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个迷人的联系。拟议的研究从两个不同的方向探索这种联系。Cuntz和Krieger在1980年引入的图代数，已经被以各种方式推广和扩展，包括大量有趣的C*-代数。研究者，沿着与合作者Gabriel Nagy，最近发现了一个广义Cuntz-Krieger唯一性定理的新证明。这项工作导致他们发现了一类C*-子代数，伪对角。这些类的定义属性之一涉及纯状态扩展，自从半个世纪前首次引入Kadison-Singer问题以来，这个主题已经引起了人们的极大兴趣。本文旨在进一步研究伪对角代数与其他特殊Cartan-like子代数之间的关系和相似之处，并回答一些与状态扩张相关的问题.另一方面，可以从有向图定义移位空间。研究者打算分析相应的图代数，以阐明著名的威廉姆斯猜想的符号动力学。这个猜想断言矩阵移位等价的概念是拓扑共轭的完全不变量，但在1997年被Kim-Wagner和瓦戈纳推翻。他们的工作留下了许多问题和调查线索。自20世纪80年代初以来，图代数已经被研究，并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Renault的定义和主要结果仅在两年前出现。拓扑共轭问题是符号动力学领域的基础，已经有大量的专家进行了研究;在这个问题上的任何进展都将是一个突破，并且研究人员的算子代数方法是新的。这两个部分的研究项目将有更广泛的影响，无论是在调查机构和整个数学界。调查员是一个数学系的成员，致力于指导研究生和本科生。她积极参与了许多活动，如协调数学科目GRE准备研讨会和运行针对研究生的研讨会。调查员将监督一个本科生和一个研究生暑期研究项目，与本提案相关的工作。最后，研究人员的出版物和讲座会议上这项研究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沟通。

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.