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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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