Stability of 𝐶*-algebras associated to graphs

We characterize stability of graph C ∗ C^* -algebras by giving five conditions equivalent to their stability. We also show that if G G is a graph with no sources, then C ∗ ( G ) C^*(G) is stable if and only if each vertex in G G can be reached by an infinite number of vertices. We use this characterization to realize the stabilization of a graph C ∗ C^* -algebra. Specifically, if G G is a graph and G ~ G̃ is the graph formed by adding a head to each vertex of G G , then C ∗ ( G ~ ) C^*(G̃) is the stabilization of C ∗ ( G ) C^*(G) ; that is, C ∗ ( G ~ ) ≅ C ∗ ( G ) ⊗