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

AbstractWe show that for a C∗-dynamical system (A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to Ĝ if and only if every nonzero closed ideal in G × αA has a nonzero intersection with A. Denote by GJ 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 GJ are discrete, then G X α A is simple if and only if A is G-simple and Γ(α) = Ĝ. 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 Â, then Λ(α)=T. The converse is only proved when A is of type I