MORITA TYPE EQUIVALENCES AND REFLEXIVE ALGEBRAS

Two unital dual operator algebras A, B are called Delta-equivalent if there exists an equivalence functor F : A(m) -> B-m which "extends" to a *-functor implementing an equivalence between the categories A(Dm) and B-Dm. Here A(m) denotes the category of normal representations of A and A(Dm) denotes the category with the same objects as A(m) and Delta(A)-module maps as morphisms (Delta(A) = A boolean AND A*). We prove that any such functor maps completely isometric representations to completely isometric representations, "respects" the lattices of the algebras and maps reflexive algebras to reflexive algebras. We present applications to the class of CSL algebras.