Spectral gap characterization of full type III factors

TL;DR

This paper characterizes fullness for type III factors using spectral gaps, extends Jones's theorem on crossed products, and proves a conjecture relating the fullness of the continuous core to the original factor and its tau invariant.

Contribution

It introduces a spectral gap criterion for type III factors, generalizes Jones's theorem on crossed products, and proves a conjecture about the fullness of the continuous core.

Findings

Spectral gap characterization of fullness for type III factors.

Fullness of crossed products under outer actions with discrete outer automorphism image.

Fullness criterion for the continuous core of type III_1 factors based on the tau invariant.

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 $\sigma : G \rightarrow \mathrm{Aut}(M)$ is an outer action of a discrete group $G$ whose image in $\mathrm{Out}(M)$ is discrete then the crossed product von Neumann algebra $M \rtimes_\sigma G$ 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 $\mathbb{R}$.