Abstract
We establish spherical variants of the Gleason–Kahane–Żelazko 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 $\operatorname{Iso}((\operatorname{Lip}(E),\|\cdot\|),(\operatorname{Lip}(F),\|\cdot\|))$ is canonical, then every weak-$2$-local $\operatorname{Iso}((\operatorname{Lip}(E)$, $\|\cdot\|),(\operatorname{Lip}(F),\|\cdot\|))$-map $\Delta$ from $\operatorname{Lip}(E)$ to $\operatorname{Lip}(F)$ is a linear map, where $\|\cdot\|$ can indistinctly stand for $\|f\|_{L} := \max\{L(f), \|f\|_{\infty} \}$ or $ \|f\|_{s} := L(f) + \|f\|_{\infty}$.
Citation
Lei Li. Antonio M. Peralta. Liguang Wang. Ya-Shu Wang. "Weak-2-local isometries on uniform algebras and Lipschitz algebras." Publ. Mat. 63 (1) 241 - 264, 2019. https://doi.org/10.5565/PUBLMAT6311908
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