{
  "abstract": "Let $A$ and $B$ be unital $C^*$-algebras, and let $\\mathrm{S}_{A^+}$ and $\\mathrm{S}_{B^+}$ denote their positive unit spheres. We first show that the diametral relation on the positive unit sphere already determines Jordan structure. More precisely, suppose that $Λ:\\mathrm{S}_{A^+}\\to\\mathrm{S}_{B^+}$ is a bijection satisfying \\[ \\|Λ(u)-Λ(v)\\|=1 \\quad\\Longleftrightarrow\\quad \\|u-v\\|=1 \\qquad (u,v\\in\\mathrm{S}_{A^+}). \\] If $A$ has no two-dimensional irreducible $^*$-representation, then $A$ and $B$ are Jordan $^*$-isomorphic. Moreover, the Jordan $^*$-isomorphism can be chosen to be compatible with $Λ$ on the peak and zero projections of positive norm-one elements. The proof is based on recovering the order and orthogonality structure of Akemann closed projections in the bidual. As a main application, we give an affirmative answer to the positive-sphere extension problem for unital $C^*$-algebras of real rank zero. Namely, if $A$ has real rank zero (but could has two-dimensional irreducible $^*$-representations) and $Λ:\\mathrm{S}_{A^+}\\to\\mathrm{S}_{B^+}$ is a surjective isometry, then $Λ$ extends to a Jordan $^*$-isomorphism from $A$ onto $B$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32029",
  "author_total": 3,
  "authors": [
    {
      "name": "Chi-Wai Leung"
    },
    {
      "name": "Chi-Keung Ng"
    },
    {
      "name": "Ngai-Ching Wong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b30trqwqtb7ex.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32029",
  "fulltext_url": null,
  "id": "b30trqwqtb7ex",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32029",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223683030",
  "pdf_url": "https://arxiv.org/pdf/2610.32029",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b30trqwqtb7ex.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32029",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32029"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Reconstructing the Jordan structure of $C^*$-algebras from the metric on positive unit spheres",
  "title_text": "Reconstructing the Jordan structure of C^*-algebras from the metric on positive unit spheres",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/b30trqwqtb7ex",
  "venue": "arXiv",
  "versions": []
}