Groupoid and Inverse Semigroup Presentations of Ultragraph $C^{*}$-Algebras

TL;DR

This paper constructs a groupoid from an ultragraph using inverse semigroup theory, showing its $C^*$-algebra is isomorphic to the ultragraph $C^*$-algebra, extending graph algebra frameworks.

Contribution

It introduces a novel groupoid and inverse semigroup approach to ultragraph $C^*$-algebras, generalizing graph algebra constructions.

Findings

The groupoid $rak{G}_rak{G}$ is canonically isomorphic to the ultragraph $C^*$-algebra.

The construction generalizes Paterson's work on directed graphs.

Provides a new algebraic framework for ultragraph $C^*$-algebras.

Abstract

Inspired by the work of Paterson on $C^{\ast}$-algebras of directed graphs, we show how to associate a groupoid $\mathfrak{G}_{\mathcal{G}}$ to an ultragraph $\mathcal{G}$ in such a way that the $C^*$-algebra of $\mathfrak{G}_{\mathcal{G}}$ is canonically isomorphic to Tomforde's $C^*$-algebra $C^*(\mathcal{G})$. The groupoid $\mathfrak{G}_{\mathcal{G}}$ is built from an inverse semigroup $S_{\mathcal{G}}$ naturally associated to $\mathcal{G}$.