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Erschienen in:
2013 | OriginalPaper | Buchkapitel
6. Dual Cones
verfasst von : Erling Størmer
Erschienen in: Positive Linear Maps of Operator Algebras
Verlag: Springer Berlin Heidelberg
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Abstract
A cone in a Hilbert space has a natural dual cone consisting of vectors with positive inner product with vectors in the given cone. In the finite dimensional case we can do the same for cones of positive maps with respect to the Hilbert-Schmidt structure. Characterizations of maps in dual cones are given both in terms of their Choi matrices and their dual functionals. Furthermore, examples of dual cones of well known mapping cones like for example k-positive maps, are shown to be well known mapping cones. The ideas of dual cones are used in Sect. 6.3 to classify positive maps of the 2×2 matrices into themselves, and in the last section we characterize maps in dual cones in terms of their positivity properties on tensor products with maps in the given cone.