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Non-Tracial Derivations and Distributions

非迹推导和分布

基本信息

批准号：
1856683
负责人：
Brent Nelson
金额：
$ 18.5万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1856683&HistoricalAwards=false
关键词：
Non Tracial Derivations Distributions

项目摘要

Von Neumann algebras are mathematical objects that offer a rigorous framework for the study of quantum physics, and can be thought of as infinite-dimensional generalizations of matrix algebras. The theory was initiated by Francis J. Murray and John von Neumann in the 1930s, and since then researchers have discovered a vast number of applications to mathematics as well as biology, physics, and engineering. In particular, one of the modern fields that studies von Neumann algebras is free probability, wherein one views the theory of von Neumann algebras as a generalization of probability theory. This perspective yields connections to random matrix theory, allows one to model the statistical behavior of large-scale data sets, and notably has been used to understand the fundamental limits of wireless communications. Furthermore, connections with the mathematical disciplines of complex analysis, dynamical systems, group theory, and harmonic analysis arise from the study of derivations on von Neumann algebras, which are analogues of the derivative in Leibniz and Newton's theory of calculus. The so-called "non-tracial" von Neumann algebras admit additional connections to physics: they have an inherit dynamical system that satisfies the Kubo-Martin-Schwinger condition from quantum statistical mechanics; and they arise naturally in conformal field theory. This project is part of the principal investigator's research efforts to better understand non-tracial von Neumann algebras through the use of free probability and an analysis of derivations.The goal of this project is to study the structure of non-tracial (type III) von Neumann algebras through three approaches, which are interconnected via free probabilistic distributions and derivations. The first approach concerns derivations on a von Neumann algebra, which lie within the scope of two major methodologies: Voiculescu's free probability theory, and Popa's deformation/rigidity theory. The former through free Fisher information and non-microstates free entropy, and the latter through deformations arising from the completely positive semigroups associated to closable derivations. The principal investigator seeks to show that derivations on non-tracial von Neumann algebras give rise to completely positive semigroups, and then use free probability and deformation/rigidity to extract structural information in the spirit of Peterson as well as Dabrowski and Ioana. The second approach concerns non-microstates regularity conditions on finite generating sets of von Neumann algebras equipped with non-tracial states, which can be regarded as non-commutative probability distributions and therefore analyzed with free probability. The principal investigator has shown that when these generators have finite free Fisher information and are well-behaved under the modular automorphism group associated to the state, then the von Neumann algebra they generate is a full factor. In recent work with Charlesworth on free Stein information, the principal investigator has found a weaker regularity condition than having finite free Fisher information that is more easily adapted to the non-tracial case. He will explore the structural consequences of this regularity condition for non-tracial von Neumann algebras, and attempt to adapt other existing regularity conditions to the non-tracial case. The third approach concerns random matrix models for non-tracial von Neumann algebras. In the tracial case, understanding the connection between the Gaussian Unitary Ensemble (GUE) and the free group factors has benefited random matrix theory by providing a clear picture of the large N limit, and has benefited free probability theory by providing finite dimensional approximations and informing the definition of microstates free entropy. The principal investigator has found a non-tracial version of the Dyson-Schwinger equation, which is a non-commutative PDE that (in the tracial case) makes explicit the connection between the GUE and the free group factors. A non-tracial Dyson-Schwinger equation gives rise to a free Araki-Woods factor, which were defined by Shlyakhtenko as the non-tracial analogue of the free group factors. The principal investigator aims to use non-tracial Dyson-Schwinger equations to develop random matrix models for free Araki-Woods factors, and more generally to develop microstates free entropy for non-tracial von Neumann algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

