Consistent coarse Baum-Connes conjecture with filtered coefficients (uniform version)

Establish that for every proper metric space X and locally finite net N_X, the evaluation-at-zero map e_* from the direct limit over k of K_*(C^*_{L,f}(⨿_N P_k(N_X), A)) to the direct limit over k of K_*(C^*_f(⨿_N P_k(N_X), A)) is an isomorphism; equivalently, prove that the coarse Baum-Connes conjecture with filtered coefficients in A holds for the separated coarse union ⨿_N X.

Background

The consistent (uniform) version considers the separated coarse union of a sequence of copies of X and the corresponding Rips complexes, forming a uniform product-type structure. This version connects to quantitative forms of the conjecture and provides a framework for uniform control across the components.

The authors note this uniform version is equivalent to a quantitative coarse Baum-Connes conjecture with coefficients (to be detailed in a subsequent work), highlighting its importance for refined analyses.