A Comparison of Takai and Treumann Dualities

TL;DR

This paper establishes a comparison between Takai duality in equivariant Kasparov theory and Treumann duality involving stable $ ext{infty}$-categories, linking two different duality frameworks in operator algebras.

Contribution

It proves a new comparison result connecting Takai duality with Treumann duality through categorical equivalences and bimodule constructions.

Findings

Demonstrates an equivalence between Takai and Treumann dualities.

Provides a categorical framework linking crossed product functors and bimodule tensoring.

Bridges two duality theories in operator algebras and KK-theory.

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 $\textbf{M}$.