The ideal center of the dual of a Banach lattice

TL;DR

This paper investigates the relationship between the ideal center of a Banach lattice and its dual, establishing conditions under which the embedding of centers is onto, and generalizing to Riesz spaces and orthomorphisms.

Contribution

It characterizes when the ideal center of a Banach lattice's dual coincides with the extended embedding, introducing the concept of a topologically full center and generalizing to Riesz spaces.

Findings

The embedding of $Z(E)$ into $Z(E')$ can be extended to a contractive algebra and lattice homomorphism.

The extension is onto $Z(E')$ if and only if $E$ has a topologically full center.

Results are generalized to the ideal center of the order dual of Archimedean Riesz spaces.

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 $Z(E)"$ 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.