Noncommutative function theory in operator algebras and operator spaces

在相对较新的算子空间领域，首席研究员提出了两条主要的研究路线，专注于新的方向和基本问题。这些内容是：1)《非对易泛函分析和非对易Banach函数论》，部分地延续了研究人员从经典泛函分析、C*-代数和von Neumann代数理论到《非对易泛函分析》的强大工具的介绍和应用；2)算子代数的一般理论和结构，例如推广到一般算子代数，作为在该领域引起广泛关注的重要的新的等价概念。研究人员和合作者还建议研究上述方法的应用。算子代数的研究最初起源于量子力学。当涉及数值变量的公式被允许作为运算符变量时，了解这些变量的行为通常至关重要。由于算子变量是不可交换的，这通常被称为“非对易数学”。正是在这个过程中，算子空间和完全有界映射理论应运而生。这一理论是研究人员帮助建立的，对于涉及线性分析的问题，它是一种新颖而令人信服的方法，这些问题出现在“非对易数学”中。这位研究者的研究部分集中在通过这种“量化”将重要的思想和工具从线性分析的经典子领域转移到解决“非对易”数学中的重要问题上。大多数项目涉及几个不同研究领域的想法的深度统一；而且往往需要极其深入和技术要求高的分析。成功的发现将使几个学科更紧密地联系在一起，吸引具有不同背景的科学家，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.