On the stability of a fixed point algebra C^*(E)^γ of a gauge action on a graph C^*-algebra
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The fixed point algebra C^*(E)^γ of a gauge action γ on a graph C^*-algebra C^*(E) and its AF subalgebras C^*(E)^γ_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)^γ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that C^*(E)^γ is stably isomorphic to a graph C^*-algebra C^*(E_Z×E) which we observe being stable. We first give an explicit isomorphism from C^*(E)^γ to a full hereditary C^*-subalgebra of C^*(E_N×E)(⊂C^*(E_Z×E) and then show that C^*(E_N×E) is stable whenever C^*(E)^γ is so. Thus C^*(E)^γ cannot be stable if C^*(E_N×E) admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue λ>1. The AF algebras C^*(E)^γ_v are shown to be nonstable whenever E is irreducible. Several examples are discussed.
Abstract
The fixed point algebra C^*(E)^γ of a gauge action γ on a graph C^*-algebra C^*(E) and its AF subalgebras C^*(E)^γ_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)^γ (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that C^*(E)^γ is stably isomorphic to a graph C^*-algebra C^*(E_Z×E) which we observe being stable. We first give an explicit isomorphism from C^*(E)^γ to a full hereditary C^*-subalgebra of C^*(E_N×E)(⊂C^*(E_Z×E) and then show that C^*(E_N×E) is stable whenever C^*(E)^γ is so. Thus C^*(E)^γ cannot be stable if C^*(E_N×E) admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue λ>1. The AF algebras C^*(E)^γ_v are shown to be nonstable whenever E is irreducible. Several examples are discussed.
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