Mathematical Sciences: C*-Algebras and the Geometry of Hilbert Space

数学科学：C*-代数和希尔伯特空间几何

• 批准号: 8801164
• 负责人: Charles Akemann
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1991-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801164&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Algebras; Geometry; Hilbert

This is mathematical research in pursuit of two famous open problems having to do with pure states on operator algebras. The term state comes from quantum mechanics. In some formulations thereof, observables are represented by selfadjoint operators in an appropriate algebra of operators, and states are certain linear functionals on the operator algebra; thus, specifying a state assigns a numerical value to each of the observables. The pure states are those not given as convex combinations of other states. In principal, they determine everything about the algebra, but pinning down the connection more precisely has led to daunting difficulties. One of the problems the present project will address is that of extending the Stone-Weierstrass theorem, which may be thought of as a result about commutative operator algebras, to the general, noncommutative situation. The other problem is to show the uniqueness of pure state extensions from maximal diagonal subalgebras of the algebra of all bounded operators on a Hilbert space.

这是数学研究追求两个著名的开放问题必须做纯状态的算子代数。态这个术语来自量子力学。在其某些公式中，观测量由适当的算子代数中的自伴算子表示，状态是算子代数上的某些线性泛函;因此，指定一个状态会为每个观测量分配一个数值。纯态是那些不作为其他态的凸组合给出的态。原则上，它们决定了关于代数的一切，但更精确地确定这种联系却导致了令人生畏的困难。本项目将解决的问题之一是，延长斯通-维尔斯特拉斯定理，这可能被认为是交换算子代数的结果，一般的，非交换的情况。另一个问题是证明Hilbert空间上所有有界算子代数的极大对角子代数的纯状态扩张的唯一性。