Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2409.09711 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917775706947584 |
|---|---|
| author | Khosravi, Aminallah Vishki, Hamid Reza Ebrahimi Faal, Ramin |
| author_facet | Khosravi, Aminallah Vishki, Hamid Reza Ebrahimi Faal, Ramin |
| contents | 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 $P(x)=\langle γ,x^k\rangle$, $(x\in \mathcal{A})$, for some $γ\in \mathcal{A}^*$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09711 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Orthogonally additive polynomials on the bidual of Banach algebras Khosravi, Aminallah Vishki, Hamid Reza Ebrahimi Faal, Ramin Functional Analysis 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 $P(x)=\langle γ,x^k\rangle$, $(x\in \mathcal{A})$, for some $γ\in \mathcal{A}^*$. 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. |
| title | Orthogonally additive polynomials on the bidual of Banach algebras |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2409.09711 |