On extremal positive maps acting between type I factors

TL;DR

This paper classifies extremal positive maps between operator algebras, showing that certain rank conditions imply maps are completely positive, and addresses a specific problem posed by Robertson.

Contribution

It provides a characterization of extremal positive maps with rank constraints and establishes when they are necessarily completely positive, advancing the understanding of positive map structure.

Findings

Maps with rank 1 image on projections are rank 1 preservers

Decomposable extremal maps satisfy rank 1 preservation

Extremal 2-positive maps are automatically completely positive

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