Operator Algebras, Operator Spaces, Frames and Applications

算子代数、算子空间、框架和应用

• 批准号: 0070376
• 负责人: Vern Paulsen
• 金额: $ 17.4万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070376&HistoricalAwards=false
• 关键词: Operator; Algebras; Spaces; Frames; Applications

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.

首席研究人员提出了与算子代数、算子空间和框架的理论和应用有关的几条主要研究路线。Blecher将学习算子代数和算子代数上的模的一般理论，Hilbert C*-模，以及与非对易选择理论有关的问题。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、函数论算子理论、内插理论。他还将与Papadakis一起研究框架和重构，以期应用再生核Hilbert空间方法和对称正交化结果。算子代数的研究最初源于量子力学。当这些变量被允许作为算子变量时，了解涉及数值变量的公式是如何表现的是很重要的。正是在这样的过程中，算子空间和完备映射理论应运而生。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，并希望用该给定的响应来设计最简单的电路。从数学上讲，这个问题变成了寻找在给定点达到一定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插补问题。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，这种交织导致了一些新的插补结果。我们发现，要回答许多常见的内插问题，需要更好地理解“量化的”即矩阵值内插。框架理论可以应用于研究如何从数据流中提取信息，以及如何从导出的信息中重建原始数据。出现这种情况的一个典型例子是CAT扫描，从大量数据中，一个人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，而是研究如何分析“好”是一个特定的框架。