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.

AbstractPaulsen/Blecher/Papadakis的主要研究人员提出了与操作员代数，操作员空间和框架的理论和应用有关的研究主体的几种主要研究线。 Blecher将研究操作员代数和模块的通用理论，对操作员代数，Hilbert C* - 模型以及与非交通性choquettheory有关的问题。 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从量子力学出发。通常，查看允许这些变量为操作变量时涉及数值变量的公式如何。这是一个过程，使操作员空间的理论和完全刺激的地图出现了。 Blecher和Paulsen的研究主要集中于各种理论在此“量化”下的行为方式。另一方面，插值理论始于一种纯粹的数学依据，并且仅在过去20年中才发现它在工程中具有重要的应用。例如，在电路设计中，一个从所需的频率响应开始，对于一些给出的频率，并希望使用iveresponse设计最简单的电路。从数学上讲，这个问题成为找到给定类型的函数的一个，该功能可实现某些给定值ATGING点。最后一个问题是我们所说的插值问题。准备好电气工程的需求使我们超越了已知的插入理论。令人惊讶的是，插值理论和操作员代数的研究是交织的，并且这种相互作用导致了一些插值结果。我们发现，对“量化”的更好理解，即矩阵值，插值是回答许多普通插值问题所需的内容。帧理论可以应用于研究我们如何从数据流中提取信息的信息以及我们如何从派生信息中重建原始数据。这种情况是猫扫描的一个典型例子，从大量数据中，一个人可以重建身体内部的图片。我们的工作不是专注于特定示例，而是一个人如何分析“好”是一个特定的框架。