Let E be a Riesz space and let E∼ denote its order dual. The
orthomorphisms Orth(E) on E, and the ideal center Z(E) of E, are
naturally embedded in Orth(E∼) and Z(E∼) respectively. We
construct two unital algebra and order continuous Riesz homomorphisms γ:((Orth(E))∼)n∼→Orth(E∼) and
m:Z(E)′′→Z(E∼) that extend the above
mentioned natural inclusions respectively. Then, the range of γ is an
order ideal in Orth(E∼) 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∼)-closure of Z(E)x contains the order ideal generated
by x for each x∈E+.) As a consequence, E has a topologically full
center Z(E) if and only if Z(E∼)=π⋅Z(E)′′ for
some idempotent $\pi\in Z(E)^{\prime\prime}.