Operator algebras between theory and application

秩序很重要。在真实的生活和科学中，某些操作的执行顺序会极大地改变结果。然而，在通常的乘法数的顺序是无关紧要的。因为乘积AB与BA相同，所以可以说因子A和B是可交换的。受量子力学基本原理的启发，数学家们研究了一种尊重运算顺序的新型乘法。在上个世纪，这导致了数学理论中包含非对易性（即，允许AB不同于BA），例如非交换几何或量子（即，非交换的）概率。在这条研究线类似的程序适用于基本概念，在经典的调和分析，如傅立叶级数和估计解微分方程在非交换空间。令人惊讶的是，在这项研究中开发的抽象工具在其他学科中也很有用。首席研究员的研究将包括对量子信息理论中量子通道的透彻分析。量子信息理论的研究通常发生在计算机科学和计算机科学部门。然而，只要量子计算机还没有大量出现，量子计算机的局限性和优势就只能通过理论和数学工具来理解。这同样适用于通过使用量子力学传输信息的设备的能力。这项工作中的跨学科研究还将包括大数据和压缩传感的数学方面。这项研究的各个方面也将有助于增强大学的教学使命，特别是培养熟悉纯数学和某些应用的学生。算子代数理论提供了许多重要的工具，对于理解经典对象的非对易方面至关重要，例如布朗运动、导数和导子、切和余切空间、拉普拉斯-贝尔特拉米算子、奇异积分核，量子信道，量子信道容量。该项目旨在将完全正映射理论的理论方面与量子信息理论和调和分析中的更多应用方面联系起来，特别是具有几何或度量风味的算子的分析性质。三重张量范数的Grothendieck程序的拟议工作属于算子空间理论的核心主题，但也受到量子信息和压缩传感的启发。首席研究员先前与量子信息理论相关的研究已经证明了与计算机科学和物理学主题相关的潜力。拟议中的关于私人渠道容量的新研究甚至可能产生超越科学的影响。

Order matters. The order in which certain operations are performed can dramatically change the outcome in real life and in the sciences. However, in the usual multiplication of numbers the order is irrelevant. Since the product AB is the same as BA one says that the factors A and B commute. Inspired by the fundamentals of quantum mechanics, mathematicians have investigated a new type of multiplication that respects the order of operations. During the last century this has led to spectacular new discoveries in mathematical theories embracing noncommutativity (i.e., allowing AB to be different from BA) such as noncommutative geometry or quantum (i.e., noncommutative) probability. In this line of research a similar program is applied to fundamental concepts in classical harmonic analysis such as Fourier series and estimates for solving differential equations in noncommutative spaces. Quite surprisingly, the abstract tools developed in this investigation are also useful in other disciplines. The research of the principal investigator will include a thorough analysis of quantum channels in quantum information theory. Research in quantum information theory usually takes place in computer science and physic departments. However, as long as quantum computers are not available in large numbers, the limitations and advantages of quantum computers can be understood only using theoretical, mathematical tools. The same applies for the capacities of devices transmitting information through the use of quantum mechanics. Interdisciplinary research in this work will also include mathematical aspects of big data and compressed sensing. All aspects of this research will also serve to enhance the teaching mission of the university, and in particular the formation of students who are familiar with pure mathematics and certain applications alike.The theory of operator algebras provides many important tools that are essential in understanding noncommutative aspects of classical objects, such as Brownian motion, derivatives and derivations, tangent and cotangent spaces, Laplace-Beltrami operators, singular integral kernels, quantum channels, and capacity of quantum channels. The project will aim to connect theoretical aspects of the theory of completely positive maps with more applied aspects in quantum information theory and harmonic analysis, in particular those analytic properties of operators that have a geometric or metric flavor. The proposed work on the Grothendieck program for triple-tensor norms belongs to the core subject in operator space theory but is also motivated by quantum information and compressed sensing. Previous research of the principal investigator related to quantum information theory has already demonstrated the potential to connect to topics in computer science and physics. The proposed new research on private capacity of channels may even have an impact beyond science.