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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=644115
• 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，它是由约翰·冯·诺依曼（John von Neumann）开创的数学家努力创造的，以适应物理学家在量子力学中所做的数学。事实上，当它已经发生并继续发生时，物理学家发现他们使用的对象在某种意义上是抽象的，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是没有意义的。冯·诺依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，算子代数提供了深入了解量子场论，* 结理论，逻辑，量子信息和量子计算等领域。* 算子代数学家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一部分。这些观点之一是关于 * 包络结构的观点。有时候，我们可以通过把一个物体放在一个更大、更易处理的物体中来说明它。对于C*-代数，这些包络结构包括双重对偶、* 乘子代数和内射包络。我的研究计划调查了最后两个物体。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。即使在小的维度上，算子系统 * 也没有得到很好的理解，并且缺乏完全序同构的分类。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 *C*-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。这个 *** 定理是关于矩阵的一个非常好理解的结果，因此它对无限维 *** 设置的推广是非平凡的。在轻微的技术术语中，舒尔-霍恩定理描述了 * 自伴矩阵在不同的正交基选择下可能的对角线。仍然在有限维中，这个定理推广到正规算子是一个没有人知道答案的问题！我的研究交换家庭的 * 自伴运营商提供了一个背景下，这可能是研究成功。多数化在量子信息中很自然地出现，我的程序也研究了这种联系，特别是所谓的王牌多数化。