A Morita Type Equivalence for Dual Operator Algebras

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,B have completely isometric normal representations α, β such that α(A) = [Mβ(B)M] ∗ and β(B) = [Mα(A)M] ∗ for a ternary ring of operators M (i.e. a linear space M such that MMM ⊂ M) if and only if there exists an equivalence functor F : AM → BM which “extends” to a ∗−functor implementing an equivalence between the categories ADM and BDM. By AM we denote the category of normal representations of A and by ADM the category with the same objects as AM and ∆(A)-module maps as morphisms (∆(A) = A ∩ A). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, that it is normal, completely isometric, maps completely isometric representations to completely isometric representations, “respects” the lattices of the algebras and maps reflexive algebras to reflexive algebras.