then ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are called *strongly Morita equivalent* (notation, ${{\mathcal{A}}}{\stackrel{\textup{SME}}{\thicksim}}{{\mathcal{B}}}$) [@BMP00 Definition 3.6]. The initials (A), (G) and (P) stand for associativity, generator and projective respectively. When ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are C\*-algebras then this definition coincides with that of [@Rie74], where $M$ *can be chosen* to be an imprimitivity bimodule and $N$ be $M^*$.

There is a concrete version of the Morita context. One of the main ingredients concerns the form of the decomposition of two algebras. (Notice here that we do not assume a priori that the operator algebras have an approximate unit.)

Let ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ be operator algebras. We say that they are *decomposable* (notation, ${{\mathcal{A}}}{\stackrel{\textup{d}}{\thicksim}}{{\mathcal{B}}}$) 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 bimodules ${}_{\al({{\mathcal{A}}})} M_{\be({{\mathcal{B}}})} \subseteq {{\mathcal{B}}}(K, H)$ and ${}_{\be({{\mathcal{B}}})} N_{\al({{\mathcal{A}}})} \subseteq {{\mathcal{B}}}(H, K)$ such that $$\al({{\mathcal{A}}}) = {\left[ M \cdot N \right]^{-\|\cdot\|}} {\quad\text{and}\quad}\be({{\mathcal{B}}}) = {\left[ N \cdot M \right]^{-\|\cdot\|}}.$$

By definition ${\stackrel{\textup{d}}{\thicksim}}$ implies a Morita context $({{\mathcal{A}}}, {{\mathcal{B}}}, M, N, {\left(\cdot, \cdot\right)}, {\left[\cdot, \cdot\right]})$ that satisfies just property (A) in the sense of Blecher-Muhly-Paulsen [@BMP00]. It is unclear (and probably not true) that ${\stackrel{\textup{d}}{\thicksim}}$ is transitive in general. Remarkably though ${\stackrel{\textup{d}}{\thicksim}}$ is an equivalence relation 