{
  "abstract": "We study two invariants of $C^*$-algebras built from positive zero divisors. First, we associate with a $C^*$-algebra $A$ the graph $Γ_+(A)$ whose vertices are the nonzero positive zero divisors of $A$, two vertices being adjacent when they are orthogonal. We show that $Γ_+$ of the Calkin algebra is connected of diameter $3$, that $Γ_+(B(H))$ is connected of diameter $4$ (for $\\dim H\\ge3$), that $Γ_+(A)$ is connected of diameter at most $4$ for every von Neumann factor $A\\neq\\mathbb C,M_2(\\mathbb C)$, and of diameter at most $3$ if $A$ is not of type~I, whereas $Γ_+(M_2(\\mathbb C))$ is disconnected. Second, we study the zero-divisor real rank $ZDRR(A)$, obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that $ZDRR(A)\\le RR(A)$. For commutative algebras $C(X)$ we prove $ZDRR(C(X))=RR(C(X))=\\dim X$ when $X$ is a compact almost-$P$-space, while $ZDRR(C(X))=0$ for separable $X$. Using Levy's connected, totally ordered almost-$P$-space we construct unital commutative $C^*$-algebras $A_n$ with $ZDRR(A_n)=n$ for every $n\\in\\mathbb N_0$. On the other hand $ZDRR(C(X\\times[0,1]))=0$ for every compact Hausdorff space $X$, so $ZDRR$ and $RR$ can differ arbitrarily.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15160",
  "author_total": 1,
  "authors": [
    {
      "name": "Ali Taghavi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fzszsh535bqm7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15160",
  "fulltext_url": null,
  "id": "fzszsh535bqm7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15160",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7221805442",
  "pdf_url": "https://arxiv.org/pdf/2610.15160",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fzszsh535bqm7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15160",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15160"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Zero-divisor graphs and zero-divisor real rank of $C^*$-algebras",
  "title_text": "Zero-divisor graphs and zero-divisor real rank of C^*-algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/fzszsh535bqm7",
  "venue": "arXiv",
  "versions": []
}