Coverings of Directed Graphs and Crossed Products of C*-algebras by Coactions of Homogeneous Spaces

The graph C*-algebra of a directed graph E is the universal C*-algebra generated by a family of partial isometries satisfying relations which reflect the path structure of E. In the first part of this paper we consider coverings of directed graphs: morphisms p:F->E which are local isomorphisms. We show that the graph algebra C*(F) can be recovered from C*(E) as a crossed product by a coaction of a homogeneous space associated to the fundamental group \pi_1 (E). These crossed products provide a good model for a general theory of crossed products by homogeneous spaces. The second part of the paper is devoted to building a framework for studying crossed products by homogeneous spaces using Rieffel's theory of proper actions.