Passing $C^*$-correspondence Relations to the Cuntz-Pimsner algebras

Abstract: We construct a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras. Applications include a generalization of the well-known result of Muhly and Solel: Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras; as well as the result of Muhly, Pask, and Tomforde: regular strong shift equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.