Noncommutative function theory in operator algebras and operator spaces

主要研究者提出了两条主要的研究路线，集中在新的方向，并围绕基础问题，在相对较新的领域的运营商空间。 这些行是：1）“非交换泛函分析和非交换Banach函数理论”，这部分地延续了研究者从经典泛函分析和C*-代数和冯诺依曼代数理论到“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，扩展到一般的算子代数的重要的新概念的等价性，吸引了广泛的关注，在该领域。研究者和合作者也提出研究上述的应用。算子代数的研究最初起源于量子力学。当这些变量被允许作为运算符变量时，了解涉及数值变量的公式的行为通常至关重要。由于算子变量不交换，这通常被称为“非交换数学”。正是在这样一个过程中，算子空间理论和完全有界映射出现了。 这一理论，其中调查人员帮助发现，是一种新颖的和令人信服的方法来解决问题，涉及线性分析中出现的“非交换矩阵”。调查员的研究重点部分转移的重要思想和工具，从经典的线性分析的子领域，通过这样的“量化”，以解决重大问题的“非交换”数学。大多数项目涉及来自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要求极高的分析。 成功的发现将使几个学科更紧密地联系在一起，吸引不同背景的科学家，并导致这些学科之间的交叉。

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.