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

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.