Research in finite von Neumann algebras

作者将试图回答关于Hilbert空间上有界算子的代数的问题，特别地，这些问题涉及有限von Neumann代数，它是Hilbert空间上的算子在伴随和强算子极限下闭的代数，并且在其上存在迹线性泛函。这位研究人员提出的主要问题被称为康奈斯嵌入问题，该问题(在许多等价公式之一中)问：具有有界迹的von Neumann代数中的所有算子是否都可以由复数上的矩阵逼近？这个问题是基本的，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相似逼近性质的群(即一个非SOFIC群)，它既是它自己的目标，也是特定的von Neumann代数研究Va-a-Vis-Connes嵌入问题的指南。在其他方面，研究者建议用“实用的Schubert演算”来学习更多关于有限von Neumann代数的知识，并用不同的方法研究有限von Neumann代数中的某些自然问题，如单交换子问题和Schur-Horn问题。群论是研究基本对称性的学科，这些对称性几乎存在于数学和自然科学的所有分支中。另一方面，冯·诺伊曼代数是由默里和冯·诺伊曼在1930年的S和1940年的S中发明的；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，正是量子力学的形式主义激发了他们的发明。在现代数学中，对这些von Neumann代数的研究与其他几个数学领域有重要的联系，例如群论和动力系统。研究人员将试图回答群论和冯·诺依曼代数中相关的基本问题，即“某些无限对象总是可以被有限对象近似吗？”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相关的许多其他公开问题有关。研究者还将研究一些关于von Neumann代数的其他自然问题。研究工作将进一步支持首席研究人员的教育努力：最直接的是教授研究生理解当前研究问题的先决条件，然后引导他们解决至关重要的研究问题，并以一种展示数学的实用和美丽的方式教育本科生，教给他们基本的数学技能和数学(和逻辑推理)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.