Mathematical Sciences: Operator Algebras

算子代数是现代分析中的一门核心学科，具有重要的历史和现实应用。它出现在本世纪初的几十年，与建立量子力学的数学框架有关。这项开创性的工作，与冯·诺伊曼和默里的名字联系在一起，被称为概率论和测度论的非交换模拟，处理算子的自伴随代数。近年来，由于在物理、工程和算符理论中的应用，数学家们把注意力转向了非自伴随算子代数理论。这方面的研究还处于起步阶段，而这类代数的结构理论正在被深入研究。皮茨教授是一位年轻的数学分析师，他的研究生涯有了一个惊人的开始。他最近完成了他的博士论文，关于某些类型的非自伴随算子代数的结构，称为巢代数，研究它们作为冯·诺伊曼（自伴随）代数的子代数。在这项工作中，他得到了关于这些代数的分解问题和与K理论的联系的几个新结果。在目前的提案中，Pitts教授计划继续他成功的研究活动，通过研究巢代数中的相似性和连通性问题的困难和长期悬而未决的问题。

Operator algebras is a central discipline in Modern Analysis which has important historical and current applications. It arose in the early decades of this century in connection with establishing a mathematical framework for quantum mechanics. This pioneering work, associated with the names of von Neumann and Murray, and referred to as the noncommutative analog of probability theory and of measure theory, treated selfadjoint algebras of operators. In recent years, again in connection with applications in physics as well as in engineering and operator theory, mathematicians have turned their attention to the theory of non-selfadjoint operator algebras. This study is in its infancy, and the structure theory of such algebras is being intensively investigated. Professor Pitts is a young mathematical analyst whose research career is off to a stunning start. He recently completed his doctoral dissertation on the structure of certain types of non-selfadjoint operator algebras called nest algebras, studying them as subalgebras of von Neumann (selfadjoint) algebras. In this work he obtained several new results on factorization problems and connections with K- theory of these algebras. In the current proposal, Professor Pitts plans to continue his successful research activity by investigating difficult and long outstanding problems concerning similarity and connectedness questions in nest algebras.