$C^{\ast }$-Rigidity of the Heisenberg Group

Abstract

We prove that the Heisenberg groups can be distinguished from the other connected and simply connected Lie groups via their group $C^{\ast }$-algebras. The main step of the proof is a characterization of the nilpotent Lie groups among the solvable Lie groups solely in terms of topological properties of their coadjoint orbits.