Extension of Isometries on Unit Sphere of $L^\infty$

Abstract

We prove that every surjective isometry between unit spheres of $L^\infty(\Sigma,\Omega, \mu)$ and a Banach space $F$ can be linearly and isometrically extended to the whole space, which means that if the unit sphere of a Banach space $F$ is isometric to the unit sphere of $L^\infty(\Sigma,\Omega, \mu)$, then $F$ is linearly isometric to $L^\infty(\Sigma,\Omega, \mu)$.