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

Title & Authors
ON THE STABILITY OF A FIXED POINT ALGEBRA C*(E)γ OF A GAUGE ACTION ON A GRAPH C*-ALGEBRA
Jeong, Ja-A.;

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