A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

Let G be a locally compact quantum group and (M,α) a G-W^⁎-algebra. The object of study of this paper is the W^⁎-category Rep^G(M) of normal, unital G-representations of M on Hilbert spaces endowed with a unitary G-representation. This category has a right action of the category Rep(G)=Rep^G(C) for which it becomes a right Rep(G)-module W^⁎-category. Given another G-W^⁎-algebra (N,β), we denote the category of normal ⁎-functors Rep^G(N)→Rep^G(M) compatible with the Rep(G)-module structure by Fun_Rep(G)(Rep^G(N),Rep^G(M)) and we denote the category of G-M-N-correspondences, studied in [5], by Corr^G(M,N). We prove that there are canonical functors P:Corr^G(M,N)→Fun_Rep(G)(Rep^G(N),Rep^G(M)) and Q:Fun_Rep(G)(Rep^G(N),Rep^G(M))→Corr^G(M,N) such that Q∘P≅id. We use these functors to show that the G-dynamical von Neumann algebras (M,α) and (N,β) are equivariantly Morita equivalent if and only if Rep^G(N) and Rep^G(M) are equivalent as Rep(G)-module-W^⁎-categories. Specializing to the case where G is a compact quantum group, we prove that moreover P∘Q≅id, so that the categories Corr^G(M,N) and Fun_Rep(G)(Rep^G(N),Rep^G(M)) are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras.