Abstract

We define the notion of a $\Lambda$system of $C^*$correspondences associated to a higherrank 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 CuntzPimsner 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 crosssectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.