A Wiener Lemma for the discrete Heisenberg group: Invertibility criteria and applications to algebraic dynamics

TL;DR

This paper establishes a Wiener Lemma for the discrete Heisenberg group, providing invertibility criteria in its algebraic structures that simplify analysis and have applications in algebraic dynamics and time-frequency analysis.

Contribution

It introduces a Wiener Lemma for the Heisenberg group that bypasses the need for detailed dual space knowledge, applicable to all countable nilpotent groups.

Findings

Provides explicit invertibility criteria in group algebras of the Heisenberg group.

Shows the lemma's applicability to algebraic dynamics.

Highlights the role of primitive ideal spaces in invertibility analysis.

Abstract

This article contains a Wiener Lemma for the convolution algebra $\ell^1(\mathbb H,\mathbb C)$ and group $C^\ast$-algebra $C^\ast(\mathbb H)$ of the discrete Heisenberg group $\mathbb H$. At first, a short review of Wiener's Lemma in its classical form and general results about invertibility in group algebras of nilpotent groups will be presented. The known literature on this topic suggests that invertibility investigations in the group algebras of $\mathbb H$ rely on the complete knowledge of $\widehat{\mathbb H}$ -- the dual of $\mathbb H$, i.e., the space of unitary equivalence classes of irreducible unitary representations. We will describe the dual of ${\mathbb H}$ explicitly and discuss its structure. Wiener's Lemma provides a convenient condition to verify invertibility in $\ell^1(\mathbb H,\mathbb C)$ and $C^\ast(\mathbb H)$ which bypasses $\widehat{\mathbb H}$. The proof of Wiener's Lemma for $\mathbb H$ relies on local principles and can be generalised to countable nilpotent groups. As our analysis shows, the main representation theoretical objects to study invertibility in group algebras of nilpotent groups are the corresponding primitive ideal spaces. Wiener's Lemma for $\mathbb H$ has interesting applications in algebraic dynamics and Time-Frequency Analysis which will be presented in this article as well.