We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{ N}\trianglelefteq\mathbb{ E}$ of locally compact groups and a twisted action $(\alpha,\tau)$ thereof on a (post)liminal $C^*$-algebra $A$ the twisted crossed product $A\rtimes_{ \alpha,\tau}\mathbb{ E}$ is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{ N}\trianglelefteq \mathbb{ E}$ is type-I as soon as $\mathbb{ E}$ is. This happens for instance if $\mathbb{ N}$ is discrete and $\mathbb{ E}$ is Lie, or if $\mathbb{ N}$ is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group $\mathbb{ G}$ type-I-preserving if all semidirect products $\mathbb{ N}\rtimes \mathbb{ G}$ are type-I as soon as $\mathbb{ N}$ is, and { \it linearly} type-I-preserving if the same conclusion holds for semidirect products $V\rtimes\mathbb{ G}$ arising from finite-dimensional $\mathbb{ G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie.
Categories: math.OA math.FA math.GR math.RT