IUMJ

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

Authors: Valentin Deaconu, Alex Kumjian, David Pask and Aidan Sims

Issue: Volume 59 (2010), Issue 5, 1687-1736

Abstract:

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