{
  "abstract": "The Atiyah-Segal homomorphism relates the representation ring of a finite group to the $K$-theory ring of its classifying space. We develop a version of this for infinite groups, but based on a ring of approximate representations rather than representations. We then use known results on the Baum-Connes conjecture to show that our homomorphism is an isomorphism for many groups: this includes all torsion-free amenable groups, and all torsion-free fundamental groups of compact manifolds of dimension at most three. The main motivation is the desire to understand approximate representations using topological tools: as a sample application we give some examples of torsion phenomena in approximate representation theory. The methods of proof are $C^*$-algebraic: the key idea is to relate our Atiyah-Segal type homomorphism to Kasparov's coassembly map. The main step realizes (Hausdorffized) $KK$-theory as approximately multiplicative ucp maps modulo stable approximate unitary equivalence for appropriate $C^*$-algebras. This is in the spirit of the stable uniqueness theorems of Dadarlat-Eilers and Lin, and may be of purely $C^*$-algebraic interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24252",
  "author_total": 1,
  "authors": [
    {
      "name": "Rufus Willett"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8nzanx7x3fc0v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24252",
  "fulltext_url": null,
  "id": "8nzanx7x3fc0v",
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  "landing_url": "https://arxiv.org/abs/2610.24252",
  "language": null,
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  "licence_label": "arXiv licence",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
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    [
      "arxiv",
      "math.GR"
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    [
      "arxiv",
      "math.KT"
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      "arxiv",
      "math.OA"
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      "openalex",
      "2610|Advanced Operator Algebra Research"
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  "openalex": "W7223005671",
  "pdf_url": "https://arxiv.org/pdf/2610.24252",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8nzanx7x3fc0v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24252",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24252"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An Atiyah-Segal homomorphism for approximate representations of infinite groups, and almost multiplicative ucp maps",
  "title_text": "An Atiyah-Segal homomorphism for approximate representations of infinite groups, and almost multiplicative ucp maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/8nzanx7x3fc0v",
  "venue": "arXiv",
  "versions": []
}