Abstract
Positive maps of ordered vector spaces into the algebra of all bounded operators acting on a Hilbert space are considered. A special class of so called nonextendible maps is introduced and investigated. This class is much smaller than the class of extreme maps.
Any positive map can be obtained from a nonextendible one by restriction.
In theC*-algebra case, the nonextendibility of a normalized positive map φ is related to the properties of the expression φ(a 2)−φ(a)2. In particular Jordan representations are non-extendible.
2-positive nonextendible maps are representations. Similar result holds for copositive maps. For abelianC*-algebras, notion of nonextendible map and that of representation coincide.
The nonextendible positive maps of the Jordan algebraM 2s of all 2×2 symmetric matrices and of the full 2×2 matrix algebra are especially investigated. Any nonextendible normalized positive map ofM 2s is a Jordan representation.M 2 admits nonextendible normalized positive maps not being Jordan representations. A large class of examples is given.
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Communicated by H. Araki
On leave of absence from: Department of Mathematical Methods in Physics, University of Warsaw, Hoża 74, 00-682 Warsaw, Poland
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Woronowicz, S.L. Nonextendible positive maps. Commun.Math. Phys. 51, 243–282 (1976). https://doi.org/10.1007/BF01617922
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DOI: https://doi.org/10.1007/BF01617922