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arXiv · 2408.12755

Guarded Fra\"iss\'e Banach spaces

Marek Cúth·Noé de Rancourt·Michal Doucha
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Abstract

We characterize separable Banach spaces having $G_\delta$ isometry classes in the Polish codings $\mathcal{P}$, $\mathcal{P}_\infty$ and $\mathcal{B}$ introduced by C\'uth-Dole\v{z}al-Doucha-Kurka [13] as those being guarded Fra\"iss\'e, a weakening of the notion of Fra\"iss\'e Banach spaces defined by Ferenczi-Lopez-Abad-Mbombo-Todorcevic [18]. We prove a Fra\"iss\'e correspondence for those spaces and make links with the notion of $\omega$-categoricity from continuous logic, showing that $\omega$-categorical Banach spaces are a natural source of guarded Fra\"iss\'e Banach spaces. Using those results, we prove that for many values of $(p, q)$, the Banach space $L_p(L_q)$ has a $G_\delta$ isometry class; we precisely characterize those values.

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Marek Cúth, Noé de Rancourt, Michal Doucha. 2024-08-22. Guarded Fra\"iss\'e Banach spaces. https://arxiv.org/abs/2408.12755

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