Noncommutative function theory in operator algebras and operator spaces

首席研究员提出了两条主要的研究路线，围绕新的方向，围绕基本问题，在相对较新的算子空间领域。这些台词是：1)“非交换泛函分析和非交换巴拿赫函数理论”，它在一定程度上继续了研究者从经典泛函分析和C*代数和冯诺伊曼代数理论到“非交换泛函分析”的强大工具的介绍和应用，2)算子代数的一般理论和结构，例如将重要的新等价概念扩展到一般算子代数，这些概念在该领域引起了广泛的关注。研究者和合作者还建议对上述应用进行研究。对算子代数的研究最初起源于量子力学。当允许数值变量作为运算变量时，了解涉及数值变量的公式的行为通常是至关重要的。由于算子变量不交换，这通常被称为“非交换数学”。正是在这样的过程中，算子空间理论和完全有界映射出现了。研究者帮助发现的这一理论，是解决“非交换数学”中出现的涉及线性分析的问题的一种新颖而引人注目的方法。研究者的研究部分集中在通过“量化”从线性分析的经典子领域转移重要思想和工具，以解决“非交换”数学中的重大问题。大多数项目涉及几个不同研究领域的思想的深度统一；并且通常需要非常深入和技术上的分析。成功的发现将使几个学科更紧密地联系在一起，吸引不同背景的科学家，并导致这些学科之间的交叉受精。

The principal investigator proposes two main lines of research, focused around new directions, and around foundational problems, in the relatively new field of operator spaces. These lines are: 1) 'noncommutative functional analysis and noncommutative Banach function theory', which in part continues the investigators introduction, and application, of powerful tools from classical functional analysis and C*-algebra and von Neumann algebra theory to 'noncommutative functional analysis', 2) the general theory and structure of operator algebras, for example extending to general operator algebras important new notions of equivalence that are attracting widespread attention in the field. The investigator and collaborators are also proposing to investigate applications of the above.The study of operator algebras originally grew out of quantum mechanics. It is often of crucial importance to see how formulas involving numerical variables behave when these variables are allowed to be operator variables. Because operator variables do not commute, this is often called 'noncommutative mathematics' It is out of such a process that the theory of operator spaces and completely bounded maps emerged. This theory, which the investigator helped to found, is a novel and compelling approach to problems involving linear analysis which arise in 'noncommutative mathematics'. The investigator's research focuses partly on transferring important ideas and tools from classical subfields of linear analysis, via such 'quantization', to solve significant problems in 'noncommutative' mathematics. Most of the projects involve a deep unification of ideas from several different research areas; and often require extremely deep and technically demanding analysis. Successful findings would bring several subjects closer together, attract scientists with disparate backgrounds, and lead to cross-fertilization between these disciplines.