Order continuity of Arens extensions of regular multilinear operators

TL;DR

This paper investigates the order continuity properties of Arens extensions of regular multilinear operators, providing counterexamples, establishing conditions for order continuity, and analyzing extensions of polynomials on Banach lattices.

Contribution

It offers new conditions under which Arens extensions are order continuous on biduals and demonstrates that all extensions are order continuous in at least one variable.

Findings

Counterexample showing limitations of recent results

Conditions for order continuity of extensions on biduals

All Arens extensions are order continuous in at least one variable

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 $E^{\sim\sim}$.