Operator Spaces and Applications to Related Areas

操作员空间和相关领域的应用

• 批准号: 0500535
• 负责人: Zhong-Jin Ruan
• 金额: $ 22.41万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-03-01 至 2009-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500535&HistoricalAwards=false
• 关键词: Operator; Spaces; Applications; Related; Areas

The PI's research is mainly in the theory of operator spaces and its applications to operator algebras, non-commutative Lp-spaces and non-commutative harmonic analysis. During the last few years, the PI, together with some other mathematicians, has made some significant contributions to these areas. He has obtained a number of important results on the local (i.e. finite dimensional) operator space properties and related approximation properties of C*-algebras and non-commutative Lp-spaces. He has also obtained some interesting applications of operator spaces to non-commutative harmonic analysis.In this proposal, he plans to continue his research in these directions and proposes the following research projects: (1) investigate some problems on non-commutative Lp-spaces, (2)investigate some problems on group C*-algebras and Fourier algebras, (3)investigate the possible generalizations to Kac algebras and locally compact quantum groups; (4)investigate some related problems in non-commutative/free probability.The theory of operator spaces is a natural quantization of functional analysis, or more precisely, a natural quantization of Banach space theory. Operator spaces were first realized by William Arveson in 1969 and were abstractly characterized by the PI in his Ph.D thesis in 1987. Since then there have been some remarkable developments in operator spaces and the theory has been quickly developed into a very active research area in modern analysis. The projects proposed here contain some important questions in operator spaces, operator algebras, non-commutative/quantum harmonic analysis and non-commutative/free probability. The progress on these projects will have significant impact in these areas, as well as in some other related mathematics research areas such as quantum group theory, non-commutative geometry and geometric group theory. Projects in this proposal also provide the outstanding resources and research problems for the PI's Ph.D students and post-docs. The support for the Wabash Seminar and Miniconference is requested in this proposal. The Wabash Seminar, together with its annual Miniconference, is devoted to the stimulation and dissemination of significant contributions to analysis in the Midwest region. This has already provided (and will continuously provide) a unique opportunity for young researchers, visitors, post-docs and graduate students from the Midwest to meet regularly and exchange ideas with leading experts in the fields.

PI的研究主要集中在算子空间理论及其在算子代数、非对易Lp-空间和非对易调和分析中的应用。在过去的几年里，PI与其他一些数学家一起，在这些领域做出了一些重大贡献。他在C*-代数和非交换LP-空间的局部(即有限维)算子空间性质和相关的逼近性质方面得到了一些重要的结果。他还得到了算子空间在非对易调和分析中的一些有趣的应用。在这个建议中，他计划在这些方向上继续他的研究，并提出以下研究项目：(1)研究非对易LP-空间的一些问题；(2)研究群C*-代数和傅立叶代数的一些问题；(3)研究Kac代数和局部紧量子群的可能推广；(4)研究非对易/自由概率中的一些相关问题。算子空间理论是泛函分析的自然量子化，或者更准确地说，是Banach空间理论的自然量子化。算子空间最早是由William Arveson在1969年提出的，并在1987年的博士论文中被PI抽象地刻画出来。从那时起，算子空间有了一些显着的发展，该理论迅速发展成为现代分析中一个非常活跃的研究领域。本文提出的方案包含了算子空间、算子代数、非对易/量子调和分析和非对易/自由概率中的一些重要问题。这些项目的进展将对这些领域以及其他一些相关的数学研究领域，如量子群论、非对易几何和几何群论产生重大影响。该提案中的项目还为PI的博士生和博士后提供了突出的资源和研究问题。本提案要求支持瓦巴什研讨会和小型会议。瓦巴什研讨会及其年度小型会议致力于促进和传播对中西部地区分析的重大贡献。这已经并将继续为来自中西部的年轻研究人员、来访者、博士后和研究生提供一个独特的机会，与该领域的顶尖专家定期会面并交流意见。