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.

摘要 Paulsen/Blecher/Papadakis 首席研究员提出了与算子代数、算子空间和框架的理论和应用相关的几条研究主线。 Blecher 将研究算子代数和算子代数模的一般理论、希尔伯特 C* 模以及与非交换 Choquet 理论相关的问题。 Paulsen 将继续从算子代数的角度研究单射算子空间和模、弱期望性质、函数论算子理论、插值理论。他还将与 Papadakis 一起研究框架和重构，以应用再现核希尔伯特空间方法和对称方法正交化结果。算子代数的研究最初源于量子力学。了解涉及数值变量的公式在允许这些变量为算子变量时如何表现通常很重要。正是在这样的过程中，算子空间和全有界映射理论应运而生。 Blecher 和 Paulsen 的研究主要集中于各种理论在这种“量子化”下如何表现的问题。另一方面，插值理论一开始只是一种纯粹的数学练习，直到过去 20 年才发现它在工程中具有重要的应用。例如，在电路设计中，人们从一些给定频率的所需频率响应开始，并希望用该给定响应设计最简单的电路。从数学上讲，这个问题变成了寻找给定类型的最简单函数，该函数在给定点达到某些给定值。最后一个问题就是我们所说的插值问题。电气工程的要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，回答许多普通插值问题需要更好地理解“量化”（即矩阵值插值）。框架理论可以应用于研究如何从数据流中提取信息，以及如何从数据流中提取信息。从导出的信息重建原始数据。这种情况的一个典型例子是 CAT 扫描，人们试图从大量数据中重建身体内部的图像。我们的工作并不关注特定的例子，而是关注如何分析特定框架的“好”程度。