Applications of the Connes spectrum to C∗-dynamical systems, II

Abstract

We show that for a C ∗-dynamical system ( A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to G ̂ if and only if every nonzero closed ideal in G × α A has a nonzero intersection with A. Denote by G J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G J are discrete, then G X α A is simple if and only if A is G-simple and Γ( α) = G ̂ . This result is applicable not only when G is discrete but also when G≃ R or G≃ T provided that A is not primitive. Specializing to single automorphisms (i.e., G= Z ) we show that if (the transposed of) α acts freely on a dense set of points in A ̂ , then Λ(α)= T . The converse is only proved when A is of type I.