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) 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 ∈ 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. Mathematics Subject Classification (2000)Primary 47B38 · 47B60 · 46B42; Secondary 46H25 · 47L10

Free and projective Banach lattices

“…The fact that Banach lattices with a topological order unit have a topologically full centre is also widely known, but finding a complete proof in the literature is not easy. The earliest is in Example 1 of [16], but that proof is more complicated than it need be. A simpler version is in Proposition 1.1 of [28] and see also Lemma 1 of [17].…”

Characterization of Riesz Spaces with Topologically Full Center

“…We will complete the result by showing that the only Riesz spaces that satisfy Corollary 2 are necessarily those with a topologically full center. For Banach lattices a proof of this was given in [11]. Initially we will give the proof of the sufficiency.…”

“…It follows that E has a topologically full center Z(E) if and only if the ideal center of its order dual E ∼ is given as Z(E ∼ ) = π · Z(E) ′′ for some idempotent π ∈ Z(E) ′′ . We point out that the proofs of the above mentioned results differ from those used in the Banach lattice case [11]. The method used in this paper owes a lot to the work and results of Huijsmans and de Pagter [8] on the bidual of an f -algebra.…”

“…Namely, for each x ∈ E + , the closure of Z(E)x is the closed ideal generated by x. Then it is shown that the homomorphism is onto Z(E ′ ) if and only if Z(E) is topologically full [11,Corollary 2].…”

“…In the Banach lattice case it is shown that if Z(E) is topologically full then it is maximal abelian [11,Corollary 3]. Wickstead [19] showed that the converse is not true.…”