Graphs of $C^{\ast }$-correspondences and Fell bundles

Abstract

We define the notion of a $\Lambda$-system of $C^{\ast }$-correspondences associated to a higher-rank graph $\Lambda$. Roughly speaking, such a system assigns to each vertex of $\Lambda$ a $C^{\ast }$-algebra, and to each path in $\Lambda$ a $C^{\ast }$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^{\ast }$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^{\ast }$-correspondences to associate a $C^{\ast }$-algebra to each $\Lambda$-system. We then construct a Fell bundle over the path groupoid $\Gg_Λ$ and show that the $C^{\ast }$-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.

To appear in Indiana Univ. Math. J

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