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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2017: 1.686

CiteScore 2017: 0.96

SCImago Journal Rank (SJR) 2017: 2.585
Source Normalized Impact per Paper (SNIP) 2017: 1.203

Mathematical Citation Quotient (MCQ) 2016: 1.28

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1435-5345
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# Spectral gap characterization of full type III factors

Amine Marrakchi
• École Normale Supérieure, 45 rue d’Ulm 75230 Paris Cedex 05, France; and Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/crelle-2016-0071

## 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 $\sigma :G\to \mathrm{Aut}\left(M\right)$ is an outer action of a discrete group G whose image in $\mathrm{Out}\left(M\right)$ is discrete, then the crossed product von Neumann algebra $M{⋊}_{\sigma }G$ 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 τ invariant is the usual topology on $ℝ$.

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Revised: 2016-11-08

Published Online: 2017-01-12

Funding Source: H2020 European Research Council

Award identifier / Grant number: GAN 637601

The research is supported by ERC Starting Grant GAN 637601.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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