An amenability-like property of finite energy path and loop groups

TL;DR

This paper demonstrates that groups of finite energy loops and paths in compact Lie groups, along with their central extensions, exhibit an amenability-like property, enabling invariant means and conjugation-invariant states for their representations.

Contribution

It introduces a new amenability-like property for finite energy path and loop groups and their central extensions, extending the concept of amenability in this context.

Findings

Finite energy loop and path groups are skew-amenable.

Existence of left-invariant means on bounded uniformly continuous functions.

Strongly continuous unitary representations admit conjugation-invariant states.

Abstract

We show that the groups of finite energy loops and paths (that is, those of Sobolev class $H^1$) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a left-invariant mean on the space of bounded functions uniformly continuous with regard to a left-invariant metric. Every strongly continuous unitary representation $\pi$ of such a group (which we call skew-amenable) has a conjugation-invariant state on $B({\mathcal H}_{\pi})$.