On 2-local nonlinear surjective isometries on normed spaces and C$^*$-algebras

TL;DR

This paper investigates 2-local nonlinear surjective isometries on normed spaces and C*-algebras, showing under certain conditions such mappings are affine and exploring their surjectivity in specific algebraic contexts.

Contribution

It establishes conditions under which 2-local nonlinear surjective isometries are affine and examines their surjectivity in C*-algebras, extending previous understanding.

Findings

Mappings with the 2-local property are affine under certain conditions.

Surjectivity of such mappings is confirmed in specific cases including C*-algebras.

The results generalize known isometry characterizations to nonlinear and local settings.

Abstract

We prove that, if the closed unit ball of a normed space $X$ has sufficiently many extreme points, then every mapping $\Phi$ from $X$ into itself with the following property is affine: For any pair of points in $X$, there exists a (not necessarily linear) surjective isometry on $X$ that coincides with $\Phi$ at the two points. We also consider surjectivity of such a mapping in some special cases including C$^*$-algebras.