Equivariant K-theory and the Chern character for discrete groups

Abstract

Let $X$ be a compact Hausdorff space, let $\Gamma$ be a discrete group that acts continuously on $X$ from the right, define $\tilde{X}=\{(x,\gamma )\in X\times \Gamma :x\cdot \gamma =x\}$, and let $\Gamma$ act on $\tilde{X}$ via the formula $(x,\gamma )\cdot \alpha =(x\cdot \alpha ,{\alpha }^{-1}\gamma \alpha )$. 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 $K_{\Gamma }^i(X)$ and $K^i(\widetilde{X}\slashΓ)$ are not isomorphic.