MORITA EQUIVALENCE OF C*-CORRESPONDENCES PASSES TO THE RELATED OPERATOR ALGEBRAS

We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong Delta-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a sigma-TRO. As an application we show that strong Morita equivalence coincides with strong Delta-equivalence for tensor algebras of aperiodic C*-correspondences.