{
  "abstract": "We exhibit a simple unital infinite-dimensional C$^*$-algebra $A$ which is not $K_1$-injective, i.e., the canonical map $ U(A)/U_0(A)\\longrightarrow K_1(A)$ is not injective. This phenomenon is well known in the non-simple case. Indeed, a natural example is the generator $u$ of $π_4(U(2))\\cong\\mathbb{Z}/2$, and we use exactly this example as the seed of our construction. The construction proceeds by developing a higher-rank matrix trapping technique in homogeneous algebras generalizing the machinery of Villadsen in higher stable rank simple C$^*$-algebras. A hypothetical homotopy from the homomorphic image of $u$ to the identity in a certain homogeneous algebra is shown, through matrix trapping, to yield a spin bordism from the class of $u$ to that of the identity matrix $\\mathrm{1}_2 \\in M_2$, and this, in turn, is obstructed by the fact that the bordism class of $u$ is not trivial. We then fit our new arguments into a generalized version of Villadsen's higher stable rank construction to arrive at a simple example.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09535",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew S. Toms"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1c8kvbak9aayr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09535",
  "fulltext_url": null,
  "id": "1c8kvbak9aayr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09535",
  "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": "W7212119868",
  "pdf_url": "https://arxiv.org/pdf/2609.09535",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1c8kvbak9aayr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09535",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09535"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A simple C$^*$-algebra which is not $K_1$-injective",
  "title_text": "A simple C^*-algebra which is not K_1-injective",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/1c8kvbak9aayr",
  "venue": "arXiv",
  "versions": []
}