An Equivariant Coarse Green-Julg Theorem for Proper Actions

We prove an equivariant coarse Green--Julg theorem for proper isometric actions of countable discrete groups on bounded geometry metric spaces. We introduce the $(Γ, X)$-controlled proper $C^{*}$-algebra, and show that the evaluation-at-zero map between the equivariant localization and Roe algebras with such coefficients induces an isomorphism on $K$-theory.