The equivariant coarse Baum-Connes conjecture for metric spaces with proper group actions

作者：Deng Jintao; Fu Benyin^*; Wang Qin

来源：Journal of Noncommutative Geometry, 2024, 18(1): 61-92.

DOI：10.4171/JNCG/519

• . Equivariant Roe algebras
• coarse embedding
• coarse Baum-Connes conjecture
• K-amenability

The equivariant coarse Baum-Connes conjecture interpolates between the Baum-Connes conjecture for a discrete group and the coarse Baum-Connes conjecture for a proper metric space. In this paper, we study this conjecture under certain assumptions. More precisely, assume that a countable discrete group F acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space XII' admits a coarse embedding into Hilbert space and F is amenable, and that the F-orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum-Connes conjecture holds for (X, F). Along the way, we prove a K-theoretic amenability statement for the F-space X under the same assumptions as above; namely, the canonical quotient map from the maximal equivariant Roe algebra of X to the reduced equivariant Roe algebra of X induces an isomorphism on K-theory.