Equivariant coarse Baum–Connes conjecture

Let $\Gamma$ be a countable discrete group acting properly on a proper metric space $X$. Let $KX_*^{\Gamma}(X)$ denote the universal $K^{\Gamma}$-group assembling the equivariant indices of objects in the relevant controlled category, and let

$$\mu_X^{\Gamma}:KX_*^{\Gamma}(X)\to K_*(C^*(X)^{\Gamma})$$

be the associated assembly map. Equivariant coarse Baum–Connes conjecture. The assembly map $\mu_X^{\Gamma}$ is an isomorphism. Equivalently, every element of $K_*(C^*(X)^{\Gamma})$ is the index of an elliptic operator, uniquely up to equivalence in $KX_*^{\Gamma}(X).$ Its injectivity part is the equivariant coarse Novikov conjecture.