Operator Algebras, Modules and Completely Bounded Maps

Paulsen/Blecher主要研究人员提出了与完全有界映射理论相关的几个主要研究方向。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空间作为算子空间的实现有关的问题。Paulsen将通过将完全有界Hochschild上同调表示为完全有界相对Yoneda上同调，多项式有界算子的一些问题，以及插值理论中的一些问题，继续研究完全有界Hochschild上同调。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“可观测”集合被描述为一个算子代数。由于这个原因，当允许数值变量作为运算符变量时，了解涉及数值变量的公式的行为通常很重要。这个过程通常被称为寻找旧理论的“量子化”版本，正是在这个过程中，完全有界映射理论得以发展。Blecher和Paulsen的研究主要集中在各种理论在这种量子化下的表现。另一方面，插补理论最初是一个纯粹的数学练习，直到最近20年才被发现在工程上有重要的应用。例如，在电路设计中，人们从几个给定频率的期望频率响应开始，并希望用该给定响应设计最简单的电路。在数学上，这个问题变成了寻找给定类型的最简单的函数，它在给定的点上达到给定的值。最后一个问题我们称之为插值问题。电气工程的需求已经超越了已知的插值理论。令人惊讶的是，插值理论和算子代数的研究是相互交织的，这种相互作用导致了一些新的插值结果。我们已经发现，更好地理解“量子化”，即矩阵值插值，是回答许多普通插值问题所需要的。

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.