Research in finite von Neumann algebras

有限冯诺依曼代数研究

• 批准号: 1202660
• 负责人: Kenneth Dykema
• 金额: $ 17.7万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1202660&HistoricalAwards=false
• 关键词: Research; finite; von; Neumann; algebras

The investigator will try to answer questions about algebras of bounded operators on Hilbert space; in particular, these questions involve finite von Neumann algebras, which are algebras of operators on Hilbert space that are closed under taking adjoints and strong--operator limits, and on which there exist tracial linear functionals. The main question the investigator proposes to work on is known as Connes' embedding problem, which asks (in one of many equivalent formulations): can all operators in a von Neumann algebra that has a bounded trace be approximated by matrices over the complex numbers? This question is fundamental and its resolution would have profound ramifications on our understanding of operator algebras and operator spaces. One important goal is to seek a group lacking certain analogous approximation properties (i.e., a non-sofic group), both as it's own goal and as a guide for particular von Neumann algebras to investigate vis-a-vis Connes' embedding problem. In other directions, the investigator proposes to use the "practical Schubert calculus" to learn more about finite von Neumann algebras, and to study certain natural questions in finite von Neumann algebras, such as the single commutator question and the Schur-Horn question, using diverse methods.Group theory is the study of fundamental symmetries that are found in nearly all branches of mathematics and natural sciences. On the other hand, von Neumann algebras were invented in the 1930's and 1940's by Murray and von Neumann; they consist of operators on infinite dimensional space and are related to the formalism of quantum mechanics, which motivated their invention. In modern mathematics, the study of these von Neumann algebras has important connections to several other areas of mathematics, such as group theory and dynamical systems. The investigator will try to answer related, fundamental problems in group theory and von Neumann algebras, namely, "can certain infinite objects always be approximated by finite ones?" Answers to these questions would have broad impact; they are related to many other open problems about groups and, respectively, operators on infinite dimensional space. The investigator will also study certain other natural questions about von Neumann algebras. The research effort will furthermore support the educational efforts of the principal investigator: most directly in teaching graduate students the prerequisites to understanding current research problems and then to guide them to vital and important research problems and but also in educating undergraduate students in a way that shows the utility and beauty of mathematics and teaches them essential mathematical skills and methods of mathematical (and logical) reasoning.

调查员将试图回答有关代数的有界算子的希尔伯特空间的问题，特别是，这些问题涉及有限冯诺依曼代数，这是代数的运营商希尔伯特空间是封闭下采取伴随和强-运营商限制，并存在tracial线性泛函。研究人员提出的主要问题是被称为康纳斯嵌入问题，它要求（在许多等价的公式之一）：可以所有运营商在冯诺依曼代数有界迹近似矩阵的复数？这个问题是根本性的，它的解决将对我们理解算子代数和算子空间产生深远的影响。一个重要的目标是寻找缺乏某些类似近似性质的群（即，一个非sofic群），既作为自己的目标，并作为一个指导特定的冯诺依曼代数研究维斯康纳斯的嵌入问题。在其他方向上，研究者建议使用“实用舒伯特演算”来学习更多关于有限冯诺依曼代数的知识，并使用不同的方法来研究有限冯诺依曼代数中的某些自然问题，如单交换子问题和舒尔-霍恩问题。群论是对几乎在数学和自然科学的所有分支中发现的基本对称性的研究。另一方面，冯·诺依曼代数是在20世纪30年代和40年代由默里和冯·诺依曼发明的;它们由无限维空间上的算子组成，与量子力学的形式主义有关，这激发了它们的发明。在现代数学中，这些冯诺依曼代数的研究与其他几个数学领域有着重要的联系，例如群论和动力系统。调查人员将试图回答相关的，基本的问题，在群论和冯诺依曼代数，即“可以某些无限对象总是近似有限的？这些问题的答案将产生广泛的影响;它们与许多其他关于群的开放问题有关，并且分别与无限维空间上的算子有关。调查员还将研究某些其他自然问题冯诺依曼代数。研究工作将进一步支持主要研究者的教育工作：最直接的教学研究生的先决条件，以了解当前的研究问题，然后引导他们的重要和重要的研究问题，而且在教育本科生的方式，显示了数学的实用性和美感，教他们基本的数学技能和方法的数学（逻辑推理）