Abstract
Let E be a row-finite directed graph, let G be a locally compact abelian group with dual group Ĝ = Γ, let ω be a labeling map from E* to Γ, and let (C*(E), G, α ω) be the C*-dynamical system defined by ω. Some mappings concerning the AF-embedding construction of \(C*(E) \times _{\alpha ^\omega } G\) are studied in more detail. Several necessary conditions of AF-embedding and some properties of almost proper labeling map are obtained. Moreover it is proved that if E is constructed by attaching some 1-loops to a directed graph T consisting of some rooted directed trees and G is compact, then ω is almost proper, that is a sufficient condition for AF-embedding, if and only if Σ k j=1 \(\omega _{\gamma _j } \ne 1_\Gamma\) for any loop γ i , γ 2, ..., γ k attached to one path in T.
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Supported by National Natural Science Foundation of China (Grant Nos. 10771161, 11071188)
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Fang, X.C. AF-Embedding of crossed products of certain graph C*-algebras by quasi-free actions (II). Acta. Math. Sin.-English Ser. 27, 1581–1590 (2011). https://doi.org/10.1007/s10114-011-8622-0
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DOI: https://doi.org/10.1007/s10114-011-8622-0