A class of ${\rm II_1}$ factors with an exotic abelian maximal amenable subalgebra
Cyril Houdayer
TL;DR
The paper proves that for any mixing orthogonal representation $π$ of $\mathbb{Z}$, the generator MASA $L(\mathbb{Z})$ is maximal amenable inside the II$_1$ factor $M = Γ(H_{\mathbb{R}})'' \rtimes_π \mathbb{Z}$ produced by Voiculescu's free Gaussian functor, via Popa's asymptotic orthogonality property. This generalizes Popa's original AOP to all mixing representations and yields uncountably many pairwise nonisomorphic $A$-$A$ bimodules disjoint from the coarse bimodule, realized as $L^2(M \ominus A)$ for exotic masas. The authors further construct a continuum of such bimodules through symmetric Rajchman measures on the circle, producing II$_1$ factors not isomorphic to interpolated free group factors and establishing strong solidity. Together, these results provide new invariants and a rich supply of exotic maximal amenable masas in II$_1$ factors, extending the landscape of known maximal amenable subalgebras.
Abstract
We show that for every mixing orthogonal representation $π: \Z \to \mathcal O(H_\R)$, the abelian subalgebra $\LL(\Z)$ is maximal amenable in the crossed product ${\rm II}_1$ factor $Γ(H_\R)\dpr \rtimes_π\Z$ associated with the free Bogoljubov action of the representation $π$. 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 ${\rm II_1}$ factor.