Research in finite von Neumann algebras

研究者将尝试回答Hilbert空间上有界算子的代数问题；特别地，这些问题涉及到有限的冯诺依曼代数，它是希尔伯特空间上的算子的代数，在带伴随和强算子极限下是封闭的，并且在其上存在迹线性泛函。研究者提出要解决的主要问题被称为Connes嵌入问题，该问题提出（在许多等效公式中的一个中）：冯·诺伊曼代数中具有有界迹的所有算子都可以用复数上的矩阵来近似吗？这个问题是根本性的，它的解决将对我们对算子代数和算子空间的理解产生深远的影响。一个重要的目标是寻找一个缺乏某些类似近似性质的群（即，非sofic群），既可以作为自己的目标，也可以作为特定的von Neumann代数的指南来研究相对于Connes的嵌入问题。在其他方向上，研究者建议使用“实用舒伯特微积分”来学习更多关于有限冯诺伊曼代数的知识，并使用多种方法研究有限冯诺伊曼代数中的某些自然问题，如单对易子问题和舒尔-霍恩问题。群论研究的是几乎在数学和自然科学的所有分支中都能找到的基本对称性。另一方面，冯·诺伊曼代数是在20世纪三四十年代由默里和冯·诺伊曼发明的；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，这促使了它们的发明。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重要的联系，如群论和动力系统。研究者将试图回答群论和冯·诺伊曼代数中相关的基本问题，即“某些无限对象是否总是被有限对象所近似？”这些问题的答案将产生广泛的影响；它们分别与无限维空间上的群和算子的许多其他开放问题有关。研究者还将研究冯·诺依曼代数的其他自然问题。研究工作将进一步支持首席研究者的教育工作：最直接的是教研究生理解当前研究问题的先决条件，然后指导他们解决至关重要的研究问题，但也在教育本科生，向他们展示数学的效用和美丽，教他们基本的数学技能和数学（和逻辑）推理的方法。

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.