| Résume | In a 2022 paper, Cúth, Doležal, Doucha and Kurka studied the descriptive complexity of isometry classes of several separable Banach spaces. They found out that the spaces Lp[0, 1] and the Gurarii space had G_\delta isometry classes. They asked if those spaces were the only ones satisfying this property.
In a 2020 paper, Ferenczi, López Abad, Mbombo and Todorcevic developed a Fraïssé correspondence for separable Banach spaces. They proved that the spaces Lp[0, 1] and the Gurarii space could be obtained as Fraïssé limits. They asked if those spaces were the only ones satisfying this property.
I'll present a joint work with Marek Cúth and Michal Doucha where we proved that those two questions are almost (but not exactly) the same. More precisely, we introduced the notion of guarded Fraïssé Banach spaces, a weakening of the Fraïssé property that is a metric version of prehomogeneity, and proved that this property is equivalent to having a G_\delta isometry class. We also proved a weak Fraïssé correspondence for those spaces. As a consequence, we could find new examples of Banach spaces having a G_\delta isometry class: the spaces Lp([0, 1], Lq[0, 1]), for many values of the pair (p, q). |
