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

TL;DR

This paper introduces a new framework linking higher-rank graph $C^*$-correspondences with Fell bundles, providing a unified approach to their associated $C^*$-algebras and illustrating its applicability through various examples.

Contribution

It defines $ ext{Lambda}$-systems of $C^*$-correspondences for higher-rank graphs and constructs a Fell bundle over the path groupoid, connecting these structures to $C^*$-algebras.

Findings

Established a correspondence between $ ext{Lambda}$-systems and Fell bundles.

Proved the $C^*$-algebra of the $ ext{Lambda}$-system equals the reduced cross-sectional algebra of the Fell bundle.

Presented examples demonstrating the construction's relevance in existing literature.

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 $\Gg_\Lambda$ 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.