{
  "abstract": "We prove that, for every countable non-exact discrete group $H$ and every nontrivial countable discrete group $K$, the reduced group $C^*$-algebra $C_r^*(K\\wr H)$ is not locally reflexive. These provide the first known examples of non-locally reflexive reduced group $C^*$-algebras, settling a longstanding existence question. Moreover, for every nontrivial countable exact discrete group $K$, we establish that local reflexivity of $C_r^*(K\\wr H)$ is equivalent to exactness of $H$, and hence to exactness of $C_r^*(K\\wr H)$. This gives a partial affirmative answer to Kirchberg's question of whether local reflexivity implies exactness for $C^*$-algebras. As applications, we construct finitely generated groups with the Haagerup property, as well as finitely generated residually finite groups, whose reduced group $C^*$-algebras are not locally reflexive. In particular, neither the Haagerup property nor coarse embeddability into Hilbert space guarantees local reflexivity of the associated reduced group $C^*$-algebra.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34695",
  "author_total": 2,
  "authors": [
    {
      "name": "Pingchuan Dong"
    },
    {
      "name": "Zhe Dong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftqe5wfwaa827.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34695",
  "fulltext_url": null,
  "id": "ftqe5wfwaa827",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34695",
  "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.GR"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7223818287",
  "pdf_url": "https://arxiv.org/pdf/2610.34695",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftqe5wfwaa827.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34695",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34695"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Local Reflexivity and Exactness for Reduced $C^*$-Algebras of Wreath Products",
  "title_text": "Local Reflexivity and Exactness for Reduced C^*-Algebras of Wreath Products",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/ftqe5wfwaa827",
  "venue": "arXiv",
  "versions": []
}