A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

GRep CorrLet be a locally compact quantum group and (M, ) a -W^*-algebra. The object of study of this paper is the W^*-category ^ (M) of normal, unital -representations of M on Hilbert spaces endowed with a unitary -representation. This category has a right action of the category () = ^ (C) for which it becomes a right () -module W^*-category. Given another -W^*-algebra (N, ), we denote the category of normal *-functors ^ (N) ^ (M) compatible with the () -module structure by Fun () (^ (N), ^ (M) ) and we denote the category of -M-N-correspondences by Corr^ (M, N). We prove that there are canonical functors P: ^ (M, N) Fun () (^ (N), ^ (M) ) and Q: Fun () (^ (N), ^ (M) ) Corr^ (M, N) such that Q P id. We use these functors to show that the -dynamical von Neumann algebras (M, ) and (N, ) are equivariantly Morita equivalent if and only if ^ (N) and ^ (M) are equivalent as () -module-W^*-categories. Specializing to the case where is a compact quantum group, we prove that moreover P Q id, so that the categories ^ (M, N) and Fun () (^ (N), ^ (M) ) are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras.