On twisted group C$^*$-algebras associated with FC-hypercentral groups and other related groups

TL;DR

This paper investigates the simplicity and trace properties of twisted group C*-algebras for FC-hypercentral and related groups, establishing conditions under which these algebras are simple or have unique tracial states, generalizing previous results.

Contribution

It extends the understanding of twisted group C*-algebras by characterizing simplicity and trace uniqueness for FC-hypercentral groups and broader classes, generalizing Kleppner's condition results.

Findings

Twisted group C*-algebra is simple if Kleppner's condition holds.

Unique tracial state in twisted group C*-algebra corresponds to Kleppner's condition.

Results generalize previous work on countable nilpotent groups.

Abstract

We show that the twisted group C$^*$-algebra associated with a discrete FC-hypercentral group is simple (resp. has a unique tracial state) if and only if Kleppner's condition is satisfied. This generalizes a result of J. 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$^*$-algebras have a unique tracial state if and only if Kleppner's condition holds.