Simplicity of ultragraph algebras

Abstract

In this paper we analyze the structure of C * -algebras associated to ultragraphs, which are generalizations of directed graphs. We characterize the simple ultragraph algebras as well as deduce necessary and sufficient conditions for an ultragraph algebra to be purely infinite and to be AF. Using these techniques we also produce an example of an ultragraph algebra that is neither a graph algebra nor an Exel-Laca algebra. We conclude by proving that the C * -algebras of ultragraphs with no sinks are Cuntz-Pimsner algebras.

Graph algebras, Exel-Laca algebras, and ultragraph algebras coincide up to Morita equivalence

“…Our use of the notation G 0 in what follows will also be in keeping with [11] rather than with [20,21].…”

Ultragraph C*-algebras via topological quivers

“…The fundamental difference between the two classes of algebras is that a graph C * -algebra is generated by a collection containing a partial isometry for each edge and a projection for each vertex, while an Exel-Laca algebra is generated by a collection containing a partial isometry for each row in the matrix (and in the non-row-finite case there are rows in the matrix corresponding to an infinite collection of edges with the same source vertex). Thus, although these two constructions agree in the row-finite case, there are C * -algebras of non-row-finite graphs that are not isomorphic to any Exel-Laca algebra, and there are Exel-Laca algebras of non-row-finite matrices that are not isomorphic to the C * -algebra of any graph [14].…”

“…Introduction. Our objective in this paper is to show how the theory of ultragraph C * -algebras, first proposed by Tomforde in [13,14], can be formulated in the context of topological graphs [6] and topological quivers [11] in a fashion that reveals the K-theory and ideal theory (for gauge-invariant ideals) of these algebras. The class of graph C * -algebras has attracted enormous attention in recent years.…”

$C^*$-algebras of labelled graphs II - simplicity results

“…In [3] we showed how to associate a C * -algebra to a labelled space, which consists of a labelled graph together with a certain collection of subsets of vertices. By making suitable choices of the labelled spaces it was shown in [3, Proposition 5.1, Theorem 6.3] that the class of labelled graph C * -algebras includes graph C * -algebras, the ultragraph C * -algebras of [21], [22] and the C * -algebras of shift spaces in the sense of [14], [4]. In this paper we shall work almost exclusively with the labelled spaces which arise in connection with shift spaces.…”