Topological Quivers

Abstract

Topological quivers are generalizations of directed graphs in which the sets of vertices and edges are locally compact Hausdorff spaces. Associated to such a topological quiver Q is a C*-correspondence, and from this correspondence one may construct a Cuntz-Pimsner algebra C*(Q). In this paper we develop the general theory of topological quiver C*-algebras and show how certain C*-algebras found in the literature may be viewed from this general perspective. In particular, we show that C*-algebras of topological quivers generalize the well-studied class of graph C*-algebras and in analogy with that theory much of the operator algebra structure of C*(Q) can be determined from Q. We also show that many fundamental results from the theory of graph C*-algebras have natural analogues in the context of topological quivers (often with more involved proofs). These include the Gauge-Invariant Uniqueness theorem, the Cuntz-Krieger Uniqueness theorem, descriptions of the ideal structure, and conditions for simplicity.