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(l(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.