Phase-isometries on the unit sphere of $C(K)$

Abstract

We say that a map $T:S_X\to S_Y$ between the unit spheres of two real normed-spaces $X$ and $Y$ is a phase-isometry if it satisfies \begin{eqnarray*} \{\|T(x)+T(y)\|, \|T(x)-T(y)\|\}=\{\|x+y\|, \|x-y\|\} \end{eqnarray*} for all $x,y\in S_X$. In the present paper, we show that there is a phase function $\epsilon :S_X\to \{-1,1\}$ such that $\epsilon \cdot T$ is an isometry which can be extended a linear isometry on the whole space $X$ whenever $T$ is surjective, $X=C(K)$ ($K$ is a compact Hausdorff space) and $Y$ is an arbitrary Banach space. Additionally, if $T$ is a phase-isometry between the unit spheres of $C(K)$ and $C(\Omega )$, where $K$ and $\Omega$ are compact Hausdorff spaces, we prove that there is a homeomorphism $\phi :\Omega \to K$ such that $T(f)\in \{f\circ \phi ,-f\circ \phi \}$ for all $f\in S_{C(K)}$. This also can be seen as a Banach-Stone type representation for phase-isometries in $C(K)$ spaces.

13 pages