We have the Pimsner-Voiculescu exact sequences and the Baum-Connes map for possible computation of the $K$-theory of the reduced group $C^*$-algebra $C^*_r(G)$ for a topological, locally compact, second-countable Hausdorff group $G$.

Up to now I have not seen much computations of $K(C^*_r(G))$.

Has anyone references to such computations, in particular in computing the left hand side of the Baum-Connes map, under the Chern map. That is, computations of the Czech cohomology groups $$\lim_{X \subseteq \underline EG} H(X,G)$$ (something like that).