Theory and applications of operator systems and operator algebras

拟议的研究是在纯数学领域，旨在进一步发展理论基础，为我们理解某些代数结构，特别是C*-代数和算子系统，出现在纯数学和理论物理。具体来说，这项研究的目的是贡献新的信息与有限维算子系统与生成群，并使用这些信息来阐明的C*-代数的属性被称为弱期望属性。这项工作在量子信息领域有潜在的应用，这是一个涉及计算科学，数学和物理学的跨学科研究领域。分析抽象数学结构的一种方法是用不同的形式来表示这个结构。理想情况下，如果一个人以某种熟悉的方式或上下文来表示所研究的抽象结构，那么他可以使用先前关于已充分理解的对象的知识来推断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要做到这一点。在技术术语中，我将寻求标准，根据这些标准可以决定一个给定的算子系统是否是矩阵算子系统的商。在这种情况下，矩阵算子系统是人们分析商算子系统的熟悉对象。在相关的静脉中，拟议的研究将检查算子系统产生的离散群，并考虑允许算子系统或生成的C*-代数检测弱期望性质的群体的属性，拟议的研究的重要性在于，首先，在事实上，数学结构的研究已经吸引了近世纪的数学家和物理学家的注意。事实上，算符代数领域最早出现在世纪早期，是为了给新兴的量子物理领域提供合适的数学工具。今天，有大量的国际研究致力于算子代数理论和泛函分析，量子信息理论领域在过去十年中有了巨大的发展。拟议的研究结果将是感兴趣的和相关的这些科学家和数学家，研究工作本身将提供一个工具，通过它可以开发和保留数学专业知识在加拿大。

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.