Guarded Fra\"iss\'e Banach spaces
Marek C\'uth, No\'e de Rancourt, Michal Doucha

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 isometry classes in the Polish codings , and 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 -categoricity from continuous logic, showing that -categorical Banach spaces are a natural source of guarded Fra\"iss\'e Banach spaces. Using those results, we prove that for many values of , the Banach space has a isometry class; we precisely characterize those values.
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Taxonomy
TopicsAdvanced Banach Space Theory
