Guarded Fra\"iss\'e Banach spaces

We characterize separable Banach spaces having G_δ isometry classes in the Polish codings P, P_∞ and B introduced by C\'uth-Dole{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 ω-categoricity from continuous logic, showing that ω-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 Lₚ(Lq) has a G_δ isometry class; we precisely characterize those values.