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

Tattwamasi Amrutam and Jacopo Bassi

Vol. 331 (2024), No. 1, 1–22

DOI: 10.2140/pjm.2024.331.1

Abstract

Let $\Gamma$ be a discrete group acting on a compact Hausdorff space $X$. Given $x\in X$ and $\mu \in \mathrm{Prob}<!--FUNCTION\; APPLICATION-->(X)$, 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 $\mathrm{SL}<!--FUNCTION\; APPLICATION-->(d,\mathbb{Z})$ for $d\ge 2$.

Keywords

relative commutants, tracial crossed product, contracting sequence

Mathematical Subject Classification

Primary: 22D25, 37A55, 46L10, 46L55, 47C15

Milestones

Received: 23 March 2024

Revised: 9 July 2024

Accepted: 1 August 2024

Published: 2 October 2024