Order continuity of Arens extensions of regular multilinear operators

TL;DR

The paper addresses when Arens extensions of regular multilinear operators and polynomials preserve order continuity in Riesz spaces and Banach lattices. It first establishes that full order continuity on the product of entire biduals cannot be deduced from existing separate continuity results by exhibiting a counterexample, then derives conditions under which order continuity holds on $E_1^{**}\times\cdots\times E_m^{**}$ or on $E^{\sim\sim}$, and shows that all Arens extensions are order continuous in at least one variable. A central result shows that finite sums of multiplicative operators yield coinciding Arens extensions that are separately order continuous, implying order continuity for all regular multilinear extensions and at the origin for regular homogeneous polynomials on $E^{\sim\sim}$. The work further provides Banach-lattice conditions (including Arens-regularity and order-continuity of dual norms) ensuring separate order continuity on bidual products, with concrete consequences for polynomials and symmetrically Arens regular spaces.

Abstract

First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{\sim\sim}$.

Theorems & Definitions (32)

• Proposition 4.1
• proof
• Corollary 4.2
• Corollary 4.3
• proof
• Lemma 4.4
• proof
• Lemma 4.5
• proof
• Theorem 4.6