Stability of 𝐶*-algebras associated to graphs

Tomforde, Mark

doi:10.1090/S0002-9939-04-07411-8

Stability of $\boldsymbol {C^*}$-algebras associated to graphs
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Proc. Amer. Math. Soc. 132 (2004), 1787-1795 Request permission

Abstract:

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

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