Spectral gap characterization of full type III factors

Abstract

We give a spectral gap characterization of fullness for type $\mathrm{III}$ factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if $M$ is a full factor and $\sigma :G\to \mathrm{Aut}(M)$ is an outer action of a discrete group $G$ whose image in $\mathrm{Out}(M)$ is discrete then the crossed product von Neumann algebra $M{⋊}_{\sigma }G$ is also a full factor. We apply this result to prove the following conjecture of Tomatsu-Ueda: the continuous core of a type ${\mathrm{III}}_1$ factor $M$ is full if and only if $M$ is full and its $\tau$ invariant is the usual topology on $\mathbb{R}$.

arXiv comment13 pages