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- W2090472795 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." @default.
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- W2090472795 date "2009-05-01" @default.
- W2090472795 modified "2023-09-24" @default.
- W2090472795 title "ON THE STABILITY OF A FIXED POINT ALGEBRA C*(E)γOF A GAUGE ACTION ON A GRAPH C*-ALGEBRA" @default.
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- W2090472795 doi "https://doi.org/10.4134/jkms.2009.46.3.657" @default.
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