Duality for crossed products of Hilbert C*-modules

Let (A,G,alpha) be a C*-dynamical system and let X be an A-Hilbert module with an alpha-compatible action eta of G. Then it is shown that there exist a coaction delta(A) of G on the reduced crossed product A x(alpha,r) G and a coaction delta(X) of G on the reduced crossed product X x(eta,r) G such that (X x (eta,r) G) x delta(X) G congruent to X (D C(L-2(G)), where C(L-2(G)) denotes the C*-algebra of all compact operators on L-2(G). Furthermore, when A has a nondegenerate coaction 6, of G on A and X is an A-Hilbert module with a nonclegenerate delta(A)-compatible coaction delta(X) of G, it is shown that there exists a dual action, of G on the crossed product X x delta(X) G such that (X x delta(X) G) x ((delta) over capX,r) G congruent to X circle times C (L-2 (G)).