On twisted group $C^{\ast }$ -algebras associated with FC-hypercentral groups and other related groups

We show that the twisted group $C^{\ast }$ -algebra associated with a discrete FC-hypercentral group is simple (respectively, has a unique tracial state) if and only if Kleppner’s condition is satisfied. This generalizes a result of Packer for countable nilpotent groups. We also consider a larger class of groups, for which we can show that the corresponding reduced twisted group $C^{\ast }$ -algebras have a unique tracial state if and only if Kleppner’s condition holds.