{
  "abstract": "We introduce a topological abelian group $\\mathrm{K}_1^Δ(A)$ for every $\\mathrm{C}^\\ast$-algebra $A$, which is functorial in $A$ and invariant under stabilisations. $\\mathrm{K}_1^Δ$ can distinguish group actions indistinguishable by equivariant (K)K-theory and traces --- we illustrate this on the irrational rotation algebra by exhibiting strongly outer actions of $\\mathbb Z$ which, via $\\mathrm{K}_1^Δ$, are not (stably) cocycle conjugate. These actions are moreover trivial in $\\mathrm{KK}^{\\mathbb Z}$ and on the Hausdorffised unitary algebraic $\\mathrm{K}_1$ group $\\overline{\\mathrm{K}}_1^{\\mathrm{alg}}(A)$. The Hausdorffisation $\\overline{\\mathrm{K}}_1^Δ(A)$ of $\\mathrm{K}_1^Δ(A)$ forms the final part of the total stable invariant $\\mathrm{\\underline{K}T}_{\\mathrm{s}}(A)$, which we also introduce. We prove that $\\mathrm{\\underline{K}T}_{\\mathrm{s}}$ classifies $*$-homomorphisms up to approximate unitary equivalence between certain stable $\\mathrm{C}^\\ast$-algebras which contain a non-zero projection. $\\mathrm{K}_1^Δ(A)$ is new even in the unital setting, but under further assumptions on $A$ recovers the unitary algebraic $\\mathrm{K}_1$ group $\\mathrm{K}_1^{\\mathrm{alg}}(A)$, whereas the Hausdorffisations $\\overline{\\mathrm{K}}_1^Δ(A)$ and $\\overline{\\mathrm{K}}_1^{\\mathrm{alg}}(A)$ agree for every unital $A$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06187",
  "author_total": 2,
  "authors": [
    {
      "name": "James Gabe"
    },
    {
      "name": "Alistair Miller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgp84cs211p9b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06187",
  "fulltext_url": null,
  "id": "dgp84cs211p9b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06187",
  "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": "W7220684079",
  "pdf_url": "https://arxiv.org/pdf/2610.06187",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgp84cs211p9b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06187",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06187"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "$\\boldsymbol{\\mathrm{K}_1^Δ}$ for $\\boldsymbol{\\mathrm{C}^\\ast}$-algebras",
  "title_text": "K_1^Δ for C^∗-algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/dgp84cs211p9b",
  "venue": "arXiv",
  "versions": []
}