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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2507.13866 |
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| _version_ | 1866910056752087040 |
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| author | Rincón-Villamizar, Michael A. Aylwin, Carlos Uzcátegui |
| author_facet | Rincón-Villamizar, Michael A. Aylwin, Carlos Uzcátegui |
| contents | We study the complementation (in $\ell_\infty$) of the Banach space $c_{0,\mathcal{I}}$, consisting of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the supremum norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that the complementation of these spaces is related to a condition requiring that the ideal is the intersection of a countable family of maximal ideals, which we refer to as $ω$-maximal ideals. We prove that if $c_{0,\mathcal{I}}$ admits a projection satisfying a certain condition, then $\mathcal{I}$ must be a special type of $ω$-maximal ideal. Additionally, we characterize when the quotient space $c_{0,\mathcal{J}} / c_{0,\mathcal{I}}$ is finite-dimensional for two ideals $\mathcal{I} \subsetneq \mathcal{J}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the complementation of spaces of $\mathcal I$-null sequences Rincón-Villamizar, Michael A. Aylwin, Carlos Uzcátegui Functional Analysis We study the complementation (in $\ell_\infty$) of the Banach space $c_{0,\mathcal{I}}$, consisting of all bounded sequences $(x_n)$ that $\mathcal{I}$-converge to $0$, endowed with the supremum norm, where $\mathcal{I}$ is an ideal of subsets of $\mathbb{N}$. We show that the complementation of these spaces is related to a condition requiring that the ideal is the intersection of a countable family of maximal ideals, which we refer to as $ω$-maximal ideals. We prove that if $c_{0,\mathcal{I}}$ admits a projection satisfying a certain condition, then $\mathcal{I}$ must be a special type of $ω$-maximal ideal. Additionally, we characterize when the quotient space $c_{0,\mathcal{J}} / c_{0,\mathcal{I}}$ is finite-dimensional for two ideals $\mathcal{I} \subsetneq \mathcal{J}$. |
| title | On the complementation of spaces of $\mathcal I$-null sequences |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.13866 |