Positive maps on matrix algebras

Positive maps on matrix algebras
Marcin Marciniak
Abstract
A linear map φ : Mm (C) → Mn (C) is called positive if it sends positive matrices
from Mm (C) into positive matrices from Mn (C). It is a great challenge to characterise extremal elements in the cone of all positive maps acting between Mm (C)
and Mn (C). The aim of our talk is to present some new results in this topic.
Among all extremal positive maps the class of exposed maps is very important.
Due to Straszewicz’s theorem it is a dense subset in the set of all extremal positive
maps. We will describe some new examples of exposed positive maps.
Further we will show that there is a link between extremal positive maps and
the theory of linear rank preservers. In particular we will discuss rank properties
of extremal positive maps.
References
[1] E. Størmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233–278.
[2] S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math.
Phys. 10 (1976), 165–183.
[3] W. A. Majewski and M. Marciniak, On the structure of positive maps between matrix algebras,
Banach Center Publ. 78 (2007), 249–263.
[4] M. Marciniak, On extremal positive maps acting on type I factors, Banach Center Publ. 89
(2010), 201–221.
Institute of Theoretical Physics and Astrophysics, Gdańsk Univeristy, Wita Stwosza
57, 80-952 Gdańsk, Poland
E-mail address: [email protected]
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