{
  "abstract": "In this paper, we construct countable uniformly locally finite metric spaces $X$ and $Y$ such that $C_u^*(X)$ is isomorphic to a hereditary $C^*$-subalgebra of $C_u^*(Y)$, but neither space coarsely embeds into the other. This gives a negative answer to the embedding rigidity problem for uniformly locally finite coarse spaces. On the positive side, we prove that a hereditary embedding $C_u^*(X)\\to C_u^*(Y)$ implies an injective coarse embedding $X\\to Y$ whenever $C_u^*(Y)$ contains no block-rank-one ghost projection. In particular, this strengthens the coarse embedding theorem of Braga, Farah, and Vignati by obtaining injectivity under the same hypothesis that every sparse subspace of $Y$ yields only compact ghost projections.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16949",
  "author_total": 1,
  "authors": [
    {
      "name": "Teng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7spbafabh3nj2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16949",
  "fulltext_url": null,
  "id": "7spbafabh3nj2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16949",
  "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",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7169869616",
  "pdf_url": "https://arxiv.org/pdf/2607.16949",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7spbafabh3nj2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2607.16949",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16949"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the embedding rigidity problem for uniformly locally finite coarse spaces",
  "title_text": "On the embedding rigidity problem for uniformly locally finite coarse spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/7spbafabh3nj2",
  "venue": "arXiv",
  "versions": []
}