Entropy Theory Methods in von Neumann Algebras

量子力学是物理学的一个领域，有着包括生物学和工程学在内的众多应用。该理论的一个基本结果是，可见项不能相互交换，因此量子力学的数学基础必须在非对易框架内。基于这一限制，S、默里和冯·诺伊曼在20世纪30年代使用希尔伯特空间上的算子代数为量子力学提供了严格的背景，现在称为冯·诺伊曼代数。近年来，这些物体为研究量子计算提供了一个精确的框架，量子计算在密码学突破方面具有巨大的潜在应用。冯·诺依曼代数的另一个领域是对随机矩阵的研究。随机矩阵本身在核物理中被用来模拟重原子核，在理论神经科学中也被用来描述大脑中神经元之间的联系。该项目还将通过多元化倡议和研究生培训为美国劳动力发展做出贡献。这个项目的目的是通过两种不同的方法来研究群von Neumann代数：Vocerescu的自由概率理论和Popa的形变/刚性理论。在沃库列斯库的自由概率论中，PI计划使用随机矩阵理论来攻击Peterson和Thom关于自由群的von Neumann代数的子代数结构的一个著名猜想。随机矩阵方法使用所谓的强收敛，它要求本征值分布收敛于弱*，并且谱收敛于Hausdorff度量。PI建立在PI和Jung的1-有界熵理论的基础上，建立了如果随机么正张量的自然强收敛结果成立，则能够推论Peterson-Thom猜想作为推论。将SOFIC熵理论引入到Popa的形变/刚性理论中，PI最近与De Santaigo-Hoff-Sclair一起证明，给定迹von Neumann代数和扩散刚性子代数的(S可延展)形变，必然可以将其推广到极大刚性子代数。一个潜在的探索途径是将这些结果推广到近似刚性子代数。这样的推广为Peterson-Thom猜想提供了第二种方法。它还将提供一种方法来证明在左正则表示中具有非平凡上同调的非顺从群不具有Cartan子代数，这是过去十年来一直悬而未决的问题。PI还计划研究这一理论对等价关系von Neumann代数的影响。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。

Quantum mechanics is an area of physics with numerous applications including biology and engineering. A basic consequence of the theory is that observables cannot commute with each other, and so the mathematical basis for quantum mechanics must be within a noncommutative framework. Based on this constraint, in the 1930's Murray and von Neumann provided a rigorous background for quantum mechanics using algebras of operators on a Hilbert space now called von Neumann algebras. Recently, these objects have provided a precise framework in which to study quantum computing, which has enormous potential application to breakthroughs in cryptography. Another area where von Neumann algebras have given tremendous insight is in the study of random matrices. Random matrices themselves are used in nuclear physics to model the nuclei of heavy atoms, as well as in theoretical neuroscience to describe the connections between neurons in the brain. This project will also contribute to US workforce development through diversity initiatives and training of graduate students. The goal of this project is to study group von Neumann algebras through two different approaches: Voiculescu's free probability theory, and Popa's deformation/rigidity theory. In Voiculescu's free probability theory, the PI plans to use the theory of random matrices to attack a well-known conjecture of Peterson and Thom on the structure of subalgebras of group von Neumann algebras of free groups. The random matrix approach uses so-called strong convergence, which demands that eigenvalue distributions converge weak*, and that the spectrum converges in the Hausdorff metric. Building on the 1-bounded entropy theory due to the PI and Jung, the PI establishes that if a natural strong convergence results holds for tensors of random unitaries, then one is able to deduce the Peterson-Thom conjecture as a corollary. Importing insight from sofic entropy theory into Popa's deformation/rigidity theory, the PI recently showed with de Santaigo-Hoff-Sinclair that given a (s-malleable) deformation of a tracial von Neumann algebra, and a diffuse, rigid subalgebra, one can necessarily extend it to a maximal rigid subalgebra. A potential avenue for exploration is in generalizing these results to approximately rigid subalgebra. Such a generalization gives a second approach to the Peterson-Thom conjecture. It would also provide a way to show that nonamenable groups with nontrivial cohomology in the left regular representation do not have Cartan subalgebras, a problem in the subject that has been open for the last decade. The PI also plans to study the implications of this theory to the case of von Neumann algebras of equivalence relations.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.