Spectral gap actions and invariant states

2013/04/26 by Han Li, Li, Han, Chi-Keung Ng +1
Mathematics · #22A10 #43A65 #46L55 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:22A10 #msc:43A65 #msc:46L55

paper · pdf · doi:10.48550/arxiv.1304.7051

10 pages; to appear in Int. Math. Res. Notices

arxiv created 2013/04/26 · arxiv updated 2013/04/29

Abstract

We define spectral gap actions of discrete groups on von Neumann algebras and study their relations with invariant states. We will show that a finitely generated ICC group Γ is inner amenable if and only if there exist more than one inner invariant states on the group von Neumann algebra L(Γ). Moreover, a countable discrete group Γ has property (T) if and only if for any action α of Γ on a von Neumann algebra N, every α-invariant state on N is a weak-^*-limit of a net of normal α-invariant states.

Related

• Strongly ergodic actions have local spectral gap
• Discrete amenable group actions on von Neumann algebras and invariant nuclear C*-subalgebras
• Spaces invariant under unitary representations of discrete groups
• Spectral gap property and strong ergodicity for groups of affine\n transformations of solenoids
• Spectral gap for products and a strong normal subgroup theorem