The Baum–Connes conjecture for locally compact groups

Let $G$ be a locally compact group. The groups $K_{\ast}^{top}(G)$ and $K_{\ast}(C_{r}^{\ast}(G))$ are the topological and analytic $K$-theory groups, respectively, and the assembly map is

$$\mu_{r}: K_{\ast}^{top}(G) \rightarrow K_{\ast}(C_{r}^{\ast}(G)).$$

Baum–Connes conjecture. The assembly map $\mu_{r}$ is an isomorphism for $\ast=0,1$.

The conjecture connects equivariant topological $K$-homology with the $K$-theory of the reduced $C^{\ast}$-algebra $C_{r}^{\ast}(G)$ and has implications for major conjectures including Novikov and Kadison–Kaplansky. It is known for several important classes of groups, but is not established for all locally compact groups.