on unital operator algebras [@BMP00 Proposition 3.3]. In fact Blecher-Muhly-Paulsen [@BMP00] show that when ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ have c.a.i.’s then being strongly Morita equivalent coincides with having $${{\mathcal{A}}}\simeq M \otimes_{{{\mathcal{B}}}}^h N {\quad\text{and}\quad}{{\mathcal{B}}}\simeq N \otimes_{{{\mathcal{A}}}}^h M.$$ This depends on an elegant decomposition of the approximate unit. Apparently the existence of approximate units is necessary. It can be shown that ${\stackrel{\textup{SME}}{\thicksim}}$ is equivalent to having ${\stackrel{\textup{d}}{\thicksim}}$ and row contractions from $M$ and $N$ that reconstruct c.a.i.’s for ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$.

The importance of Morita Theory appears through the Morita Theorems. Let us give a description that depicts the essential points. Morita Theorem I suggests that Morita equivalence implies an equivalence of the representations by tensoring (within the category) with the bimodules of the Morita context. Morita Theorem II suggests that an equivalence of representations gives rise to a Morita context, whereas Morita Theorem III implies that such equivalences are given just by tensoring with appropriate bimodules. Finally Morita Theorem IV suggests that if the algebras satisfy a countability condition (the c.a.i.’s are countable) then Morita equivalence coincides with stable isomorphism. Recall that ${{\mathcal{A}}}$ and ${{\mathcal{B}}}$ are *stably isomorphic* if ${{\mathcal{A}}}\otimes {{\mathcal{K}}}\simeq {{\mathcal{B}}}\otimes {{\mathcal{K}}}$ by a completely isometric isomorphism, where ${{\mathcal{K}}}$ is the 