OA seminar: Jon P Bannon

Speaker: Jon Bannon (Sienna College).

Title: Full factors and co-amenable inclusions

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 co-amenable 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