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C*-Envelopes and Other Themes in Operator Algebras

C*-包络线和算子代数中的其他主题

基本信息

批准号：
2054781
负责人：
Elias Katsoulis
金额：
$ 13.44万
依托单位：
East Carolina University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-15 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054781&HistoricalAwards=false
关键词：
Envelopes Other Themes Operator Algebras

项目摘要

This research project seeks to resolve various problems in operator algebra theory, an area of mathematical analysis. The general study of operator algebras began in the 1930’s in an effort to provide a mathematical formulation of particle physics and quantum mechanics, notably to model algebras of physical observables. Since then, this vibrant area of mathematics has become an independent discipline bringing valuable insight to other areas of mathematics (knot theory, dynamical systems, group theory), physics (statistical mechanics) and engineering (signal processing). The award will contribute to US workforce development through training of students at East Carolina University. There are three major areas of study in this project: the C*-envelope of an operator algebra, the structure theory of crossed products and isomorphism invariants for operator algebras. The methods employed in the study of these areas are that of Functional Analysis with a strong emphasis on modern dilation theory and representation theory of associative algebras. A particular feature of the project is the use, for the first time, of non-selfadjoint techniques in an attempt to resolve issues in the selfadjoint theory, including the Hao-Ng isomorphism problem and the description of co-universal C*-algebras for covariant representations of product systems. Conversely, this project seeks to incorporate in the toolkit of non-selfadjoint operator algebraists techniques which are standard in the selfadjoint world. For instance, classifying Arveson’s semicrossed products up to stable isomorphism relates directly to the problem of classification of crossed products via a novel use of non-selfadjoint Takai duality. Such a synergy has not been explored before and seems to open exciting possibilities for both the selfadjoint and non-selfadjoint theory of operator algebras. Towards this end, the PI intends to continue organizing seminars and workshops aimed at further facilitating collaboration between experts in these two fields while at the same time promoting the exposure of students and early-career mathematicians to this promising area of mathematics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本研究项目旨在解决数学分析领域算子代数理论中的各种问题。算子代数的一般研究开始于20世纪30年代，目的是为粒子物理学和量子力学提供一个数学公式，特别是为物理观测量的代数建模。从那时起，这个充满活力的数学领域已经成为一个独立的学科，为数学（纽结理论，动力系统，群论），物理（统计力学）和工程（信号处理）的其他领域带来了有价值的见解。该奖项将通过培训东卡罗莱纳大学的学生，为美国劳动力发展做出贡献。本计画主要研究三个领域：算子代数的C*-包络、交叉积的结构理论及算子代数的同构不变量。在这些领域的研究所采用的方法是，功能分析与大力强调现代膨胀理论和代表性理论的结合代数。该项目的一个特点是首次使用非自伴技术来解决自伴理论中的问题，包括Hao-Ng同构问题和乘积系统协变表示的协泛C*-代数的描述。相反，该项目旨在纳入非自伴算子代数学家技术的工具包中，这些技术在自伴世界中是标准的。例如，分类Arveson的semicrossed产品稳定同构直接涉及到问题的分类交叉产品通过一个新的使用非自伴高井对偶。这样的协同作用还没有被探索过，似乎为算子代数的自伴和非自伴理论开辟了令人兴奋的可能性。为此，PI打算继续组织研讨会和讲习班，旨在进一步促进这两个领域的专家之间的合作，同时促进学生和早期职业数学家接触这一有前途的数学领域。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。