Nonextendible positive maps
Nonextendible positive maps
复制标题
不可扩展的正图
DOI：
10.1007/bf01617922
复制
发表时间：
1976
影响因子：
2.4
通讯作者：
S. Woronowicz
中科院分区：
文献类型：
--
作者：
S. Woronowicz
讨论了序向量空间到Hilbert空间上所有有界算子代数的正映射。引入并研究了一类特殊的不可扩张映射。这类映射比极端映射小得多。任何正映射都可以通过限制从一个不可扩张映射得到。在C *-代数的情况下，正规化正映射φ的不可扩张性与表达式φ（a 2）−φ（a）2的性质有关。特别地，Jordan表示是不可扩张的。2-正不可扩张映射是表示。类似的结果也适用于共正映射。对于交换C *-代数，不可扩映射与表示的概念是一致的，特别研究了全2×2对称矩阵的Jordan代数M2和全2×2矩阵代数的不可扩正映射. M2的任何不可扩张的归一化正映射都是Jordan表示，M2允许不可扩张的归一化正映射不是Jordan表示。一个大类的例子。
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(a2)−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 algebraM2s of all 2×2 symmetric matrices and of the full 2×2 matrix algebra are especially investigated. Any nonextendible normalized positive map ofM2s is a Jordan representation.M2 admits nonextendible normalized positive maps not being Jordan representations. A large class of examples is given.