Finite Dimensional Operator Systems, Completely Positive Maps, and Majorization

算符代数是数学的一个领域，它是由约翰*冯·诺伊曼开创的数学家的努力发展而来的，目的是创造符合物理学家在量子力学中所做的数学。事实上，正如它已经*发生并继续发生的那样，物理学家发现他们使用的对象在某种意义上是*数学的--这符合他们关于他们的模型如何工作的直觉--但从当时公认的数学的观点来看*是没有意义的。冯·诺伊曼创造的数学领域并没有直接实现成为量子力学语言的目标，但它本身就成为了一个数学世界。*在过去的60年里，算符代数提供了对量子场论、*纽结理论、逻辑、量子信息和量子计算等多个领域的见解。*算符代数学家考虑的代数自然是无限维的，因此它们不太*服从我们的直觉。这使得研究人员除了开发一些直觉外，还创造了无数的*技巧和观点来理解这些巨大物体的部分。这些观点之一是*包络结构的观点。有时，通过考虑一个*更大、更易处理的对象来描述该对象是可能的。对于C*-代数，这些包络结构包括双对偶、*乘子代数和内射包络。我的研究项目调查了最后两个物体。对于*算子系统，最自然的包络对象是Arveson在1972年定义的C*-包络，这个*对象也是我的研究项目的一部分。*算子系统是包含其所有算子的恒等式和伴随的算子的子空间。*它们是研究完全正地图的自然对象。即使在很小的维度上，算子系统*也不被很好地理解，并且缺乏一个直到完全序同构的分类。我的程序旨在*填补这一空白，努力实现并有效地分类有限维算子系统是它们的*C*-包络。*我的研究计划的另一个分支是关于优选性和Schur-Horn定理的研究。这个*定理是关于矩阵的一个非常好理解的结果，因此它在无限维*环境中的推广是非平凡的。用轻微的专业术语来说，Schur-Horn定理刻画了*自伴矩阵在不同的正交基选择下可能的对角线。仍然是在有限维中，将这个定理推广到正规算子是一个没有人知道答案的问题！我对*自伴算子族的研究提供了一个可以成功研究这一点的背景。优势化自然出现在*量子信息中，我的程序也研究了这种联系，特别是与所谓的王牌优势化。**

Operator Algebras is an area of Mathematics that grew out of the efforts of mathematicians-pioneered by John ***von Neumann-to create mathematics that fit what physicists were doing in Quantum Mechanics. Indeed, as it has ***happened and continues to happen, physicists found themselves using objects that in some sense were ***mathematical-and that fit their intuition on how their models were working-but did not make sense from the point ***of view of the accepted mathematics of the time. The mathematical area created by von Neumann did not directly ***fulfill the goal of becoming the language of Quantum Mechanics, but it became a mathematical world on its own. ***Over the last 60 years, Operator Algebras have provided insight into areas as diverse as quantum field theory, ***knot theory, logic, quantum information and quantum computing, among others. ******The algebras considered by operator algebraists are naturally infinite-dimensional, and so they are not very ***amenable to our intuition. This has led researchers to, besides developing some intuition, create a myriad of ***tricks and points of view to understand parts of these immense objects. One of these points of view is that of ***enveloping structures. Sometimes it is possible to say something about an object by considering it inside a ***bigger, more tractable object. For C*-algebras, some of these enveloping structures include the double dual, the ***multiplier algebra, and the injective envelope. My research program investigates these last two objects. For ***Operator Systems, the most natural enveloping object is the C*-envelope, defined by Arveson in 1972, and this ***object is also part of my research program.***Operator systems are subspaces of operators that contain the identity and the adjoints of all its operators. ***They are the natural objects on which to study completely positive maps. Even in small dimensions, operators systems ***are not well-understood, and a classification up to complete order isomorphism is lacking. My program aims to ***fill this gap, by working towards and effective classification of finite-dimensional operator systems are their ***C*-envelopes. ****Another branch of my research program consists of the study of majorization and the Schur-Horn theorem. This***theorem is a very well understood result about matrices, such that its generalizations to an infinite-dimensional ***setting are non-trivial. In slight technical terms, the Schur-Horn theorem characterizes the possible diagonals of ***a self-adjoint matrix under different choices of an orthonormal basis. Still in finite-dimension, a generalization ***of this theorem to normal operators is a question no one knows the answer to! My research on commuting families of ***selfadjoint operators provides a context where this may be studied successfully. Majorization appears naturally in ***Quantum Information, and my program also investigates this connection, in particular with the so-called trumping majorization.**