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 〈T (x), T (y)〉 = τ(〈x, y〉) for all x, y ∈ E. 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 a dynamical system (G, η,E) and the set of all (u′, u)-covariant τ̃ -maps from the crossed product E ×η G to F , where τ and τ̃ are completely positive.