Spectral gap characterization of full type III factors

Abstract

Abstract We give a spectral gap characterization of fullness for type III {\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 σ : G → Aut ⁢ ( M ) {\sigma:G\rightarrow\mathrm{Aut}(M)} is an outer action of a discrete group G whose image in Out ⁢ ( M ) {\mathrm{Out}(M)} is discrete, then the crossed product von Neumann algebra M ⋊ σ G {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 III 1 {\mathrm{III}_{1}} factor M is full if and only if M is full and its τ invariant is the usual topology on ℝ {\mathbb{R}} .