On extremal positive maps acting between type I factors

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 $\rank ϕ(P)\leq 1$ 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 $\rank ϕ(P)\leq 1$ 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.