Operator Algebras, Modules and Completely Bounded Maps

算子代数、模和全有界图

• 批准号: 9706996
• 负责人: Vern Paulsen
• 金额: $ 21.38万
• 依托单位: University of Houston
• 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: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706996&HistoricalAwards=false
• 关键词: Operator; Algebras; Modules; Completely; Bounded

Abstract Paulsen/Blecher The Principal Investigators propose several main lines of research on topics related to the theory of completely bounded maps. Blecher will be studying the general theory of operator algebras and modules over operator algebras, Hilbert C*-modules, and questions relating to realizations of Banach spaces as operator spaces. Paulsen will continue to study the completely bounded Hochschild cohomology through its presentation as a completely bounded relative Yoneda cohomology, some problems concerning polynomially bounded operators, and questions in interpolation theory. The study of operator algebras originally grew out of quantum mechanics. The set of "observables" in a quantum mechanical system is described as an algebra of operators. For this reason, it is often important to see how formulas involving numerical variables behave when these variables are allowed to be operator variables. This process is often referred to as finding "quantized" versions of the old theories and it was out of this process that the theory of completely bounded maps grew. Blecher and Paulsen's research focuses mainly on questions of how various theories behave under this type of quantization. On the other hand, interpolation theory started as a purely mathematical exercise, and only in the past 20 years has it been found to have important applications in engineering. For example, in electrical circuit design, one starts with a desired frequency response, for a few given frequencies, and wishes to design the simplest circuit with that given response. Mathematically, this problem becomes one of finding the simplest function of a given type that achieves some given values at given points. This last problem is what we call an interpolation problem. Already the demands of electrical engineering take us beyond the known interpolation theories. Surprisingly, interpolation theory and the study of operator algebras is interwoven, and this interplay has lead to some new interpolation results. We ha ve found that a better understanding of the "quantized", i.e., matrix-valued, interpolation is what is needed to answer many ordinary interpolation questions.

摘要Paulsen/Blecher首席研究人员提出了有关与完全有限的地图理论相关的主题的几个主要研究行。 Blecher将研究运营商代数和模块的一般理论，对操作员代数，Hilbert C* - 模型以及与Banach空间作为操作员空间的实现有关的问题。 Paulsen将通过将其作为完全有限的相对Yoneda的同谋的演示来研究完全有限的Hochschild共同体，有关多项式限制的操作员的一些问题以及插值理论中的问题。 最初对操作员代数的研究是由量子力学生长的。量子机械系统中的一组“可观察力”被描述为操作员的代数。因此，通常重要的是要查看涉及数值变量的公式如何在允许这些变量为操作器变量时表现。这个过程通常被称为发现旧理论的“量化”版本，而完全有限的地图理论越来越多。 Blecher和Paulsen的研究主要关注有关各种理论在这种类型的量化下的行为方式的问题。另一方面，插值理论始于纯粹的数学练习，仅在过去的20年中，才发现它在工程学中具有重要的应用。例如，在电路设计中，一个从所需的频率响应开始，以一些给定的频率开始，并希望通过给定响应设计最简单的电路。从数学上讲，此问题成为找到给定类型的最简单功能之一，该功能在给定点处实现了一些给定的值。最后一个问题是我们称之为插值问题。电气工程的需求已经使我们超越了已知的插值理论。令人惊讶的是，插值理论和操作员代数的研究是交织的，这种相互作用导致了一些新的插值结果。我们发现，对“量化”的更好理解，即矩阵值，插值是回答许多普通插值问题所需的。