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Spectral gap characterization of full type III factors

  • Amine Marrakchi EMAIL logo
Published/Copyright: January 12, 2017
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2019 Issue 753

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:G→Aut⁢(M) is an outer action of a discrete group G whose image in Out⁢(M) is discrete, then the crossed product von Neumann algebra M⋊σG 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 ℝ.

Funding source: H2020 European Research Council

Award Identifier / Grant number: GAN 637601

Funding statement: The research is supported by ERC Starting Grant GAN 637601.

Acknowledgements

We are very grateful to our advisor Cyril Houdayer for attracting our attention to this problem and for his help and suggestions throughout this work. We also thank Yoshimichi Ueda for explaining to us [15, Lemma 6] and for his useful comments.

References

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Received: 2016-06-08
Revised: 2016-11-08
Published Online: 2017-01-12
Published in Print: 2019-08-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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  1. Frontmatter
  2. Homotopy techniques for tensor decomposition and perfect identifiability
  3. Schottky groups acting on homogeneous rational manifolds
  4. Limit linear series for curves not of compact type
  5. Uniform congruence counting for Schottky semigroups in SL2(𝐙)
  6. The dualizing sheaf on first-order deformations of toric surface singularities
  7. Embedded minimal surfaces of finite topology
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https://doi.org/10.1515/crelle-2016-0071

Articles in the same Issue

  1. Frontmatter
  2. Homotopy techniques for tensor decomposition and perfect identifiability
  3. Schottky groups acting on homogeneous rational manifolds
  4. Limit linear series for curves not of compact type
  5. Uniform congruence counting for Schottky semigroups in SL2(𝐙)
  6. The dualizing sheaf on first-order deformations of toric surface singularities
  7. Embedded minimal surfaces of finite topology
  8. Spectral gap characterization of full type III factors
  9. On the universal cover and the fundamental group of an RCD*(K,N)-space
  10. Calabi–Yau and fractional Calabi–Yau categories
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