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Register a new userStable isomorphism and strong Morita equivalence of operator algebras
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Authors
G. K. Eleftherakis
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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.
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We define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case.
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We prove that for every group $G$ and any two sets $I,J$, the Brandt semigroup algebras $ell(B(I,G))$ and $ell(B(J,G))$ are Morita equivalent with respect to the Morita theory of self-induced Banach algebras introduced by Gronbaek. As applications, we show that if $G$ is an amenable group, then for a wide class of Banach $ell(B(I,G))$-bimodules $E$, and every $n>0$, the bounded Hochschild cohomology groups $H^n(ell(B(I,G)),E^*)$ are trivial, and also, the notion of approximate amenability is not Morita invariant.
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