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 $\rank \phi(P)\leq 1$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2008
- DOI:
- 10.48550/arXiv.0812.2311
- arXiv:
- arXiv:0812.2311
- Bibcode:
- 2008arXiv0812.2311M
- Keywords:
-
- Mathematics - Operator Algebras;
- Mathematics - Functional Analysis;
- Quantum Physics;
- 46L05;
- 15A30
- E-Print:
- 21 pages