Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

TL;DR

This paper establishes criteria for when AF-algebras can be represented as graph, Exel-Laca, or ultragraph C*-algebras, revealing that all stable AF-algebras are both graph and Exel-Laca algebras, and characterizing certain AF-algebras.

Contribution

It provides necessary and sufficient conditions for AF-algebras to be realized as various classes of graph-related C*-algebras, advancing the understanding of their structure.

Findings

All stable AF-algebras are graph and Exel-Laca algebras.

All simple AF-algebras are either graph or Exel-Laca algebras.

Characterization of AF-algebras isomorphic to row-finite graph C*-algebras with no sinks.

Abstract

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C*-algebra, an Exel-Laca algebra, and an ultragraph C*-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C*-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C*-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C*-algebra of a row-finite graph with no sinks.