On extremal positive maps acting between type I factors

• Autorzy: Marcin Marciniak
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46L05: General theory of C * -algebras
• 15A30: Algebraic systems of matrices

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2010
• Tom: 89
• Numer: 1
• Strony: 201-221

Daty

wydano

2010

Twórcy

autor

Marcin Marciniak

• Institute of Theoretical Physics and Astrophysics, Gdańsk University, Wita Stwosza 57, 80-954 Gdańsk, Poland

Identyfikatory

DOI

10.4064/bc89-0-12