Exactness of the Cuntz-Pimsner Construction

TL;DR

This paper demonstrates that key results in Cuntz-Pimsner algebra theory follow from the functorial nature of the construction, establishing its exactness and exploring related categorical sequences.

Contribution

It shows the Cuntz-Pimsner construction is a functor that is exact, providing a new categorical perspective and simplifying proofs of fundamental results.

Findings

Fundamental results are consequences of functoriality.

The Cuntz-Pimsner functor is exact.

Descriptions of exact sequences in categories.

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.