### 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) |
---|---|

Pages (from-to) | 411-433 |

Number of pages | 23 |

Journal | Journal of Operator Theory |

Volume | 46 |

Issue number | 2 |

State | Published - Sep 2001 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Operator Theory*,

*46*(2), 411-433.

**Skew products and crossed products by coactions.** / Kaliszewski, Steven; Quigg, John; Raeburn, Iain.

Research output: Contribution to journal › Article

*Journal of Operator Theory*, vol. 46, no. 2, pp. 411-433.

}

TY - JOUR

T1 - Skew products and crossed products by coactions

AU - Kaliszewski, Steven

AU - Quigg, John

AU - Raeburn, Iain

PY - 2001/9

Y1 - 2001/9

N2 - 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(ℓ2(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.

AB - 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(ℓ2(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.

KW - C-algebra

KW - Coaction

KW - Directed graph

KW - Duality

KW - Groupoid

KW - Skew product

UR - http://www.scopus.com/inward/record.url?scp=0037815940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037815940&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0037815940

VL - 46

SP - 411

EP - 433

JO - Journal of Operator Theory

JF - Journal of Operator Theory

SN - 0379-4024

IS - 2

ER -