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 ...

Abstract. For each piecewise monotonic map τ of [0, 1], we associate a pair of C*-algebras Fτ and Oτ and calculate their K-groups. The algebra Fτ is an AI-algebra. We characterize when Fτ and Oτ are simple. In those cases, Fτ has a unique trace, and Oτ is purely infinite with a unique KMS-state. In the case that τ is Markov, these algebras include the Cuntz-Krieger algebras OA, and the associated AF-algebras FA. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and β-transformations. For the case of interval exchange maps and of β-transformations, the C*-algebra Oτ coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani respectively. 1.