Linear functionals preserving the units of a Riesz space

TL;DR

This paper characterizes unital linear functionals on Riesz spaces that preserve strong units, showing they act like Riesz homomorphisms on certain subspaces and linking this property to the space being e-clean.

Contribution

It establishes a characterization of unital linear functionals that preserve strong units as Riesz homomorphisms on e-clean subspaces, connecting functional properties to the structure of the Riesz space.

Findings

Unital linear functionals that do not vanish on strong units behave like Riesz homomorphisms.

The space is e-clean if and only if such functionals are Riesz homomorphisms.

Provides a structural criterion linking functionals and the e-clean property.

Abstract

Let $L$ be a Riesz space with a strong unit $e>0$.$\mathfrak{\ }$We show that a unital linear functional $H:L\rightarrow \mathbb{R}$ satisfies $% H\left( u\right) \neq 0$ for any strong unit $u\in L$ if and only if $H$ acts like a Riesz homomorphism on every $e$-clean vector subspace of $L.$ We deduce that $L$ is $e$-clean if and only if any unital linear functional $% H:L\rightarrow \mathbb{R}$ such that $H\left( u\right) \neq 0$ for any strong unit $u\in L$ is a Riesz homomorphism.