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

## TL;DR

This paper proves that weak-2-local isometries between uniform and Lipschitz algebras are linear maps, solving longstanding problems and extending classical theorems in functional analysis.

## Contribution

It establishes spherical variants of key theorems and applies them to show weak-2-local isometries are linear, addressing open questions in the field.

## Key findings

- Weak-2-local isometries are linear maps between uniform algebras.
- Solved problems posed by Hatori, Miura, Oka, and Takagi in 2007.
- Extended results to Lipschitz algebras with canonical isometry sets.

## 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}.$