Intermediate subalgebras for reduced crossed products of discrete groups

Let $\alpha :\mathrm{\Gamma }\curvearrowright A$ be an action of a discrete group $\mathrm{\Gamma }$ on a unital C*-algebra $A$ by -automorphisms and let $A{⋊}_{\alpha ,\lambda }\mathrm{\Gamma }$ denote the corresponding reduced crossed product C-algebra. Assuming that $\mathrm{\Gamma }$ satisfies the approximation property, we establish a sufficient and (almost always) necessary condition on the action $\alpha$ for the existence of a Galois correspondence between intermediate C*-algebras for the inclusion $A\subseteq A{⋊}_{\alpha ,\lambda }\mathrm{\Gamma }$ and partial subactions of $\alpha$. This condition, which we refer to as pointwise residual proper outerness, is a natural noncommutative generalization of freeness.