The coarse Baum–Connes conjecture with coefficients

Let $X$ be a proper metric space, let $N_X$ be a locally finite net in $X$, and let $A$ be a $C^*$-algebra. Let $P_k(N_X)$ denote the Rips complex, and let $e_A$ be the evaluation-at-zero homomorphism from the localization algebra to the Roe algebra.

Coarse Baum–Connes conjecture with coefficients. The homomorphism

$$e_{A,*}:\lim_{k\rightarrow\infty}K_*(C_L^*(P_k(N_X),A))\longrightarrow\lim_{k\rightarrow\infty}K_*(C^*(P_k(N_X),A))$$

is an isomorphism of abelian groups.

This is the coefficient-valued coarse assembly conjecture that underlies the paper's quantitative refinement. The source recalls it as a conjecture and uses known cases to derive the quantitative conjecture.