Graphs of $C^*$-correspondences and Fell bundles

Abstract

We define the notion of a �-system of C � -correspondences associated to a higher-rank graph �. Roughly speaking, such a system assigns to each vertex ofa C � - algebra, and to each path ina C � -correspondence in a way which carries compositions of paths to balanced tensor products of C � -correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of C � -correspondences to associate a C � -algebra to each �-system. We then construct a Fell bundle over the path groupoid Gand show that the C � -algebra of the �-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.