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 $M^{\ast }\beta (B)M,$ (resp. $M\alpha (A)M^{\ast }$). 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.

We present some shorter proofs using references from the literature. Also example 3.7 is new and provides a new proof of the fact our notion of strong Morita equivalence is stronger than "BMP strong Morita equivalence " introduced by Blecher, Muhly and Paulsen

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