J. Korean Math. Soc. 2019; 56(1): 1-23
Online first article November 9, 2018 Printed January 1, 2019
https://doi.org/10.4134/JKMS.j170579
Copyright © The Korean Mathematical Society.
Hossein Larki
Shahid Chamran University of Ahvaz
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.
Keywords: ultragraph $C^*$-algebra, primitive ideal, pure infiniteness
MSC numbers: 46L55
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