Operator Algebras and Representations

算子代数和表示

• 批准号: 9700130
• 负责人: Palle Jorgensen
• 金额: $ 9.6万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-01 至 2001-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700130&HistoricalAwards=false
• 关键词: Operator; Algebras; Representations

Abstract Jorgensen Spectpal pairs were studied first by Palle E.T. Jorgensen and Steen Pedersen in connection with their work on the Fuglede conjecture, but they turned out later to also have connections to wavelet theory, fractal iterations, Julia sets, and quasi-crystals. The harmonic analysis which is needed will be project 1 of the proposal, and it is based on analysis of certain representations of the Cuntz algebras, and on endomorphisms of von Neumann algebras. This representation theory is further expected to be useful in connection with the study of concrete geometric tiling questions. But the representations needed are type III (generically), while some of the methods from earlier work are type I. Nonetheless they have already proved amenable for adaptation to the infinite case. Jorgensen also plans to collaborate with his colleague Florin Radulescu on dynamical systems of endomorphisms which arise in Berezin quantization. The index theory which is available for the time-independent systems is likely to help us understand the time-dependent case which is the result of the operator algebraic approach to Berezin quantization. The proposal is in the interface of basic mathematics with its technology applications. The latter refer to mathematical tools from harmonic analysis and wavelet theory which are applied in current data-compression algorithms. These application spin-offs include "fingerprint electronic storage" and high-resolution television; but the proposal is on the theoretical side of the equation. There are two basic operations from mathematics (translation and scaling) which go into the filters that govern algorithms, and the repetition in the recursive process refers to the preservation of the same minimal data from one step to the next. Hence the term "filter bank". Surprisingly, and noted by the proposer in recent research papers, these filter-constructions may be understood as representations of the kinds of algebras which form the basis for the proposed researc h.

摘要 Jorgensen Spectpal 对首先由 Palle E.T. 研究。 Jorgensen 和 Steen Pedersen 与他们关于 Fuglede 猜想的工作有关，但后来证明他们也与小波理论、分形迭代、Julia 集和准晶体有关。所需的调和分析将是该提案的项目 1，它基于对 Cuntz 代数的某些表示以及冯诺依曼代数的自同态的分析。这种表示理论有望在具体几何平铺问题的研究中发挥作用。但所需的表示形式是 III 型（一般而言），而早期工作中的一些方法是 I 型。尽管如此，它们已经被证明适合适应无限情况。 Jorgensen 还计划与他的同事 Florin Radulescu 合作研究别雷津量子化中出现的自同态动力系统。适用于时间无关系统的指数理论可能会帮助我们理解时间相关的情况，这是 Berezin 量化的算子代数方法的结果。 该提案涉及基础数学与其技术应用的接口。 后者指的是当前数据压缩算法中应用的调和分析和小波理论的数学工具。 这些应用副产品包括“指纹电子存储”和高分辨率电视；但该提议是在等式的理论方面。 数学中有两个基本操作（平移和缩放）进入控制算法的过滤器，递归过程中的重复是指从一步到下一步保存相同的最小数据。 因此术语“滤波器组”。令人惊讶的是，提议者在最近的研究论文中指出，这些过滤器结构可以被理解为构成所提议研究基础的代数类型的表示。