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Decomposition of Completely Positive Maps
Author(s) -
Paul Michael
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3211850114
Subject(s) - mathematics , section (typography) , hilbert space , generalization , decomposition , pure mathematics , decomposition theorem , operator (biology) , algebra over a field , discrete mathematics , mathematical analysis , ecology , biochemistry , chemistry , repressor , advertising , transcription factor , gene , business , biology
The paper is concerned with completely positive maps on the algebra of unbounded operatore L +( D ) and on its completion L (D, D + ). A decomposition theorem for continuous positive functionals is proved in [Tim. Loef.), and [Scholz 91] contains a generalization to maps into operator algebra on finite dimensional Hilbert spaces H 0 . The aim of the present paper is to construct an analogous decomposition without the assumption that H 0 is finite dimensional. Moreover, the Kraus ‐ theorem [Kraus] is proved for normal completely positive mappings on L(D, D + ). The paper is organized as follows. Section 1 contains the necessary definitions and notations. In Section 2 we prove the decomposition theorem. Section 3 deal with the structure of the normal completely positive mappings.