Strongly ergodic actions have local spectral gap

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We show that an ergodic measure preserving action Γ (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 II₁ factors as well as its recent generalization to full type III factors.