Geometric Structure of Operator Spaces

算子空间的几何结构

• 批准号: 0500957
• 负责人: Timur Oikhberg
• 金额: --
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500957&HistoricalAwards=false
• 关键词: Geometric; Structure; Operator; Spaces

The current project aims to investigate geometric properties of operator spaces, with the emphasis on two topics: the existence of operator spaces with prescribed properties, and the structure of the Fermion algebra and spaces related to it. The problem of constructing spaces with given properties was first posed by Grothendieck, and later attracted attention of many accomplished mathematicians, such as Gowers and Maurey. However, many questions in this area remain open, chief among them the so-called ``square-cube problem'': the existence of a space which is not isomorphic to its square, but such that its square is isomorphic to its cube. We plan to approach this class of problems using operator space ``building blocks''. In investigating the Fermion algebra, we plan to discover what properties it shares with spaces of functions. In particular, we will investigate the complemented and completely complemented subspaces of this algebra.One of the key advances in physics over the past century was the creation and development of quantum mechanics. The main mathematical tool of quantum mechanics is replacing scalars (numbers) by operators on a Hilbert space (they can be thought of as infinite matrices). Initially, the mathematicians and physicists investigated only single operators, but later the need to consider whole sets of operators became apparent. Research by von Neumann, Gelfand, and others led to the development of the theory of C*-algebras. More recently, operator spaces (also called ``non-commutative'' or ``quantized Banach spaces'') arose from the study of maps on C*-algebras. It turns out that operator spaces provide an appropriate framework for studying algebras of operators. In fact, several long-standing problems in operator theory have been solved using operator space techniques. The current project deals with two topics: the existence of operator spaces with prescribed properties; and an investigation of the operator space structure of classical spaces, such as the Fermion algebra (related to the behavior of sub-atomic particles such as protons or neutrons). If successful, this research will advance our understanding of operator spaces, and potentially, enhance our knowledge of physical phenomena.

当前项目旨在研究算子空间的几何性质，重点关注两个主题：具有规定性质的算子空间的存在性，以及费米子代数及其相关空间的结构。构造具有给定属性的空间的问题首先由格洛滕迪克提出，后来引起了许多有成就的数学家的关注，例如高尔斯和莫雷。然而，这一领域的许多问题仍然悬而未决，其中最主要的是所谓的“平方立方问题”：是否存在与其正方形同构的空间，但其正方形与其立方体同构。我们计划使用运算符空间“构建块”来解决此类问题。在研究费米子代数时，我们计划发现它与函数空间共有哪些属性。特别是，我们将研究该代数的补子空间和完全补子空间。上个世纪物理学的关键进展之一是量子力学的创建和发展。量子力学的主要数学工具是用希尔伯特空间上的运算符替换标量（数字）（它们可以被认为是无限矩阵）。最初，数学家和物理学家仅研究单个运算符，但后来考虑整组运算符的需要变得显而易见。冯·诺依曼、盖尔范德等人的研究促进了 C* 代数理论的发展。最近，算子空间（也称为“非交换”或“量化巴拿赫空间”）源于对 C* 代数映射的研究。事实证明，算子空间为研究算子代数提供了一个合适的框架。事实上，算子理论中的几个长期存在的问题已经通过算子空间技术得到了解决。当前的项目涉及两个主题：具有规定属性的算子空间的存在；以及对经典空间的算子空间结构的研究，例如费米子代数（与质子或中子等亚原子粒子的行为有关）。如果成功，这项研究将增进我们对算子空间的理解，并有可能增强我们对物理现象的了解。