{
  "abstract": "In this paper we establish two main results concerning the comparison theory of positive elements in approximately homogeneous $C^{*}$-algebras with diagonal maps. The first is an existence theorem which interfaces with the rank problem; given a Villadsen algebra $A$, we construct a convex cone that induces an embedding of a semigroup into $Cu(A)$. This semigroup is identified with the set of strictly positive, lower-semicontinuous affine maps on the trace simplex of $A$. The second is a construction of an AH algebra that produces an obstacle to classifying the approximate unitary orbits of positive elements in terms of their spectral data. Our counterexample sheds light on a critical topological obstruction when attempting to understand equivalence classes of $*$-homomorphisms from $C_{0}((0,1])$ to an arbitrary $C^{*}$-algebra of stable rank one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38142",
  "author_total": 2,
  "authors": [
    {
      "name": "Alejandro Cano"
    },
    {
      "name": "Andrew Toms"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1e6jjpanq1gee.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38142",
  "fulltext_url": null,
  "id": "1e6jjpanq1gee",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38142",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38142",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1e6jjpanq1gee.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38142",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38142"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "New structure for Villadsen algebras of the first type",
  "title_text": "New structure for Villadsen algebras of the first type",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/1e6jjpanq1gee",
  "venue": "arXiv",
  "versions": []
}