Free Probability Studies in von Neumann Algebras

冯诺依曼代数的免费概率研究

• 批准号: 0501178
• 负责人: Dan-Virgil Voiculescu
• 金额: $ 53.6万
• 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: 2005
• 资助国家: 美国
• 起止时间: 2005-03-01 至 2011-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501178&HistoricalAwards=false
• 关键词: Free; Probability; Studies; von; Neumann

The PI will work on free probability and use it to study von Neumann algebras. Motivated by problems arising from the free probability analogue of Shannon's differential entropy, analysis questions involving free difference quotient derivations,the natural derivations for noncommutative variables, will be studied. Extensions of the free probability parallel to classical probability, such as exploring free analogues of extreme value theory and of the Wasserstein distance will be continued. The focus on the operator algebra side will be on structural features of von Neumann algebras of free groups such as the automorphism group.Free probability is a probabilistic framework adapted to quantities with the highest degree of noncommutativity. This parallels a large area of classical probability from the Gauss law ( whose free probability counterpart is the semicircle law) to information theoretic entropy. The proposal is to develop new tools in free probability theory aiming at problems on operator algebras and random matrices. The applications of free probability to the asymptotics of large random matrices have also been of interest in connection with the random matrices in certain physics models and in models of multiuser telecommunication systems.

PI将致力于自由概率，并用它来研究冯诺依曼代数。出于所产生的问题从自由概率模拟香农的微分熵，分析问题涉及自由差商推导，自然推导的非交换变量，将进行研究。扩展的自由概率平行于经典概率，如探索自由类似物的极值理论和瓦瑟斯坦距离将继续。在算子代数方面的重点将是冯诺依曼代数的自由群的结构特征，如自同构群。自由概率是一个概率框架，适用于具有最高程度的非交换性的量。这与从高斯定律（其自由概率对应物是循环定律）到信息论熵的大量经典概率相似。本文的目的是针对算子代数和随机矩阵的问题，发展自由概率论中的新工具。 自由概率在大型随机矩阵渐近性中的应用，在某些物理模型和多用户电信系统模型中的随机矩阵方面也引起了人们的兴趣。