{
  "abstract": "Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\\colon X\\to\\mathbb{R}$ induces a one-parameter group $σ_h$ of automorphisms of $\\mathcal{B}(\\ell_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of $σ_h$ are precisely the uniform Roe algebra of the pseudo-metric induced by $h$. If we moreover assume that $X$ is a uniformly locally finite metric space and $h$ is allowed to range over all coarse maps, this gives dynamical characterizations of both algebras: $\\mathrm{C}^*_{ql}(X)$ consists exactly of the operators which are continuous for all such flows, while $\\mathrm{C}^*_u(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to $\\mathrm{C}^*$-algebras between $\\mathrm{C}^*_u(X)$ and $\\mathrm{C}^*_{ql}(X)$ which are invariant under bijective coarse equivalence, and we show that this scale is not degenerate: if $X$ is a coarse disjoint union of expander graphs, the algebra generated by operators which are analytic on a strip for every diagonal flow sits strictly between $\\mathrm{C}^*_u(X)$ and $\\mathrm{C}^*_{ql}(X)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39762",
  "author_total": 1,
  "authors": [
    {
      "name": "Bruno de Mendonça Braga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/07kqaf3zda8dw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39762",
  "fulltext_url": null,
  "id": "07kqaf3zda8dw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39762",
  "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.FA"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7215041999",
  "pdf_url": "https://arxiv.org/pdf/2609.39762",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/07kqaf3zda8dw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39762",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39762"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Dynamical $\\mathrm C^*$-algebras and coarse geometry",
  "title_text": "Dynamical C^*-algebras and coarse geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/07kqaf3zda8dw",
  "venue": "arXiv",
  "versions": []
}