Full factors and co-amenable inclusions

We show that if $M$ is a full factor and $N \subset M$ is a co-amenable subfactor with expectation, then $N$ is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if $M$ is a full factor and $\sigma \colon G \curvearrowright M$ is an outer action of a compact group $G$, then $\sigma$ is automatically minimal and $M^G$ is a full factor which has w-spectral gap in $M$. Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion $N\subset M$ of von Neumann algebras are equivalent, thus closing the cycle of implications given in Anantharaman-Delaroche's paper in 1995.