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arXiv · 2410.16886

Trimming the Johnson bonsai

Félix Cabello Sánchez·Jesús M. F. Castillo·Yolanda Moreno
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Abstract

We show that if $p>1$ every subspace of $\ell_p(\Gamma)$ is an $\ell_p$-sum of separable subspaces of $\ell_p$, and we provide examples of subspaces of $\ell_p(\Gamma)$ 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 $\ell_p(\Gamma)\to L_1(2^{\Gamma})$ with $\Gamma$ 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 $\ell_p(\Gamma)\to X$ is a complemented subspace of a space with the SCP and, consequently, has the SEP.

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Félix Cabello Sánchez, Jesús M. F. Castillo, Yolanda Moreno. 2024-10-22. Trimming the Johnson bonsai. https://arxiv.org/abs/2410.16886

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