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A note on Lebesgue type decomposition for covariant completely positive maps on $C^*$-algebras. (English) Zbl 1201.46050
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.
46L05General theory of $C^*$-algebras
46L51Noncommutative measure and integration
46L40Automorphisms of $C^*$-algebras
46L55Noncommutative dynamical systems
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