Equivariant (co)module nuclearity of C^*-crossed products

We define an equivariant and equicovariant versions of the notion of module nuclearity. More precisely, for a discrete group Γ and operator A-Γ-(co)module B, E over a Γ-C^*-algebra A, we define E-Γ-nuclearity of B, as an equivariant version of the notion of E-nuclearity, in which the identity map on B is required to be approximately factored through matrix algebras on E with module structures coming both from the original module structure of E and the Γ-action on E. For trivial actions of Γ, this is shown to reduce to the notion of module nuclearity, introduced and studied by the first author. As a concrete example, for a discrete group Γ acting amenably on a unital C^*-algebra A, we show that the reduced crossed product Aᵣ Γ is A-Γ-nuclear. Conversely, if A is a nuclear C^*-algebra with a Γ-invariant state ρ and Aᵣ Γ is A-Γ-nuclear, then we deduce that Γ is amenable. We show that when Aᵣ Γ is A-Γ-nuclear and A has the completely bounded approximation property (resp., is exact), then so is Aᵣ Γ. We prove similar results for Aᵣ Γ, regarded as an A-Γ-comodule.