The ℓᵖ-localization-algebra formulation of the coarse Baum–Connes conjecture

Let $X$ be a locally finite metric space with bounded geometry, let $P_r(X)$ be its Rips complex at scale $r$, let $C^p_L(P_r(X))$ be the $\ell^p$-localization algebra, and let $C^p(X)$ be the $\ell^p$-Roe algebra. The evaluation map

$$\operatorname{ev}\colon C^p_L(P_r(X))\to C^p(X)$$

that sends $f$ to $f(0)$ induces

$$\operatorname{ev}_*\colon \lim_{r\to\infty}K_*(C^p_L(P_r(X)))\to K_*(C^p(X)).$$

The $\ell^p$-coarse Baum–Connes conjecture. The induced map $\operatorname{ev}_*$ is a K-theoretic isomorphism. This is the $\ell^p$ localization-algebra formulation of the conjecture; the source notes coarse-equivalence invariance of the relevant K-theory but supplies no resolution status for the general assertion.