compacts on $\ell^2$ and the tensor product is the spatial one.

Strong Morita equivalence for approximately unital operator algebras satisfies the first three parts of Morita Theory [@Ble01; @BMP00]. In particular, two approximately unital operator algebras are strongly Morita equivalent if and only if their categories of left operator modules are equivalent (via a completely contractive functor) [@BMP00].

Eleftherakis [@Ele14] introduced a stronger notion of equivalence. Two operator algebras ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are *strongly $\Delta$-equivalent* (notation, ${{\mathcal{A}}}{\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}{{\mathcal{B}}}$) if there are non-degenerate completely isometric representations $$\al \colon {{\mathcal{A}}}\to {{\mathcal{B}}}(H) {\quad\text{and}\quad}\be \colon {{\mathcal{B}}}\to {{\mathcal{B}}}(K)$$ and a TRO $M \subseteq {{\mathcal{B}}}(K, H)$ such that $$\al({{\mathcal{A}}}) = {\left[ M \be({{\mathcal{B}}}) M^* \right]^{-\|\cdot\|}} {\quad\text{and}\quad}\be({{\mathcal{B}}}) = {\left[ M^* \al({{\mathcal{A}}}) M \right]^{-\|\cdot\|}}.$$ In [@Ele14] it is shown that ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ is an equivalence relation. Even though ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ is originally defined on approximately unital operator algebras in [@Ele14], the elements we record here still hold for non-unital cases. This is exhibited in [@EleKak16] where ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ is considered for operator spaces.

As noted ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ and ${\stackrel{\textup{SME}}{\thicksim}}$ coincide with the usual strong Morita equivalence when restricted to C\*-algebras. However a fundamental difference between ${\, \stackrel{\textup{$\Delta$}}{\thicksim} \,}$ and ${\stackrel{\textup{SME}}{\thicksim}}$ concerns stable isomorphisms 