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Decomposability of extremal positive unital maps on M 2 (). (English) Zbl 1116.47033
Bożejko, Marek (ed.) et al., Quantum probability. Papers presented at the 25th QP conference on quantum probability and related topics, Będlewo, Poland, June 20–26, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 73, 347-356 (2006).

Let M n () be the algebra of matrices with complex entries. For m,n in , we have

M m (M n ())M m ()M n ()M mn ()·

Thus M m (𝕄 n ()) has the structure of a C * -algebra. A matrix AM m (M n ()) is positive iff

i,j=1 m μ i ¯μ j <ϑ i ,A ij ϑ j 0foreveryϑ 1 ,,ϑ m n andμ 1 ,,μ m ·

A linear map ϕ:M m ()M n () is called positive if ϕ(A) is a positive matrix for every positive matrix AM m (). If k, ϕ is called k-positive map (resp., k-copositive map) whenever [ϕ(A) ij )] i,j=1 k (resp., [ϕ(A) j,i )] i,j=1 k ) is a positive element in M k (M n ()) for every positive [A ij ] i,j=1 k in M k (M m ()). If ϕ is k-positive (resp., k-copositive) for every k, then ϕ is called completely positive (resp., completely copositive).

A positive map which is a sum of completely positive and completely copositive maps is called decomposable. It is known that if m=n=2, then every positive map is decomposable. Given an extremal unital positive map ϕ:M 2 ()M 2 (), the authors construct concrete maps ϕ 1 and ϕ 2 which give a decomposition of ϕ. They also show that in most cases this decomposition is unique.

MSC:
47B65Positive and order bounded operators
47L07Convex sets and cones of operators