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 H₀. The aim of the present paper is to construct an analogous decomposition without the assumption that H₀ 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.