when passing to general operator spaces. Due to [@BGR77], Morita Theorem IV holds when ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are $\si$-unital C\*-algebras. This is not true for ${\stackrel{\textup{SME}}{\thicksim}}$ on nonselfadjoint operator algebras, even when the operator algebras are unital [@BMP00 Example 8.2]. On the other hand it is shown in [@Ele14 Theorem 3.2] that ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ on operator algebras coincides with stable isomorphism under the appropriate $\si$-unital condition. As a consequence ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ is strictly stronger than ${\stackrel{\textup{SME}}{\thicksim}}$.

Bimodule structure
==================

A key role in Morita Theory is played by the linking algebra. This construction induces an operator bimodule structure on C\*-correspondences. In this section we use it to show that isomorphism of C\*-correspondences is preserved when passing to the category of operator bimodules, and vice versa.

Every right Hilbert module $X_A$ comes with an operator module structure. This can be verified by seeing $X_A$ inside its *augmented linking algebra* $$\begin{bmatrix} A & X^* \\ X & {{\mathcal{L}}}(X) \end{bmatrix} :=
\{ \begin{bmatrix} a & y^* \\ x & u \end{bmatrix} \mid a \in A, x, y \in X, u \in {{\mathcal{L}}}(X) \}$$ where $X^*$ is the adjoint module ${{\mathcal{K}}}(X,A)$ of $X$ and the multiplication rule is given by $$\begin{bmatrix} a_1 