Adding tails to $C^{\ast }$-correspondences

Abstract

We describe a method of adding tails to $C^{\ast }$-correspondences which generalizes the process used in the study of graph $C^{\ast }$-algebras. We show how this technique can be used to extend results for augmented Cuntz-Pimsner algebras to $C^{\ast }$-algebras associated to general $C^{\ast }$-correspondences, and as an application we prove a gauge-invariant uniqueness theorem for these algebras. We also define a notion of relative graph $C^{\ast }$-algebras and show that properties of these $C^{\ast }$-algebras can provide insight and motivation for results about relative Cuntz-Pimsner algebras.