Classificatios of C*- Algebras and Ultra products of Von Neumann Algebras

Abstract

We prove that a graph C^*-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C^*-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the Restorff and Ruiz. As a key step in the argument, we show to produce stability for certain full hereditary sub algebras associated to such graph C^*-algebras. We further prove that, except under Circumstance, an unique proper nontrivial idea in a graph C^*-algebra is stable.