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A class of $$II_{1}$$ factors with an exotic abelian maximal amenable subalgebra. (English) Zbl 1303.46044
Summary: We show that for every mixing orthogonal representation $$\pi : \mathbb{Z} \to \mathcal O(H_{\mathbb{R}})$$, the abelian subalgebra $$\mathrm {L}(\mathbb{Z})$$ is maximal amenable in the crossed product $$\mathrm{II}_1$$ factor $$\Gamma (H_{\mathbb{R}})^{\prime \prime } \rtimes _\pi \mathbb{Z}$$ associated with the free Bogoljubov action of the representation $$\pi$$. This provides uncountably many non-isomorphic $$A$$-$$A$$-bimodules which are disjoint from the coarse $$A$$-$$A$$-bimodule and of the form $$\mathrm {L}^2(M \ominus A)$$, where $$A \subset M$$ is a maximal amenable masa in a $$\mathrm{II}_1$$ factor.

##### 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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