RUI: Semigroups of endomorphisms on von Neumann algebras

RUI：冯·诺依曼代数上的自同态半群

• 批准号: 0100407
• 负责人: Geoffrey Price
• 金额: $ 7.56万
• 依托单位: United States Naval Academy
• 依托单位国家: 美国
• 项目类别: Interagency Agreement
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2004-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100407&HistoricalAwards=false
• 关键词: RUI; Semigroups; endomorphisms; von; Neumann

AbstractPrice This proposal is directed towards the investigation of some problems on dynamical systems on von Neumann algebras, and is related to recent work by a number of authors on the subject of semigroups of endomorphisms on von Neumann algebra factors. This research project involves a number of areas of mathematics, including operator algebras, finite fields, number theory, and combinatorics. In contrast to semigroups of automorphisms, a semigroup of endomorphisms may be viewed as a dynamical system that may proceed forward but not backward in time. It is remarkable how challenging this subject has proven to be in light of the relatively simpler theory of one-parameter semigroups of automorphisms on a type I factor. A principal goal of this proposal is to make additional progress in the classification of one-parameter semigroups of unital endomorphisms, analogous to Wigner's characterization of groups of automorphisms acting on factors of type I. The study of operator algebras traces its origins back to the work of von Neumann and others. Their goal was to construct mathematical models that capture the behavior of quantum mechanical systems, and to use these models to make predictions about the time evolution of such systems. As knowledge has grown and techniques in the field have been refined, connections have been established between operator algebras and a number of other areas of mathematics and science. This proposal involves connections among the fields of operator algebras, commutative algebra, number theory, and combinatorics. The principal objects of study in this project are known as binary shifts on a certain type of operator algebra and are defined using bitstreams of 0's and 1's such as one studies in the theory of linear recurring sequences. A major goal of this project is to complete the classification of the binary shifts and to relate this classification to the analysis of linear recurring sequences. Binary shifts will also be studied for their potential applications to the theory of quantum dynamical systems, specifically those systems that may proceed forward, but not backward, in time.

这个建议是为了研究von Neumann代数上的动力系统的一些问题，并与一些作者最近关于von Neumann代数因子上的自同态半群的工作有关。这项研究项目涉及多个数学领域，包括算子代数、有限域、数论和组合学。与自同构的半群不同，自同态的半群可以被看作是一个在时间上可以向前而不是向后的动力系统。值得注意的是，鉴于相对简单的关于类型I因子的单参数自同构半群的理论，这一主题已被证明是多么具有挑战性。这一建议的一个主要目的是在单位自同态的单参数半群的分类方面取得额外的进展，类似于Wigner对作用于第一型因子的自同构群的刻画。对算子代数的研究可以追溯到von Neumann等人的工作。他们的目标是构建捕捉量子力学系统行为的数学模型，并使用这些模型来预测此类系统的时间演化。随着知识的增长和该领域技术的改进，算子代数与数学和科学的许多其他领域之间建立了联系。这一建议涉及算子代数、交换代数、数论和组合学等领域之间的联系。这个项目的主要研究对象被称为某种类型的算子代数上的二进制移位，并且使用0‘S和1’S的比特流来定义，例如研究线性递归序列理论的人。这个项目的一个主要目标是完成二进制移位的分类，并将这种分类与线性递归序列的分析联系起来。还将研究二进制移位在量子动力系统理论中的潜在应用，特别是那些在时间上可能向前推进但不会倒退的系统。