Primitive ideals and pure infiniteness of ultragraph $C^*$-algebras
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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-R{\o}rdam) via Fell bundles.
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-R{\o}rdam) via Fell bundles.
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