Morita equivalence of C*-correspondences passes to the related operator algebras

TL;DR

This paper demonstrates that Morita equivalence of C*-correspondences extends to their associated operator algebras, establishing stable isomorphism and equivalence conditions for various related algebraic structures.

Contribution

It proves that Morita equivalence of C*-correspondences implies Morita equivalence of their operator algebras, including Cuntz-Pimsner algebras and tensor algebras, under broad conditions.

Findings

Morita equivalence passes to relative Cuntz-Pimsner C*-algebras

Stable isomorphism of operator algebras when equivalence is via a σ-TRO

Equivalence of strong Morita and strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences

Abstract

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.