Graphs, groupoids and Cuntz-Krieger algebras

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 {0, 1}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.