### Abstract

If a locally compact group G acts on a C∗-algebra B, we have both full and reduced crossed products and each has a coaction of G. We investigate 'exotic' coactions in between the two, which are determined by certain ideals E of the Fourier-Stieltjes algebra B(G); an approach that is inspired by recent work of Brown and Guentner on new C∗-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C∗-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain 'E-crossed product duality', intermediate between full and reduced duality. We give partial results concerning exotic coactions with the ultimate goal being a classification of which coactions are determined by ideals of B(G).

Original language | English (US) |
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Pages (from-to) | 411-434 |

Number of pages | 24 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 59 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2016 |

### Keywords

- coaction
- C∗-bialgebra
- Fourier-Stieltjes algebra
- group C∗-algebra

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*59*(2), 411-434. https://doi.org/10.1017/S0013091515000164