Coarse Baum–Connes conjecture

Statement

Let $X$ be a proper metric space with bounded geometry. This means that there are a subset $Z \subseteq X$ and a constant $c > 0$ such that every point of $X$ lies within distance $c$ of $Z$. Moreover, for every $r > 0$, the number of points of $Z$ in a ball of radius $r$ is bounded independently of the centre. For $d > 0$, let $P_d(Z)$ be the Rips complex. Its vertex set is $Z$, and its simplices are the finite subsets of $Z$ of diameter at most $d$. Let $C^*(X)$ be the Roe algebra of $X$. Fix an ample Hilbert space module over $C_0(X)$. Call a bounded operator $T$ on it locally compact if $fT$ and $Tf$ are compact operators for every compactly supported continuous function $f$ on $X$. Then $C^*(X)$ is the norm closure of the locally compact operators of finite propagation. The conjecture states that the coarse assembly map $$\mu \colon \lim_{d \to \infty} K_*(P_d(Z)) \to K_*(C^*(X))$$ is an isomorphism for $* = 0, 1$. Here $K_*(P_d(Z))$ is K-homology and $K_*(C^*(X))$ is the K-theory of the C*-algebra $C^*(X)$.