Bimodules in crossed products of von Neumann algebras

In this paper, we study bimodules over a von Neumann algebra M in the context of an inclusion M⊆M⋊αG, where G is a discrete group acting on a factor M by outer ⁎-automorphisms. We characterize the M-bimodules X⊆M⋊αG that are closed in the Bures topology in terms of the subsets of G. We show that this characterization also holds for w⁎-closed bimodules when G has the approximation property (AP), a class of groups that includes all amenable and weakly amenable ones. As an application, we prove a version of Mercer's extension theorem for certain w⁎-continuous surjective isometric maps on X.