Geometry of Operator Spaces

算子空间的几何

• 批准号: 9970369
• 负责人: Timur Oikhberg
• 金额: $ 6.24万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970369&HistoricalAwards=false
• 关键词: Geometry; Operator; Spaces

AbstractOikhbergThe PI plans to continue his work on several aspects of geometry of operator spaces. One line of work deals with separable extension properties. Here the most intriguing problem is whether K (the space of compact operators on a separable Hilbert space) is complemented in every separable operator space containing it (this is a "non-commutative" version of a classical result of Sobczyk). A related issue is giving a complete description of separable locally reflexive operator spaces which are completely complemented in every separable locally reflexive operator superspace (an operator space counterpart of Zippin's characterization of the space of convergent sequences). Another direction of research is the study of maximal operator spaces (quotients of duals of commutative C*-algebras). The PI plans to determine the cardinality of the set of n-dimensional subspaces of maximal spaces, as well as the structure of n-dimensional subspaces of the dual of a 2n-dimensional commutative C*-algebra (in the spirit of Kashin's work). Possible applications of the proposed research include better understanding of the structure of function spaces, of the space of operators on a Hilbert space, and Banach space geometry in general (extensions of local reflexivity).One of the key mathematical tools of quantum mechanics is replacing scalars (numbers) by operators on a Hilbert space (they can be thought of as infinite matrices). Further exploration in this direction led to the development of the theory of C*-algebras, and, later, to the introduction of the notion of operator space (also called "non-commutative" or "quantized Banach space"). One can view an operator space as a Banach space with additional structure, induced by its embedding into the C*-algebra B(H) of bounded linear operators on a Hilbert space H. The investigation of operator spaces has advanced very rapidly over the last ten years, combining ideas and techniques from Banach space theory and the theory of C*-algebras. It has already produced answers to some long-standing problems of Operator Theory. The proposed research deals with two themes: the existence of non-commutative analogs of classical Banach space results; and the use of Banach space methods in the operator space case. If successful, this research will advance our understanding of connections between Banach and operator spaces, and potentially, enhance our knowledge of physical phenomena.

Abstractoikhbergthe Pi计划继续在操作员空间几何方面的几个方面进行工作。一条工作与可分开的扩展特性有关。在这里，最有趣的问题是，在包含它的每个可分开的操作员空间中，K（可分离的希尔伯特空间中紧凑型运算符的空间）是否得到补充（这是Sobczyk的经典结果的“非交换性”版本）。 一个相关的问题是对可分开的本地反身操作员空间进行完整描述，这些空间在每个可分开的本地反身操作员超空空间中都完全补充（Zippin对收敛序列空间的表征的操作员空间对应物）。研究的另一个方向是对最大操作员空间的研究（可交换C* - 代数的双重的商）。 PI计划确定最大空间的N维子空间的基数，以及2n维交换性C*-Algebra（以Kashin的工作精神）的双重二维子空间的结构。拟议的研究的可能应用包括更好地理解功能空间的结构，希尔伯特空间上运营商的空间以及一般的Banach空间几何形状（局部反射性的扩展）。量子力学的关键数学工具之一是代替希尔伯特空间上的操作员（可以认为是无限制的量表）。朝这个方向进行进一步的探索导致了C* - 代数理论的发展，后来引入了操作员空间的概念（也称为“非交通”或“量化的Banach空间”）。 人们可以将操作员空间视为带有附加结构的BANACH空间，这是由于其嵌入Hilbert Space H上有界线性算子的C*-Algebra B（H）。对操作员空间的研究在过去十年中的研究非常迅速，结合了Banach Space理论和C*-Algebera的理论的思想和技术。它已经为操作者理论的一些长期问题提供了答案。拟议的研究涉及两个主题：古典Banach空间结果的非共同类似物的存在；以及在操作员空间案例中使用Banach空间方法。如果成功，这项研究将提高我们对Banach和运营商空间之间联系的理解，并可能增强我们对身体现象的了解。