A covariant Stinespring type theorem for τ-maps

Let τ be a linear map from a unital C*-algebra A to a von Neumann algebra B and let C be a unital C*-algebra. A map T from a Hilbert A-module E to a von Neumann C-B module F is called a τ-map if\ud < T(x), T(y) >=τ( < x, y > ) for all x,y∈ E.\ud A Stinespring type theorem for τ-maps and its covariant version are obtained when τ is completely positive. We show that there is a bijective correspondence between the set of all τ-maps from E to F which are (u',u)-covariant with respect to the dynamical system (G,η,E) and the set of all (u' ,u)-covariant τ~-maps from the crossed product E×η G to F, where τ and τ~ are completely positive