# Full factors and coamenable inclusions

### Jon Bannon (Siena College)

Thursday 23rd May, 2019 16:00-17:00 Maths 311B

#### Abstract

If a factor is full then every approximately inner automorphism is inner. We have the following generalization: If a factor $M$ is full, then the any correspondence of $M$ weakly equivalent to the trivial correspondence must contain the trivial correspondence. This generalization provides a perspective that allows us to prove that if $N\subset M$ is a coamenable inclusion with faithful, normal conditional expectation and $M$ a full factor, then there is a nonzero $p\in N'\cap M$ so that $pN$ is full. As a corollary, this affirmatively answers Problem 3.3.2 of Popa's correspondences preprint from 1986. Furthermore, we use this to prove that if a compact group $G$ acts on a full factor $M$ then the action is minimal, i.e. $(M^{G})'\cap M=\mathbb{C}1$. Consequently, for any such action by results of Izumi, Longo and Popa there is a Galois correspondence between the intermediate subalgebras of $M^{G}\subset M$ and fixed point subalgebras of the closed subgroups of $G$