Let $Y$ be a (locally compact) non-Hausdorff topological space. I want to know if there is a necessary and/or sufficient condition for $Y=X/G$, that is, $Y$ is the orbit space of a locally compact Hausdorff $G$-space $X$. I'd then indeed be able to call $Y$ an NC Space, with associated C*-algebra $C_0(X)\rtimes G$.
I couldn't find this "inverse" question in Connes' NCG or in Williams' Crossed Products of C*-algebras, where I might expect to find it (nor in Bourbaki, although there were similar things), though on pg. 112 Connes does state that when we have this situation (X/G being non-Hausdorff when X is LCH), the appropriate C*-algebra to assign is $C_0(X)\rtimes G$.
It seems like a natural question since many examples of nc spaces arise as orbit spaces.