Adding Tails to $C^*$-Correspondences

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

2000 Mathematics Subject Classification: 46L08, 46L55

Keywords and Phrases: $C^*$-correspondence, Cuntz-Pimsner algebra, relative Cuntz-Pimsner algebra, graph $C^*$-algebra, adding tails, gauge-invariant uniqueness

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