Type I orbits in the pure states of a \(C^ *\)-dynamical system. II
For a \(C^*\)-algebra A with an action \(\alpha\) of a locally compact abelian group G, one considers the pure states of A, or rather the spectrum of A with the associated action. Type I orbits are defined and studied in the previous paper [ibid. 23, 321-336 (1987; Zbl 0633.46062)]. We continue this study and show that if A is separable and simple and G is separable and if there is a type I orbit through a pure state f with trivial stabilizer \(\{t\in G:\pi_ f\circ \alpha_ t\sim \pi_ f\}=\{0\}\), then there is a type I orbit with stabilizer equal to any given closed subgroup of G. This in particular shows that if the system is asymptotically abelian in the sense that there is an automorphism \(\gamma\) of A such hat \(\gamma \circ \alpha_ t=\alpha_ 1\circ \gamma\), \(t\in G\), and \([\gamma^ n(x),y]\to 0\) as \(n\to \infty\) for any x,y\(\in A\), there is an \(\alpha\)-covariant irreducible representation.
scientific article; zbMATH DE number 4046426