Twisted coarse Baum–Connes conjecture with coefficients

Establish that for every metric space X with bounded geometry and every coarse X-algebra (X, A), the twisted assembly map mu_{(X,A)}: lim_{d→∞} K_*(C^*_{L,(X,A)}(P_d(X), A)) → K_*(C^*_{(X,A)}(X, A)) is an isomorphism.

Background

The paper introduces twisted Roe algebras C^*_{(X,A)}(X,A) for metric spaces X with bounded geometry using coarse X-algebras (X, A), and defines a twisted localization algebra C^*_{L,(X,A)}(P_d(X),A) on the Rips complex P_d(X). These fit into a canonical twisted assembly map from the localization side to the twisted Roe algebra.

The conjecture generalizes the classical coarse Baum–Connes conjecture: when (X,A) = ℓ^∞(X, K) it reduces to the usual coarse Baum–Connes conjecture, and when (X,A) = ℓ^∞(X, A') it matches the coarse Baum–Connes conjecture with coefficients in A'. The authors prove the conjecture in several significant cases, including spaces that coarsely embed into Hilbert space and spaces admitting coarse fibration structures whose base and fiber satisfy the twisted conjecture, thus motivating the global validity of the conjecture.