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

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.