equivalence for C\*-correspondences and formulate the following programme for the tensor algebras of C\*-correspondences:

> **Rigidity of SME.** Let ${{\mathcal{T}}}_E^+$ and ${{\mathcal{T}}}_F^+$ be the tensor algebras of the C\* - correspondences $E$ and $F$. When is it true that $E {\stackrel{\textup{SME}}{\thicksim}}F$ is equivalent to ${{\mathcal{T}}}_E^+ {\stackrel{\textup{SME}}{\thicksim}}{{\mathcal{T}}}_F^+$?

The origins of this programme can be traced in the work of Arveson [@Arv67] on classifying dynamics by nonselfadjoint operator algebras. In this category ${\stackrel{\textup{SME}}{\thicksim}}$ is seen as a generalized similarity, rather than a decomposition[^3]. Muhly and Solel [@MuhSol00] provide an affirmative answer for injective and aperiodic C\*-correspondences. The main tool for the forward direction is to establish ${\stackrel{\textup{SME}}{\thicksim}}$ for the Toeplitz-Pimsner C\*-algebras ${{\mathcal{T}}}_E$ and ${{\mathcal{T}}}_F$. An elegant construction of matrix representations for ${{\mathcal{T}}}_E$ and ${{\mathcal{T}}}_F$ is used in [@MuhSol00] to accomplish this. Aperiodicity is used for the converse to ensure that an induced Morita context is implemented fiber-wise.

Our first motivation for the current paper was to remove the injectivity assumption for the forward direction of SME-rigidity. In fact we accomplish more by directly showing an equivalence implemented by the same TRO between representations. As a consequence ${\stackrel{\textup{SME}}{\thicksim}}$ on C\*-correspondences implies equivalence of all related operator algebras, i.e. their tensor algebras and their 