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