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

Abstract

We define the notion of a $Λ$-system of $C^*$-correspondences associated to a higher-rank graph $Λ$. Roughly speaking, such a system assigns to each vertex of $Λ$ a $C^*$-algebra, and to each path in $Λ$ a $C^*$-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 $Λ$-system. We then construct a Fell bundle over the path groupoid $\Gg_Λ$ and show that the $C^*$-algebra of the $Λ$-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.