{
  "annotation_id": "99e4b3f0-68b9-4cff-b837-dfcfc69412aa",
  "date_created": "2026-02-17T05:53:20.513000Z",
  "date_modified": "2026-02-17T05:53:20.513000Z",
  "file_hash": "1337b4702262661b93d4d78f7f9ab7149cb0b2508a72d77e0382b3d024235ed6",
  "private": false,
  "record": {
    "abstract": "We characterize projections among positive norm-one elements in unital C$^*$-algebras in pure geometric terms determined by the norm of the underlying Banach space. Concretely, let $A$ be a C$^*$-algebra (or a JB$^*$-algebra) whose positive cone and unit sphere are denoted by ${A}^+$ and $\\mathrm{S}_{A}$, respectively. The positive portion of the unit sphere in $A$, denoted by $\\mathrm{S}_{{A}^+}$, is the set ${A}^+ \\cap \\mathrm{S}_{A}$, while the unit sphere of positive norm-one elements around a subset $\\mathscr{S}$ in $\\mathrm{S}_{A^+}$ is the set $$\\hbox{Sph}_{_{\\mathrm{S}_{{A}^+}}} (\\mathscr{S}) :=\\Big\\{ x\\in \\mathrm{S}_{{A}^+} : \\|x-s\\|=1 \\hbox{ for all } s\\in \\mathscr{S} \\Big\\}.$$ Assuming that $A$ is unital, we establish that an element $a\\in \\mathrm{S}_{{A}^+}$ is a projection if, and only if, it satisfies the double sphere property, that is, $ \\hbox{Sph}_{_{\\mathrm{S}_{{A}^+}}} \\left(\\hbox{Sph}_{_{\\mathrm{S}_{{A}^+}}} \\left(\\{a\\}\\right) \\right) = \\{a\\}.$",
    "arxiv_id": "2601.09669",
    "authors": [
      "Antonio M. Peralta",
      "Pedro Saavedra"
    ],
    "categories": [
      "math.OA",
      "math.FA"
    ],
    "license": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
    "title": "A metric characterization of projections among positive norm-One elements in unital C$^*$-algebras",
    "url": "https://arxiv.org/abs/2601.09669",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "2e75a806-4920-4e20-8d32-ce4c4b246a90",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}