{
  "abstract": "Let $(\\mathcal N,ν)$ be a finite tracial von Neumann algebra and set $\\mathcal{K}_r(P,Q)=ν((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\\ge0, \\qquad 1/2\\le r\\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\\le r<1$. This single mechanism resolves two problems. First, if $(\\mathcal R,ω)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\\in M_3(\\mathcal R)_+$, then $$ [ω(|H_{ij}|^p)]_{i,j=1}^3\\ge0, \\qquad 1\\le p\\le2, $$ settling the Lin--van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\\mathcal M,τ)$, define $$ d_t(A,B)^2=\\frac{τ(A)+τ(B)}2- \\|B^{t/4}A^{t/4}\\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\\mathcal M,τ)_+$ for $1\\le t\\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\\text{ is a metric on }L^1(\\mathcal M,τ)_+ \\quad\\Longleftrightarrow\\quad t\\in[1,2]\\ \\text{or}\\ \\mathcal M\\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11021",
  "author_total": 2,
  "authors": [
    {
      "name": "Trung Dung Vuong"
    },
    {
      "name": "Hiroyuki Osaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a4rhh3exfjra8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11021",
  "fulltext_url": null,
  "id": "a4rhh3exfjra8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11021",
  "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"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7221128039",
  "pdf_url": "https://arxiv.org/pdf/2610.11021",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a4rhh3exfjra8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:06Z",
      "native_id": "2610.11021",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11021"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics",
  "title_text": "Three-Point Trace Positivity:Schatten Norm Compression and Interpolating Metrics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/a4rhh3exfjra8",
  "venue": "arXiv",
  "versions": []
}