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

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 \Gamma acts properly and isometrically on a discrete metric space X with bounded geometry, not necessarily cocompact. We show that if the quotient space X/\Gamma admits a coarse embedding into Hilbert space and \Gamma is amenable, and that the \Gamma -orbits in X are uniformly equivariantly coarsely equivalent to each other, then the equivariant coarse Baum–Connes conjecture holds for (X,\Gamma) . Along the way, we prove a K -theoretic amenability statement for the \Gamma -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.