{
  "abstract": "Let $R$ be a unital ring equipped with a Sylvester matrix rank function $\\rk$. A harmonic function $α$ on a Bratteli diagram $B$ defines a weighted matrix rank on the associated algebraic direct limit $A(B,R)$. We prove that, if $α$ is extreme and the total weight of blocks of any fixed bounded size tends to zero, then the rank completion of $A(B,R)$ is isomorphic to $\\mathcal M_{R,\\rk}$, the rank completion of the direct system $\\Mat_{2^k}(R)$ with connecting maps $x\\mapsto\\diag(x,x)$ and normalized ranks $2^{-k}\\rk$. The isomorphism preserves the unital $R$-algebra structure and the ranks on all rectangular matrices. The coefficient ring need not be regular, and the specified rank need not be induced from a regular ring. We recover factor-sequence uniqueness and construct corners of every prescribed rank in $(0,1]$ that are isomorphic to $\\mathcal M_{R,\\rk}$ with their normalized ranks. Examples show that the coefficient rank can affect the isomorphism type and that the completion can be non-regular and non-simple. For complex coefficients, the trace determined by $α$ gives a rank completion of the associated AF $C^*$-algebra canonically isomorphic to the affiliated-operator ring of its GNS closure. The rank completion of the algebraic direct limit can be a proper subring of this ring.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12924",
  "author_total": 1,
  "authors": [
    {
      "name": "Baojie Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ajjtbwxggfy68.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12924",
  "fulltext_url": null,
  "id": "ajjtbwxggfy68",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12924",
  "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.RA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7213047571",
  "pdf_url": "https://arxiv.org/pdf/2609.12924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ajjtbwxggfy68.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12924"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Uniqueness of Rank-Metric Completions of Bratteli Systems",
  "title_text": "Uniqueness of Rank-Metric Completions of Bratteli Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/ajjtbwxggfy68",
  "venue": "arXiv",
  "versions": []
}