Operator Algebras, Operator Spaces, Frames and Applications

【摘要】paulsen /Blecher/ papadakis主要研究者提出了与算子代数、算子空间和框架的理论和应用相关的几个主要研究方向。Blecher将学习算子代数的一般理论和算子代数上的模，Hilbert C*模，以及与非交换Choquettheory有关的问题。Paulsen将继续从算子代数的角度研究内射算子空间和模，弱期望性质，函数理论算子理论，插值理论。与Papadakis一起，他还将研究框架和重建，以期应用再现核希尔伯特空间方法和对称正交化结果。对算子代数的研究最初起源于量子力学。当允许数值变量作为运算符变量时，了解包含数值变量的公式的行为通常是很重要的。正是在这样的过程中，算子空间理论和完全有界映射理论应运而生。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，插值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函数，它在给定的点上达到给定的某些值。最后一个问题我们称之为插值问题。电气工程的要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交织在一起的，这种相互作用导致了一些新的插值结果。我们发现，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。框架理论可以应用于研究我们如何从数据流中提取信息，以及如何从衍生信息中重构原始数据。出现这种情况的一个典型例子是CAT扫描，从大量数据中，人们试图重建身体内部的图像。我们的工作不是集中在特定的例子上，而是集中在如何分析一个特定的框架有多“好”。

ABSTRACTPaulsen/Blecher/PapadakisThe Principal Investigators propose several main lines of research ontopics related to the theory and applications of operator algebras,operator spaces, and frames. Blecher will be studying the generaltheory of operator algebras and modules over operator algebras,Hilbert C*-modules, and questions relating to noncommutative Choquettheory. Paulsen will continue to study injective operator spaces andmodules, the weak expectation property, function theoretic operatortheory, interpolation theory from an operator algebra point of view.With Papadakis he will also be studying and frames and reconstructionswith a view to applying reproducing kernel Hilbert space methods andsymmetric orthogonalization results.The study of operator algebras originally grew out of quantum mechanics.It is often important to see how formulas involving numerical variablesbehave when these variables are allowed to be operator variables. It isout of such a process that the theory of operator spaces and completelybounded maps emerged. Blecher and Paulsen's research focuses mainly onquestions of how various theories behave under this `quantization'.On the other hand, interpolation theory started as a purely mathematicalexercise, and only in the past 20 years has it been found to haveimportant applications in engineering. For example, in electrical circuitdesign, one starts with a desired frequency response, for a few givenfrequencies, and wishes to design the simplest circuit with that givenresponse. Mathematically, this problem becomes one of finding thesimplest function of a given type that achieves certain given values atgiven points. This last problem is what we call an interpolation problem.Already the demands of electrical engineering take us beyond the knowninterpolation theories. Surprisingly, interpolation theory and the studyof operator algebras is interwoven, and this interplay has lead to somenew interpolation results. We have found that a better understanding ofthe "quantized", i.e., matrix-valued, interpolation is what is neededto answer many ordinary interpolation questions.Frame theory can be applied to the study of how we extract informationout of streams of data, and how we reconstruct the original data fromthe derived information. A typical example of a situation where thisarises is the CAT scan, where from a large quantity of data, one istrying to reconstruct a picture of the inside of a body. Our work isnot focused on particular examples, but on how one analyzes how "good"is a particular frame.