The Effros–Hahn-induction conjecture for separable dynamical systems

Let $(A,G,\alpha)$ be a separable dynamical system, where $A$ is a $C^*$-algebra, $G$ is a locally compact group, and $\alpha$ is an action of $G$ on $A$. The system satisfies the Effros–Hahn-induction property (EHI) if, for every $P\in\operatorname{Prim}(A)$ and every primitive ideal $J$ of $A\rtimes_{\alpha}G_P$ with $\operatorname{Res}J=P$, the induced ideal $\operatorname{Ind}_{G_P}^{G}J$ is primitive in $A\rtimes_{\alpha}G$. Effros–Hahn-induction conjecture. Every separable dynamical system $(A,G,\alpha)$ satisfies EHI. Earlier results establish EHI under additional hypotheses, including commutativity or continuous trace together with suitable freeness assumptions, and amenability with a free action on $\operatorname{Prim}A$; the conjecture asks whether separability alone suffices.