On phase-isometries between the positive cones of continuous function spaces
On phase-isometries between the positive cones of continuous function spaces
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DOI：
10.1007/s43034-022-00242-0
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发表时间：
2022
影响因子：
1
通讯作者：
Duanxu Dai
中科院分区：
文献类型：
--
作者：
Longfa Sun;Yinghua Sun;Duanxu Dai
Let $K$ be a compact Hausdorff perfectly normal space and $T$ be a compact Hausdorff space, $C_+(K)=\{f\in C(K): f(k)\geq0\; {\rm for\; all\;}\; k\in K\}$ be the positive cone of $C(K)$..In this paper, we show that if $F:C_+(K)\rightarrow C_+(T)$ is a phase-isometry, i.e.,.\begin{equation}\nonumber.\{\|F(f)+F(g)\|,\|F(f)-F(g)\|\}=\{\|f+g\|,\|f-g\|\},\;\forall f,g\in C_+(K),.\end{equation}.then there exists a nonempty closed subset $S\subset T$ such that $F(\cdot)|_S: C_+(K)\rightarrow C_+(S)$ (restriction of $F(\cdot)$ to $S$ ) is an additive isometry (the restriction of a linear isometry between $C(K)$ and $C(S)$). Moreover, if $F$ is almost surjective, then $K$ and $T$ are homeomorphic and $F$ is the restriction of a surjective linear isometry between $C(K)$ and $C(T)$ induced by the homeomorphism.
