Endomorphisms on Von Neumann Algebras

摘要von Neumann代数上的价格自同态是非对易动力系统的一个例子，它可以在时间上向前推进，但不能向后推进。离散自同态的上循环共轭类的分析与子因子的Jones指数理论有关，与迹超有限因子上自同构的外共轭类的Connes分类理论有许多联系。自同态的一个重要的族是幂移动族，它自然地出现在子因子的Jones隧道上的移位的构造中，并且是具有有限相对交换指数的自同态的限制。利用幂移位和线性移位寄存器系统之间的联系，普赖斯将试图完成他对有理幂移位的分类。普赖斯还计算了所有有理幂变换的熵，并期望与E·斯托默一起研究在所有幂变换的家族中确定康纳斯-斯托默熵的值范围的问题。此外，在研究幂移位和移位寄存器系统之间的相互作用中出现了一些组合问题，作为本项目的一部分，普赖斯将与一些本科生一起回答一些关于有限域上Toeplitz和Hankel矩阵的秩的相关问题。这一研究项目代表了一种尝试，利用普赖斯已经证明存在于两个数学领域之间的联系，其中一个领域相对抽象，另一个领域相对具体。自20世纪30年代以来，S算子代数一直被用作模拟量子力学系统行为的数学构造。另一方面，移位寄存器系统领域是密码学的一个分支，是对安全消息进行编码和传输的手段之一。利用线性移位寄存器系统的一些数论方面，并在本科生研究助理的参与下，普赖斯将继续他的映射族分类程序，称为幂移位。幂平移是可以被视为决定特定物理可观测代数的向前时间演化的映射。虽然这个代数被用来模拟量子力学系统，但它也因其在纽结理论中的应用以及在理解DNA结构方面的潜在应用而受到研究。

Abstract Price Endomorphisms on von Neumann algebras are examples of noncommutative dynamical systems which may proceed forward, but not backward, in time. The analysis of the cocycle conjugacy classes of discrete endomorphisms is related to the Jones theory of index for subfactors and has numerous connections to the classification theory of Connes of the outer conjugacy classes of automorphisms on the tracial hyperfinite factor. An important family of endomorphisms is the family of Powers shifts, which appear naturally in the construction of shifts on Jones tunnels of subfactors and as restrictions of endomorphisms with finite relative commutant index. Using a connection between Powers shifts and linear shift register systems, Price will attempt to complete his classification of the rational Powers shifts. Price has also computed the entropy of all rational Powers shifts and expects to work with E. Stormer on the problem of determining the range of values of the Connes-Stormer entropy among the family of all Powers shifts. In addition, a number of combinatorial problems have arisen from the interplay between the study of Powers shifts and shift register systems, and as part of this project Price will work with some undergraduate students to answer some related questions about the ranks of Toeplitz and Hankel matrices over finite fields. This research project represents an attempt to exploit connections which Price has shown to exist between two fields of mathematics, one of which is relatively abstract and one of which is comparatively concrete. Since the 1930's operator algebras have been used as mathematical constructs to model the behavior of quantum mechanical systems. The field of shift register systems, on the other hand, is a branch of cryptology, and is one of the means by which secure messages are encoded and transmitted. Using some number-theoretic aspects of linear shift register systems, and with the participation of undergraduate research assistants, Price will con tinue his program of classifying families of mappings known as Powers shifts. Powers shifts are mappings which may be viewed as determining the forward time evolution of a particular algebra of physical observables. Although this algebra has been used in modeling quantum mechanical systems, it has also been studied for its applications to knot theory and for its potential applications towards the understanding of the structure of DNA.