BaumConnes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$equivariant Khomology of the universal example for proper actions of $G$ and Ktheory of $C^*_r(G)$ (the reduced $C^*$algebra of the group) is an isomorphism. Is there an example of the group $G$ such that:
1. It is not known whether BaumConnes conjecture for this group holds but
2. Groups $K_G(\underline{E}G)$ and $K(C^*_r(G))$ are abstracly isomorphic and nontrivial.
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2$\begingroup$ I don't know, but I'd be a little surprised  there aren't actually that many techniques for calculating $K(C_r^*(G))$ other than the BaumConnes conjecture. $\endgroup$ – Paul Siegel Oct 10 '17 at 16:20