A class of 𝐼𝐼₁ factors with an exotic abelian maximal amenable subalgebra

Houdayer, Cyril

doi:10.1090/S0002-9947-2014-05964-3

A class of $\textrm {II_1}$ factors with an exotic abelian maximal amenable subalgebra
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Trans. Amer. Math. Soc. 366 (2014), 3693-3707 Request permission

Abstract:

We show that for every mixing orthogonal representation $\pi : \mathbf {Z} \to \mathcal O(H_{\mathbf {R}})$, the abelian subalgebra $\mathrm {L}(\mathbf {Z})$ is maximal amenable in the crossed product $\mathrm {II}_1$ factor $\Gamma (H_{\mathbf {R}})'' \rtimes _\pi \mathbf {Z}$ 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.

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