{
  "abstract": "For a $C^*$-algebra $A$ let $\\mathcal I(A)$ be the set of all elements $a+b$ with $a,b\\in A_+$, $ab=0$ and $\\|a\\|=\\|b\\|=1$. We study the algebraic, metric and topological properties of $\\mathcal I(A)$. We describe $\\mathcal I(A\\oplus B)$ and show that $\\mathcal I(A/J)\\subseteqπ(\\mathcal I(A))$ for the quotient map $π:A\\to A/J$. For finite-dimensional $A$ we prove $\\mathcal I(A)=\\operatorname{Sph}^+_A(E(A))$, where $E(A)$ is the set of minimal projections of $A$ and $\\operatorname{Sph}^+_A(E)$ is the set of positive norm-one elements at distance one from every element of $E$; we also show that the analogous statement with the minimal projections of the bidual fails in general. When $\\mathcal I(A)\\neq\\varnothing$ the distance from $1$ to $\\mathcal I(A)$ is $0$ or $1$, according to whether $A$ has a nontrivial projection. Moreover, $\\mathcal I(A)$ is norm compact if and only if $A$ is finite-dimensional, closedness of $\\mathcal I(A)$ passes to hereditary subalgebras, $\\mathcal I(A)$ is norm closed for every von Neumann algebra, and the extreme points of $\\mathcal I(A)$ are exactly its projections. For a compact, locally path connected space $X$ the set $\\mathcal I(C(X))$ is closed, whereas $\\mathcal I(\\ell_c(A))$ fails to be closed when $A$ is unital without nontrivial projections.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15167",
  "author_total": 1,
  "authors": [
    {
      "name": "Ali Taghavi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/37nz6z1crmgrb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15167",
  "fulltext_url": null,
  "id": "37nz6z1crmgrb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15167",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7221833659",
  "pdf_url": "https://arxiv.org/pdf/2610.15167",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/37nz6z1crmgrb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15167",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15167"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the set of orthogonal positive sums in a $C^*$-algebra",
  "title_text": "On the set of orthogonal positive sums in a C^*-algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/37nz6z1crmgrb",
  "venue": "arXiv",
  "versions": []
}