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

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