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 γ G, A α G,ψ) is the induced C∗-correspondence and β is the induced action on OX, then OX β G∼= OX γ G. 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].