ON THE STABILITY OF A FIXED POINT ALGEBRA C*(E)γOF A GAUGE ACTION ON A GRAPH C*-ALGEBRA

Abstract

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