2014/01/14 by Jan Cameron, Cameron, Jan, Roger R. Smith +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.1401.3355
In this paper, we study bimodules over a von Neumann algebra M in two related contexts. The first is an inclusion M ⊆ M \rtimesαG, where G is a discrete group acting on a factor M by outer automorphisms. The second is a regular inclusion M ⊆ N of finite factors. In the case of crossed products, we characterize the M-bimodules X that lie between M and M \rtimesαG and 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 isometric maps on X. We establish a similar theorem for bimodules arising from regular inclusions of finite factors, which generalizes the crossed product situation when G acts on a finite factor. In the final section we apply these ideas to provide new examples of singly generated finite factors.