Group actions, groupoids, and their $C^{\ast }$-algebras

Abstract

$C^{\ast }$-algebras were first introduced in order to model physical observables in quantum mechanics, but are now studied more abstractly in pure mathematics. Much of the current research of $C^{\ast }$-algebraists involves constructing interesting classes of $C^{\ast }$-algebras from various mathematical objects---such as groups, groupoids, and directed graphs---and studying their properties. Groupoid $C^{\ast }$-algebras were introduced by Renault in 1980, and provide a unifying model for $C^{\ast }$-algebras associated to groups, group actions, and graphs. In this talk, I will define topological groupoids and examine Renault's construction of groupoid $C^{\ast }$-algebras. I will discuss several examples of groupoids, including group actions and graph groupoids, and will conclude with a brief description of my PhD research.