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

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