Extension properties of order continuous functionals and applications to the theory of Banach lattices

Abstract

Let E be a Banach lattice and E 0 an ideal of E. Let f be a positive norm-bounded order continuous functional on E 0. Then f has a norm preserving positive order continuous extension to E. Moreover, there exists a minimum extension among all such extensions. This result enables us to obtain an isometric lattice homomorphism from ( E 0) ∗ n into E ∗ n. As an application, we prove that if F is a Banach lattice and if E 0 is the ideal generated by F in E (= F ∗∗), then ( E 0) ∗ n can be identified with F ∗ in the canonical sense. Certain extension properties of order continuous functionals thus follow. Banach lattices with strictly monotone norm are also investigated.