Primitive ideals and pure infiniteness of ultragraph $C^{\ast }$-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 approach the quotient $C^{\ast }$-algebra $C^{\ast }(\mathcal{G})/I_{(H,B)}$ by the $C^{\ast }$-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 $C^{\ast }$-algebras (in the sense of Kirchberg-R$ø$rdam) via Fell bundles.

arXiv commentThis is modified and improved version of the paper "Quotient Ultragraph $C^{\ast }$-algebras"; [arXiv:1512.00346] some results modified, and purely infinite ultragraph $C^{\ast }$-algebras described in the sense of Kirchberg-Rordam