Amenable actions and weak containment of certain representations of discrete groups

Abstract

Abstract. We consider a countable discrete group F acting ergodically on a standard Borel space S with quasi-invariant measure p . Let n be a unitary representation of T on L2(S, dp, «#") "nicely" related with S. We prove that if T acts amenably on S then n is weakly contained in the regular representation.

Transference for amenable actions

“…When p = 2 and G acts amenably on X the same inequalities are true: see [1], Theorem 3.2.3 (for discrete groups see also [5] for ergodic actions and [6] for general amenable actions including semisimple algebraic groups).…”

“…When G is discrete and ergodic on X, the existence of such a sequence was essentially the proof (given in [5]) that the quasi-regular representation of G on X is weakly contained into the regular representation of G. In [1] it is proved that the above condition is in fact equivalent to the amenabilty of the G action on X for any locally compact second-countable group.…”

A Spectral Gap Property for Random Walks Under Unitary Representations

“…The following remarkable result was shown in [Anan03, Corollary 3.2.2]: r spec (λ X (µ)) = r spec (λ G (µ)) for any adapted probability measure µ on G (the case of a discrete group was previously treated in [Kuhn94]). In particular, if G is non amenable, then r spec (λ X (µ)) < 1, by Corollary 6.…”

Sharp lower bounds for the asymptotic entropy of symmetric random walks

“…A theorem of Kuhn [Kuh94] (valid for ergodic amenable actions) implies that the representation π is weakly contained in the regular representation π reg , which means that for every f ∈ L 2 (B 0 , ν 0 ), there exists a sequence {f n } n∈N ⊂ ℓ 2 (Γ) such that lim n π reg (γ)f n , f n = π(γ)f, f for every γ ∈ Γ.…”

“…Acknowledgements: The authors gladly thank Bachir Bekka for pointing out to them the references [Zim78] and [Kuh94], and Peter Haïssinsky for letting them reproduce the above short proof of the fundamental inequality.…”