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A decomposition theorem for positive maps, and the projection onto a spin factor. (English) Zbl 1345.46059

The paper is devoted to the structure of positive maps between \( C^\ast\)-algebras in the finite-dimensional case. The author shows that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal (so-called bi-optimal), i.e., it majorizes neither a non-zero completely positive map nor a co-positive map. In order to obtain a deeper understanding of this decomposition he considers it in detail for the trace invariant positive projection of the full matrix algebra \(M_{2^n}\) onto a spin factor inside it. Explicit formulas are obtained for the decomposable map and the bi-optimal map in the decomposition when the spin factor is irreducible and contained in the \(2^n\) by \(2^n\) matrices over the quaternions.

MSC:

46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
47B65 Positive linear operators and order-bounded operators
46L05 General theory of \(C^*\)-algebras
46L70 Nonassociative selfadjoint operator algebras
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