On Extremal Positive Maps Acting between Type I Factors

The paper is devoted to the problem of classification of extremal positive maps acting between B(K) and B(H) where K and H are Hilbert spaces. It is shown that every positive map with the property that rankφ(P ) ≤ 1 for any one-dimensional projection P is a rank 1 preserver. It allows to characterize all decomposable extremal maps as those which satisfy the above condition. Further, we prove that every extremal positive map which is 2positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rankφ(P ) ≤ 1 for some one-dimensional projection P and satisfy the condition of local complete positivity. It allows us to give a negative answer for Robertson’s problem in some special cases.