A generalization of the Kowalski -S\{l} odkowski theorem and its application to 2-local maps on function spaces

TL;DR

This paper extends a spherical version of the Kowalski-S extl{}odkowski theorem and proves that 2-local maps on certain function spaces are actually surjective isometries, answering a longstanding open problem.

Contribution

It generalizes the Kowalski-S extl{}odkowski theorem and establishes that 2-local isometries are genuine surjective isometries on specific function spaces.

Findings

Extended the spherical Kowalski-S extl{}odkowski theorem.

Proved 2-local maps are surjective isometries.

Resolved the open problem on 2-local isometries by Molnár.

Abstract

In this paper, we extend a spherical variant of the Kowalski-S\{l}odkowski theorem due to Li, Peralta, Wang and Wang. 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 the problem on 2-local isometries posed by Moln\'ar.