Stable isomorphism and strong Morita equivalence of operator algebras

We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Δ$-equivalent if they have completely isometric representations $α$ and $β$ respectively and there exists a ternary ring of operators $M$ such that $α(A)$ (resp. $β(B)$) is equal to the norm closure of the linear span of the set $M^*β(B)M,$ (resp. $Mα(A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Δ$-equivalent, then the operator algebras $A\otimes \cl K$ and $B\otimes \cl K$ 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 $A\otimes \cl K$ and $B\otimes \cl K$ are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $Δ$-equivalent.

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