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 $\text{SL}(d,\mathbb{Z})$, $d\ge 2$.

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

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Cited by in corpus (1)

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