Strongly ergodic actions have local spectral gap

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 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.

6 pages