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arXiv · 1810.12252

Linear functionals preserving the units of a Riesz space

Fethi Benamor
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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.

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Fethi Benamor. 2018-10-29. Linear functionals preserving the units of a Riesz space. https://arxiv.org/abs/1810.12252

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