### 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 |
---|---|

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

### Cite this

**Strength of convergence and multiplicities in the spectrum of a C*-dynamical system.** / Archbold, Robert J; an Huef, Astrid.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. 96, no. 3, pp. 545-581. https://doi.org/10.1112/plms/pdm038

}

TY - JOUR

T1 - Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

AU - Archbold, Robert J

AU - an Huef, Astrid

PY - 2008/5

Y1 - 2008/5

N2 - 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.

AB - 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.

KW - star-algebras

KW - transformation groups

KW - continuous trace

KW - crossed-products

KW - bounded trace

KW - irreducible representations

KW - automorphism-groups

KW - integrable actions

KW - orbit space

KW - lie-groups

U2 - 10.1112/plms/pdm038

DO - 10.1112/plms/pdm038

M3 - Article

VL - 96

SP - 545

EP - 581

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 3

ER -