The coarse Baum–Connes and coarse Novikov conjectures

Let $X$ be a countable discrete metric space with bounded geometry, and let $P_d(X)$ be its Rips complex at scale $d$. The associated assembly maps are

$$\mu:\lim_{d\to\infty}K_*(P_d(X))\to K_*(C^*(X)),\qquad \mu_{\max}:\lim_{d\to\infty}K_*(P_d(X))\to K_*(C^*_{\max}(X)).$$

Coarse Baum–Connes and coarse Novikov conjectures. The coarse Baum–Connes conjecture claims that $\mu$ is an isomorphism, while the coarse Novikov conjecture claims that $\mu$ is injective. Their maximal versions assert respectively that $\mu_{\max}$ is an isomorphism and that $\mu_{\max}$ is injective. These conjectures concern whether coarse geometric $K$-homology is detected, or completely computed, by the Roe-algebra assembly map. The supplied text gives no resolution status.