Mathematical Sciences: Studies in Semigroups of Endomorphisms, Operator Algebras, and Numerical Quantum Mechanics

数学科学：自同态半群、算子代数和数值量子力学的研究

• 批准号: 9212893
• 负责人: William Arveson
• 金额: $ 8.85万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-07-01 至 1995-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9212893&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Studies; Semigroups; Endomorphisms

Arveson will investigate several areas of non-commutative analysis involving von Neumann algebras and C*-algebras, quantized index theory, and the role of the non-commutative spheres of Bratteli, Elliott, Evans and Kishimoto in numerical quantum mechanics. The research will involve the index theory and classification theory of semigroups of endomorphisms of factors, the structure of the C*-algebras associated to such semigroups, and the properties of C*-algebras associated with the discretized canonical commutation relations. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.

Arveson将研究非交换分析的几个领域，包括von Neumann代数和C*-代数，量子化指标理论，以及Bratteli， Elliott， Evans和Kishimoto的非交换球在数值量子力学中的作用。本研究将涉及因子自同态半群的指标理论和分类理论，与半群相关的C*-代数的结构，以及与离散正则交换关系相关的C*-代数的性质。这个项目的一般数学领域在希尔伯特空间算子的代数理论中有其基础。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些抽象对象有各种各样的应用。例如，它们在结理论中发挥着关键作用，而结理论目前正被用于研究DNA的结构，它们在非交换几何中具有重要的基础意义，而非交换几何在物理学中正变得越来越重要。