Representations of the Polydisc Algebra

摘要Ferguson利用上同调工具，结合算子理论和算子代数的方法，研究了Hilbert空间上多盘代数的表示。在有界上同调的一般框架下，可以表述和研究膨胀理论中的问题以及交换n元算子的联合相似问题。上同群可以具体地实现为有界算子的商，因此这些群，以及用于计算它们的技术，应该引起算子理论家和算子代数家的兴趣。一般来说，Banach模的上同调理论的一个重要部分不适用于算子代数上Hilbert模的研究。因此，没有标准的技术可以用来计算上同群。迄今为止所采用的方法包括同调代数和算子理论技术，这两个看似不同的数学领域。因此，这些群的计算导致了算子理论的新见解和新技术。同样重要的是，当在有界上同调的一般背景下表述算子理论中的某些问题时，它们变得更加透明，简单的代数计算通常会导致问题的显著减少。仅仅因为这个原因，Hilbert模的有界上同调很可能成为那些研究算子理论的人的有力工具。

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.