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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-12,
author = {Marcin Marciniak},
title = {On extremal positive maps acting between type I factors},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {201-221},
zbl = {1214.46035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-12}
}
Marcin Marciniak. On extremal positive maps acting between type I factors. Banach Center Publications, Tome 89 (2010) pp. 201-221. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc89-0-12/