论文标题
$ c（k）$的单位球体上的相对图
Phase-isometries on the unit sphere of $C(K)$
论文作者
论文摘要
我们说的是，在两个真实范围的空间$ x $和$ y $之间的单位球之间的地图$ t：s_x \ rightarrow s_y $，如果满足\ begin \ begin \ begin {eqnarray*} \ | t（x）+t（x）+t（x）+t（y） \ | | x-y \ | \} \ end {eqnarray*}对于s_x $中的所有$ x，y \。在本文中，我们表明存在一个阶段函数$ \ varepsilon：s_x \ rightarrow \ { - 1,1 \} $，使得$ \ varepsilon \ cdot t $是一个等距，可以将其扩展到整个空间$ x $的整个空间$ x $ a Surpive and cuns $ x $ a $ x = c（$ x = c（k）$ x = c（k）$ x = c（k）$ x = c（k）$ x = c（k） $ y $是任意的Banach空间。此外，如果$ t $是$ c（k）$和$ c（ω）$之间的相位均衡，其中$ k $ and $ k $和$ω$是紧凑的Hausdorff空间，我们证明同型同型$φ：ω\ rightarrow k $ t（f）$ t（f）$ t（f）$ t（f） $ f \ in s_ {c（k）} $。这也可以看作是$ c（k）$空间中相层的Banach-Stone类型表示。
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 $C(Ω)$, where $K$ and $Ω$ are compact Hausdorff spaces, we prove that there is a homeomorphism $φ: Ω\rightarrow K$ such that $T(f)\in\{f\circ φ,-f\circ φ\}$ 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.
