AI 中文总结
该研究构建实超巴拿赫代数的谱层面分次K-理论，采用范畴与同伦论方法，定义相关谱并证明其性质，建立Bott周期性与Morita等价的联系，还给出经典K群的谱细化及非分次拓扑环K-理论的泛性质刻画。
AI 中文摘要
我们为实超巴拿赫代数构建了谱层面的分次K-理论。我们的构造采用范畴论与同伦论的方法，遵循代数K-理论的范式：分次K-理论谱由有限生成投射分次模的拓扑群胚的(∞,1)-范畴群完备化通过(上)纤维序列得到，而非源自Fredholm算子空间或Kasparov循环。我们定义了连通谱k^ABS_A（作为上纤维细化了Atiyah–Bott–Shapiro构造）及其周期化版本K^gr_A，并证明二者均为松弛对称幺半群的，且关于双模（而非仅同态）具有函子性。研究表明，分次A-模无法表示K^gr_0(A)中所有上循环的现象，纯粹是k^ABS_A上的π_0现象。我们得到了自然等价K^gr_{A⊗̂Cl_{p,q}} ≃ Σ^{p−q}K^gr_A，该等价将拓扑Bott周期性与Cl_8和ℝ之间的Morita等价联系起来。限制到可逆有限维半单超代数时，可得到对称幺半群函子Pic(Bim(sBan_ℝ)^fd) → Pic(Mod(KO))，该函子分裂出Pic(Mod(KO))的底部三层Postnikov层，建立了超可除代数与可逆KO-模之间的直接联系。我们还给出了Karoubi和van Daele分次K群的谱细化版本，并附带显式比较等价，由此建立了与KK-理论的关联。此外，我们对非分次拓扑环的K-理论进行了广泛的一般性研究，该研究本身可能具有独立价值。具体而言，我们通过泛性质刻画了非分次巴拿赫代数的连通拓扑K-理论。
英文摘要
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.
Comments83+15 pages
