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

Geraldo Botelho, Luis Alberto Garcia, Vinícius C. C. Miranda

TL;DR

This paper investigates when Arens extensions of regular disjointness preserving $m$-linear operators between Riesz spaces and Banach lattices preserve disjointness. It proves two main sufficient conditions: (a) finite lattice rank of the operator, and (b) the dual of the codomain has a Schauder basis of disjointness preserving functionals, which guarantee that all AR extensions $AR_m^{rho}(A)$ preserve disjointness. Linear-case results show that the second adjoint $T''$ of a disjointness preserving operator is disjointness preserving when the target is Archimedean (and hence for Banach lattices). The work extends prior results on Riesz multimorphisms and Arens regularity and provides applicable criteria for common Banach lattices such as $c_0$, $\, ext{l}_p$, etc.

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.

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

TL;DR

This paper investigates when Arens extensions of regular disjointness preserving -linear operators between Riesz spaces and Banach lattices preserve disjointness. It proves two main sufficient conditions: (a) finite lattice rank of the operator, and (b) the dual of the codomain has a Schauder basis of disjointness preserving functionals, which guarantee that all AR extensions preserve disjointness. Linear-case results show that the second adjoint of a disjointness preserving operator is disjointness preserving when the target is Archimedean (and hence for Banach lattices). The work extends prior results on Riesz multimorphisms and Arens regularity and provides applicable criteria for common Banach lattices such as , , etc.

Abstract

Let be (non necessarily Archimedean) Riesz spaces, let be an Archimedean Riesz space and let be a regular disjointness preserving -linear operator. We prove that all Arens extensions of are disjointness preserving if either has finite lattice rank or the spaces are Banach lattices and has a Schauder basis consisting of disjointness preserving functionals.

Paper Structure

This paper contains 3 sections, 13 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Disjointness preserving linear and multilinear operators
  3. Arens extensions of disjointness preserving multilinear operators

Key Result

Theorem 2.1

Let $T \colon E \to F$ be an order bounded linear operator between two Riesz spaces. (1) If $F$ is Archimedean and $T$ is disjointness preserving, then $T" \colon E^{\sim \sim} \to F^{\sim \sim}$ is disjointness preserving. (2) If $J_E(E) = E^{\sim \sim}$ and $T$ is disjointness preserving, then $T

Theorems & Definitions (26)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 16 more