On relative commutants of subalgebras in group and tracial crossed product von Neumann algebras

Abstract

Let $\Gamma$ be a discrete group acting on a compact Hausdorff space $X$. Given $x\in X$, and $\mu \in \text{Prob}(X)$, we introduce the notion of contraction of $\mu$ towards $x$ with respect to unitary elements of a group von Neumann algebra not necessarily coming from group elements. Using this notion, we study relative commutants of subalgebras in tracial crossed product von Neumann algebras. The results are applied to negatively curved groups and $\text{SL}(d,\mathbb{Z})$, $d\ge 2$.

This is the final version. All the suggestions of the referees have been implemented. To appear in the Pacific Journal of Mathematics

References in corpus (4)

• Charmenability of arithmetic groups of product type
• Quasi-regular representations of discrete groups and associated C*-algebras
• A generalized Powers averaging property for commutative crossed products
• Subalgebras, subgroups, and singularity

Cited by in corpus (1)

• Invariant $C^{\ast }$-subalgebras of the reduced group $C^{\ast }$-algebra