Fullness of crossed products of factors by discrete groups

Abstract: Let $M$ be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes_\sigma \Gamma$. When $\Gamma$ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes_\sigma \Gamma$ is full if and only if $M$ is full and the quotient map $\overline{\sigma} : \Gamma \rightarrow \mathrm{Out}(M)$ has finite kernel and discrete image. This answers a question of Jones from 1981. When $M$ is full and $\Gamma$ is arbitrary, we give a sufficient condition for $M \rtimes_\sigma \Gamma$ 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 $\Gamma$ is a non-inner amenable group, then the crossed product $M \rtimes_\sigma \Gamma$ is full.