On labeled graph $C^*$-algebras

Abstract

Given a directed graph $E$ and a labeling $\mathcal{ℒ}$, one forms the labeled graph $C^{\ast }$-algebra by taking a weakly left-resolving labeled space $\left(E,\mathcal{ℒ},\mathcal{ℬ}\right)$ and considering a universal generating family of partial isometries and projections. We work on ideals for a labeled graph $C^{\ast }$-algebra when the graph contains sinks. Using some of the tools we build, we compute $C^{\ast }\left(E,\mathcal{ℒ},\mathcal{ℬ}\right)$ when $E$ is a finite graph.