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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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