Quotients of Ultragraph C*-Algebras

TL;DR

This paper studies the structure of quotient ultragraph C*-algebras, introducing quotient ultragraphs and proving key theorems to understand their ideal structure and uniqueness properties.

Contribution

It introduces quotient ultragraphs and their C*-algebras, establishing isomorphisms with quotients of original algebras and proving gauge invariant and Cuntz-Krieger theorems.

Findings

Established isomorphism between $C^*( ext{quotient ultragraph})$ and quotient of $C^*( ext{original ultragraph})$

Proved gauge invariant and Cuntz-Krieger uniqueness theorems for quotient ultragraph C*-algebras

Described primitive gauge invariant ideals of quotient ultragraph C*-algebras

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 analyze the structure of the quotient $C^*$-algebra $C^*(\mathcal{G})/I_{(H,B)}$. For simplicity's sake, we first introduce the notion of quotient ultragraph $\mathcal{G}/(H,B)$ and an associated $C^*$-algebra $C^*(\mathcal{G}/(H,B))$ such that $C^*(\mathcal{G}/(H,B))\cong C^*(\mathcal{G})/I_{(H,B)}$. We then prove the gauge invariant and the Cuntz-Krieger uniqueness theorems for $C^*(\mathcal{G}/(H,B))$ and describe primitive gauge invariant ideals of $C^*(\mathcal{G}/(H,B))$.