{
  "abstract": "We show how a simple metric condition in terms of the C$^*$-norm of an arbitrary C$^*$-algebra $A$ can be employed to determine those norm-one positive elements which are projections. Concretely, if we write $S_{A^+}$ for the intersection of the positive cone and the unit sphere of $A$, we establish that an element $a \\in S_{A^+}$ is a projection if, and only if, $$\\{a\\} = \\left\\{x\\in S_{A^+}: \\|x-z\\| = 1 \\hbox{ for all } z\\in S_{A^+}\\hbox{ with } \\|a-z\\|=1 \\right\\}.$$ This proves a conjecture which has been previously confirmed for von Neumann algebras and unital C$^*$-algebras. In the setting of not necessarily unital JB$^*$-algebras, the Shirshov--Cohn theorem is invoked to obtain a similar metric characterisation of projections. In the wider setting of JB$^*$-triples, we show how a metric characterisation in terms of inner ideals can be established to determine non-zero tripotents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35290",
  "author_total": 2,
  "authors": [
    {
      "name": "Antonio M. Peralta"
    },
    {
      "name": "Pedro Saavedra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5m59mjyaj6q0a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35290",
  "fulltext_url": null,
  "id": "5m59mjyaj6q0a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35290",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223867882",
  "pdf_url": "https://arxiv.org/pdf/2610.35290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5m59mjyaj6q0a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra",
  "title_text": "The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/5m59mjyaj6q0a",
  "venue": "arXiv",
  "versions": []
}