$AF$ Embedding of Crossed Products of Certain Graph $C^*$-Algebras by Quasi-free Actions

Details $AF$ Embedding of Crossed Products of Certain Graph $C^*$-Algebras by Quasi-free Actions $AF$ Embedding of Crossed Products of Certain Graph $C^*$-Algebras by Quasi-free Actions

2005-10-26

arxiv.org/abs/math/0510551v1

Mathematics

Operator Algebras

Scientific paper

We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph whose each weak connected component is a rooted (possibly infinite) directed tree, and the labelling map is almost proper, which is proved to be a reasonable generalization of the earlier case, then the crossed product can be embedded to an $AF$ algebra.

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