von Neumann Algebras, Free Probability and Free Entropy

本计画的研究目标是进一步发展“非交换”（或“自由”）机率论，并将其应用于冯诺依曼代数。约翰·冯·诺依曼在20世纪30年代引入了冯·诺依曼代数的概念，为研究非对易对象提供了一个自然的框架，最初是在量子力学的背景下。冯·诺依曼代数的研究通常被看作是“非交换测度空间”的研究。在过去的70年里，这已经成为一个丰富而肥沃的研究领域，有大量的具体对象，意想不到的属性和具有挑战性的中心开放问题。非对易测度空间中的非对易概率论（或自由概率论）是由Dan Voiculescu在1980年代提出的。有趣的是，它能够解决几个重要的问题，在该地区的冯诺依曼代数。我们希望进一步发展自由概率论，以期解决冯诺依曼代数领域的其他问题。特别地，我们希望通过计算vonNeumann代数的自由轨道维数来计算它们的自由熵维数;寻找新的不含Cartan子代数的vonNeumann代数（DanVoiculescu证明了自由群因子不存在Cartan子代数）;从自由概率论的角度研究了vonNeumann代数的极大内射子代数问题;利用G. Pisier相似长度引入冯诺依曼代数的主要动机是为了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，自由概率论在冯诺依曼代数领域有一些令人惊讶的应用。自由概率理论使我们对冯·诺依曼代数领域中的许多其他开放问题有了深入的了解，如冯·诺依曼代数的分类问题、生成元问题和可分解性问题。 进一步调查这两个领域之间的联系，自由概率论和冯诺依曼代数，是必要的和紧迫的。该项目旨在针对冯诺依曼代数领域的问题开发自由概率论的新工具。这些问题的解决将为我们对von Neumann代数进行分类提供新的途径。该项目的意义在于，自由概率论和冯·诺依曼代数的新发展一直对数学和物理学的几个领域，如统计学和量子力学有着深刻的应用。拟议活动的智力价值：该项目是首席研究员在自由概率论，冯诺依曼代数的生成器问题和冯诺依曼代数的最大内射子代数方面的先前工作的延续。拟议活动的更广泛影响：该项目将在能够加强新罕布什尔州大学研究开发更广泛影响的地点进行。新罕布什尔州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。

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.