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Structure of $$\mathrm{II}_{1}$$ factors arising from free Bogoljubov actions of arbitrary groups. (English) Zbl 1297.46042
Summary: In this paper, we investigate several structural properties for crossed product $$\mathrm{II}_{1}$$ 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 $$M$$.

##### MSC:
 46L10 General theory of von Neumann algebras 46L54 Free probability and free operator algebras 46L55 Noncommutative dynamical systems 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations
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