Extremal Positive Maps on M 3 (ℂ) and Idempotent Matrices

A new method of analysing positive bistochastic maps on the algebra of complex matrices M3 has been proposed. By identifying the set of such maps with a convex set of linear operators on ℝ8, one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been shown that all positive bistochastic maps, extremal in the set of all positive maps of M3, that are not Jordan isomorphisms of M3 are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one-dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.