Structure of ${\rm II_1}$ factors arising from free Bogoljubov actions of arbitrary groups

Abstract

In this paper, we investigate several structural properties for crossed product ${\rm II_1}$ factors $M$ arising from free Bogoljubov actions associated with orthogonal representations $\pi : G \to \mathcal O(H_\mathbf R)$ of arbitrary countable discrete groups. Under fairly general assumptions on the orthogonal representation $\pi : G \to \mathcal O(H_{\mathbf R})$, we show that $M$ does not have property Gamma of Murray and von Neumann. Then we show that any regular amenable subalgebra $A \subset M$ can be embedded into $L(G)$ inside $M$. Finally, when $G$ is assumed to be amenable, we locate precisely any possible amenable or Gamma extension of $L(G)$ inside $M$.