{
  "abstract": "We embed skew fields of meromorphic functions in several non-commuting variables into the topological $*-$algebra of closed linear operators affiliated to a finite von Neumann algebra, $\\mathfrak{M}$, with faithful, normal tracial state, $τ$. Namely, let $\\mathscr{O} _d (r)$, $r \\in (0, +\\infty]$, denote the rings of formal power series in $d \\in \\mathbb{N}$ non-commuting variables with \"radius of convergence\" at least $r$ and let $Ξ\\in \\mathfrak{M} ^{1\\times d}$ be a tuple of row-norm $<r$ with the property that there exist no non-zero trace-class operators $T_1, \\cdots, T_d \\in \\mathscr{B} (L^2 (\\mathfrak{M}, τ))$ obeying $\\sum _j [T_j , Ξ_j] =0$. Then the unital homomorphism $\\mathscr{O} _d (r) \\rightarrow \\mathfrak{M}$ defined by the evaluation, $\\mathfrak{z} _j \\mapsto Ξ_j$, extends to an embedding of the universal skew field of fractions of $\\mathscr{O} _d (r)$ into the $*-$algebra of closed operators affiliated to $\\mathfrak{M}$. In their $L^2-$homology theory of von Neumann algebras, Connes and Shlyakhtenko defined a certain quantity, $0 \\leq Δ(Ξ) \\leq d$, as the Murray-von Neumann co-dimension of the Hilbert module generated by all $d-$tuples of finite rank operators obeying the above commutator condition with $Ξ\\in \\mathfrak{M} ^{1\\times d}$. As demonstrated by Mai, Speicher and Yin, this $Δ$ measures, in a certain precise sense, the algebraic independence of the components of $Ξ$. Connes-Shlyakhtenko proved that $Δ$ is an upper bound for Voiculescu's non-microstates free entropy dimension, $δ^* (Ξ)$, and remark that it is of interest to determine whether these two quantities are equal. Using non-commutative function theory, we show that this is generally not the case, by constructing a triple of self-adjoint operators, $Ξ= (Ξ_1,Ξ_2,Ξ_3)$, for which this inequality is strict, $δ^*(Ξ) < Δ(Ξ)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21004",
  "author_total": 4,
  "authors": [
    {
      "name": "Méric L. Augat"
    },
    {
      "name": "Robert T. W. Martin"
    },
    {
      "name": "Maximilian Tornes"
    },
    {
      "name": "Eli Shamovich"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8nb4n1ggg46mr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21004",
  "fulltext_url": null,
  "id": "8nb4n1ggg46mr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21004",
  "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.OA"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7222510810",
  "pdf_url": "https://arxiv.org/pdf/2610.21004",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8nb4n1ggg46mr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.21004",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21004"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras",
  "title_text": "Skew fields of non-commutative meromorphic functions affiliated to finite operator algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8nb4n1ggg46mr",
  "venue": "arXiv",
  "versions": []
}