RUI: Semigroups of endomorphisms on von Neumann algebras

摘要：本文针对冯·诺依曼代数上的动力系统的一些问题进行了研究，并与最近一些作者关于冯·诺依曼代数因子上的自同态半群的研究工作有关。这个研究项目涉及许多数学领域，包括算子代数、有限域、数论和组合学。与自同构的半群相反，自同构的半群可以被看作是一个在时间上可以向前而不能向后进行的动力系统。值得注意的是，鉴于I型因子上自同构的单参数半群的相对简单的理论，这个主题被证明是多么具有挑战性。这个建议的一个主要目标是在一元自同态的单参数半群的分类上取得额外的进展，类似于维格纳对作用于i型因子的自同态群的描述。算子代数的研究可以追溯到冯·诺伊曼和其他人的工作。他们的目标是建立数学模型来捕捉量子力学系统的行为，并利用这些模型来预测这些系统的时间演化。随着知识的增长和该领域技术的改进，在算子代数和许多其他数学和科学领域之间建立了联系。这一建议涉及算子代数、交换代数、数论和组合学领域之间的联系。本项目的主要研究对象是在某种类型的算子代数上的二进制移位，并使用0和1的比特流来定义，例如线性循环序列理论中的一个研究。本项目的主要目标是完成二元移位的分类，并将这种分类与线性循环序列的分析联系起来。二进制位移也将研究其在量子动力系统理论中的潜在应用，特别是那些可能在时间上向前而不是向后进行的系统。

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.