Atomic positive linear maps in matrix algebras. (English) Zbl 0963.46042
By using a result of M.H. Eom and S.-H. Kye [Math. Scand. 86, No. 1, 130-142 (2000; Zbl 0964.46036)] concerned with the facial structure of cones in the space \(B(M_m(\mathbb{C}), M_n(\mathbb{C}))\) of linear mappings from the matrix algebra \(M_m(\mathbb{C})\) to \(M_n(\mathbb{C})\), the author examines the atomicity of positive linear mappings from \(M_n(\mathbb{C})\) to itself. A positive linear mapping is said to be atomic if it cannot be decomposed into the sum of a 2-positive linear mapping and a 2-co-positive linear mapping. In particular, he shows that that the family \(\tau_{n,k}\) of maps, introduced by Choi, are atomic for \(n\) greater than 2 and for \(1\leq k\leq n-2\). He also shows that mappings of a certain class on \(M_3(\mathbb{C})\) are atomic if and only if they cannot be decomposed as the sum of a completely positive linear mapping and a completely co-positive linear mapping.
Reviewer: C.M.Edwards (Oxford)
MSC:
| 46L07 | Operator spaces and completely bounded maps |
| 46L05 | General theory of \(C^*\)-algebras |
| 15A30 | Algebraic systems of matrices |
Keywords:
facial structure of cones; atomicity of positive linear mappings; completely positive linear mapping; completely co-positive linear mappingCitations:
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