Quasi-product actions of a compact group on a \(C^*\)-algebra (Q1316373)

Let \(\alpha\) be an action of a compact group on a separable prime \(C^*\)-algebra \(A\). The authors prove several conditions on \(\alpha\) to be equivalent, among which are the following two: there exists a faithful irreducible representation of \(A\) which is also irreducible on \(A^ \alpha\); there exists an \(\alpha\)-invariant pure state \(\omega\) of \(A\) such that \(\pi_{\omega\mid A^ \alpha}\) is faithful (it follows also that \(\pi_ \omega\) is faithful). These two conditions mean, roughly speaking, that there exists, respectively, free orbits and fixed points in the space of equivalence classes of irreducible representations of \(A\), acted upon by \(G\).