A remark on fullness of some group measure space von Neumann algebras

Abstract

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.