Strongly ergodic actions have local spectral gap

Amine Marrakchi

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arXiv · 1707.00438

Strongly ergodic actions have local spectral gap

Amine Marrakchi

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Abstract

We show that an ergodic measure preserving action $Γ\curvearrowright (X,μ)$ of a discrete group $Γ$ on a $σ$-finite measure space $(X,μ)$ satisfies the local spectral gap property (introduced by Boutonnet, Ioana and Salehi Golsefidy) 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 and elementary proof of Connes' spectral gap theorem for full $\mathrm{II}_1$ factors as well as its recent generalization to full type $\mathrm{III}$ factors.

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BibTeX RIS

Amine Marrakchi. 2017-09-15. Strongly ergodic actions have local spectral gap. https://arxiv.org/abs/1707.00438

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