von Neumann Algebras, Free Probability and Free Entropy

冯诺依曼代数、自由概率和自由熵

• 批准号: 0600887
• 负责人: Junhao Shen
• 金额: $ 6.78万
• 依托单位: University of New Hampshire
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600887&HistoricalAwards=false
• 关键词: von; Neumann; Algebras; Free; Probability

The research objective of this project is to further develop "noncommutative" (or "free") probability theory and to find its applications to the subject of von Neumann algebras. John von Neumann introduced the concept of a von Neumann algebra in the 1930's to provide a natural framework for the study of noncommutative objects, initially in the context of quantum mechanics. The study of von Neumann algebras is often viewed as the study of "noncommutative measure spaces". Over the last seventy years, this has turned out to be a rich and fertile field of study, with a large number of concrete objects, unexpected properties and challenging and central open problems. Noncommutative probability theory (or free probability theory) in the context of noncommutative measure spaces was developed by Dan Voiculescu in the 1980's. Interesting for its sake, it was able to solve several important problems in the area of von Neumann algebras. We wish to further develop free probability theory; with a view to attacking other problems in the area of von Neumann algebras. In particular, we wish to compute the free entropy dimensions of von Neumann algebras by computing their free orbit dimensions; to search for new von Neumann algebras that have no Cartan subalgebras (Dan Voiculescu showed that free group factors have no Cartan subalgebras); to investigate maximal injective subalgebra problems of von Neumann algebras from the point view of free probability theory; and to find full factors that have Kadison's Similarity Property by using G. Pisier's similarity length. Main motivation for the introduction of von Neumann algebras is to obtain a more rigorous mathematical formulation of the basics of quantum mechanics. From a probabilistic point view, free probability theory has some surprising applications in the area of von Neumann algebras. Free probability theory gives us deep insights into many other open problems in the area of von Neumann algebras, such as the classification problem of von Neumann algebras, the generator question and decomposability. Further investigation of the links between these two fields, free probability theory and von Neumann algebras, is both necessary and urgent. This project aims to develop new tools in free probability theory aiming at problems in the area of von Neumann algebras. The solutions to these problems will provide us with new ways to classify von Neumann algebras. The significance of the project lies in the fact that new developments in the theory of free probability and von Neumann algebra have always had profound applications to several fields in mathematics and physics such as statistics and quantum mechanics. Intellectual Merit of the Proposed Activities: The project is the continuation of principal investigator's prior work in free probability theory, in generator problem of von Neumann algebras and in maximal injective subalgebras of von Neumann algebras. Broader Impact of the Proposed Activities: The project will take place at a location that strengthens the broader impacts of research development in the University of New Hampshire. The research group of Operator Algebra and Operator Theory in the University of New Hampshire consists of several well-established mathematicians. Joint-workshops will be organized to focus on helping graduate students to learn the subjects.

本项目的研究目标是进一步发展“非对易”(或“自由”)概率理论，并寻找其在von Neumann代数中的应用。约翰·冯·诺伊曼在20世纪30年代的S中引入了冯·诺伊曼代数的概念，为研究非对易物体提供了一个自然的框架，最初是在量子力学的背景下进行的。对von Neumann代数的研究通常被看作是对“非对易度量空间”的研究。在过去的70年里，这被证明是一个丰富而丰富的研究领域，有大量的具体对象，意想不到的性质和具有挑战性的中心开放问题。非对易度量空间中的非对易概率论(或称自由概率论)是由Dan Voulescu在20世纪80年代发展起来的。有趣的是，它能够解决von Neumann代数领域中的几个重要问题。我们希望进一步发展自由概率理论，以期解决von Neumann代数领域的其他问题。特别地，我们希望通过计算von Neumann代数的自由轨道维度来计算von Neumann代数的自由熵维；寻找没有Cartan子代数的新的von Neumann代数(Dan Voulescu证明了自由群因子没有Cartan子代数)；从自由概率论的角度研究von Neumann代数的极大内射子代数问题；利用G.Pisier的相似长度找到具有Kadison相似性质的全因子。引入von Neumann代数的主要动机是获得量子力学基础的更严格的数学公式。从概率的观点来看，自由概率论在von Neumann代数领域有一些令人惊讶的应用。自由概率理论使我们对von Neumann代数领域中的许多其他公开问题有了更深入的了解，如von Neumann代数的分类问题、生成元问题和可分解性。进一步研究自由概率论和von Neumann代数这两个领域之间的联系是必要和紧迫的。该项目旨在开发自由概率论的新工具，以解决von Neumann代数领域的问题。这些问题的解决将为我们提供对von Neumann代数进行分类的新途径。该项目的意义在于，自由概率理论和冯·诺依曼代数的新发展一直在统计学和量子力学等数学和物理领域有着深刻的应用。拟议活动的智力价值：该项目是主要研究人员在自由概率论、von Neumann代数的生成元问题和von Neumann代数的极大内射子代数方面先前工作的继续。拟议活动的更广泛影响：该项目将在新汉普郡大学加强研究发展的更广泛影响的地点进行。新汉普郡大学的算子代数和算子理论研究小组由几位著名的数学家组成。将组织联合讲习班，重点帮助研究生学习这些科目。