Graph C*-algebras, special subalgebras, and applications

图代数在算子代数和动力系统领域之间提供了一条引人入胜的纽带。这项拟议的研究从两个不同的方向探讨了这种联系。由Cuntz和Krieger于1980年提出的图代数已被用各种方法推广和推广，形成了一大批有趣的C*-代数。这位研究人员与合作者加布里埃尔·纳吉最近发现了一个推广的坎茨-克里格唯一性定理的新证明。这项工作导致了他们发现了一类C*-子代数，伪对角线。这些类的定义属性之一涉及纯状态扩展，自半个世纪前首次引入Kadison-Singer问题以来，这个话题引起了极大的兴趣。这一建议旨在进一步研究伪对角线与其他特殊的Cartan-like子代数之间的关系和相似之处，并回答有关状态扩张的一些相关问题。另一方面，人们可以从有向图定义移位空间。为了阐明著名的符号动力学的Williams猜想，研究者打算分析相应的图代数。这一猜想断言矩阵移位等价的概念是拓扑共轭的一个完全不变量，但在1997年被Kim-Roush和Wager证明。他们的工作留下了许多悬而未决的问题和调查路线。图代数自20世纪80年代初被研究以来，出现在算子代数领域的许多领域。C*-代数的Cartan子代数理论是一个很有潜力的领域，因为雷诺在这方面的定义和主要结果是两年前才出现的。拓扑共轭问题是符号动力学领域的基础问题，已被大量专家研究，这一问题的任何进展都将是一个突破，研究者的算子-代数方法是新的。这项研究项目的两个部分都将在研究人员所在机构和整个数学界产生更广泛的影响。这位调查员是数学系的一名成员，致力于指导研究生和本科生。她积极参与了一系列活动，如协调一个数学科目GRE准备研讨会和举办一个针对研究生的研讨会。调查员将指导一名本科生和一名研究生的暑期研究项目，与这项提案相关的工作。最后，研究人员的出版物和会议上关于这项研究的演讲将加强动力学和算子代数之间的桥梁，并促进这些领域的专家之间的交流。

Graph algebras provide a fascinating link between the fields of operator algebras and dynamical systems. The proposed research explores this connection in two different directions. The graph algebras, introduced by Cuntz and Krieger in 1980, have been generalized and extended in a variety of ways to comprise a large collection of interesting C*-algebras. The investigator, along with collaborator Gabriel Nagy, has recently found a new proof of a generalized Cuntz-Krieger uniqueness theorem. This work has led to their discovery of a class of C*-subalgebras, pseudo-diagonals. One of the defining properties of these classes involves pure state extensions, a topic that has seen a great deal of interest since the Kadison-Singer Problem was first introduced half a century ago. This proposal seeks to further investigate the relationship and parallels between pseudo-diagonals and the other special Cartan-like subalgebras, as well as to answer some related questions about state extensions. On the other hand, one can define a shift space from a directed graph. The investigator intends to analyze the corresponding graph algebras in order to shed light on the famous Williams Conjecture of symbolic dynamics. This conjecture, which asserted that the notion of matrix shift equivalence was a complete invariant for topological conjugacy, was disproved by Kim-Roush and Wagoner in 1997. Their work leaves open many questions and lines of enquiry. Graph algebras have been studied since the early 1980s and appear in many areas of the field of operator algebras. The theory of Cartan subalgebras of C*-algebras is an area with much potential, as Renault's definition and major results on the subject appeared only two years ago. The topological conjugacy problem is fundamental to the area of symbolic dynamics and has been studied by a large number of experts; any progress on this problem will be a breakthrough, and the investigator's operator-algebraic approach is new. Both parts of this research project will have broader impacts both within the investigator's institution and in the mathematical community as a whole. The investigator is a member of a mathematics department with a strong commitment to mentoring graduate and undergraduate students. She is actively involved in a number of activities, such as coordinating a math subject GRE preparation workshop and running a seminar aimed at graduate students. The investigator will supervise an undergraduate and a graduate summer research project on work related to this proposal. Finally, the investigator's publications and lectures at conferences on this research will strengthen the bridge between dynamics and operator algebras and foster communication between the specialists in these fields.