Decomposition of Completely Positive Maps

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.