Fixed points for bounded orbits in Hilbert spaces

TL;DR

This paper characterizes amenability of locally compact groups via fixed point properties of affine actions on Hilbert spaces, introduces a new induction method, and extends key theorems to broader group classes.

Contribution

It establishes a fixed point characterization of amenability, introduces a 'moderate' induction variant, and generalizes the Gaboriau--Lyons theorem to non-amenable groups.

Findings

Amenability characterized by fixed point property in Hilbert space actions

Introduced a 'moderate' induction method for representations

Extended Gaboriau--Lyons theorem to non-amenable groups

Abstract

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact groups (e.g. countable groups). Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau--Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for locally compact groups. We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.