Stable isomorphism and strong Morita equivalence of operator algebras

We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Delta $-equivalent if they have completely isometric representations $alpha $ and $beta $ respectively and there exists a ternary ring of operators $M$ such that $alpha (A)$ (resp. $beta (B)$) is equal to the norm closure of the linear span of the set $M^*beta (B)M, $ (resp. $Malpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Delta $-equivalent, then the operator algebras $Aotimes cl K$ and $Botimes cl K$ are isomorphic. Here $cl K$ is the set of compact operators on an infinite dimensional separable Hilbert space and $otimes $ is the spatial tensor product. Conversely, if $Aotimes cl K$ and $Botimes cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Delta $-equivalent.