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

TL;DR

This paper explores how certain nilpotent Lie groups are uniquely identified by their $C^*$-algebras, revealing specific classes where the algebraic structure determines the group.

Contribution

It demonstrates that within exponential Lie groups, some are uniquely determined by their $C^*$-algebras or unitary dual, expanding understanding of the isomorphism problem.

Findings

Exponential Lie groups with Heisenberg components are uniquely determined by their unitary dual.

Nilpotent Lie groups of dimension ≤ 5 are uniquely identified by the Morita class of their $C^*$-algebras.

Filiform and 6-dimensional free two-step nilpotent Lie groups share this uniqueness property.

Abstract

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.