Noncommutative Harmonic Analysis, Operator Algebras, and Interpolation

非交换调和分析、算子代数和插值

• 批准号: 0098157
• 负责人: Gelu Popescu
• 金额: $ 7.54万
• 依托单位: University of Texas at San Antonio
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098157&HistoricalAwards=false
• 关键词: Noncommutative; Harmonic; Analysis; Operator; Algebras

AbstractPopescuThe proposed research considers problems in noncommutative harmonic analysis, operator algebras, and interpolation in several variables. The framework of this proposal is mainly the full Fock space, certain noncommutative (resp. commutative) analytic Toeplitz algebras, and the algebra of all bounded linear operators on a Hilbert space. Noncommutative dilation theory, Poisson transforms on $C^*$-algebras generated by isometries, and commutant lifting theorems are considered in order to find noncommutative (resp. commutative) multivariable analogues to some classical results. The main directions of this proposed research are the following: harmonic analysis on Fock spaces; power bounded sequences of operators, structure, and numerical invariants; central intertwining lifting, suboptimization, and analytic interpolation in several variables; dilation theory for tuples of operators (noncontractions) and non-analytic interpolation in several variables. The motivation of this research is the recent worldwide interest in the noncommutative aspect of harmonic analysis originated from the concept of quantization which links together several branches of mathematics and is closely related to mathematical physics. The objective of this research is to advance the understanding of the relatively new area of multivariable operator theory and apply some of these results to the study of completely positive maps and their invariants, function theory and interpolation in several variables, multivariable linear systems, scattering, control theory, and model theory for tuples of operators. AbstractPopescuThe proposed research considers problems in noncommutative harmonic analysis, operator algebras, and interpolation in several variables. The framework of this proposal is mainly the full Fock space, certain noncommutative (resp. commutative) analytic Toeplitz algebras, and the algebra of all bounded linear operators on a Hilbert space. Noncommutative dilation theory, Poisson transforms on $C^*$-algebras generated by isometries, and commutant lifting theorems are considered in order to find noncommutative (resp. commutative) multivariable analogues to some classical results. The main directions of this proposed research are the following: harmonic analysis on Fock spaces; power bounded sequences of operators, structure, and numerical invariants; central intertwining lifting, suboptimization, and analytic interpolation in several variables; dilation theory for tuples of operators (noncontractions) and non-analytic interpolation in several variables. The motivation of this research is the recent worldwide interest in the noncommutative aspect of harmonic analysis originated from the concept of quantization which links together several branches of mathematics and is closely related to mathematical physics. The objective of this research is to advance the understanding of the relatively new area of multivariable operator theory and apply some of these results to the study of completely positive maps and their invariants, function theory and interpolation in several variables, multivariable linear systems, scattering, control theory, and model theory for tuples of operators.

该研究考虑了非对易调和分析、算子代数和多元插值法中的问题。该方案的框架主要是全Fock空间，一定的非对易空间。可交换的)解析Toeplitz代数，以及Hilbert空间上所有有界线性算子的代数。非对易膨胀理论，由等距生成的$C^*-代数上的Poisson变换，以及交换提升定理被考虑以求出非对易的。交换的)多变量类似于一些经典的结果。本文的主要研究方向如下：Fock空间上的调和分析；算子、结构和数值不变量的幂有界序列；中心缠绕提升、次优化和多元解析内插；算子组(非压缩)的膨胀理论和多元非解析内插。这项研究的动机是最近全世界对调和分析的非对易方面的兴趣源于量子化的概念，量子化将数学的几个分支联系在一起，并与数学物理密切相关。这项研究的目的是促进对多变量算子理论这一相对较新的领域的理解，并将其中一些结果应用于完全正映射及其不变量、多元函数论和多元插值论、多变量线性系统、散射、控制理论和算子元组的模型理论的研究。该研究考虑了非对易调和分析、算子代数和多元插值法中的问题。该方案的框架主要是全Fock空间，一定的非对易空间。可交换的)解析Toeplitz代数，以及Hilbert空间上所有有界线性算子的代数。非对易膨胀理论，由等距生成的$C^*-代数上的Poisson变换，以及交换提升定理被考虑以求出非对易的。交换的)多变量类似于一些经典的结果。本文的主要研究方向如下：Fock空间上的调和分析；算子、结构和数值不变量的幂有界序列；中心缠绕提升、次优化和多元解析内插；算子组(非压缩)的膨胀理论和多元非解析内插。这项研究的动机是最近全世界对调和分析的非对易方面的兴趣源于量子化的概念，量子化将数学的几个分支联系在一起，并与数学物理密切相关。这项研究的目的是促进对多变量算子理论这一相对较新的领域的理解，并将其中一些结果应用于完全正映射及其不变量、多元函数论和多元插值论、多变量线性系统、散射、控制理论和算子元组的模型理论的研究。