The Mazur-Ulam property for the space of complex null sequences

TL;DR

This paper proves that the space of complex null sequences $c_0( ext{Gamma})$ and finite-dimensional $ ext{ell}_ ext{infty}^m$ satisfy the Mazur-Ulam property, meaning isometries on their unit spheres extend uniquely to linear isometries.

Contribution

It establishes the Mazur-Ulam property for $c_0( ext{Gamma})$ spaces and finite-dimensional $ ext{ell}_ ext{infty}^m$, extending the class of spaces known to have this property.

Findings

The space $c_0( ext{Gamma})$ satisfies the Mazur-Ulam property.

Finite-dimensional $ ext{ell}_ ext{infty}^m$ also satisfies the property.

Surjective isometries on the unit sphere extend uniquely to linear isometries.

Abstract

Given an infinite set $\Gamma$, we prove that the space of complex null sequences $c_0(\Gamma)$ satisfies the Mazur-Ulam property, that is, for each Banach space $X$, every surjective isometry from the unit sphere of $c_0(\Gamma)$ onto the unit sphere of $X$ admits a (unique) extension to a surjective real linear isometry from $c_0(\Gamma)$ to $X$. We also prove that the same conclusion holds for the finite dimensional space $\ell_{\infty}^m$.