DECOMPOSITION OF COMPLETELY POSITIVE MAPS

The paper is concerned with completely positive maps on the algebra of unbounded operators L+(D) and on its completion L(D, D+). A decomposi tion theorem for continuous positive functionals is proved in [Tim. Lo ef.], and [Scholz 91] contains a generalization to maps into operator algebras on finite dimensional Hilbert spaces H-0. The aim of the pres ent paper is to construct an analogous decomposition without the assum ption that H-0 is finite dimensional. Moreover, the Kraus-theorem [Kra us] is proved for normal completely positive mappings on L(D, D+). The paper is organized as follows. Section I contains the necessary defin itions and notations. Ln Section 2 we prove the decomposition theorem. Section 3 deal with the structure of the normal completely positive m appings.