Characterization of Riesz spaces with topologically full center

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 $γ$ 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 $σ(E,E^{\sim})$-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 $Z(E^{\sim})=π\cdot Z(E)^{\prime\prime}$ for some idempotent $π\in Z(E)^{\prime\prime}.$

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