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Journal für die reine und angewandte Mathematik

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

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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


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 σ:GAut(M) is an outer action of a discrete group G whose image in Out(M) is discrete, then the crossed product von Neumann algebra MσG is also a full factor. We apply this result to prove the following conjecture of Tomatsu–Ueda: the continuous core of a type III1 factor M is full if and only if M is full and its τ invariant is the usual topology on .


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About the article

Received: 2016-06-08

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, DOI: https://doi.org/10.1515/crelle-2016-0071.

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