Primitive ideals and pure infiniteness of ultragraph $C^*$-algebras

TL;DR

This paper investigates the structure of ultragraph $C^*$-algebras, focusing on primitive ideals and pure infiniteness, by analyzing quotients, gauge invariance, and Fell bundles to extend existing theorems.

Contribution

It introduces new descriptions of primitive gauge invariant ideals and characterizes purely infinite ultragraph $C^*$-algebras using Fell bundles, extending prior theoretical frameworks.

Findings

Established gauge invariant and Cuntz-Krieger uniqueness theorems for ultragraph $C^*$-algebra quotients.

Described primitive gauge invariant ideals in ultragraph $C^*$-algebras.

Characterized purely infinite ultragraph $C^*$-algebras via Fell bundles.

Abstract

Let $\mathcal{G}$ be an ultragraph and let $C^*(\mathcal{G})$ be the associated $C^*$-algebra introduced by Mark 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.