Applications of operator algebra theory to certain problems in analysis

算子代数理论在某些分析问题中的应用

• 批准号: 0901457
• 负责人: Marius Junge
• 金额: $ 31.5万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901457&HistoricalAwards=false
• 关键词: Applications; operator; algebra; theory; certain

AbstractJungeThe aim of this proposal is to apply methods from the operator of algebras to problems in noncommutative analysis. This includes developing a theory of singular integrals and Fourier multipliers for noncommutative function spaces. The new insights gained from this work should also reflect back on the noncommutative spaces and their differential geometric properties. Another field of applications of tools from the theory of operator spaces lies in quantum information theory. Here violation of Bell inequalities and quantum error correction seems to relate well to the theory of cb-maps from operator space theory.In real life the order in which certain operations are executed can make a big difference. For example first boiling water and the adding oil is very different from first boiling oil and then adding water. The mathematical community has now fully accepted that one should allow the main object of interest to be performed in a certain order. Noncommutative analysis is about functions and their properties in the realm of non-commuting variables. The most important example here are matrix-valued functions. The theory of classical analysis, such as Fourier analysis or harmonic analysis has a lot to say about functions, and as this proposal intends to show, also about matrix-valued functions. Another natural application of this circle of ideas (the mathematical theory called functional analysis) lies in quantum information theory and the theoretical analysis of channels. Channels are the operation performed signals used to transport data in quantum information theory.

本文的目的是将代数算子的方法应用于非交换分析问题。这包括发展非交换函数空间的奇异积分和傅立叶乘数理论。从这项工作中获得的新见解也应该反映在非对易空间及其微分几何性质上。算子空间理论工具的另一个应用领域是量子信息理论。在这里违反贝尔不等式和量子纠错似乎涉及到理论的cb-地图从算子空间theory.In真实的生活中的顺序，其中某些操作的执行可以有很大的不同。例如，先煮水，然后加入油与先煮油，然后加水是非常不同的。数学界现在已经完全接受，人们应该允许主要的感兴趣的对象以一定的顺序进行。非交换分析是关于非交换变量领域中的函数及其性质。这里最重要的例子是矩阵值函数。经典分析的理论，如傅立叶分析或调和分析，有很多关于函数的内容，正如本提案所要展示的，也有关于矩阵值函数的内容。这个思想循环（称为泛函分析的数学理论）的另一个自然应用在于量子信息理论和通道的理论分析。信道是量子信息理论中用于传输数据的信号执行的操作。