Fullness of crossed products of factors by discrete groups

Abstract

Let $M$ be an arbitrary factor and $σ: Γ\curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes_σΓ$. When $Γ$ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes_σΓ$ is full if and only if $M$ is full and the quotient map $\overlineσ : Γ\rightarrow \mathrm{Out}(M)$ has finite kernel and discrete image. This answers a question of Jones from 1981. When $M$ is full and $Γ$ is arbitrary, we give a sufficient condition for $M \rtimes_σΓ$ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if $M$ is any full factor (possibly of type $\mathrm{III}$) and $Γ$ is a non-inner amenable group, then the crossed product $M \rtimes_σΓ$ is full.