A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W∗W∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,BA,B have completely isometric normal representations α,βα,β such that α(A)=[M∗β(B)M]−w∗α(A)=[M∗β(B)M]−w∗ and β(B)=[Mα(A)M∗]−w∗β(B)=[Mα(A)M∗]−w∗ for a ternary ring of operators MM (i.e. a linear space MM such that MM∗M⊂MMM∗M⊂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 AA and by ADM the category with the same objects as AM and Δ(A)Δ(A)-module maps as morphisms (Δ(A)=A∩A∗Δ(A)=A∩A∗). We prove that this functor is equivalent to a functor “generated” by a B,AB,A bimodule, and that it is normal and completely isometric.