The ideal separation property for reduced group $C^{\ast }$-algebras

We say that an inclusion of an algebra $A$ into a $C^{\ast }$-algebra $B$ has the ideal separation property if closed ideals in $B$ can be recovered by their intersection with $A$. Such inclusions have attractive properties from the point of view of harmonic analysis and noncommutative geometry. We establish several permanence properties of locally compact groups for which $L^1(G)\subseteq C_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\ast }(G)$ has the ideal separation property.