Full Factors and Co-amenable Inclusions

Abstract

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 $$\sigma :G \curvearrowright M$$ 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.