标题： 一个离散的海森堡群的维纳引理：可逆性准则及其在代数动力学中的应用 显示英文标题

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

摘要： 本文包含离散Heisenberg群$\mathbb H$的卷积代数$\ell^1(\mathbb H,\mathbb C)$和群$C^\ast$-代数$C^\ast(\mathbb H)$的Wiener引理。首先，将简要回顾Wiener引理的经典形式以及关于幂零群的群代数中可逆性的普遍结果。该主题的现有文献表明，$\mathbb H$的群代数中的可逆性研究依赖于$\widehat{\mathbb H}$的完全知识——即$\mathbb H$的对偶，也就是不可约酉表示的酉等价类的空间。 我们将明确描述${\mathbb H}$的对偶，并讨论其结构。 维纳引理提供了在$\ell^1(\mathbb H,\mathbb C)$和$C^\ast(\mathbb H)$中验证可逆性的方便条件，该条件绕过了$\widehat{\mathbb H}$。 对于$\mathbb H$的维纳引理的证明依赖于局部原理，并可以推广到可数幂零群。 根据我们的分析，研究幂零群的群代数中可逆性的主要表示理论对象是相应的原始理想空间。 对于$\mathbb H$的维纳引理在代数动力系统和时频分析中有有趣的应用，这些应用也将在这篇文章中介绍。 显示英文摘要

摘要： 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.