The ideal center of the dual of a Banach lattice

Let $E$ be a Banach lattice. Its ideal center $Z(E)$ is embedded naturally in the ideal center $Z(E^{\mathrm{\prime }})$ of its dual. The embedding may be extended to a contractive algebra and lattice homomorphism of $Z(E)\mathrm{"}$ into $Z(E^{\mathrm{\prime }})$. We show that the extension is onto $Z(E^{\mathrm{\prime }})$ if and only if $E$ has a topologically full center. (That is, for each $x\in E$, the closure of $Z(E)x$ is the closed ideal generated by $x$.) The result can be generalized to the ideal center of the order dual of an Archimedean Riesz space and in a modified form to the orthomorphisms on the order dual of an Archimedean Riesz space.