TYPE-I PERMANENCE

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 (Formula Presented) of locally compact groups and a twisted action (α, τ) thereof on a (post)liminal C*-algebra A the twisted crossed product (Formula Presented),τ E is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup (Formula Presented) is type-I as soon as E is. This happens for instance if N is discrete and E is Lie, or if 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 G type-I-preserving if all semidirect products (Formula Presented) are type-I as soon as N is, and linearly type-Ipreserving if the same conclusion holds for semidirect products (Formula Presented) arising from finite-dimensional G-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie.