# Spectral gap characterization of full type III factors

Spectral gap characterization of full type III factors AbstractWe 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⋊σG{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 III1{\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}}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik de Gruyter

# Spectral gap characterization of full type III factors

Journal für die reine und angewandte Mathematik, Volume 2019 (753): 18 – Aug 1, 2019
18 pages

/lp/de-gruyter/spectral-gap-characterization-of-full-type-iii-factors-ygtG7H08e5
Publisher
de Gruyter
© 2019 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5345
eISSN
1435-5345
DOI
10.1515/crelle-2016-0071
Publisher site
See Article on Publisher Site

### Abstract

AbstractWe 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⋊σG{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 III1{\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}}.

### Journal

Journal für die reine und angewandte Mathematikde Gruyter

Published: Aug 1, 2019

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