Operator Algebras, Operator Theory and Free Probability Investigations

希尔伯特空间上的算子为量子力学观测提供了无限维线性代数框架。由于两个线性映射的组合取决于哪个映射先起作用，这是一个非交换的数学设置，其中正在开发基本数学理论的非交换类似物。根据他最近的研究成果，首席研究员将在涉及希尔伯特空间算子的两个方向上工作。一方面，对算子的n元组在一定扰动下的不变性的探索，导致了在一定意义上与n元组几乎交换的算子环的研究。与算子理论问题的联系，以及与阿兰·康尼斯意义上的非交换几何的联系正在出现。第二个方向是在自由概率领域，它是非交换概率设置中最不可交换的。自由概率是由首席研究员提出的，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，在某些物理模型和多用户通信系统模型中都有应用。最近，首席研究员发现自由概率可以扩展到所谓的三自由度概率论，这是一个快速发展的新领域，有许多开放的问题，也将成为自由概率问题研究的一部分。更详细地说，主要研究者将一方面探索交换子的模赋范理想，小于紧算子的理想，它们构成了一个新的算子代数框架，用于研究算子的赋范理想摄动，另一方面是自由概率对左右不可交换变量系统的三自由度扩展。算子理论问题涉及到准中心近似单位相对于一个与几乎正规算子、熵和超适群有不同关系的赋范理想的障碍。交换子模赋范理想是Hilbert空间上算子对合的Banach代数，它与C*-代数有着意想不到的许多关系，就像C*-代数是非C*-Banach代数的冕一样。这些代数的k理论方面也令人感兴趣，例如，它们为摄动下绝对连续谱的不变性结果提供了一个新的角度。在自由概率及其三自由度扩展方面，重点将放在分析上，而其他研究人员则强调组合学。厄米算子的迹类对易子既出现在模赋范理想的对易子问题中，也出现在自由概率和三自由度概率问题中，并有可能是二者相交的点。首席研究员希望吸引年轻的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，这与其他数学领域有许多联系，并为年轻数学家提供了良好的培训。

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.