{
  "id": "1804.10674",
  "version": "v1",
  "published": "2018-04-27T20:19:45.000Z",
  "updated": "2018-04-27T20:19:45.000Z",
  "title": "Mankiewicz's theorem and the Mazur--Ulam property for C*-algebras",
  "authors": [
    "Michiya Mori",
    "Narutaka Ozawa"
  ],
  "comment": "13 pages",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "We prove that every unital C*-algebra $A$, possibly except for the $2$ by $2$ matrix algebra, has the Mazur--Ulam property. Namely, every surjective isometry from the unit sphere $S_A$ of $A$ onto the unit sphere $S_Y$ of another normed space $Y$ extends to a real linear map. This extends the result of A. M. Peralta and F. J. Fernandez-Polo who have proved the same under the additional assumption that both $A$ and $Y$ are von Neumann algebras. In the course of the proof, we strengthen Mankiewicz's theorem and prove that every surjective isometry from a closed unit ball with enough extreme points onto an arbitrary convex subset of a normed space is necessarily affine.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-27T20:19:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46B04",
      "46L05"
    ],
    "keywords": [
      "mazur-ulam property",
      "unit sphere",
      "surjective isometry",
      "real linear map",
      "arbitrary convex subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}