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

Robert J Archbold, Astrid an Huef

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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 languageEnglish
Pages (from-to)545-581
Number of pages36
JournalProceedings of the London Mathematical Society
Volume96
Issue number3
Early online date23 Nov 2007
DOIs
Publication statusPublished - May 2008

Keywords

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

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