Arens extensions of disjointness preserving multilinear operators on Riesz spaces and Banach lattices

TL;DR

This paper investigates conditions under which Arens extensions of disjointness preserving multilinear operators between Riesz spaces and Banach lattices remain disjointness preserving, expanding understanding of operator extensions in lattice theory.

Contribution

It establishes that all Arens extensions preserve disjointness if the operator has finite lattice rank or the spaces are Banach lattices with specific dual space properties.

Findings

All Arens extensions are disjointness preserving under finite lattice rank.

Extensions preserve disjointness when spaces are Banach lattices with a Schauder basis of disjointness preserving functionals.

Provides new conditions ensuring the stability of disjointness in multilinear operator extensions.

Abstract

Let $E_1, \ldots, E_m$ be (non necessarily Archimedean) Riesz spaces, let $F$ be an Archimedean Riesz space and let $A \colon E_1 \times \cdots \times E_m \to F$ be a regular disjointness preserving $m$-linear operator. We prove that all Arens extensions of $A$ are disjointness preserving if either $A$ has finite lattice rank or the spaces are Banach lattices and $F^*$ has a Schauder basis consisting of disjointness preserving functionals.