Coarse Baum-Connes and coarse Novikov conjectures

Establish that for every metric space X with bounded geometry, the coarse assembly map μ: lim_{d→∞} K_*(P_d(X)) → K_*(C^*(X)) is an isomorphism; equivalently, show that the coarse Novikov conjecture holds by proving μ is injective.

Background

The paper recalls Roe algebras C^*(X) and localization algebras, and defines the coarse assembly map μ: lim_{d→∞} K_(P_d(X)) → K_(C^*(X)), where P_d(X) denotes the Rips complex of a bounded geometry metric space X. The coarse Baum-Connes conjecture predicts this assembly map is an isomorphism, while the coarse Novikov conjecture asks for injectivity only.

These conjectures connect large-scale geometry to operator K-theory and index theory. Throughout the paper, they serve as a benchmark and motivation for the relative versions formulated later, which use quotients of Roe algebras to detect refined obstructions to positive scalar curvature at infinity.