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.

当前的项目旨在研究操作员空间的几何特性，重点是两个主题：具有规定属性的操作员空间的存在，以及Fermion代数的结构以及与之相关的空间。 Grothendieck首先提出了具有给定特性的空间的问题，后来引起了许多成熟的数学家的关注，例如Gowers和Maurey。但是，该领域的许多问题仍然开放，其中包括所谓的``Square Cube问题''：存在与正方形同构的空间的存在，但其正方形对其立方体是同构的。我们计划使用操作员空间``构建块''解决此类问题。在研究费米昂代数时，我们计划发现其与功能空间共享的属性。特别是，我们将调查该代数的补充且完全补充的子空间。过去一个世纪的物理学的主要进步之一是量子力学的创造和发展。量子力学的主要数学工具是在希尔伯特空间上替换操作员的标量（数字）（可以将其视为无限矩阵）。最初，数学家和物理学家仅研究了单个操作员，但是后来考虑整个运营商的需求变得显而易见。冯·诺伊曼（von Neumann），盖凡德（Gelfand）和其他人的研究导致了C* - 代数理论的发展。最近，运营商空间（也称为``非交通''或``量化的banach space''）来自于对C*-Algebras上的地图的研究。事实证明，操作员空间为研究操作员代数提供了适当的框架。实际上，使用操作员空间技术解决了操作者理论中的一些长期存在问题。当前的项目涉及两个主题：具有规定属性的操作员空间的存在；以及对经典空间的操作员空间结构的研究，例如费米昂代数（与质子或中子等亚原子颗粒的行为有关）。如果成功，这项研究将提高我们对操作员空间的理解，并可能增强我们对身体现象的了解。