This work is a generalization of Rieffel’s characterization of Morita equivalence for W∗ - algebras to the case of unital dual operator algebras.

We provide characterizations of TRO equivalence of two unital dual operator algebras A and B that is, of the existence of completely isometric normal representations α and β such that α(A) = [M∗β(B)M] −w∗ and β(B) = [Mα(A)M∗ ] −w∗ for a ternary ring of operators M.

The first characterization is in terms of the equivalence between appropriate categories of completely contractive normal representations of the algebras, where the morphisms are required to intertwine the representations as well as their restrictions to the diagonals. The second characterization, obtained jointly with V.I. Paulsen, is in terms of stable isomorphism. We present applications of this theory to the class of reflexive algebras, especially to CSL algebras.