冯·诺依曼代数是为量子物理研究提供严格框架的数学对象，可以被认为是矩阵代数的无限维推广。这一理论是由弗朗西斯·J·默里和约翰·冯·诺伊曼在20世纪30年代提出的，从那时起，研究人员发现了大量在数学以及生物学、物理学和工程学中的应用。特别是，研究von Neumann代数的现代领域之一是自由概率，其中人们将von Neumann代数理论视为概率论的推广。这种观点产生了与随机矩阵理论的联系，允许人们对大规模数据集的统计行为进行建模，特别是已经被用于理解无线通信的基本限制。此外，与复分析、动力系统、群论和调和分析等数学学科的联系源于对von Neumann代数导数的研究，von Neumann代数类似于莱布尼茨和牛顿微积分理论中的导数。所谓的“非踪迹”冯诺依曼代数承认了与物理学的额外联系：它们有一个继承的动力系统，满足量子统计力学中的Kubo-Martin-Schwinger条件；它们自然地出现在保形场论中。这个项目是主要研究人员通过使用自由概率和分析导数来更好地理解非迹von Neumann代数的研究工作的一部分。本项目的目标是通过三种方法来研究非迹(类型III)von Neumann代数的结构，这三种方法通过自由概率分布和导数相互联系。第一种方法涉及冯·诺依曼代数上的推导，它属于两种主要方法的范围：沃库列斯库的自由概率理论和波帕的形变/刚性理论。前者通过自由Fisher信息和非微态自由熵，后者通过与可闭导子相关的完全正半群所产生的变形。主要研究人员试图证明非迹von Neumann代数上的导子产生完全正半群，然后根据Peterson、Dabrowski和Ioana的精神，利用自由概率和形变/刚性来提取结构信息。第二种方法涉及具有非迹状态的von Neumann代数的有限生成集上的非微态正则性条件，这些条件可以看作是非对易概率分布，因此可以用自由概率来分析。主要研究者证明了当这些生成元具有有限的自由Fisher信息并且在与态相关的模自同构群下表现良好时，则它们生成的von Neumann代数是一个满因子。在最近与查尔斯沃思关于自由斯坦信息的工作中，首席研究员发现了一种比有限自由Fisher信息更弱的正则性条件，这种条件更容易适应非跟踪情况。他将探索这种正则性条件对于非迹von Neumann代数的结构后果，并试图使其他现有的正则性条件适用于非迹情况。第三种方法涉及非迹von Neumann代数的随机矩阵模型。在追踪的情况下，理解高斯么正系综(GUE)和自由群因子之间的联系通过提供大N极限的清晰图景而有益于随机矩阵理论，并且通过提供有限维近似和告知微态自由熵的定义而有益于自由概率论。首席研究人员发现了Dyson-Schwinger方程的一个非对易版本，它是一个非对易的偏微分方程组，它(在可跟踪的情况下)明确地说明了GUE和自由群因子之间的联系。非迹Dyson-Schwinger方程产生了自由Araki-Wood因子，Shlyakhtenko将其定义为自由群因子的非迹类似物。这位首席研究员的目标是使用非迹Dyson-Schwinger方程来开发自由Araki-Wood因子的随机矩阵模型，并更广泛地开发非迹von Neumann代数的微态自由熵。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Free Stein irregularity and dimension

免费斯坦因不规则性和尺寸

DOI：
10.7900/jot.2019aug29.2271
发表时间：
2021
期刊：
Journal of operator theory
影响因子：
0.8
作者：
Charlesworth, Ian;Nelson, Brent
通讯作者：
Nelson, Brent

A Random Matrix Approach to Absorption in Free Products

免费产品吸收的随机矩阵方法

DOI：
10.1093/imrn/rnaa191
发表时间：
2020
期刊：
International Mathematics Research Notices
影响因子：
1
作者：
Hayes, Ben;Jekel, David;Nelson, Brent;Sinclair, Thomas
通讯作者：
Sinclair, Thomas

Quantum edge correspondences and quantum Cuntz–Krieger algebras

量子边对应和量子 Cuntz Krieger 代数

DOI：
10.1112/jlms.12702
发表时间：
2023
期刊：
Journal of the London Mathematical Society
影响因子：
0
作者：
Brannan, Michael;Hamidi, Mitch;Ismert, Lara;Nelson, Brent;Wasilewski, Mateusz
通讯作者：
Wasilewski, Mateusz

Non-tracial Free Graph von Neumann Algebras

非踪迹自由图冯诺依曼代数