Type-I permanence

Abstract

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.