Super $K$-theory and group completion

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.