The relative coarse Baum–Connes and relative coarse Novikov conjectures

Let $X$ be a metric space with bounded geometry and let $Y\subseteq X$ be a subspace. The relative coarse assembly map is

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

Relative coarse Baum–Connes and relative coarse Novikov conjectures. The relative coarse Baum–Connes conjecture for $(X,Y)$ claims that $\mu_{Y,\infty}$ is an isomorphism, while the relative coarse Novikov conjecture for $(X,Y)$ claims that $\mu_{Y,\infty}$ is injective. These are relative analogues of the coarse Baum–Connes and coarse Novikov conjectures, formulated using localization algebras at infinity. The supplied text gives no resolution status.