### Abstract

We consider separable C*-dynamical systems (A, G, alpha) for which the induced action of the group G on the primitive ideal space Prim A of the C*-algebra A is free. We study how the representation theory of the associated crossed product C*-algebra A x(alpha) G depends on the representation theory of A and the properties of the action of G on Prim A and the spectrum (A) over cap. Our main tools involve computations of upper and lower bounds on multiplicity numbers associated to irreducible representations of A x(alpha) G. We apply our techniques to give necessary and sufficient conditions, in terms of A and the action of G, for A x(alpha) G to be (i) a continuous-trace C*-algebra, (ii) a Fell C*-algebra and (iii) a bounded-trace C*-algebra. When G is amenable, we also give necessary and sufficient conditions for the crossed product C*-algebra A x(alpha) G to be (iv) a liminal C*-algebra and (v) a Type T C*-algebra. The results in (i), (iii)-(v) extend some earlier special cases in which A was assumed to have the corresponding property.

Original language | English |
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Pages (from-to) | 545-581 |

Number of pages | 36 |

Journal | Proceedings of the London Mathematical Society |

Volume | 96 |

Issue number | 3 |

Early online date | 23 Nov 2007 |

DOIs | |

Publication status | Published - May 2008 |

### Keywords

- star-algebras
- transformation groups
- continuous trace
- crossed-products
- bounded trace
- irreducible representations
- automorphism-groups
- integrable actions
- orbit space
- lie-groups