Theory and applications of operator systems and operator algebras

算子系统和算子代数的理论与应用

• 批准号: RGPIN-2014-05708
• 负责人: Farenick, Douglas
• 金额: $ 1.68万
• 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=611796
• 关键词: Theory; applications; operator; systems; algebras

The proposed research is in the area of pure mathematics and aims to further develop the theoretical basis for our understanding of certain algebraic structures, particularly C*-algebras and operator systems, that arise in both pure mathematics and theoretical physics. Specifically, this research aims to contribute new information about finite-dimensional operator systems associated with finitely generated groups and to use this information to shed light on a property of C*-algebras known as the weak expectation property. There is potential for applications of this work in the field of quantum information, which is an interdisciplinary research area involving computing science, mathematics, and physics. One way to analyse an abstract mathematical structure is to represent the structure in a different guise. Ideally, if one represents the abstract structure under study in some familiar manner or context, then one can use previous knowledge about well-understood objects to deduce properties or information about the abstract structure under study. The first goal of the proposed research is aimed at doing precisely that. In technical terms, I will be seeking criteria upon which one can decide if a given operator system is a quotient of a matrix operator system. In this scenario the matrix operator system is the familiar object through which one analyses a quotient operator system. In a related vein, the proposed research will examine operator systems arising from finitely generated discrete groups and consider properties of the groups that allow the operator system or generated C*-algebra to detect the weak expectation property. The importance of the proposed research lies, first of all, in the fact that the mathematical structures to be studied have attracted the attention of mathematicians and physicists for nearly a century. Indeed, the field of operator algebras first arose in the early twentieth century out of a need to provide the appropriate mathematical tools for the emerging field of quantum physics. Today, there is a significant amount of international research devoted to operator algebra theory and functional analysis, and the field of quantum information theory has grown enormously in the last decade. The results of the proposed research will be of interest and relevance to such scientists and mathematicians, and the research work itself will provide a vehicle through which mathematical expertise can be developed and retained within Canada.

拟议的研究属于纯数学领域，旨在进一步发展我们理解纯数学和理论物理中出现的某些代数结构，特别是 C* 代数和算子系统的理论基础。具体来说，这项研究旨在提供有关与有限生成群相关的有限维算子系统的新信息，并利用这些信息来阐明 C* 代数的弱期望性质。这项工作在量子信息领域具有应用潜力，这是一个涉及计算科学、数学和物理的跨学科研究领域。 分析抽象数学结构的一种方法是以不同的形式表示该结构。理想情况下，如果一个人以某种熟悉的方式或上下文表示所研究的抽象结构，那么就可以使用有关易于理解的对象的先前知识来推断有关所研究的抽象结构的属性或信息。拟议研究的第一个目标就是为了做到这一点。用技术术语来说，我将寻求一个标准，根据该标准可以确定给定的算子系统是否是矩阵算子系统的商。在这种情况下，矩阵算子系统是分析商算子系统的常见对象。与此相关，拟议的研究将检查由有限生成的离散群产生的算子系统，并考虑允许算子系统或生成的 C* 代数检测弱期望属性的组的属性。 这项研究的重要性首先在于，近一个世纪以来，所要研究的数学结构引起了数学家和物理学家的关注。事实上，算子代数领域最初出现于二十世纪初，是出于为新兴的量子物理领域提供适当数学工具的需要。如今，国际上有大量研究致力于算子代数理论和泛函分析，并且量子信息理论领域在过去十年中得到了巨大发展。拟议研究的结果将引起这些科学家和数学家的兴趣和相关性，研究工作本身将提供一个工具，通过该工具可以在加拿大发展和保留数学专业知识。