A Morita Type Equivalence for Dual Operator Algebras

Abstract

We generalize the main theorem of Rieffel for Morita equivalence of W*-algebras to the case of unital dual operator algebras: two unital dual operator algebras A and B have completely isometric normal representations alpha, beta such that alpha(A) is the w*-closed span of M*beta(B)M and beta(B) is the w*-closed span of Malpha(A)M* for a ternary ring of operators M (i.e. a linear space M such that MM*M \subset M if and only if there exists an equivalence functor $F:_{A}M\to_{B}M$ which "extends" to a *-functor implementing an equivalence between the categories $_{A}DM$ and $_{B}DM.$ By $_{A}M$ we denote the category of normal representations of A and by $_{A}DM$ the category with the same objects as $_{A}M$ and $Δ(A)$-module maps as morphisms ($Δ(A)=A\cap A^*$). We prove that this functor is equivalent to a functor "generated" by a B, A bimodule, that it is normal and completely isometric.