Asymptotic Abelianness, weak mixing, and property T

Abstract

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.

Turbulence, representations, and trace-preserving actions

“…, n and a ∈ Ω, Proof. By [35,Thm. 3.8], the G δ subset WM(G, R) of Act(G, R) is dense precisely when G does not have property T. Thus, since meagerness of orbits is part of the definition of generic turbulence, it suffices to show (1)⇒(2), and this follows by Lemma 5.10.…”

“…Note that when G does not have property T the set of weakly mixing actions in Act(G, R) is a dense G δ [35], and so when G is countably infinite and amenable the free weakly mixing actions form an Aut(R)-invariant dense G δ subset of Fr(G, R) seeing that the latter is a dense G δ in Act(G, R) by Lemma 5.1.…”

The Haagerup property for locally compact quantum groups

“…The following lemma abstracts calculations used in the classical situation for establishing density of mixing representations in [8] and weak mixing representations in [46]. (ii) is stable under tensoring with another representation, i.…”

Mixing actions of groups with the Haagerup approximation property

“…The next lemma, with its cutting and splicing along almost invariant sets, has obvious parallels with the arguments of [8].…”

“…In answer to a question of Bergelson and Rosenblatt: Theorem 1.4 (Kerr-Pichot; [8]). A countable group Γ does not have property (T) if and only if the weak mixing actions on (X, µ) are a dense G δ in the space of all actions.…”