Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 345-368.

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

Authors: Mark Tomforde
Author institution: Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA

Summary: 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 provid es 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.

Keywords: Ultragraph, graph $C^*$-algebra, Exel-Laca algebra, Cuntz-Krieger algebra, desingularization

Contents Full-Text PDF