We revisit a central result of Muhly and Solel on operator algebras ofC*-correspondences. We prove that (possibly non-injective) strongly Moritaequivalent C*-correspondences have strongly Morita equivalent relativeCuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in thesense of Blecher, Muhly and Paulsen) and strong $\Delta$-equivalence (in thesense of Eleftherakis) for the related tensor algebras. In particular, weobtain stable isomorphism of the operator algebras when the equivalence isgiven by a $\sigma$-TRO. As an application we show that strong Moritaequivalence coincides with strong $\Delta$-equivalence for tensor algebras ofaperiodic C*-correspondences.
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