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

$$\begin{aligned} \left\{ \Vert T(x)+T(y)\Vert , \Vert T(x)-T(y)\Vert \right\} =\left\{ \Vert x+y\Vert , \Vert x-y\Vert \right\} \end{aligned}$$

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 real linear isometry on X whenever T is surjective, $X=C(K)$ and Y is an arbitrary Banach space. Additionally, if T is a surjective phase-isometry between the unit spheres of C(K) and $C(\varOmega )$, where K and $\varOmega$ are compact Hausdorff spaces, we prove that there are a homeomorphism $\varphi : \varOmega \rightarrow K$ and a continuous unimodular function h on $\varOmega$ such that

$$\begin{aligned} T(f)\in \{h\cdot f\circ \varphi ,-h\cdot f\circ \varphi \} \end{aligned}$$

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.