The real Baum–Connes conjecture

Let $\Gamma$ be a discrete group, let $E(\Gamma,\mathrm{fin})$ be the classifying space for proper $\Gamma$-actions, and let $C^*_{\mathbb{R},r}\Gamma$ be the real reduced group $C^*$-algebra. The real Baum–Connes assembly map in degree $n$ is

$$\mu_n\colon KO_n^\Gamma(E(\Gamma,\mathrm{fin}))\to KO_n(C^*_{\mathbb{R},r}\Gamma).$$

Real Baum–Connes conjecture. The assembly map $\mu_n$ is an isomorphism. This is the real, 8-periodic analogue of the Baum–Connes conjecture, with equivariant KO-homology on the left and real K-theory of the reduced group algebra on the right. Its resolution is not supplied in the source.