Invariant Integrals on Topological Groups

TL;DR

This paper extends the fixed-point property from discrete groups to topological groups, introduces a new class of normed Riesz spaces, and explores applications of this property in a functional analysis context.

Contribution

It generalizes the fixed-point property to topological groups and introduces a novel class of normed Riesz spaces linked to group representations.

Findings

Identification of groups with the fixed-point property

Development of a new class of normed Riesz spaces

Applications in functional analysis and group theory

Abstract

We generalize the fixed-point property for discrete groups acting on convex cones given by Monod in \cite{monod} to topological groups. At first, we focus on describing this fixed-point property from a functional point of view, and then we look at the class of groups that have it. Finally, we go through some applications of this fixed-point property. To accomplish these tasks, we introduce a new class of normed Riesz spaces that depend on group representation.