Extending surjective isometries defined on the unit sphere of ${\ell }_{\infty }(\Gamma )$

Abstract

Let $\Gamma$ be an infinite set equipped with the discrete topology. We prove that the space ${\ell }_{\infty }(\Gamma ),$ of all complex-valued bounded functions on $\Gamma$, satisfies the Mazur-Ulam property, that is, every surjective isometry from the unit sphere of ${\ell }_{\infty }(\Gamma )$ onto the unit sphere of an arbitrary complex Banach space $X$ admits a unique extension to a surjective real linear isometry from ${\ell }_{\infty }(\Gamma )$ to $X$.

Cited by in corpus (1)

• Extension of isometries from the unit sphere of a rank-2 Cartan factor