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 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.