On extremal positive maps acting between type I factors

Marcin Marciniak. "On extremal positive maps acting between type I factors." Banach Center Publications 89.1 (2010): 201-221. <http://eudml.org/doc/286487>.

@article{MarcinMarciniak2010,
abstract = {The paper is devoted to the problem of classification of extremal positive linear 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) ≤ 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 rank ϕ(P) ≤ 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.},
author = {Marcin Marciniak},
journal = {Banach Center Publications},
keywords = {positive maps; extremal; decomposable; completely positive},
language = {eng},
number = {1},
pages = {201-221},
title = {On extremal positive maps acting between type I factors},
url = {http://eudml.org/doc/286487},
volume = {89},
year = {2010},
}

TY - JOUR
AU - Marcin Marciniak
TI - On extremal positive maps acting between type I factors
JO - Banach Center Publications
PY - 2010
VL - 89
IS - 1
SP - 201
EP - 221
AB - The paper is devoted to the problem of classification of extremal positive linear 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) ≤ 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 rank ϕ(P) ≤ 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.
LA - eng
KW - positive maps; extremal; decomposable; completely positive
UR - http://eudml.org/doc/286487
ER -