Extending surjective isometries defined on the unit sphere of $\ell_\infty(\Gamma)$

TL;DR

This paper proves that every surjective isometry on the unit sphere of the space of bounded functions on an infinite set extends uniquely to a linear isometry on the entire space, confirming the Mazur-Ulam property.

Contribution

It establishes the Mazur-Ulam property for ll_ty(mma) spaces, extending isometries from the sphere to the whole space in a unique way.

Findings

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

The ll_ty(mma) space satisfies the Mazur-Ulam property.

Extension of isometries holds for arbitrary complex Banach spaces.

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 $X$.