$C^*$-correspondence functoriality of Cuntz-Pimsner algebras

TL;DR

This paper develops a functorial framework linking $C^*$-correspondences to their Cuntz-Pimsner algebras, demonstrating that Morita equivalence of correspondences implies Morita equivalence of their associated algebras.

Contribution

It introduces a functorial approach to $C^*$-correspondences and extends known results to a more general Morita equivalence context.

Findings

Established a functor from $C^*$-correspondences to Cuntz-Pimsner algebras.

Proved Morita equivalence of correspondences implies Morita equivalence of their Cuntz-Pimsner algebras.

Generalized a known result of Muhly and Solel.

Abstract

We construct a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras. The objects in our domain category are $C^*$-correspondences, and the morphisms are the isomorphism classes of $C^*$-correspondences satisfying certain conditions. As an application, we recover a well-known result of Muhly and Solel. In fact, we show that functoriality leads us to a more generalized result: strongly Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.