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