Asymptotic Galois correspondence for discrete amenable group actions on factors

Abstract

If R is an AFD factor of non-type I and α is an ultrafree action of a discrete amenable group G on R , then for an automorphism θ of R the following two conditions are equivalent: 1. (i) θ is of the form θ = α g0 for some g 0 ϵ G; 2. (ii) If { x n } is a bounded sequence in R such that { α g ( x n ) − x n } converges to zero σ ∗-strongly for every g ϵ G, then { θ( x n ) − x n } also converges to zero σ ∗-strongly.