Theory and applications of operator systems and operator algebras

我的研究重点是泛函分析的一部分，包括作用于希尔伯特空间、C*代数和冯·诺伊曼代数以及抽象算子系统的有界线性算子理论。这些特定领域是非常活跃的研究领域，吸引了世界各地大量高质量的数学家。事实上，这些领域也代表了加拿大在国际上特别强大的数学分支。本研究计划的长期目标是全面推进算子代数理论，并通过这些进展，使算子系统和算子代数理论在相关领域，如单算子理论、量子信息论、自由凸性（也称为算子凸性、矩阵凸性和非交换凸性）和非交换函数理论中做出新的、有用的贡献。作为这个项目的一部分，我的目标是加强这些看似不同的子领域之间的相互作用，并在这些研究方面聘请高素质的人员，这些人员有潜力开发一个知识库，该知识库具有超出当前提议工作范围的持续价值。在短期内，我建议继续在我目前的NSERC发现基金期间进行的研究轨迹，特别强调实现以下目标：(I)理解算子系统张量积的矩阵状态空间的性质；（ii）提出了自由拓扑，确定了自由测度理论和算子-凸函数在自由凸集上的积分表示的框架；（三）费米子代数上的量子通道谱分析；（iv）发展量子噪音的新量度。这项研究的预期成果包括：(a)培训具有强大数学技能的新人员，特别强调发展妇女、少数民族或残疾人的数学才能；(b)数学家、理论物理学家和计算机科学家在世界范围内研究和应用的数学分支知识的进步。

The focus of my research program concerns the part of functional analysis that encompasses the theory of bounded linear operators acting on Hilbert spaces, C*-algebras and von Neumann algebras, and abstract operator systems. These particular fields are very active areas of research that engage a significant number of high-quality mathematicians worldwide. Indeed, these areas also represent branches of mathematics in which Canada is known internationally to be particularly strong. The long-term objectives of the present research proposal are to make advances in operator algebra theory overall and, through those advances, to make novel, useful contributions of the theory of operator systems and operator algebras to related fields such as single-operator theory, quantum information theory, free convexity (also known as operator convexity, matrix convexity, and noncommutative convexity), and noncommutative function theory. As part of this program, I aim to strengthen the interactions between these seemingly diverse subfields, and to engage highly qualified personnel in those aspects of this research that have the potential to develop a knowledge base that has sustained value beyond the scope of the presently proposed work. In the near term, I am proposing to continue the trajectory of the research undertaken during my current NSERC Discovery Grant, with a particular emphasis on achieving the following objectives: (i) understanding the nature of the matrix-state spaces of operator system tensor products; (ii) advancing free topology, and determining a framework for free measure theory and integral representations of operator-convex functions on free convex sets; (iii) the analysis of the spectra of quantum channels on the fermion algebra; (iv) developing new measures of quantum noise. Expected outcomes of this research include: (a) the training of new personnel equipped with a powerful skill set in mathematics, with a particular emphasis on developing the mathematical talents of women, minority peoples, or people with disabilities; and (b) the advancement of knowledge in a branch of mathematics that is studied and applied worldwide by mathematicians, theoretical physicists, and computer scientists.