Weak-2-local isometries on uniform algebras and Lipschitz algebras

Abstract

We establish spherical variants of the Gleason–Kahane–Zelazko and Kowalski–S lodkowski theorems, and we apply them to prove that every weak-2-local isometry between two uniform algebras is a linear map. Among the consequences, we solve a couple of problems posed by O. Hatori, T. Miura, H. Oka, and H. Takagi in 2007. Another application is given in the setting of weak-2-local isometries between Lipschitz algebras by showing that given two metric spaces E and F such that the set Iso((Lip(E), k·k), (Lip(F), k·k)) is canonical, then every weak-2-local Iso((Lip(E), k · k), (Lip(F), k · k))-map ∆ from Lip(E) to Lip(F) is a linear map, where k · k can indistinctly stand for kfkL := max{L(f), kfk∞} or kfks := L(f) + kfk∞.