Strongly ergodic actions have local spectral gap

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2018-09, Vol.146 (9), p.3887-3893
Main Author: MARRAKCHI, AMINE
Format: Article
Language:English
Subjects:
B. ANALYSIS
Mathematics
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/proc/14034