Mathematical Sciences: Semigroup of Endomorphisms of Operator Algebras

数学科学：算子代数自同态半群

• 批准号: 9500291
• 负责人: William Arveson
• 金额: $ 10.5万
• 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: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500291&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Semigroup; Endomorphisms; Operator

9500291 Arveson The research proposed here addresses problems related to the structure and classification of semigroups of automorphisms of operator algebras and their relationship to several areas of modern analysis, quantum physics, and probability theory. A principal component of the project is to establish a connection between certain semigroups and continuous tensor products of Hilbert spaces. more precisely, the classification of E_0-semigroups up to cocycle conjugacy was shown to be equivalent to the problem of classifying these continuous tensor product systems. In turn, the latter structures give rise to computable invariants of various types: numerical (an index invariant), topological (a non-commutative topological space, the spectral C*-algebra). Moreover, such continuous tensor products occur naturally in probability theory; for example, they can be associated with random distributions having stationary independent increments, and with other martingale-type processes as well. A secondary project is to continue the proposer's recent work on the role of operator algebras in the problem of computing the spectra of infinite dimensional self-adjoint operators. The general area of mathematics of this project has its basis in the theory of algebras of operators on Hilbert space. 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 seemingly 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. ***

小行星9500291 这里提出的研究解决了与算子代数的自同构半群的结构和分类及其与现代分析，量子物理和概率论的几个领域的关系有关的问题。该项目的一个主要组成部分是建立某些半群和希尔伯特空间的连续张量积之间的联系。更准确地说，证明了E_0-半群在上循环共轭下的分类等价于这些连续张量积系统的分类问题。反过来，后一种结构又产生了各种类型的可计算不变量：数值（指数不变量），拓扑（非交换拓扑空间，谱C*-代数）。此外，这种连续张量积在概率论中自然出现;例如，它们可以与具有平稳独立增量的随机分布以及其他鞅型过程相关联。第二个项目是继续提议者最近的工作中的作用，算子代数的问题，计算频谱的无限维自伴算子。 这个项目的一般数学领域有其基础的理论代数的运营商在希尔伯特空间。运算符可以被认为是复数的有限或无限矩阵。特殊类型的算子通常被放在一个代数中，自然称为算子代数。这些看似抽象的物体有着令人惊讶的多种应用。例如，它们在纽结理论中起着关键作用，而纽结理论目前正被用于研究DNA的结构。 ***