Strongly ergodic actions have local spectral gap

2017/07/03 by Marrakchi, Amine · 1 citation
#37A05 #37A30 #46L10 #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1707.00438

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 II₁ factors as well as its recent generalization to full type III factors.

Cited by

• Ergodic Generative Flows

Related

• Strongly ergodic equivalence relations: spectral gap and type III invariants
• A Spectral Strong Approximation Theorem for Measure Preserving Actions
• Local spectral gap in simple Lie groups and applications
• Spectral rigidity of group actions on homogeneous spaces
• Spectral gap actions and invariant states