Orthogonally additive polynomials on the bidual of Banach algebras

TL;DR

This paper explores the extension and properties of orthogonally additive polynomials on Banach algebras and their biduals, revealing conditions under which these polynomials can be represented in a standard form.

Contribution

It investigates the extension of k-homogeneous polynomials to biduals and the relationship between their orthogonally additive properties in Banach algebras and their biduals, especially for dual Banach algebras.

Findings

Extension of k-homogeneous polynomials to biduals analyzed

Relationship between OA properties of algebra and bidual established

The bidual of  enjoys the k-OA property

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 $P:\mathcal{A}\to \mathbb{C}$ can be expressed in the standard form $P(x)=\langle \gamma,x^k\rangle$, $(x\in \mathcal{A})$, for some $\gamma\in \mathcal{A}^*$. In this paper we first investigate the extensions of a $k$-homogeneous polynomial from $\mathcal{A}$ to the bidual $\mathcal{A}^{**}$; equipped with the first Arens product. We then study the relationship between $k$-OA properties of $\mathcal{A}$ and $\mathcal{A}^{**}$: This relation is specially investigated for a dual Banach algebra. Finally we examine our results for the dual Banach algebra $\ell^{1}$, with pointwise product, and we show that the Banach algebra $(\ell^{1})^{**}$ enjoys k-OA property.