Guarded Fra\"iss\'e Banach spaces

TL;DR

This paper introduces guarded Fra"iss"e Banach spaces, characterizes their properties, and links them with $ ext{omega}$-categoricity, applying these concepts to analyze the isometry classes of certain $L_p(L_q)$ spaces.

Contribution

It defines guarded Fra"iss"e Banach spaces, establishes a Fra"iss"e correspondence, and connects these with $ ext{omega}$-categoricity, providing new insights into the structure of Banach spaces.

Findings

Characterization of $G_\delta$ isometry classes in certain Polish codings.

Identification of conditions under which $L_p(L_q)$ spaces have $G_\delta$ isometry classes.

Establishment of a link between guarded Fra"iss"e spaces and $ ext{omega}$-categoricity.

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.