Representations of the Polydisc Algebra

多圆盘代数的表示

• 批准号: 9706837
• 负责人: Sarah Ferguson
• 金额: $ 5.33万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 1999-05-10
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706837&HistoricalAwards=false
• 关键词: Representations; Polydisc; Algebra

Abstract Ferguson The proposed research is to study representations of the polydisc algebra on Hilbert space using cohomological tools together with techniques from operator theory and operator algebras. Problems in dilation theory, as well as, joint similarity problems for commuting N-tuples of operators, can be formulated and studied in the general framework of bounded cohomology. The cohomological groups can be realized concretely as quotients of bounded operators and thus these groups, as well as, the techniques used to compute them, should be of interest to both operator theorists and operator algebraists. A significant portion of the cohomology theory developed for Banach modules is not, in general, applicable in the study of Hilbert modules over operator algebras. Consequently, there are no standard techniques one can use to compute cohomology groups. The methods employed so far involve homological algebra together with operator theoretic techniques, two seemingly disparate areas of mathematics. Consequently, computations of these groups leads to new insight and new techniques in operator theory. Also important is that certain problems in operator theory when formulated in the general context of bounded cohomology become more transparent and simple algebraic computation often leads to a significant reduction in the problem. For this reason alone, bounded cohomolgy for Hilbert modules will likely become a powerful tool for those working in operator theory.

摘要Ferguson提出的研究是利用上同调工具，结合算子理论和算子代数的技巧，研究Hilbert空间上多圆盘代数的表示。膨胀理论中的问题，以及交换的N元组算子的联合相似问题，都可以在有界上同调的一般框架下表示和研究。上同调群可以具体地实现为有界算子的商，因此这些群以及计算它们的技巧应该是算子理论家和算子代数学家都感兴趣的。一般地，为Banach模发展的上同调理论的很大一部分不适用于算子代数上的Hilbert模的研究。因此，没有标准的技术可以用来计算上同调群。到目前为止，所使用的方法涉及同调代数和算符理论技术，这两个似乎完全不同的数学领域。因此，对这些群的计算为算子理论带来了新的见解和新的技术。同样重要的是，当算子理论中的某些问题在有界上同调的一般上下文中表述时，变得更加透明，简单的代数计算通常会导致问题的显着减少。仅因为这个原因，Hilbert模的有界上同调很可能成为从事算子理论工作的人的有力工具。