Amenable groups and a geometric view on unitarisability

TL;DR

This paper explores the unitarisability of groups through geometric actions on positive invertible operators, establishing fixed point theorems and extending results about amenable and unitarisable groups with universal constants.

Contribution

It provides a geometric proof of unitarisability results, constructs barycenters in geodesic spaces, and links amenability with unitarisability.

Findings

Extensions of unitarisable groups by amenable groups are unitarisable.

Virtual unitarisability implies unitarisability with universal constant estimates.

Fixed point theorem for amenable group actions on geodesic metric spaces.

Abstract

We investigate unitarisability of groups by looking at actions on the cone of positive invertible operators of a Hilbert space. This way, we give a geometric prove to a result by Gilles Pisier on the existence of some universal constants for unitarisable groups. By constructing barycenters for finite sets in a some class of geodesic metric spaces, we prove a fixed point theorem for actions of amenable groups by isometries. In particular, we show that extensions of unitarisable groups by amenable groups are unitarisable and virtual unitarisability implies unitarisability and estimate the corresponding universal constants.