A Comparison of Takai and Treumann Dualities
Vikram Nadig
TL;DR
This work situates Takai duality, classically expressed as (A ⋊ G) ⋊ Ĝ ≅ A ⊗ K(L^2(G)), within the modern framework of stable ∞-categories by constructing the stable ∞-category KK^G for G-equivariant KK-theory and proving that the crossed-product functor descends to KK^G → KK^{Ĝ}. It then juxtaposes this with Treumann duality, which provides an exotic equivalence between KU_p[G]-module categories and KU_p[Ĝ]-module categories via a canonical M bimodule, yielding an equivalence on modules of finite type. The main achievement is a natural equivalence that fills a square of functors, showing that Takai duality and Treumann duality are compatible refinements in a common higher-categorical setting after p-completion. This unifies operator-algebraic and higher-algebraic dualities and clarifies how classical Takai duality corresponds to Treumann’s exotic equivalence through a precise categorical bridge involving K-theory and module categories over KU_p. The results illuminate how dualities in noncommutative geometry can be understood as manifestations of universal properties in stable ∞-categories, with potential implications for comparisons across C*-algebraic and chromatic settings.
Abstract
We prove a comparison result between two duality statements - Takai duality, which is implemented by the crossed product functor $- \rtimes G: KK^{G} \to KK^{\hat G}$ on equivariant Kasparov categories; and Treumann duality, which asserts the existence of an exotic equivalence of stable $\infty$-categories $\text{Mod}(KU_p[G])^{ft} \simeq \text{Mod}(KU_p[\hat G])^{ft}$ given by tensoring with a particular $(G,\hat G)$-bimodule $\textbf{M}$.