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Strong $C^*$-rigidity of the Heisenberg groups

Ingrid Beltita, Daniel Beltita

TL;DR

The paper proves a strong rigidity phenomenon: if a 1-connected Lie group G has a unitary dual homeomorphic to that of a Heisenberg group H, then G is isomorphic to H, with the Morita-equivalence class of the group C*-algebra providing a distinguishing invariant. The authors develop a two-pronged strategy: first reduce to the solvable case when the radical is co-compact, and then establish rigidity within solvable type I groups using the Pukánszky/Auslander-Kostant framework together with induced representations and deformations to control limit points in the unitary dual. A key technical contribution is the deformation analysis of induced representations, which links coadjoint orbits to spectral limits and enables precise topological comparisons of unitary duals. The work also exhibits nilpotent-counterexamples to underscore the scope and limits of dual rigidity and ends with open problems on Morita-invariant classifications of group C*-algebras beyond Heisenberg groups.

Abstract

We prove a strong rigidity property of the Heisenberg groups, that is, they can be distinguished from any other 1-connected Lie groups via their unitary dual spaces, in particular via the Morita equivalence class of their group $C^*$-algebras.

Strong $C^*$-rigidity of the Heisenberg groups

TL;DR

The paper proves a strong rigidity phenomenon: if a 1-connected Lie group G has a unitary dual homeomorphic to that of a Heisenberg group H, then G is isomorphic to H, with the Morita-equivalence class of the group C*-algebra providing a distinguishing invariant. The authors develop a two-pronged strategy: first reduce to the solvable case when the radical is co-compact, and then establish rigidity within solvable type I groups using the Pukánszky/Auslander-Kostant framework together with induced representations and deformations to control limit points in the unitary dual. A key technical contribution is the deformation analysis of induced representations, which links coadjoint orbits to spectral limits and enables precise topological comparisons of unitary duals. The work also exhibits nilpotent-counterexamples to underscore the scope and limits of dual rigidity and ends with open problems on Morita-invariant classifications of group C*-algebras beyond Heisenberg groups.

Abstract

We prove a strong rigidity property of the Heisenberg groups, that is, they can be distinguished from any other 1-connected Lie groups via their unitary dual spaces, in particular via the Morita equivalence class of their group -algebras.

Paper Structure

This paper contains 9 sections, 23 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Heisenberg partitions
  3. Results on induced representations
  4. On the topology of unitary duals of solvable Lie groups
  5. Remarks on metabelian Lie groups
  6. Remarks on solvable Lie groups of type I
  7. Rigidity within the class of solvable Lie groups
  8. Lie groups with co-compact radical
  9. Proof of the main theorem and open problems

Key Result

Theorem 1.1

If $G$ is a 1-connected Lie group and $H$ is a Heisenberg group with a homeomorphism $\widehat{G}\simeq \widehat{H}$, then there exists a Lie group isomorphism $G\simeq H$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • ...and 45 more