Super K-theory and group completion

@misc{aretz2026super,
  title = {{Super $K$-theory and group completion}},
  author = {David Aretz and Luuk Stehouwer},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02407},
  eprint = {2609.02407},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02407},
}

TY  - UNPB
TI  - Super K-theory and group completion
AU  - David Aretz
AU  - Luuk Stehouwer
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02407
UR  - https://arxiv.org/abs/2609.02407
AB  - 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 (∞,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^ABS_A refining the Atiyah--Bott--Shapiro construction as a cofiber, together with its periodification K^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^gr_0(A) is shown to be purely a π_0-phenomenon on k^ABS_A. We obtain a natural equivalence K^gr_A⊗̂ Cl_p,q ≃ Σ^p−qK^gr_A, which links topological Bott periodicity with the Morita equivalence between Cl_8 and ℝ. Restricting to invertible finite-dimensional semisimple super algebras yields a symmetric monoidal functor Pic(Bim(sBan_ℝ)^fd) → Pic(Mod(KO)) which splits off the bottom three Postnikov layers of Pic(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.
ER  -