Finite Dimensional Operator Systems, Completely Positive Maps, and Majorization

算子代数是数学的一个领域，由约翰·冯·诺伊曼（John von neumann）为首的数学家们的努力发展而来，他们创造了适合物理学家在量子力学中所做的数学。事实上，正如它已经发生和继续发生的那样，物理学家发现他们使用的对象在某种意义上是数学的，并且符合他们对模型如何工作的直觉，但从当时公认的数学观点来看没有意义。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，但它自己成为了一个数学世界。***在过去的60年里，算子代数为量子场论、***结理论、逻辑、量子信息和量子计算等领域提供了见解。******算子代数学家所考虑的代数自然是无限维的，因此它们不太符合我们的直觉。这使得研究人员除了发展一些直觉之外，还创造了无数的技巧和观点来理解这些巨大物体的部分。其中一种观点是包络结构。有时候，我们可以通过把一个对象放在一个更大、更容易处理的对象中来描述它。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。我的研究计划是调查后两个对象。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***对象也是我研究计划的一部分。***算子系统是包含其所有算子的单位和伴随的算子的子空间。它们是研究完全正图的自然对象。即使在小维度中，算子系统***也没有被很好地理解，并且缺乏达到完全有序同构的分类。我的计划旨在填补这一空白，通过研究有限维算子系统的有效分类，即它们的***C -信封。****我研究计划的另一个分支包括对多数化和舒尔-霍恩定理的研究。这个定理是一个很容易理解的关于矩阵的结果，因此它推广到一个无限维的集合是非平凡的。用简单的专业术语来说，舒尔-霍恩定理描述了一个自伴随矩阵在不同的标准正交基选择下的可能对角线。仍然是在有限维，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***自伴随算子的通勤族的研究为这一问题的成功研究提供了一个背景。在***量子信息中，多数化是自然出现的，我的程序也研究了这种联系，特别是与所谓的王牌多数化

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.**