an:05702414 Zbl 1201.46050 Joi\c ta, Maria A note on Lebesgue type decomposition for covariant completely positive maps on $C^*$-algebras. EN Banach J. Math. Anal. 4, No. 2, 75-86, electronic only (2010). ISSN 1735-8787/e 2010

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*46L05 46L51 46L40 46L55 covariant completely positive map; Radon-Nikod\'ym derivative; Lebesgue decomposition Summary: 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 $(G, \alpha , A)$ and $u$-covariant completely positive maps from the crossed product $A\times \alpha G$ to $L(H)$, which preserves the Lebesgue decomposition.