Equivariant K-theory and the Chern character for discrete groups

Let $X$ be a compact Hausdorff space, let $Γ$ be a discrete group that acts continuously on $X$ from the right, define $\widetilde{X} = \{(x,γ) \in X \times Γ: x\cdotγ= x\}$, and let $Γ$ act on $\widetilde{X}$ via the formula $(x,γ)\cdotα= (x\cdotα, α^{-1}γα)$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $K^i_Γ(X) \otimes \mathbb{C} \cong K^i(\widetilde{X}\slashΓ) \otimes \mathbb{C}$ for $i = 0, -1$. In this note, we present an example where the groups $K^i_Γ(X)$ and $K^i(\widetilde{X}\slashΓ)$ are not isomorphic.

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