Type I orbits in the pure states of a \(C^ *\)-dynamical system (Q1096144)

Groups of automorphisms on \(C^*\) algebras are considered: \(\alpha\) : \(G\to Aut(A)\). For such an action \(\alpha\) and a subgroup H of G, define the fixed-point-algebras, \(A^ G=\{a\in A:\alpha_ g(a)=a\), \(g\in G\}\) and \(A^ H=\{a\in A:\alpha_ h(a)=a\), \(h\in H\}\). For states f, let \(\pi_ f\) denote the GNS representation. For states f on the crossed product \(A\ltimes G\), the restriction \(\pi\) of \(\pi_ f\) to A is considered, and \(\rho =\int^{\oplus}_{G}\pi \circ \alpha_ tdt\). The relative commutant of \(\rho\) (A) is denoted \(\rho\) (A)'. Theorem 3.5 is reproduced below: Let A be a separable prime \(C^*\)-algebra and \(\alpha\) a faithful continuous action of a (separable) compact abelian group G on A. Let H be an arbitrary closed subgroup of G. Then the following conditions are equivalent: 1. \(A^ G\) is prime and there exists a G-invariant pure state f of A such that \(\pi_ f\) is faithful. 2. \(A^ H\) is prime and there exists an H-invariant pure state \(\phi\) of A such that \(\pi_{\phi}\) is faithful and \(\rho_{\phi}(A)''\cap \rho_{\phi}(A)'=\{p:\) \(p\in H^{\perp}\}''\otimes 1\), where \(A^ H=A^{\alpha | H}(0)\), etc.