Classify Fraïssé Banach spaces beyond the known examples

Determine whether there exist infinite-dimensional Fraïssé Banach spaces other than the Gurariĭ space and the Lebesgue spaces L_p([0,1]) for 1 ≤ p < ∞ with p not an even integer greater than 2; equivalently, establish whether these known examples exhaust all infinite-dimensional Fraïssé Banach spaces.

Background

Fraïssé Banach spaces form an approximate homogeneity class introduced to mirror classical Fraïssé theory in metric structures. Known infinite-dimensional examples include the Gurariĭ space and L_p([0,1]) for p not in {4,6,8,...}; the spaces L_{2n}([0,1]) with n ≥ 2 are known not to be Fraïssé. Whether any other infinite-dimensional Banach spaces are Fraïssé remains a central classification issue.

Resolving this would clarify the scope of Fraïssé homogeneity in Banach space theory and connect to broader questions about isometric rigidity and structure of isometry groups.

References

Examples of infinite-dimensional Fraïssé Banach spaces include the Gurari space and, as shown in , the spaces L_p([0, 1]) for 1 \leqslant p < \infty with p \notin 4, 6, 8, \ldots (the spaces L_{2n}[0, 1] for integers n\geqslant 2 are known not to be Fraïssé, as a consequence of a result of Randrianantoanina ). It is an important open question whether those examples are the only ones.

— Isometric rigidity and Fraïssé properties of Orlicz sequence spaces  (2604.02080 - Rancourt et al., 2 Apr 2026) in Section 1.1 (Fraïssé theory for Banach spaces)