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

Abstract

We establish spherical variants of the Gleason-Kahane-Zelazko and Kowalski-Słodkowski 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$((Lip(E),\Vert .\Vert ),(Lip(F),\Vert .\Vert ))$ is canonical, then every\hyphenation{every} weak-2-local Iso$((Lip(E),\Vert .\Vert ),(Lip(F),\Vert .\Vert ))$-map $\Delta$ from $Lip(E)$ to $Lip(F)$ is a linear map, where $\Vert .\Vert$ can indistinctly stand for $\Vert f{\Vert }_{{}_L}:=\max \{L(f),\Vert f{\Vert }_{\infty }\}$ or $\Vert f{\Vert }_{{}_s}:=L(f)+\Vert f{\Vert }_{\infty }.$