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

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.