Spectral gap characterization of full type III factors

Amine Marrakchi 1
  • 1 École Normale Supérieure, 45 rue d’Ulm 75230, Paris, France
Amine Marrakchi
  • Corresponding author
  • 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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Abstract

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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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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