Identifying AF-algebras that are graph C⁎C⁎-algebras

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C⁎C⁎-algebra. We prove that any separable, unital, Type I C⁎C⁎-algebra with finitely many ideals is isomorphic to a graph C⁎C⁎-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C⁎C⁎-algebra if and only if it is a Type I C⁎C⁎-algebra with finitely many ideals. We also consider nonunital AF-algebras that have a largest ideal with the property that the quotient by this ideal is the only unital quotient of the AF-algebra. We show that such an AF-algebra is isomorphic to a graph C⁎C⁎-algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to MkMk for some natural number k  . All of these results provide vast supporting evidence for the conjecture that an AF-algebra is isomorphic to a graph C⁎C⁎-algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the extension problem for graph C⁎C⁎-algebras.