{
  "abstract": "A positive functional is a trace if and only if it satisfies any one of many inequalities, the first being Gardner's $|φ(X)|\\leφ(|X|)$; positive central elements are characterised similarly. We ask how strongly these characterisations fail. For $φ=\\mathrm{Tr}(ρ\\,\\cdot)$ on a matrix algebra the natural scale is the Michelson contrast $δ$ of the density. For Gardner's inequality the sharp constant depends on $δ$ and on the contrast $δ'$ of $|X|$ only through their product: it is $(1-δ^2δ'^2)^{-1/2}$, the secant of the angle whose sine is $δδ'$. On a finite von Neumann algebra the contrast is the wrong parameter, since every central density gives a trace; the governing parameter is the noncentrality, defined through the central condition number $κ_Z(ρ)$ of the central envelopes of $ρ$ (the largest and smallest elements of Halpern's central spectrum). A transfer principle lifts sharp matrix constants to exact constants in terms of $κ_Z$ in every finite von Neumann algebra. For 38 trace characterisations the constant is exact in 23 cases, has proved bounds in 13 and is known only numerically in two; for instance it is $\\cosh(\\frac12\\lnκ_Z)$ for Gardner's inequality and for the subadditivity of the modulus. On matrix algebras the finite exact violations vanish at the trace to the orders $δ$, $δ^2$ or $δ^4$, and outside factors and abelian algebras no function of the contrast describes the constants. We also find the sharp constants of the Novikov-Tikhonov characterisations of central elements, sharp bounds for three unnormalised trace functionals, and sharp contrast thresholds below which subadditivity of the square root and the power-mean inequality cannot fail. A numerical search, certified as a lower bound for each supremum, supports conjectured values for the remaining inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34388",
  "author_total": 3,
  "authors": [
    {
      "name": "Sami Abdullah Abed"
    },
    {
      "name": "Irina Nikolaeva"
    },
    {
      "name": "Andrej Novikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75xxqa0xstaqz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34388",
  "fulltext_url": null,
  "id": "75xxqa0xstaqz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34388",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223909387",
  "pdf_url": "https://arxiv.org/pdf/2610.34388",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75xxqa0xstaqz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.34388",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34388"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "How characterisations of traces and central elements fail: sharp violation constants via michelson contrast and noncentrality",
  "title_text": "How characterisations of traces and central elements fail: sharp violation constants via michelson contrast and noncentrality",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/75xxqa0xstaqz",
  "venue": "arXiv",
  "versions": []
}