On Lebesgue type decomposition for covariant completely positive maps onC*-algebras

Abstract

We show that there is an affine order isomorphism between completely positive maps from a C*-algebra $A$ to the C*-algebra $L(H)$ of all bounded linear operators on a Hilbert space $H,$ $u$-covariant with respect to a C*-dynamical system $\left( G,\alpha ,A\right)$ and $u$-covariant completely positive maps from the crossed product $A\times _{\alpha }G$ to $L(H)$, which preserves the Lebesgue decomposition.