Finite Dimensional Operator Systems, Completely Positive Maps, and Majorization

算符代数是数学的一个领域，它是由约翰开创的数学家的努力发展而来的 冯·诺伊曼--创造出符合物理学家在量子力学中所做的事情的数学。事实上，正如它所做的那样 发生并继续发生，物理学家发现他们使用的对象在某种意义上是 数学上的--这符合他们对模型如何工作的直觉--但从根本上讲并不合理 对当时公认的数学的看法。冯·诺伊曼创造的数学领域并没有直接 完成了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，算符代数提供了从量子场论到量子场论的各种领域的洞察， 纽结理论、逻辑、量子信息和量子计算等。 算符代数学家所考虑的代数自然是无限维的，因此它们不是很 顺从我们的直觉。这使得研究人员除了开发一些直觉外，还创造了无数的 理解这些巨大物体的部分的技巧和观点。其中一个观点是 包络结构。有时可以通过考虑对象内部的内容来描述该对象 更大、更易操控的物体。对于C*-代数，这些包络结构中的一些包括对偶、 乘子代数和内射包络。我的研究项目调查了最后两个物体。为 算子系统，最自然的包络对象是C*-包络，由Arveson在1972年定义，这 对象也是我研究计划的一部分。 算子系是包含其所有算子的恒等式和伴随的算子的子空间。 它们是研究完全正值地图的自然对象。即使在很小的维度上，操作员系统 没有被很好地理解，并且缺乏一个直到完全序同构的分类。我的计划旨在 填补这一空白，通过致力于有限维算子系统的有效分类是他们的 C*-信封。 我的研究计划的另一个分支是研究优选化和Schur-Horn定理。这 定理是关于矩阵的一个非常好理解的结果，使得它的推广到无限维上 设置不是无关紧要的。用简单的技术术语来说，Schur-Horn定理刻画了可能的对角线 不同正交基选择下的自伴矩阵。仍然在有限维中，这是一个推广 这个定理对于正规算子来说是一个没有人知道答案的问题！我对老年人通勤家庭的研究 自伴算符提供了一个可以成功研究这一点的背景。优势化自然地出现在 量子信息，我的程序也调查这种联系，特别是与所谓的王牌优势。

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.