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      Absolute continuity for operator valued completely positive maps on C *-algebras

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      Author(s)
      Gheondea, A.
      Kavruk, A. Ş.
      Date
      2009
      Source Title
      Journal of Mathematical Physics
      Print ISSN
      0022-2488
      Electronic ISSN
      1089-7658
      Volume
      50
      Issue
      2
      Pages
      022102-1 - 022102-29
      Language
      English
      Type
      Article
      Item Usage Stats
      151
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      Abstract
      Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C* -algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34-54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon-Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon-Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi's matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained. © 2009 American Institute of Physics.
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      http://hdl.handle.net/11693/22807
      Published Version (Please cite this version)
      http://dx.doi.org/10.1063/1.3072683
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