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

Abstract: We say that a map $T: S_X\rightarrow S_Y$ 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 $\varepsilon:S_X\rightarrow {-1,1}$ such that $\varepsilon \cdot T$ 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 $\varphi: \Omega\rightarrow K$ such that $T(f)\in{f\circ \varphi,-f\circ \varphi}$ 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.