A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

Abstract

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 RepG(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)=RepG(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 RepG(N)→RepG(M) compatible with the Rep(G)-module structure by FunRep(G)(RepG(N),RepG(M)) and we denote the category of G-M-N-correspondences, studied in [5], by CorrG(M,N). We prove that there are canonical functors P:CorrG(M,N)→FunRep(G)(RepG(N),RepG(M)) and Q:FunRep(G)(RepG(N),RepG(M))→CorrG(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 RepG(N) and RepG(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 CorrG(M,N) and FunRep(G)(RepG(N),RepG(M)) are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras.