$C^*$-rigidity of the Heisenberg group

TL;DR

This paper demonstrates that the Heisenberg groups are uniquely identifiable among connected, simply connected Lie groups through their group $C^*$-algebras, using topological properties of coadjoint orbits.

Contribution

It provides a $C^*$-rigidity result for the Heisenberg group, characterizing nilpotent Lie groups among solvable ones via coadjoint orbit topology.

Findings

Heisenberg groups are distinguished by their $C^*$-algebras.

Characterization of nilpotent Lie groups through coadjoint orbit topology.

The main step involves topological analysis of coadjoint orbits.

Abstract

We prove that the Heisenberg groups can be distinguished from the other connected and simply connected Lie groups via their group $C^*$-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.