Applications of operator algebra theory to certain problems in analysis

摘要这个建议的目的是将代数算子的方法应用于非对易分析中的问题。这包括发展非对易函数空间的奇异积分和傅立叶乘子理论。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，贝尔不等式的违反和量子纠错似乎与算子空间理论中的CB映射理论有很好的联系。在现实生活中，某些操作的执行顺序可能会产生很大的不同。例如，先烧开水和加油与先烧油再加水有很大不同。数学界现在已经完全接受了这样一种观点，即人们应该允许主要感兴趣的对象以特定的顺序执行。非对易分析是关于函数及其在非对易变量领域的性质的分析。这里最重要的例子是矩阵值函数。经典分析理论，如傅立叶分析或调和分析，有很多关于函数的内容，正如本提案想要展示的那样，也关于矩阵值函数。这一思想圈(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渠道的理论分析。在量子信息论中，通道是用来传输数据的操作信号。

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.