Stable isomorphism and strong Morita equivalence of operator algebras

Abstract

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. $M\alpha (A)M^*$). 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 $A\otimes \cl K$ 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 $\Delta $-equivalent.