Operator Algebras, Operator Theory and Free Probability Investigations

算子代数、算子理论和自由概率研究

• 批准号: 1665534
• 负责人: Dan-Virgil Voiculescu
• 金额: $ 19.7万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-05-15 至 2021-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665534&HistoricalAwards=false
• 关键词: Operator; Algebras; Theory; Free; Probability

Operators on Hilbert space provide the infinite-dimensional linear algebra framework for quantum mechanical observables. Since composition of two linear maps depends on which of the maps acts first, this is a noncommutative mathematics setting where noncommutative analogues of basic mathematical theories are being developed. The principal investigator will work in two directions involving Hilbert space operators, following recent realizations in his research. On one hand the exploration of invariant properties of n-tuples of operators under certain types of perturbations leads to the study of the rings of operators which almost commute in a certain sense with the n-tuple. Connections to operator theory questions as well as to noncommutative geometry in the sense of Alain Connes are appearing. The second direction is in the area of free probability, the most noncommutative among noncommutative probability settings. Free probability, which was initiated by the principal investigator, has important connections to random matrices, combinatorics and to operator algebras. Via the random matrix connection, there are applications to certain models in physics and to models of multiuser communication systems. Recently the principal investigator found that free probability has an extension to a so-called bifree probability theory, which is a fast developing new area, with many open problems and which will also be part of the free probability problems studied. In more detail, the principal investigator will explore on one hand commutants modulo normed ideals of operators, smaller than the ideal of compact operators, which constitute a new operator algebra framework for the study of normed ideal perturbations of operators and on the other hand the bifree extension of free probability, to systems with noncommuting left and right variables. The operator theory questions involve obstructions to quasicentral approximate units relative to a normed ideal with diverse relations to almost normal operators, entropy and supramenable groups. The commutants modulo normed ideals are Banach algebras with involution of operators on Hilbert space, which have unexpectedly many relations to C*-algebras, like in the appearance of C*-algebras which are coronas of non-C*-Banach-algebras. The K-theory aspects of these algebras are also of interest, for instance they provide a new angle on results about invariance of absolutely continuous spectra under perturbations . On the side of free probability and of its bifree extension the emphasis will be on the analysis, while other researchers emphasize the combinatorics. Trace-class commutators of hermitian operators appear both in questions about commutants modulo normed ideals and in free and bifree probability, with the possibility of being a point where the two subjects meet. The principal investigator expects to attract young mathematicians to research in areas of his investigations and in particular to continue organizing conferences and seminars about free probability, which has many connections to other fields in mathematics and provides good training for young mathematicians.

希尔伯特空间上的算子为量子力学的观测量提供了无限维线性代数框架。由于两个线性映射的合成取决于哪一个映射首先起作用，这是一个非交换数学环境，其中正在开发基本数学理论的非交换类似物。首席研究员将在两个方向工作，涉及希尔伯特空间算子，在他的研究最近实现。一方面，在某些类型的扰动下，对算子的n元组的不变性质的探索导致了对在某种意义上几乎与n元组交换的算子环的研究。连接到运营商理论的问题，以及非交换几何意义上的阿兰康纳斯正在出现。第二个方向是在自由概率领域，在非交换概率设置中最具非交换性。自由概率，这是发起的主要调查员，有重要的联系，以随机矩阵，组合数学和算子代数。通过随机矩阵连接，可以应用于物理学中的某些模型和多用户通信系统的模型。最近，主要研究者发现自由概率有一个所谓的双自由概率理论的扩展，这是一个快速发展的新领域，有许多开放的问题，这也将是自由概率问题研究的一部分。 更详细地说，首席研究员将探讨一方面交换模赋范理想的运营商，小于理想的紧凑运营商，这构成了一个新的运营商代数框架研究赋范理想扰动的运营商和另一方面的bifrefree扩展的自由概率，系统与noncommuting左，右变量。算子理论的问题涉及到相对于赋范理想的拟中心近似单位的障碍，与几乎正规算子、熵和超群有着不同的关系。模赋范理想交换子是Hilbert空间上算子对合的Banach代数，它与C*-代数有着意想不到的联系，例如C*-代数的出现是非C *-Banach-代数的冠。 这些代数的K-理论方面也很有趣，例如它们提供了一个新的角度，关于扰动下绝对连续谱的不变性的结果。在自由概率及其双自由扩展方面，重点将放在分析上，而其他研究人员则强调组合学。厄米特算子的迹类算子出现在关于交换子模赋范理想的问题中，也出现在自由和双自由概率中，有可能是两个主题相遇的点。首席研究员希望吸引年轻的数学家在他的调查领域进行研究，特别是继续组织关于自由概率的会议和研讨会，这与数学的其他领域有许多联系，并为年轻的数学家提供良好的培训。