Exactness of the Cuntz-Pimsner Construction

Abstract

In prior work we described how the Cuntz-Pimsner construction may be viewed as a functor. The domain of this functor is a category whose objects are $C^*$-correspondences and morphisms are isomorphism classes of certain pairs comprised of a $C^*$-correspondence and an isomorphism. The codomain is the well-studied category whose objects are $C^*$-algebras and morphisms are isomorphism classes of $C^*$-correspondences. In this paper we show that certain fundamental results in the theory of Cuntz-Pimsner algebras are direct consequences of the functoriality of the Cuntz-Pimsner construction. In addition, we describe exact sequences in the target and domain categories, and prove that the Cuntz-Pimsner functor is exact.