Comparison of completely positive maps on a C* algebra and a Lebesgue decomposition theorem

Abstract

From the expositions in [M], [BP], [P2] it is now well understood that Status: Runital completely positive maps on unital C* algebras are the quantum probabilistic analogues of transition probability operators in Markov processes. In [BS], V.P. Belavkin and P.Staszewski introduced three different notions of absolute continuity of one completely positive (c.p.) map with respect to another on a C* algebra and proved the existence of a Radon-Nikodym density under the condition of ‘strong complete absolute continuity’. Here we combine their approach with the Hilbert space - theoretic proof of the classical RadonNikodym theorem (See the exercises in Section 31 of [H] or Section 47 of [P1] and obtain the Lebesgue decomposition of a unit al c.p. map into its absolutely continuous and singular parts with respect to another such map. Analogues of chain rule and martingale properties of Radon-Nikodym derivatives and some examples are also included.