Orthogonally additive polynomials on the bidual of Banach algebras

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 $P(x)=⟨\gamma ,x^k⟩$, $(x\in \mathcal{A})$, for some $\gamma \in {\mathcal{A}}^{\ast }$. In this paper we first investigate the extensions of a $k$-homogeneous polynomial from $\mathcal{A}$ to the bidual ${\mathcal{A}}^{\ast \ast }$; equipped with the first Arens product. We then study the relationship between $k$-OA properties of $\mathcal{A}$ and ${\mathcal{A}}^{\ast \ast }$: 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 $({\ell }^1{)}^{\ast \ast }$ enjoys k-OA property.