Localised Baum–Connes surjectivity conjecture

Let $G$ be a group, let $\underline{E}G$ be a universal example for proper $G$-actions, and let $K_*^G(\underline{E}G)_{\operatorname{loc}}$ denote its localised equivariant $K$-homology. The localised equivariant index is the map

$$\operatorname{index}_G^{\operatorname{loc}}\colon K_*^G(\underline{E}G)_{\operatorname{loc}}\to K_*(C^*_{\operatorname{red}}(G)).$$

Localised Baum–Connes surjectivity conjecture. The map

$$\operatorname{index}_G^{\operatorname{loc}}\colon K_*^G(\underline{E}G)_{\operatorname{loc}}\to K_*(C^*_{\operatorname{red}}(G))$$

is surjective.

This is a localised version of the surjectivity part of the Baum–Connes conjecture, allowing equivariant indices for non-cocompact actions. It is implied by ordinary Baum–Connes surjectivity, but remains open in general.