Journal of the
Korean Mathematical Society JKMS

The fixed point algebra $C^*(E)^\gamma$ of a gauge action $\gamma$ on a graph $C^*$-algebra $C^*(E)$ and its AF subalgebras $C^*(E)^\gamma_v$ associated to each vertex $v$ do play an important role for the study of dynamical properties of $C^*(E)$. In this paper, we consider the stability of $C^*(E)^\gamma$ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that $C^*(E)^\gamma$ is stably isomorphic to a graph $C^*$-algebra $C^*(E_{\mathbb Z}\times E)$ which we observe being stable. We first give an explicit isomorphism from $C^*(E)^\gamma$ to a full hereditary $C^*$-subalgebra of $C^*(E_{\mathbb N}\times E)\,(\subset C^*(E_{\mathbb Z}\times E))$ and then show that $C^*(E_{\mathbb N}\times E)$ is stable whenever $C^*(E)^\gamma$ is so. Thus $C^*(E)^\gamma$ cannot be stable if $C^*(E_{\mathbb N}\times E)$ admits a trace. It is shown that this is the case if the vertex matrix of $E$ has an eigenvector with an eigenvalue $\lambda>1$. The AF algebras $C^*(E)^\gamma_v$ are shown to be nonstable whenever $E$ is irreducible. Several examples are discussed.

Keywords: graph $C^*$-algebra, stable $C^*$-algebra, fixed point algebra, full hereditary $C^*$-subalgebra

MSC numbers: 46L05, 46L55