MIN-PHASE-ISOMETRIES ON THE UNIT SPHERE OF $L^1$ SPACES

Abstract Let $(\Omega , \mathfrak {M}, \mu )$ be a measure space. We show that if a surjective map $f: S_{L^1(\mu )} \to S_Y$ between the unit spheres of real $L^1(\mu )$ and of an arbitrary real normed space Y satisfies $$ \begin{align*} \min\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\min\{\|x+y\|, \|x-y\|\},\quad x,y\in S_{L^1(\mu)}, \end{align*} $$ then there exists a phase function $\varepsilon : S_{L^1(\mu )} \to \{-1, 1\}$ such that $\varepsilon \cdot f$ is a surjective isometry from $S_{L^1(\mu )}$ onto $S_Y$ , and furthermore, this isometry can be extended to a linear isometry on the whole space $L^1(\mu )$ .