Theory and applications of operator systems and operator algebras

我的研究项目的重点是泛函分析的一部分，它包括作用在Hilbert空间上的有界线性算子理论，C*-代数和von Neumann代数，以及抽象算子系统。这些特定的领域是非常活跃的研究领域，吸引了世界各地大量高素质的数学家。事实上，这些领域也代表了数学的分支，加拿大在这方面在国际上被认为是特别强大的。本研究方案的长期目标是在算子代数理论方面取得全面进展，并通过这些进展，使算子系统和算子代数理论在单算子理论、量子信息论、自由凸性(也称为算子凸性、矩阵凸性和非对易凸性)和非对易函数理论等相关领域做出新的、有用的贡献。作为这一计划的一部分，我的目标是加强这些看似不同的子领域之间的互动，并在这项研究的那些方面聘请高素质的人员，这些方面有可能开发出一个具有持续价值的知识库，其价值超出了目前拟议的工作范围。在短期内，我建议继续在我目前的NSERC发现拨款期间进行的研究轨迹，特别强调实现以下目标：(I)了解算子系统张量积的矩阵-状态空间的性质；(Ii)推进自由拓扑，并确定自由测量理论的框架和自由凸集上的算子-凸函数的积分表示；(Iii)在费米子代数上分析量子通道的频谱；(Iv)开发量子噪声的新测量。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天赋；(B)促进一个数学分支的知识进步，该分支由数学家、理论物理学家和计算机科学家在全世界范围内研究和应用。

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.