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

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 ${\ell }_{\infty }^m$.