A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

GRep CorrLet be a locally compact quantum group and (M, α) 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, β), 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, α) and (N, β) 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.