Characterization of Riesz spaces with topologically full center

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 \[ γ:((Orth(E))^{\sim})_{n}^{\sim}\rightarrow Orth(E^{\sim})\text{ }% \] 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 $\pi \in Z(E{)}^{\prime \prime }.$