LP-coarse Baum-Connes conjecture isomorphism

Establish that for every Riemannian manifold X with bounded geometry and any maximal 1-net N in X, the evaluation map ev: lim_{d→∞} CP(P_d(N)) → CP(X), from the LP-localization algebras of the Rips complexes P_d(N) to the LP-Roe algebra CP(X) of X, induces an isomorphism on K-theory ev_*: lim_{d→∞} K_*(CP(P_d(N))) → K_*(CP(X)).

Background

The coarse Baum-Connes conjecture relates the K-homology of a space to the K-theory of its Roe algebra via an assembly map; its injectivity is often referred to as the coarse Novikov conjecture. In the LP setting considered here, the authors define the LP-Roe algebra CP(X) for a space X of bounded geometry and use Rips complexes P_d(N) built from a maximal 1-net N to formulate a localization framework.

Conjecture 2.4 states that the evaluation map from the directed system of LP-localization algebras CP(P_d(N)) to CP(X) yields an isomorphism on K-theory for any bounded-geometry Riemannian manifold X. While this paper proves the conjecture for spaces that coarsely embed into l^q (for all q in [1,∞) and all p in [1,∞)), the general conjecture remains open beyond those classes.