### Abstract

Given a labeling c of the edges of a directed graph E by elements of a discrete group G, one can form a skew-product graph E ×_{c} G. We show, using the universal properties of the various constructions involved, that there is a coaction δ of G on C* (E) such that C* (E ×_{c} G) is isomorphic to the crossed product C* (E) ×_{δ} G. This isomorphism is equivariant for the dual action δ̂ and a natural action γ of G on C* (E ×_{c} G); following results of Kumjian and Pask, we show that C* (E ×_{c} G) ×_{γ} G ≅ C* (E ×_{c} G) ×_{γ,r} G ≅ C* (E) ⊗ K(ℓ^{2}(G)), and it turns out that the action γ is always amenable. We also obtain corresponding results for r-discrete groupoids Q and continuous homomorphisms c : Q → G, provided Q is amenable. Some of these hold under a more general technical condition which obtains whenever Q is amenable or second-countable.

Original language | English (US) |
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Pages (from-to) | 411-433 |

Number of pages | 23 |

Journal | Journal of Operator Theory |

Volume | 46 |

Issue number | 2 |

State | Published - Sep 1 2001 |

### Keywords

- C*-algebra
- Coaction
- Directed graph
- Duality
- Groupoid
- Skew product

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Operator Theory*,

*46*(2), 411-433.