Asymptotic Abelianness, weak mixing, and property T

Abstract

Let G be a second countable locally compact group and H a closed subgroup. We characterize the lack of Kazhdan's property T for the pair (G,H) by the genericity of G-actions on the hyperfinite II1 factor with a certain asymptotic Abelianness property relative to H, as well as by the genericity of measure-preserving G-actions on a nonatomic standard probability space that are weakly mixing for H. The latter furnishes a definitive generalization of a classical theorem of Halmos for single automorphisms and strengthens a recent result of Glasner, Thouvenot, and Weiss on generic ergodicity. We also establish a weak mixing version of Glasner and Weiss's characterization of property T for discrete G in terms of the invariant state space of a Bernoulli shift and show that on the CAR algebra a type of norm asymptotic Abelianness is generic for G-actions when G is discrete and admits a nontorsion Abelian quotient.