Skew products and crossed products by coactions

Given a labeling c of the edges of a directed graph E by elements of a discrete group G, one can form a skew-product graph E ×_c G. We show, using the universal properties of the various constructions involved, that there is a coaction δ of G on C* (E) such that C* (E ×_c G) is isomorphic to the crossed product C* (E) ×_δ G. This isomorphism is equivariant for the dual action δ̂ and a natural action γ of G on C* (E ×_c G); following results of Kumjian and Pask, we show that C* (E ×_c G) ×_γ G ≅ C* (E ×_c G) ×_γ,r G ≅ C* (E) ⊗ K(ℓ²(G)), and it turns out that the action γ is always amenable. We also obtain corresponding results for r-discrete groupoids Q and continuous homomorphisms c : Q → G, provided Q is amenable. Some of these hold under a more general technical condition which obtains whenever Q is amenable or second-countable.