Representations and Semisimplicity of Ordered Topological Vector Spaces

TL;DR

This paper extends classical representation theorems for ordered topological vector spaces, focusing on semisimple spaces that admit injective positive representations, and explores their geometric, algebraic, and topological properties.

Contribution

It introduces a new class of semisimple spaces, characterizes them through various properties, and links them to classical regularly ordered spaces.

Findings

Semisimple spaces admit injective positive representations.

The class of semisimple spaces is characterized by multiple equivalent properties.

Semisimple spaces form a natural topological analogue of regularly ordered spaces.

Abstract

This paper studies ways to represent an ordered topological vector space as a space of continuous functions, extending the classical representation theorems of Kadison and Schaefer. Particular emphasis is put on the class of semisimple spaces, consisting of those ordered topological vector spaces that admit an injective positive representation to a space of continuous functions. We show that this class forms a natural topological analogue of the regularly ordered spaces defined by Schaefer in the 1950s, and is characterized by a large number of equivalent geometric, algebraic, and topological properties.