The ideal center of the dual of a Banach lattice

Abstract

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)ʺ 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.