RUI: Quantum Dynamical Systems and Marginal States

这一建议是针对调查的一些问题的动力系统上冯诺依曼代数，并涉及到最近的工作，由一些作者的主题半群的自同态冯诺依曼代数的因素。 这个研究项目涉及数学的许多领域，包括算子代数，有限域，数论和组合学。 与自同构半群相反，自同态半群可以被看作是一个动力系统，它可以在时间上向前但不能向后发展。 值得注意的是如何具有挑战性的这一主题已被证明是在相对简单的理论，单参数半群的自同构的第一类因素。 这个建议的一个主要目标是在单位自同态的单参数半群的分类方面取得进一步的进展，类似于维格纳对作用于I型因子的自同构群的刻画。 算子代数的研究可以追溯到冯·诺依曼等人的工作。 他们的目标是构建数学模型来捕捉量子力学系统的行为，并使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完善，算子代数与其他一些数学和科学领域之间建立了联系。 这个建议涉及算子代数，交换代数，数论和组合学领域之间的联系。 在这个项目中的主要研究对象被称为在某种类型的算子代数上的二进制移位，并且使用0和1的比特流来定义，例如在线性递归序列理论中的研究。 本计画的主要目标是完成二进制移位的分类，并将此分类与线性循环序列的分析联系起来。 二进制移位也将研究其潜在的应用量子动力学系统的理论，特别是那些系统，可能会前进，但不是向后，在时间。

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.