Identifying AF-algebras that are graph C*-algebras

TL;DR

This paper characterizes exactly which AF-algebras are isomorphic to graph C*-algebras, focusing on conditions involving Type I properties and finitely many ideals, and provides evidence for a broader conjecture.

Contribution

It establishes necessary and sufficient conditions for AF-algebras to be isomorphic to graph C*-algebras, advancing understanding of their structural classification.

Findings

Separable, unital, Type I AF-algebras with finitely many ideals are graph C*-algebras.

Unital AF-algebras are graph C*-algebras iff they are Type I with finitely many ideals.

Certain nonunital AF-algebras are graph C*-algebras if their unital quotient is a matrix algebra.

Abstract

We consider the problem of identifying exactly which AF-algebras are isomorphic to a graph C*-algebra. We prove that any separable, unital, Type I C*-algebra with finitely many ideals is isomorphic to a graph C*-algebra. This result allows us to prove that a unital AF-algebra is isomorphic to a graph C*-algebra if and only if it is a Type I 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*-algebra if and only if its unital quotient is Type I, which occurs if and only if its unital quotient is isomorphic to M_k 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*-algebra if and only if each unital quotient of the AF-algebra is Type I with finitely many ideals, and bear relevance for the intrigiung question of finding K-theoretical criteria for when an extension of two graph C*-algebras is again a graph C*-algebra.