On extremal positive maps acting between type I factors
Volume 89 / 2010
Abstract
The paper is devoted to the problem of classification of extremal positive linear maps acting between ${\mathfrak B}({\cal K})$ and ${\mathfrak B}({\cal H})$ where ${\cal K}$ and ${\cal H}$ are Hilbert spaces. It is shown that every positive map with the property that $\mathop{\rm rank}\phi(P)\leq 1$ for any one-dimensional projection $P$ is a rank 1 preserver. This allows us 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 be automatically completely positive. Finally, we get the same conclusion for extremal positive maps such that $\mathop{\rm rank}\phi(P)\leq 1$ for some one-dimensional projection $P$ and satisfy the condition of local complete positivity. This allows us to give a negative answer to Robertson's problem in some special cases.
