Crossed products of C∗-correspondences by amenable group actions

Let (X,A,ϕ) be a C∗-correspondence with a continuous action (γ,α) by an amenable locally compact group G. We will show that if (X⋊γG,A⋊αG,ψ) is the induced C∗-correspondence and β is the induced action on OX, then OX⋊βG≅OX⋊γG. As applications, we obtain [B. Abadie, Takai duality for crossed products by Hilbert C∗-modules, preprint arXiv:0709.1122v1, Proposition 4.5] and [A. Kumjian, D. Pask, C∗-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19 (6) (1999) 1503–1519, Corollary 2.5].