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The Connes spectrum for actions of Abelian groups on continuous-trace algebras

  • Steven Hurder (a1), Dorte Olesen (a2), Iain Raeburn (a3) and Jonathan Rosenberg (a4)
Abstract

We study the various notions of spectrum for an action α of a locally compact abelian group G on a type IC*-algebra A, and discuss how these are related to the structure of the crossed product AαG. In the case where A has continuous trace and the action of G on  is minimal, we completely describe the ideal structure of the crossed product. A key role is played by the restriction of α to a certain ‘symmetrizer subgroup’ S of the common stabilizer in G of the points of Â. We show by example that, contrary to a conjecture of Bratteli, it is possble for AG to be primitive but not simple, provided that S is not discrete. In such cases, the Connes spectrum Γ(α) differs from the strong Connes spectrum of Kishimoto. The counterexamples come from subtle phenomena in topological dynamics.

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References
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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