Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ );(b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ .Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

中文翻译：

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KOWALSKI-SŁODKOWSKI 定理的球面版本及其应用

李等。['均匀代数和 Lipschitz 代数上的弱 2-局部等距'，出版。垫。63(2019), 241–264] 通过建立以下球面变量来推广 Kowalski-Słodkowski 定理：让一种是一个单位复数 Banach 代数并让$\Delta : A \to \mathbb {C}$是满足以下性质的映射：（一种）$\三角洲$是 1-齐次的（即，$\Delta (\lambda x)=\lambda \Delta (x)$对所有人$x \in 澳元,$\lambda \in \mathbb C$);(二)$\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (xy), \quad x,y \in A$.然后$\三角洲$是线性的并且存在$\lambda _{0} \in \mathbb {T}$这样$\lambda_{0}\Delta $是乘法的。在本笔记中，我们证明如果 (a) 放宽到$\Delta (0)=0$， 然后$\三角洲$是复线性或共轭线性并且$\overline {\Delta (\mathbf {1})}\Delta $是乘法的。我们将 Kowalski-Słodkowski 定理扩展为一个结论。作为推论，我们证明了某个函数空间上所有满射等距（不假设线性）的集合中的每个 2-局部映射实际上是一个满射等距。这对 Molnár ['On 2-local *-automorphisms and 2-local isometry ofB（H）'，J.数学。肛门。应用程序。479(1) (2019), 569–580] 以及 Molnár 和 O. Hatori 之间的私人通信，2018 年。