Abstract.To a compact Hausdorff space which covers itself, we associate an r-discrete locally compact Hausdorff groupoid. Its C*-algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra Op . We consider examples related to coverings of the circle and of a Heisenberg 3-manifold.

Abstract.To a compact Hausdorff space which covers itself, we associate an r-discrete locally compact Hausdorff groupoid. Its C*-algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra Op . We consider examples related to coverings of the circle and of a Heisenberg 3-manifold.

Given a continuous open surjective morphism π : G → H of étale groupoids with amenable kernel, we construct a Fell bundle E over H and prove that its C * -algebra C * r (E) is isomorphic to C * r (G). This is related to results of Fell concerning C * -algebraic bundles over groups. The case H = X, a locally compact space, was treated earlier by Ramazan. We conclude that C * r (G) is strongly Morita equivalent to a crossed product, the C * -algebra of a Fell bundle arising from an action of the groupoid H on a C * -bundle over H 0 . We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles.

We present the continuous graph approach for some generalizations of the Cuntz-Krieger algebras. These algebras are simple, nuclear, and purely infinite, with rich K-theory. They are tied with the dynamics of a shift on an infinite path space. Interesting examples occur when the vertex spaces are unions of tori, and the shift is not necessarily expansive. We also show how the algebra of a continuous graph could be thought as a Pimsner algebra. Introduction. Recent papers are dealing with different generalizations of the Cuntz-Krieger algebrasThe exact relationship between these approaches remains to be explored, but certainly there are overlaps. In [Pi], the author considers a Hilbert bimodule H over a C*-algebra, and creation operators on a corresponding Fock space. These operators generate the Toeplitz algebra T H and, taking a quotient of this, one obtains the algebra O H . If the Hilbert bimodule is projective and finitely generated over an abelian, finite dimensional C*-algebra, then one recovers the algebras O A .In [P1], the starting point is a Smale space (a compact metric space endowed with an expansive homeomorphism with canonical coordinates), on which one defines the stable and unstable equivalence relations. The associated C*-algebras have natural shift automorphisms, and the crossed products are the so called Ruelle algebras. These are strongly Morita equivalent to particular Cuntz-Krieger algebras if the Smale space is a topological Markov shift.Our point of view is to start with a continuous oriented graph (or diagram) E, to consider the space of one-sided infinite paths (obtained by concatenation of edges in E), and to associate a groupoid (à la Renault) using the unilateral shift on this path space. The C*-algebra of this groupoid plays the role of a continuous version of the Cuntz-Krieger algebras, since these could be obtained by the same construction from a finite graph defined by a 0-1 matrix. In many cases, this groupoid algebra is simple, purely infinite, with computable K-theory. This approach offers more freedom for constructing easy, concrete examples, with prescribed K-theory. It should 247 248 VALENTIN DEACONU be mentioned that C*-algebras associated with discrete graphs were studied in [KPRR], [KPR], [KP]. See also the survey [K2].The continuous graph approach is very similar to the point of view of polymorphisms or correspondences, introduced earlier in a measure theoretical context by Vershik and Arzumanian (see [AR] for a precise definition and references).Even though our groupoid algebras could be obtained also by using the Pimsner approach, with a right choice of the Hilbert bimodule, we feel that the present point of view has certain advantages, beeing tied with the dynamics of a shift. For example, even in a case where this shift is not expansive, so the space of two-sided infinite paths has no obvious Smale space structure, we will prove that the corresponding algebra is simple and purely infinite.In the particular case when the vertex space is a disjoint union of tori, we ...

Abstract. We define the action of a locally compact group G on a topological graph E.