{
  "abstract": "Christensen and Pedersen proved that every properly infinite $\\mathrm{AW}^*$-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of $\\mathrm{AW}^*$-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a $*$-homomorphism between $\\mathrm{AW}^*$-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about $κ$-join-preserving $*$-homomorphisms. We also show that a finite $\\mathrm{AW}^*$-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20718",
  "author_total": 1,
  "authors": [
    {
      "name": "Jananan Arulseelan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8j67dzna5sfpx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20718",
  "fulltext_url": null,
  "id": "8j67dzna5sfpx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20718",
  "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.LO"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7213593062",
  "pdf_url": "https://arxiv.org/pdf/2609.20718",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8j67dzna5sfpx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.20718",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20718"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Revisiting the Transfinite Christensen-Pedersen Argument",
  "title_text": "Revisiting the Transfinite Christensen-Pedersen Argument",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/8j67dzna5sfpx",
  "venue": "arXiv",
  "versions": []
}