Operator Algebras, Modules and Completely Bounded Maps

主要研究人员就与完全有界映射理论相关的主题提出了几条主要的研究路线。Blecher将学习算子代数和算子代数上的模的一般理论，Hilbert C*-模，以及与Banach空间实现为算子空间有关的问题。Paulsen将继续研究完全有界的Hochschild上同调，通过将其表示为完全有界的相对Yoneda上同调，多项式有界算子的一些问题，以及内插理论中的问题。对算子代数的研究最初起源于量子力学。量子力学系统中的一组“可观测”被描述为一个算符代数。因此，查看涉及数值变量的公式在这些变量被允许作为运算符变量时的行为通常很重要。这个过程通常被称为寻找旧理论的“量化”版本，正是在这个过程中，完全有界映射理论发展起来。Blecher和Paulsen的研究主要集中在各种理论在这种量子化下如何表现的问题上。另一方面，内插理论最初是一种纯粹的数学练习，直到最近20年才发现它在工程上有重要的应用。例如，在电路设计中，对于几个给定的频率，人们从期望的频率响应开始，并希望设计具有该给定响应的最简单的电路。从数学上讲，这个问题变成了寻找在给定点达到某些给定值的给定类型的最简单函数的问题。最后一个问题就是我们所说的内插问题。电气工程的需求已经使我们超越了已知的插补理论。令人惊讶的是，插补理论和算子代数的研究是交织在一起的，这种相互作用导致了一些新的插补结果。我们已经发现，更好地理解“量化”即矩阵值插值是回答许多普通插值问题所必需的。

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.