Strictly outer actions of locally compact groups: beyond the full factor case

TL;DR

This paper investigates the structure of relative commutants in crossed product factors under group actions, revealing conditions under which these commutants are contained in certain subalgebras and when they are trivial.

Contribution

It extends the understanding of outer actions of locally compact groups on factors, especially beyond the full factor case, by characterizing the relative commutant structure.

Findings

The relative commutant is contained in the crossed product with the subgroup acting without spectral gap.

If the action is not approximately inner for all non-identity elements, the relative commutant is trivial.

Answers a question of Marrakchi and Vaes regarding the structure of relative commutants in this setting.

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 $M'\cap (M\rtimes_{\alpha} G)=\mathbb{C}$.