Decomposition of Completely Positive Maps

✍ Scribed by Michael Paul

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 518 KB
• Volume: 185
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.3211850114

✦ Synopsis

Abstract

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.