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

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

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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}$.

Article information

Source
Publ. Mat., Volume 63, Number 1 (2019), 241-264.

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

Permanent link to this document
https://projecteuclid.org/euclid.pm/1544151637

Digital Object Identifier
doi:10.5565/PUBLMAT6311908

Mathematical Reviews number (MathSciNet)
MR3908793

Zentralblatt MATH identifier
07040968

Subjects
Primary: 46B04: Isometric theory of Banach spaces 46B20: Geometry and structure of normed linear spaces 46J10: Banach algebras of continuous functions, function algebras [See also 46E25] 46E15: Banach spaces of continuous, differentiable or analytic functions
Secondary: 30H05: Bounded analytic functions 32A38: Algebras of holomorphic functions [See also 30H05, 46J10, 46J15] 46J15: Banach algebras of differentiable or analytic functions, Hp-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30] 47B48: Operators on Banach algebras 47B38: Operators on function spaces (general) 47D03: Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}

Keywords
2-local isometries uniform algebras Lipschitz functions, spherical Gleason–Kahane–Żelazko theorem spherical Kowalski–Słodkowski theorem weak-2-local isometries

Citation

Li, Lei; Peralta, Antonio M.; Wang, Liguang; Wang, Ya-Shu. Weak-2-local isometries on uniform algebras and Lipschitz algebras. Publ. Mat. 63 (2019), no. 1, 241--264. doi:10.5565/PUBLMAT6311908. https://projecteuclid.org/euclid.pm/1544151637


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