Strictly outer actions of locally compact groups: beyond the full factor case
Abstract
We show that, given a continuous action $\alpha$ of a locally compact group $G$ on a factor $M$, the relative commutant $M'\cap(M\rtimes_{\alpha} G)$ is contained in $M\rtimes_{\alpha} H$ where $H$ is the subgroup of elements acting without spectral gap. As a corollary, we answer a question of Marrakchi and Vaes by showing that if $M$ is semifinite and $\alpha_g$ is not approximately inner for all $g\neq 1$, then $M'\cap (M\rtimes_{\alpha} G)=\mathbb{C}$.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.11738
- arXiv:
- arXiv:2407.11738
- Bibcode:
- 2024arXiv240711738M
- Keywords:
-
- Mathematics - Operator Algebras
- E-Print:
- 12 pages