Coarse Baum-Connes conjecture with filtered coefficients

Establish that for every proper metric space X and every locally finite net N_X in X, the evaluation-at-zero map e_* from the direct limit over k of the K-theory groups K_*(C^*_{L,f}(P_k(N_X), A)) of localization algebras with filtered coefficients A to the direct limit over k of the K-theory groups K_*(C^*_f(P_k(N_X), A)) of Roe algebras with filtered coefficients A is an isomorphism.

Background

The paper introduces Roe and localization algebras with filtered coefficients to generalize the original coarse Baum-Connes conjecture. For a proper metric space X, Rips complexes P_k(N_X) built from a locally finite net N_X form a coarsening system used to define inductive limits of K-theory groups. The conjecture asserts that the evaluation-at-zero map from localization to Roe K-theory induces an isomorphism at the level of the corresponding limits.

This formulation extends the classical coarse Baum-Connes conjecture (recovered when A = C with the trivial filtration) and aims to provide a computable topological model for the K-theory of Roe algebras with filtered coefficients.