Graph C*-algebras, special subalgebras, and applications

Kansas State University, Manhattan KS

Investigators

Abstract

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.