstract. We compute the K-theory groups of the Cuntz-Krieger C ∗-algebraOA associated to an infinite matrix A of zeros and ones.

Abstract. Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C ∗-algebra, C ∗ (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C ∗ (Λ). The K-groups of C ∗ (Λ) for k> 2 can be calculated under certain circumstances. We state that for all k, the torsion-free rank of K0(C ∗ (Λ)) and K1(C ∗ (Λ)) are equal when C ∗ (Λ) is unital, and we determine the position of the class of the unit of C ∗ (Λ) in K0(C ∗ (Λ)). 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.

Abstract. This paper is comprised of two related parts. First we discuss which k-graph algebras have faithful gauge invariant traces, where the gauge action of T k is the canonical one. We give a sufficient condition for the existence of such a trace, identify the C ∗-algebras of k-graphs satisfying this condition up to Morita equivalence, and compute their K-theory. For k-graphs with faithful gauge invariant trace, we construct a smooth (k, ∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the T k action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra. 1.

Abstract. We describe a class of C ∗-algebras which simultaneously generalise the ultragraph algebras of Tomforde and the shift space C ∗-algebras of Matsumoto. In doing so we shed some new light on the different C ∗-algebras that may be associated to a shift space. Finally, we show how to associate a simple C ∗-algebra to an irreducible sofic shift.