$C^{\ast }$-correspondence functoriality of Cuntz-Pimsner algebras

Abstract

We construct a functor that maps $C^{\ast }$-correspondences to their Cuntz-Pimsner algebras. The objects in our domain category are $C^{\ast }$-correspondences, and the morphisms are the isomorphism classes of $C^{\ast }$-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 $C^{\ast }$-correspondences have Morita equivalent Cuntz-Pimsner algebras.

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