Graphs, Groupoids, and Cuntz–Krieger Algebras

We associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of paths emanating from a distinguished vertex ★. We show that under certain conditions theirC*-algebras are Morita equivalent; the groupoidC*-algebraC*(G) is the Cuntz–Krieger algebra of an infinite {0, 1} matrix defined byG, and that the algebrasC*(G(★)) contain theC*-algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoidC*-algebras using the general theory of Renault and calculate theirK-theory.