  Let $E$ be a Banach lattice. Its ideal center $Z(E)$ is embedded naturally in
the ideal center $Z(E')$ of its dual. The embedding may be extended to a
contractive algebra and lattice homomorphism of $Z(E)"$ into $Z(E')$. We show
that the extension is onto $Z(E')$ 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.
