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

Abstract

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 $\Gamma ⋉L^{\infty }(X)$ 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 $\Gamma ⋉L^{\infty }(G/H)$ is full.

8 pages; minor corrections (v2); more corrections (v3); Compos. Math., to appear

Cited by in corpus (6)

• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• Stationary characters on lattices of semisimple Lie groups
• Strongly ergodic equivalence relations: spectral gap and type III invariants
• The approximation property and exactness of locally compact groups
• On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras
• Fullness of crossed products of factors by discrete groups