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Strongly ergodic actions have local spectral gap


Author: Amine Marrakchi
Journal: Proc. Amer. Math. Soc. 146 (2018), 3887-3893
MSC (2010): Primary 37A05, 37A30, 46L10
DOI: https://doi.org/10.1090/proc/14034
Published electronically: May 24, 2018
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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 $ \textup {II}_1$ factors as well as its recent generalization to full type $ \textup {III}$ factors.


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Additional Information

Amine Marrakchi
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
Email: amine.marrakchi@math.u-psud.fr

DOI: https://doi.org/10.1090/proc/14034
Keywords: Spectral gap, strongly ergodic, group action, full factor, maximality argument
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
Article copyright: © Copyright 2018 American Mathematical Society