Orthogonally additive polynomials on the bidual of Banach algebras

2024/09/15 by Amin A. Khosravi, Khosravi, Aminallah, Hamid Reza Ebrahimi Vishki +3
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Mathematics #Matrix Theory and Algorithms #Pure mathematics

paper · pdf · doi:10.48550/arxiv.2409.09711

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2024/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We say that a Banach algebra A has k-orthogonally additive property (k-OA property, for short) if every orthogonally additive k-homogeneous polynomial P:A→ ℂ can be expressed in the standard form P(x)=⟨ γ,x^k⟩, (x∈ A), for some γ∈ A^*. In this paper we first investigate the extensions of a k-homogeneous polynomial from A to the bidual A^**; equipped with the first Arens product. We then study the relationship between k-OA properties of A and A^**: This relation is specially investigated for a dual Banach algebra. Finally we examine our results for the dual Banach algebra ℓ¹, with pointwise product, and we show that the Banach algebra (ℓ¹)^** enjoys k-OA property.

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