Structure of $\mathrm{I}{\mathrm{I}}_1$ factors arising from free Bogoljubov actions of arbitrary groups

Abstract

In this paper, we investigate several structural properties for crossed product $\mathrm{I}{\mathrm{I}}_1$ 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 $M$.

34 pages; v3: final version, simplified exposition and proof of Theorem B corrected

References in corpus (1)

• W*-superrigidity for group von Neumann algebras of left-right wreath products

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