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

Abstract

AbstractWe study the various notions of spectrum for an action α of a locally compact abelian groupGon a typeIC*-algebraA, and discuss how these are related to the structure of the crossed productA⋊αG. In the case whereAhas continuous trace and the action ofGon  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’Sof the common stabilizer inGof the points of Â. We show by example that, contrary to a conjecture of Bratteli, it is possble forA⋊Gto be primitive but not simple, provided thatSis not discrete. In such cases, the Connes spectrum Γ(α) differs from the strong Connes spectrumof Kishimoto. The counterexamples come from subtle phenomena in topological dynamics.