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

Lei Li; Antonio M. Peralta; Liguang Wang; Ya-Shu Wang

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

Functional Analysis 2017-05-11 v1 Spectral Theory

Authors: Lei Li , Antonio M. Peralta , Liguang Wang , Ya-Shu Wang

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 $\|f\|_{_s} := L(f) + \|f\|_{\infty}.$

Keywords

weak-2-local isometries uniform algebras lipschitz algebras gleason-kahane-zelazko theorem kowalski-slodkowski theorem linear maps metric spaces

Cite

@article{arxiv.1705.03619,
  title  = {Weak-2-local isometries on uniform algebras and Lipschitz algebras},
  author = {Lei Li and Antonio M. Peralta and Liguang Wang and Ya-Shu Wang},
  journal= {arXiv preprint arXiv:1705.03619},
  year   = {2017}
}

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

Abstract

Keywords

Cite

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