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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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