Operator Spaces and Applications to Related Areas

PI的研究主要集中在算子空间理论及其在算子代数、非交换Lp-空间和非交换调和分析中的应用。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。他获得了一些重要的结果，当地（即有限维）运营商空间的性质和相关的逼近性质的C*-代数和非交换Lp-空间。他还获得了算子空间在非对易调和分析中的一些有趣应用，在本提案中，他计划在这些方向继续研究，并提出以下研究项目：（1）研究了非交换Lp-空间上的一些问题，（2）研究了群C ~*-代数和Fourier代数上的一些问题，（3）研究Kac代数和局部紧量子群的可能推广;（4）研究非交换/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，Banach空间理论的自然量子化。 算子空间首先由William Arveson在1969年实现，并在1987年由PI在他的博士论文中抽象地描述。 从那时起，算子空间有了一些显着的发展，该理论已迅速发展成为现代分析中一个非常活跃的研究领域。本文所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。 本计划书中的项目也为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。该瓦巴什研讨会，连同其年度小型会议，致力于刺激和传播的重大贡献，分析在中西部地区。这已经为来自中西部的年轻研究人员、访问者、博士后和研究生提供了（并将继续提供）一个独特的机会，让他们定期与该领域的顶尖专家会面并交流想法。

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.