Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
“Classification of simple graph algebras”
This summer we will be continuing our study of graph C*-algebras: to a directed graph E, one can associate a C*-algebra C∗(E) whose ∗-algebraic properties can be determined through graphical properties of E. For example, C∗(E) is an AF-algebra if and only if E is acyclic. Graph C*-algebras form a useful class of examples for C*-algebraists.
Through the famous and general classification results of Elliott and Kirchberg–Phillips, isomorphism of graph C*-algebras can be determined through a pair of groups known as K- theory (with some other data attached). We’ll start the summer by describing this classification theorem: defining the K-theory groups and the various hypotheses in the classification. Graph C*-algebras have sisters in ring theory called Leavitt path algebras, but there is no analogue of the Kirchberg–Phillips result — so there are still some algebraic open questions.
I hope this topic will be of interest to analysis and algebra students alike, and we’ll begin with a low prereq ceiling. Everyone is welcome!