Quotients of Ultragraph C*-Algebras

Abstract

Let $\mathcal{G}$ be an ultragraph and let $C^{\ast }(\mathcal{G})$ be the associated $C^{\ast }$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^{\ast }(\mathcal{G})$, we analyze the structure of the quotient $C^{\ast }$-algebra $C^{\ast }(\mathcal{G})/I_{(H,B)}$. For simplicity's sake, we first introduce the notion of quotient ultragraph $\mathcal{G}/(H,B)$ and an associated $C^{\ast }$-algebra $C^{\ast }(\mathcal{G}/(H,B))$ such that $C^{\ast }(\mathcal{G}/(H,B))\cong C^{\ast }(\mathcal{G})/I_{(H,B)}$. We then prove the gauge invariant and the Cuntz-Krieger uniqueness theorems for $C^{\ast }(\mathcal{G}/(H,B))$ and describe primitive gauge invariant ideals of $C^{\ast }(\mathcal{G}/(H,B))$.

arXiv commentV2: Minor changes are made and some typos are corrected