2013/08/22 by Soren Eilers, Eilers, Soren, Takeshi Katsura +5 · 1 citation
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA) #math.OA
paper · pdf · doi:10.48550/arxiv.1308.5014
24 pages
arxiv created 2013/08/22 · arxiv updated 2013/08/26
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 Mk 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.