Strongly ergodic actions have local spectral gap
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- by Amine Marrakchi PDF
- Proc. Amer. Math. Soc. 146 (2018), 3887-3893 Request permission
Abstract:
We show that an ergodic measure preserving action $\Gamma \curvearrowright (X,\mu )$ of a discrete group $\Gamma$ on a $\sigma$-finite measure space $(X,\mu )$ satisfies the local spectral gap property introduced in Invent. Math. 208 (2017), 715–802, if and only if it is strongly ergodic. In fact, we prove a more general local spectral gap criterion in arbitrary von Neumann algebras. Using this criterion, we also obtain a short proof of Connes’ spectral gap theorem for full $\mathrm {II}_1$ factors as well as its recent generalization to full type $\mathrm {III}$ factors.References
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Additional Information
- Amine Marrakchi
- Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
- MR Author ID: 1198217
- Email: amine.marrakchi@math.u-psud.fr
- Received by editor(s): September 29, 2017
- Received by editor(s) in revised form: November 17, 2017
- Published electronically: May 24, 2018
- Additional Notes: The author was supported by ERC Starting Grant GAN 637601
- Communicated by: Adrian Ioana
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 3887-3893
- MSC (2010): Primary 37A05, 37A30, 46L10
- DOI: https://doi.org/10.1090/proc/14034
- MathSciNet review: 3825842