The Mazur-Ulam property for uniform algebras

TL;DR

This paper establishes a sufficient condition under which a surjective isometry between unit spheres of certain Banach spaces extends to a real-linear isometry, with applications to uniform algebras and specific function spaces.

Contribution

It introduces a new sufficient condition for the homogeneous extension of surjective isometries to be real-linear, applicable to uniform algebras and certain regular function spaces.

Findings

The condition applies to uniform algebras.

It extends isometries to real-linear maps under specified conditions.

Applicable to extremely C-regular spaces of continuous functions.

Abstract

We give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.