A class of $\mathrm{I}{\mathrm{I}}_1$ factors with an exotic abelian maximal amenable subalgebra

Abstract

We show that for every mixing orthogonal representation $\pi :\mathbb{Z}\to \mathcal{O}(H_{\mathbb{R}})$, the abelian subalgebra $\LL(\Z)$ is maximal amenable in the crossed product ${\mathrm{II}}_1$ factor $Γ(H_\R)\dpr \rtimes_π\Z$ 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 $\LL^2(M \ominus A)$ where $A\subset M$ is a maximal amenable masa in a $\mathrm{I}{\mathrm{I}}_1$ factor.

16 pages

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