Operator Algebras and Representations

约根森光谱对最早是由Palle E.T.Jorgensen和Steen Pedersen在他们关于Fuglede猜想的工作中研究的，但后来发现它们也与小波理论、分形迭代、Julia集和准晶体有关。所需的调和分析将是提案的项目1，它基于对Cuntz代数的某些表示的分析，以及von Neumann代数的自同态。这一表示理论可望进一步用于研究具体的几何平铺问题。但所需要的表示是类型III(一般)，而早期工作中的一些方法是类型I。尽管如此，它们已经被证明是服从于适应无限情况的。约根森还计划与他的同事弗洛林·拉杜列斯库合作，研究贝雷津量子化中出现的自同态的动力学系统。指数理论适用于时间无关的系统，这可能有助于我们理解Berezin量子化的算符代数方法的结果与时间相关的情况。该提案涉及基础数学及其技术应用的接口。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。这些应用衍生产品包括“指纹电子存储”和高分辨率电视；但这项提议还停留在理论层面。有两个数学上的基本运算(平移和缩放)进入管理算法的过滤器，递归过程中的重复指的是从一个步骤到下一个步骤保存相同的最小数据。因此有了“滤波器组”这一术语。令人惊讶的是，正如提出者在最近的研究论文中指出的那样，这些滤子结构可以被理解为构成所提议的研究的基础的各种代数的表示。

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.