Endomorphisms on Von Neumann Algebras

摘要：von Neumann代数上的价格自同态是在时间上可以向前而不能向后进行的非交换动力系统的例子。离散自同构的环共轭类的分析与子因子的琼斯指数理论有关，并与迹超有限因子上自同构的外共轭类的锥的分类理论有许多联系。一个重要的自同态族是幂变换族，它自然地出现在子因子的琼斯隧道上的变换的构造中，并作为有限相对交换指标的自同态的限制。利用异能移位和线性移位寄存器系统之间的联系，Price将尝试完成他对理性异能移位的分类。Price还计算了所有有理异能移位的熵，并希望与E. Stormer合作确定所有异能移位家族中cones -Stormer熵的取值范围。此外，由于Powers移位和移位寄存器系统之间的相互作用，出现了一些组合问题，作为这个项目的一部分，Price将与一些本科生一起回答关于有限域上Toeplitz矩阵和Hankel矩阵的秩的一些相关问题。这个研究项目代表了一种探索Price已经证明存在于两个数学领域之间的联系的尝试，其中一个是相对抽象的，另一个是相对具体的。自20世纪30年代以来，算子代数已被用作数学结构来模拟量子力学系统的行为。另一方面，移位寄存器系统领域是密码学的一个分支，是对安全信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，在本科生研究助理的参与下，Price将继续他的分类映射族的计划，这些映射族被称为鲍尔移位。幂次变换是一种映射，它可以被看作是确定特定物理可观测物代数的正向时间演化。虽然这种代数已被用于量子力学系统的建模，但它也被用于研究结理论的应用，以及它在理解DNA结构方面的潜在应用。

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.