Operator Algebras, Operator Theory and Free Probability Investigations

希尔伯特空间上的算符为量子力学可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，这是一个非对易的数学环境，正在发展基本数学理论的非对易类比。首席研究人员将在两个方向上工作，涉及希尔伯特空间算子，遵循他最近的研究成果。一方面，对n元组算子在某些类型扰动下的不变性的探索导致了对几乎在一定意义上与n元组可交换的算子环的研究。与算子论问题以及阿兰·康尼斯意义上的非对易几何的联系正在出现。第二个方向是在自由概率领域，这是非对易概率设置中最非对易的。自由概率是由首席研究者提出的，它与随机矩阵、组合学和算子代数有着重要的联系。通过随机矩阵连接，可以应用于物理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，自由概率是所谓的二叉树概率理论的扩展，它是一个快速发展的新领域，有许多公开的问题，也将是自由概率问题研究的一部分。更详细地，主要的研究者将一方面探索比紧算子的理想小的模范理想的交换子，这构成了一个新的算子代数框架，用于研究算子的赋范理想扰动，另一方面自由概率的二叉树推广到具有非交换左右变量的系统。算子论问题涉及到关于赋范理想的拟中心近似单位的障碍，它与几乎正规的算子、熵和可超群有不同的关系。模赋范理想的交换子是Hilbert空间上具有算子对合的Banach代数，它们与C*-代数有着出乎意料的许多联系，就像C*-代数是非C*-Banach-代数的冠冕一样。这些代数的K理论方面也很有趣，例如，它们为关于绝对连续谱在扰动下的不变性的结果提供了一个新的角度。在自由概率及其二叉树扩展方面，重点将放在分析上，而其他研究者则侧重于组合学。厄米算子的迹类交换子既出现在关于模范理想的交换子的问题中，也出现在自由和二叉树概率中，有可能是这两个主题相遇的点。首席研究人员希望吸引青年数学家在他的研究领域进行研究，特别是继续组织关于自由概率的会议和研讨会，这与数学中的其他领域有许多联系，并为年轻数学家提供良好的培训。

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.