Analytic and Geometric Aspects of Operator Algebras

本文的主要研究者Anderson将继续发展有限von Neumann代数中n元自伴算子的几何谱理论。一个目标是使用这种新的方法，试图发展一种关于这类算子的联合谱的连贯理论。另一个目的是利用它加深我们对无理旋转代数和由自由群决定的von Neumann代数等算子代数的结构的理解。本课题主要研究算子代数的某些方面。事实证明，这些对象在研究各种各样的问题方面取得了丰硕成果，包括数学物理、量子力学和纽结理论等领域的问题。主要研究人员和他的同事发现了一种查看某些分析数据(矩阵的特征值及其无穷维类似物--算符的谱)的新方法。该项目的一个主要目标是解决一个难题(非通勤运营商的联合频谱)，到目前为止，它经受住了反复的猛烈攻击。众所周知，特征值和其他光谱数据对于理解工程和科学中的许多现实世界问题是必不可少的。因此，这个项目可能会产生一些方法来解决应用数学中的问题，而这些问题以前是无法获得的。

Abstract Anderson The principal investigator will continue to develop a geometric spectral theory for n-tuples of self-adjoint operators in finite von Neumann algebras. One goal is to use this new approach to try to develop a coherent theory of the joint spectrum of such operators. Another aim is to use it to deepen our understanding of the structure of such operator algebras as irrational rotation algebras and von Neumann algebras determined by free groups. This project is concerned with the study of certain aspects of operator algebras. These objects have proved fruitful in the study of a wide variety of questions, including problems in such fields as mathematical physics, quantum mechanics and knot theory. The principal investigator and his coworkers have discovered a new way to view certain analytic data (eigenvalues of matrices and their infinite dimensional analogues - spectra of operators). A major goal of the project is to solve a difficult problem (joint spectrum for non-commuting operators), which thus far has withstood repeated vigorous attacks. It is well known that eigenvalues and other spectral data are essential in understanding many real world problems in engineering and science. Thus, it is possible that this project could produce methods for attacking questions in applied mathematics that have not been previously accessible.