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.