Bimodules in crossed products and regular inclusions of finite factors

In this paper, we study bimodules over a von Neumann algebra $M$ in two related contexts. The first is an inclusion $M \subseteq M \rtimes_\alpha G$, where $G$ is a discrete group acting on a factor $M$ by outer automorphisms. The second is a regular inclusion $M \subseteq N$ of finite factors. In the case of crossed products, we characterize the $M$-bimodules $X$ that lie between $M$ and $M \rtimes_\alpha G$ 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.