Frame Vector Group Representations and Amenability Properties

2025/08/27 by Dutkay, Dorin Ervin, Georgescu, Catalin, Picioroaga, Gabriel
#20F65 #22D10 #42C15 #46C05 #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2508.20061

Abstract

We provide a new characterization of amenability for countable groups, based on frame representations admitting almost invariant vectors. By relaxing the frame inequalities, thereby weakening amenability, we obtain a large class of countable groups which we call \it framenable. We show that this class has some permanence properties, stands in contrast with property (T), and contains, for example, all free groups \mathbbF_n, \textupAut(\mathbbF₂) and \textupAut(\mathbbF₃), all (countable) lattices of SL(2,ℝ), the Baumslag-Solitar groups BS_p,q, the braid groups B_n, and Thompson's group F.

Citations

• Parseval Frames from Compressions of Cuntz Algebras
• Non-inner amenability of the Thompson groups T and V
• Subgroup dynamics and C^∗-simplicity of groups of homeomorphisms
• Ordered groups and topology
• Atomic representations of Cuntz algebras
• Relative amenability
• Cantor systems, piecewise translations and simple amenable groups
• Linear Representations of the Automorphism Group of a Free Group
• On simplicity of reduced C*-algebras of groups
• A Direct Integral Decomposition of the Wavelet Representation

Related

• Overcompleteness of coherent frames for unimodular amenable groups
• Frames generated by actions of countable discrete groups
• Hyperlinearity via amenable near representations
• Cantor systems, piecewise translations and simple amenable groups
• On the Jewett-Krieger theorem for amenable groups