Characterization of Riesz spaces with topologically full center

TL;DR

This paper characterizes Riesz spaces with topologically full centers by constructing algebra and order continuous homomorphisms that relate the centers and orthomorphisms of the space and its dual, establishing conditions for topologically full centers.

Contribution

It introduces new homomorphisms linking the centers and orthomorphisms of a Riesz space and its dual, providing a characterization of topologically full centers.

Findings

Range of gamma is an order ideal iff m is surjective

m is surjective iff E has a topologically full center

Z(E) equals Z(E^{ ilde{}}) under certain conditions

Abstract

Let $E$ be a Riesz space and let $E^{\sim}$ denote its order dual. The orthomorphisms $Orth(E)$ on $E,$ and the ideal center $Z(E)$ of $E,$ are naturally embedded in $Orth(E^{\sim})$ and $Z(E^{\sim})$ respectively. We construct two unital algebra and order continuous Riesz homomorphisms \[ \gamma:((Orth(E))^{\sim})_{n}^{\sim}\rightarrow Orth(E^{\sim})\text{ }% \] and \[ m:Z(E)^{\prime\prime}\rightarrow Z(E^{\sim}) \] that extend the above mentioned natural inclusions respectively. Then, the range of $\gamma$ is an order ideal in $Orth(E^{\sim})$ if and only if $m$ is surjective. Furthermore, $m$ is surjective if and only if $E$ has a topologically full center. (That is, the $\sigma(E,E^{\sim})$-closure of $Z(E)x$ contains the order ideal generated by $x$ for each $x\in E_{+}.$) As a consequence, $E$ has a topologically full center $Z(E)$ if and only if $Z(E^{\sim})=\pi\cdot Z(E)^{\prime\prime}$ for some idempotent $\pi\in Z(E)^{\prime\prime}.$