  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 $\mathbb{R}$.
