Abstract

Abstract. 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 rank φ(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 2-positive turns out to automatically completely positive. Finally we get the same conclusion for such extremal positive maps that rank φ(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. 1.

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