Finite Dimensional Operator Systems, Completely Positive Maps, and Majorization

有限维算子系统、完全正映射和主要化

• 批准号: RGPIN-2015-03762
• 负责人: Argerami, Martin
• 金额: $ 0.8万
• 依托单位: University of Regina
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612553
• 关键词: Finite; Dimensional; Operator; Systems; Completely

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.

算子代数是数学的一个领域，它是由数学家们的努力发展起来的， 冯·诺伊曼--创造出符合物理学家在量子力学中所做的数学。事实上，正如它所做的那样 物理学家发现他们使用的物体在某种意义上是 这符合他们的直觉，他们的模型是如何工作的，但从这一点上讲， 当时公认的数学观点。冯·诺依曼创建的数学领域并没有直接 实现了成为量子力学语言的目标，但它自己变成了一个数学世界。 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。 算子代数学家所考虑的代数自然是无限维的，因此它们并不十分复杂。 服从于我们的直觉这使得研究人员除了发展一些直觉外，还创造了无数的 来理解这些巨大物体的一部分。其中一个观点是， 包络结构。有时候，通过将一个对象考虑在一个 更大更易处理的物体对于C*-代数，这些包络结构中的一些包括双对偶， 乘子代数和内射包络。我的研究计划调查了最后两个物体。为 算子系统中，最自然的包络对象是Arveson在1972年定义的C* 包络， 这也是我研究计划的一部分。 算子系统是包含所有算子的恒等式和伴随的算子的子空间。 它们是研究完全正映射的自然对象。即使在很小的维度上，算子系统 还没有被很好地理解，并且缺乏直到完全序同构的分类。我的计划旨在 填补这一空白，通过努力和有效的分类有限维算子系统是他们的 信封。 我的研究计划的另一个分支包括优控制和舒尔-霍恩定理的研究。这 定理是关于矩阵的一个非常好理解的结果，因此它的推广到无限维 设置是不平凡的。在轻微的技术术语中，舒尔-霍恩定理描述了 一个自伴矩阵在不同的选择下的正交基。仍然在有限维中， 这一定理对正规算子的应用是一个没有人知道答案的问题！我对通勤家庭的研究 自伴算子提供了一个可以成功研究的背景。多数化自然出现在 量子信息，我的程序也研究了这种联系，特别是所谓的王牌优势。