Studies in Free Probability and Operator Algebras

该项目的主要主题是自由概率的分析方面和模为希尔伯特-施密特类的几乎正常的算子。这一方面涉及对“自由黎曼球”的进一步研究，“自由黎曼球”是黎曼球的一个高度非对易的类似物，以及相应的高度非对易函数。另一方面，几乎正规算子问题涉及到对Hilbert空间上某些Banach算子代数及其K-理论的研究。该项目的算符理论部分也与非对易概率有关，因为所考虑的算符代数自然出现在非对易位势理论中，并且在最简单的情况下，自由概率中的Fisher信息的类似涉及几乎正规的算符。自由概率是一种概率框架，适用于处理表现出最高非对易程度的随机变量。这一理论与算子代数理论、随机矩阵理论以及组合学中的某些主题有着重要的联系。对自由概率的研究往往受益于这些联系，反过来，也可能对这些其他领域感兴趣。具体地，在多用户无线通信的某些模型中和在某些物理模型中使用随机矩阵的用户已经发现随机矩阵的自由概率方法是有用的。

The main themes of the project are the analytic side of free probability and almost-normal operators modulo the Hilbert-Schmidt class. This will involve on the one hand further study of the "free Riemann sphere," a highly noncommutative analogue of the Riemann sphere, and of corresponding highly noncommutative functions. On the other hand, the almost-normal operator questions are related to the study of certain Banach algebras of operators on Hilbert space and their K-theory. The operator theory part of the project also has connections to noncommutative probability, since the operator algebras considered appear naturally in noncommutative potential theory and the analogue of Fisher information in free probability involves, in the simplest cases, almost-normal operators. Free probability is a probability framework adapted to deal with random variables that exhibit the highest degree of noncommutativity. The theory has important connections with operator algebra theory, with random matrix theory, and with certain topics in combinatorics. Studies in free probability often benefit from these connections and, in return, may be of interest in these other fields. In particular, users of random matrices in certain models of multiuser wireless communications and in certain physics models have found the free probability approach to random matrices useful.