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2019 Weak-2-local isometries on uniform algebras and Lipschitz algebras
Lei Li, Antonio M. Peralta, Liguang Wang, Ya-Shu Wang
Publ. Mat. 63(1): 241-264 (2019). DOI: 10.5565/PUBLMAT6311908

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

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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

Information

Received: 10 May 2017; Revised: 17 November 2017; Published: 2019
First available in Project Euclid: 7 December 2018

zbMATH: 07040968
MathSciNet: MR3908793
Digital Object Identifier: 10.5565/PUBLMAT6311908

Subjects:
Primary: 46B04, 46B20, ‎46E15, 46J10

Rights: Copyright © 2019 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.63 • No. 1 • 2019
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