A unified approach to Exel-Laca algebras and C*-algebras associated to graphs

TL;DR

This paper introduces ultragraphs, a generalization of directed graphs, and develops a unified framework for associating C*-algebras to them, encompassing graph and Exel-Laca algebras, with key theorems adapted to this setting.

Contribution

It defines ultragraphs and their C*-algebras, unifying graph and Exel-Laca algebras, and extends fundamental theorems to this broader context.

Findings

Ultragraph algebras include graph and Exel-Laca algebras.

Techniques from graph algebras apply to ultragraph algebras.

Versions of key theorems are established for ultragraph algebras.

Abstract

We define an ultragraph, which is a generalization of a directed graph, and describe how to associate a C*-algebra to it. We show that the class of ultragraph algebras contains the C*-algebras of graphs as well as the Exel-Laca algebras. We also show that many of the techniques used for graph algebras can be applied to ultragraph algebras and that the ultragraph provides a useful tool for analyzing Exel-Laca algebras. Our results include versions of the Cuntz-Krieger Uniqueness Theorem and the Gauge-Invariant Uniqueness Theorem for ultragraph algebras.