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

We show that, given a continuous action $\alpha$ of a locally compact group $G$ on a factor $M$, the relative commutant $M^{\mathrm{\prime }}\cap (M{⋊}_{\alpha }G)$ is contained in $M{⋊}_{\alpha }H$ 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 $g\mathrm{\ne }1$, then $M^{\mathrm{\prime }}\cap (M{⋊}_{\alpha }G)=\mathbb{C}$.