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

TL;DR

This work addresses when a continuous action $\alpha$ of a locally compact group $G$ on a factor $M$ induces a strictly outer crossed product, by locating the relative commutant $M'\cap(M\rtimes_{\alpha} G)$ within a subgroup crossed product $M\rtimes_{\alpha} H$ where $H=\{g\in G : L^{2}(\mathrm{id}) \prec L^{2}(\alpha_g)\}$. The authors develop a bimodule-based framework and a uniform spectral-gap principle for weak containment of $L^{2}(\mathrm{id})$ in $L^{2}(\alpha_g)$ across neighborhoods in $G$, and use this to prove the key containment. Consequently, they obtain that for a semifinite $M$, if each $\alpha_g$ with $g\neq 1$ is not approximately inner, then $M'\cap(M\rtimes_{\alpha} G)=\mathbb{C}$, and they deduce a corollary for topologically outer actions on σ-finite type II factors, extending Connes-Takesaki type results beyond the full-factor setting. The results provide new tools to analyze the rigidity of dynamical inclusions and establish strict outerness in broader contexts, linking spectral properties of automorphisms to centralizers in crossed products.

Abstract

We show that, given a continuous action $α$ of a locally compact group $G$ on a factor $M$, the relative commutant $M'\cap(M\rtimes_α G)$ is contained in $M\rtimes_α H$ 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 $α_g$ is not approximately inner for all $g\neq 1$, then $M'\cap (M\rtimes_α G)=\mathbb{C}$.

Theorems & Definitions (25)

• Theorem 1.1: marrakchivaesSpectral
• Theorem 1
• Corollary 2
• Theorem 3
• Proposition 2.1
• proof
• Proposition 2.2: popa1986correspondences
• Proposition 2.3: connes1994noncommutative
• Proposition 2.4
• proof