# PRIMITIVE IDEALS AND PURE INFINITENESS OF ULTRAGRAPH C⁎-ALGEBRAS

Larki, Hossein

• Accepted : 2018.09.11
• Published : 2019.01.01
• 6 1

#### Abstract

Let ${\mathcal{G}}$ be an ultragraph and let $C^*({\mathcal{G}})$ be the associated $C^*$-algebra introduced by Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*({\mathcal{G}})$, we approach the quotient $C^*$-algebra $C^*({\mathcal{G}})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-Rørdam) via Fell bundles.

#### Keywords

ultragraph $C^*$-algebra;primitive ideal;pure infiniteness