{
  "abstract": "We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\\infty,1)$-categorical group completion of topological groupoids of finitely generated projective graded modules, rather than from spaces of Fredholm operators or Kasparov cycles. We define a connective spectrum $k^{\\mathrm{ABS}}_A$ refining the Atiyah--Bott--Shapiro construction as a cofiber, together with its periodification $K^{\\mathrm{gr}}_A$, and show that both are lax symmetric monoidal and functorial in bimodules, not merely in homomorphisms. The failure of graded $A$-modules to present all cocycles in $K^{\\mathrm{gr}}_0(A)$ is shown to be purely a $π_0$-phenomenon on $k^{\\mathrm{ABS}}_A$. We obtain a natural equivalence $K^{\\mathrm{gr}}_{A\\widehat{\\otimes} \\mathrm{Cl}_{p,q}} \\simeq Σ^{p-q}K^{\\mathrm{gr}}_A$, which links topological Bott periodicity with the Morita equivalence between $\\mathrm{Cl}_8$ and $\\mathbb{R}$. Restricting to invertible finite-dimensional semisimple super algebras yields a symmetric monoidal functor $\\operatorname{Pic}(\\operatorname{Bim}(\\mathrm{sBan}_{\\mathbb{R}})^{\\mathrm{fd}}) \\to \\operatorname{Pic}(\\mathrm{Mod}(KO))$ which splits off the bottom three Postnikov layers of $\\operatorname{Pic}(\\mathrm{Mod}(KO))$, giving a direct link between super division algebras and invertible $KO$-modules. We also give spectral refinements of Karoubi's and van Daele's graded $K$-groups, with explicit comparison equivalences, therefore connecting to $KK$-theory. We also provide an extensive general treatment for $K$-theory of ungraded topological rings that might be of independent interest. In particular, we characterize connective topological $K$-theory of ungraded Banach algebras by a universal property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02407",
  "author_total": 2,
  "authors": [
    {
      "name": "David Aretz"
    },
    {
      "name": "Luuk Stehouwer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8gw1csmbh2shy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02407",
  "fulltext_url": null,
  "id": "8gw1csmbh2shy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02407",
  "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.AT"
    ],
    [
      "arxiv",
      "math.KT"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7207617270",
  "pdf_url": "https://arxiv.org/pdf/2609.02407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8gw1csmbh2shy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.KT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Super $K$-theory and group completion",
  "title_text": "Super K-theory and group completion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/8gw1csmbh2shy",
  "venue": "arXiv",
  "versions": []
}