A Nakano carrier type-theorem for orthogonally additive polynomials in Riesz spaces

TL;DR

The paper addresses extending the Nakano Carrier theorem to order-continuous, homogeneous orthogonally additive polynomials on Archimedean Riesz spaces. It develops carrier theory, proving that the null ideal $N_{P}$ is an ideal and becomes a band when $P$ is order continuous, and establishes equivalences between disjointness of polynomials and disjointness of their carriers, including fiberwise extensions to $C(Y)$-valued polynomials and a counterexample highlighting limitations in general settings. It then introduces the order polynomial dual $(^{n}E)^{\sim_p}$ and the canonical embedding $\widehat{x}$, showing the embedding is an order-continuous orthogonally additive $n$-homogeneous polynomial with order-dense range in the order-continuous dual $(((^{n}E)^{\sim_p})^{\sim})_{n}$, providing a duality framework for a Nakano-type analysis beyond Banach lattices. These results advance structural understanding and duality tools for nonlinear orthogonally additive maps on Riesz spaces, with potential for representation-type analysis in a broader lattice-theoretic setting.

Abstract

The purpose of the present paper is to prove the Nakano theorem for orthogonally additive polynomials in Riesz spaces

Theorems & Definitions (12)

• Lemma 1
• Theorem 2
• Theorem 3
• Proposition 4
• Theorem 5
• Theorem 6
• Example 7
• Theorem 8
• Proposition 9
• Lemma 10