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Spectral gap characterization of full type III factors. (English) Zbl 1433.46040
Summary: We give a spectral gap characterization of fullness for type 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 $$\text{III}_1$$ factor $$M$$ is full if and only if $$M$$ is full and its $$\tau$$ invariant is the usual topology on $$\mathbb{R}$$.

##### MSC:
 46L36 Classification of factors 46L40 Automorphisms of selfadjoint operator algebras 46L55 Noncommutative dynamical systems
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