A Comparison of Takai and Treumann Dualities

We prove a comparison result between two duality statements - Takai duality, which is implemented by the crossed product functor $-⋊G:K{K}^G\to K{K}^{\widehat{G}}$ on equivariant Kasparov categories; and Treumann duality, which asserts the existence of an exotic equivalence of stable $\mathrm{\infty }$-categories $\text{Mod}(K{U}_p[G]{)}^{ft}\simeq \text{Mod}(K{U}_p[\widehat{G}]{)}^{ft}$ given by tensoring with a particular $(G,\widehat{G})$-bimodule $\mathbf{\text{M}}$.