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 subset of M x(alpha), G, where G is a discrete group acting on a factor M by outer *-automorphisms. We characterize the M-bimodules X subset of M x(alpha) G 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. (C) 2015 Elsevier Inc. All rights reserved.