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$\mathcal{Z}$-stability of twisted group C*-algebras of nilpotent groups

Ulrik Enstad, Eduard Vilalta

TL;DR

This work characterizes when twisted group C*-algebras $C^*_r(G,\sigma)$ of finitely generated nilpotent groups are $\mathcal{Z}$-stable by linking regularity properties to cocycle non-rationality. It proves the equivalence among nowhere scatteredness, pureness, and $\mathcal{Z}$-stability, with non-rationality of $\sigma$ providing a concrete criterion, and it gives explicit, verifiable conditions for abelian and 2-step nilpotent cases. The analysis uses a 2-cocycle decomposition, fiber bundle descriptions, and known finite nuclear dimension results to deduce purity and $\mathcal{Z}$-stability, including a Hirsch-length based dimension reduction. Finally, the results are applied to generalized time-frequency analysis, yielding partial converses to the Balian–Low theorem for lattices in nilpotent Lie groups and concrete frame/Riesz sequence criteria depending on $d_\pi\mathrm{covol}(\Gamma)$ and cocycle non-rationality.

Abstract

We prove that the twisted group C*-algebra of a finitely generated nilpotent group is $\mathcal{Z}$-stable if and only if it is nowhere scattered, a condition that we characterize in terms of the given group and 2-cocycle. As a main application, we prove new converses to the Balian-Low Theorem for projective, square-integrable representations of nilpotent Lie groups.

$\mathcal{Z}$-stability of twisted group C*-algebras of nilpotent groups

TL;DR

This work characterizes when twisted group C*-algebras of finitely generated nilpotent groups are -stable by linking regularity properties to cocycle non-rationality. It proves the equivalence among nowhere scatteredness, pureness, and -stability, with non-rationality of providing a concrete criterion, and it gives explicit, verifiable conditions for abelian and 2-step nilpotent cases. The analysis uses a 2-cocycle decomposition, fiber bundle descriptions, and known finite nuclear dimension results to deduce purity and -stability, including a Hirsch-length based dimension reduction. Finally, the results are applied to generalized time-frequency analysis, yielding partial converses to the Balian–Low theorem for lattices in nilpotent Lie groups and concrete frame/Riesz sequence criteria depending on and cocycle non-rationality.

Abstract

We prove that the twisted group C*-algebra of a finitely generated nilpotent group is -stable if and only if it is nowhere scattered, a condition that we characterize in terms of the given group and 2-cocycle. As a main application, we prove new converses to the Balian-Low Theorem for projective, square-integrable representations of nilpotent Lie groups.

Paper Structure

This paper contains 11 sections, 27 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Abelian and 2-step nilpotent groups
  3. Generalized time-frequency analysis
  4. Nowhere scatteredness and pureness
  5. Twisted group C*-algebras
  6. Group C*-algebras
  7. Twisted group C*-algebras
  8. Finitely generated nilpotent groups
  9. Abelian groups
  10. 2-step nilpotent groups and generalized discrete Heisenberg groups
  11. Applications to generalized time-frequency analysis

Key Result

Theorem 2

Let $G$ be a finitely generated nilpotent group, and let $\sigma$ be a $2$-cocycle on $G$. Then, the following conditions are equivalent:

Theorems & Definitions (57)

  • Theorem 2: cf. \ref{['prp:StrIrrCharNSCA']}
  • Theorem 3: \ref{['prp:AbeCharStComp']}
  • Theorem 4: \ref{['prp:NSCAGenHeis']}
  • Theorem 5
  • theorem 2.3
  • proof
  • lemma 2.4
  • proof
  • proposition 2.5
  • proof
  • ...and 47 more