高维情形下具有映射锥对称性的锥
Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case
摘要：
在有限维的情况下，我们提出了一种新的方法，锥理论的映射锥对称性，首先介绍了Størmer。我们的方法是基于一个内积的定义在空间的线性映射之间的两个代数的运营商和事实，Jamiovenkowski-Choi同构是一个等距。我们考虑一个稍微修改类的锥，虽然没有实质性的不同，从原来的映射锥的Størmer。使用新的方法，几个已知的结果被证明更快，往往比以前更具有一般性。例如，一个映射锥的对偶证明也是一个映射锥，而不需要任何额外的假设。本文的主要结果是一个特征的锥与映射锥对称性，说，一个给定的地图是一个元素，这样的锥当且仅当组成的地图与共轭的任意元素的对偶锥是完全积极的。一个类似的结果是已知的情况下，地图从一个代数的运营商到本身和锥是一个对称的映射锥。我们的结果是证明没有额外的假设之间的对称性和平等的域和目标空间。我们展示了它是如何给出一些旧的结果作为推论，包括一个示例性的应用程序。
英文摘要：
In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Størmer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamiołkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Størmer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.
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