Guarded Fraïssé Banach spaces
Fraïssé theory, originally developed in the context of model theory, establishes a bijective correspondence between classes of finite structures having good amalgamation properties and so-called Fraïssé structures, i.e. countable structures satisfying a strong homogeneity property. This correspondence has recently been extended to Banach spaces by Ferenczi, Lopez-Abad, Mbombo et Todorcevic, who proved that the spaces \(L_p\), \(1 \leq p \neq 4, 6, 8, … < \infty\), and the Gurarij space, are Fraïssé. They asked whether those are the only examples; this question is related to Mazur’s rotation problem.
In a work in progress with Marek Cúth and Michal Doucha, we introduce a weak version of the Fraïssé correspondence, the so-called guarded Fraïssé correspondence, extending results obtained by Krawczyk and Kubiś in the discrete setting. We prove that a separable Banach space is guarded Fraïssé if and only if its isometry class is \(G_\delta\) for a natural topological coding of separable Banach spaces introduced by Cúth, Doležal, Doucha et Kurka. This links to the above-mentioned question of Ferenczi, Lopez-Abad, Mbombo et Todorcevic with descriptive set theory of Banach
I will present those results and their links with some questions in continuous logic. I will also discuss the existence of new examples of guarded Fraïssé Banach spaces.