On the isomorphism problem for $C^*$-algebras of nilpotent Lie groups

We investigate to what extent a nilpotent Lie group is determined by its $C^*$-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension $\le 5$ are uniquely determined by the Morita equivalence class of their $C^*$-algebras. We also find that this last property is shared by the filiform Lie groups and the $6$-dimensional free two-step nilpotent Lie group.