{
  "id": "2008.09282",
  "version": "v1",
  "published": "2020-08-21T03:09:47.000Z",
  "updated": "2020-08-21T03:09:47.000Z",
  "title": "Phase-isometries on the unit sphere of $C(K)$",
  "authors": [
    "Dongni Tan",
    "Yueli Gao"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-21T03:09:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit sphere",
      "phase-isometry",
      "compact hausdorff space",
      "banach-stone type representation",
      "arbitrary banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}