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

• Larki, Hossein (Department of Mathematics Faculty of Mathematical Sciences and Computer Shahid Chamran University of Ahvaz)
• 투고 : 2017.09.05
• 심사 : 2018.09.11
• 발행 : 2019.01.01
• 235 17

#### 초록

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.

#### 키워드

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

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