{
  "abstract": "In 2014, K.Guo and the second author constructed the first example of an infinite Blaschke product $B$ (a thin Blaschke product), for which the associated multiplication operator $M_B$ is irreducible on the Bergman space $L_a^2(\\mathbb{D})$ over the unit disk $\\mathbb D$. We take a substantial step forward toward characterizing the irreducibility of the operator $M_B$ on $L_a^2(\\mathbb{D})$, or equivalently, the trivialness of the von Neumann algebra $\\{M_B,M_B^*\\}'$. This characterization indicates a close connection with the primeness of $B$ and with the two-component property of the Riemann surface $\\mathcal{S}_B$ associated with $B$. The approach is built upon a nested family of connected components of the sublevel set $\\{z\\in \\mathbb{D}:|B(z)|<t\\}$ for $0<t<1$, which captures the function-theoretic and geometric feature of $B$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00933",
  "author_total": 4,
  "authors": [
    {
      "name": "Danni Guo"
    },
    {
      "name": "Hansong Huang"
    },
    {
      "name": "Shan Li"
    },
    {
      "name": "Xianfeng Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9pk2ebz2j7xf3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00933",
  "fulltext_url": null,
  "id": "9pk2ebz2j7xf3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00933",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00933",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9pk2ebz2j7xf3.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00933",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00933"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A new perspective on Riemann surfaces and von Neumann algebras via thin Blaschke products",
  "title_text": "A new perspective on Riemann surfaces and von Neumann algebras via thin Blaschke products",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9pk2ebz2j7xf3",
  "venue": "arXiv",
  "versions": []
}