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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.16886 |
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| _version_ | 1866910660616519680 |
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| author | Sánchez, Félix Cabello Castillo, Jesús M. F. Moreno, Yolanda |
| author_facet | Sánchez, Félix Cabello Castillo, Jesús M. F. Moreno, Yolanda |
| contents | We show that if $p>1$ every subspace of $\ell_p(Γ)$ is an $\ell_p$-sum of separable subspaces of $\ell_p$, and we provide examples of subspaces of $\ell_p(Γ)$ for $0<p\leq 1$ that are not even isomorphic to any $\ell_p$-sum of separable spaces, notably the kernel of any quotient map $\ell_p(Γ)\to L_1(2^Γ)$ with $Γ$ uncountable. We involve the separable complementation property (SCP) and the separable extension property (SEP), showing that if $X$ is a Banach space of density character $\aleph_1$ with the SCP then the kernel of any quotient map $\ell_p(Γ)\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16886 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trimming the Johnson bonsai Sánchez, Félix Cabello Castillo, Jesús M. F. Moreno, Yolanda Functional Analysis 46B26 We show that if $p>1$ every subspace of $\ell_p(Γ)$ is an $\ell_p$-sum of separable subspaces of $\ell_p$, and we provide examples of subspaces of $\ell_p(Γ)$ for $0<p\leq 1$ that are not even isomorphic to any $\ell_p$-sum of separable spaces, notably the kernel of any quotient map $\ell_p(Γ)\to L_1(2^Γ)$ with $Γ$ uncountable. We involve the separable complementation property (SCP) and the separable extension property (SEP), showing that if $X$ is a Banach space of density character $\aleph_1$ with the SCP then the kernel of any quotient map $\ell_p(Γ)\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP. |
| title | Trimming the Johnson bonsai |
| topic | Functional Analysis 46B26 |
| url | https://arxiv.org/abs/2410.16886 |