Studies in Free Probability and Operator Algebras

自由概率和算子代数研究

• 批准号: 1301727
• 负责人: Dan-Virgil Voiculescu
• 金额: $ 38.09万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301727&HistoricalAwards=false
• 关键词: Studies; Free; Probability; Operator; Algebras

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.

该项目的主要主题是自由概率的分析方面和对Hilbert-Schmidt类模的几乎正规算子。这一方面将涉及进一步研究“自由黎曼球”，黎曼球的高度非对易类比，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Hilbert空间上某些Banach算子代数及其k理论的研究。该项目的算子理论部分也与非对易概率有关，因为所考虑的算子代数自然地出现在非对易势理论中，并且在最简单的情况下，自由概率中的Fisher信息的模拟涉及到几乎正常的算子。自由概率是一种概率框架，适用于处理表现出最高程度非交换性的随机变量。该理论与算子代数理论、随机矩阵理论以及组合学中的某些课题有着重要的联系。自由概率的研究经常受益于这些联系，反过来，也可能对这些其他领域产生兴趣。特别是，在某些多用户无线通信模型和某些物理模型中使用随机矩阵的用户发现随机矩阵的自由概率方法很有用。