ON THE STRUCTURE OF POSITIVE MAPS

A natural characterization of the structure of positive maps will be present. Our arguments will be based on the concept of exposed points, links between tensor products and mapping spaces, and smoothness of unit balls in certain Banach spaces. The emphasize will be put on the geometrical characterization of the set of unital positive maps (given by the norm dual to the certain projective norm) and analytical aspects of smoothness of unit balls in certain Banach spaces. The last will play the crucial role in the characterization of exposed unital positive maps. These results seem to be an answer to old open problems studied both in Operator Algebras and Quantum Information.