Approximating the group algebra of the lamplighter by infinite matrix products

TL;DR

This paper develops a new approximation technique for group algebras, particularly the lamplighter group algebra, using crossed product algebras and Fourier transform, with applications to $ ext{l}^2$-Betti numbers.

Contribution

Introduces a novel approximation method for group algebras via crossed product algebras and Fourier transform, enhancing understanding of the lamplighter group's algebraic structure.

Findings

Provides explicit approximation for lamplighter group algebra

Connects crossed product algebras with group algebras via Fourier transform

Enables computation of transcendental $ ext{l}^2$-Betti numbers

Abstract

In this paper, we introduce a new technique in the study of the $*$-regular closure of some specific group algebras $KG$ inside $\mathcal{U}(G)$, the $*$-algebra of unbounded operators affiliated to the group von Neumann algebra $\mathcal{N}(G)$. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form $C_K(X) \rtimes_T \mathbb{Z}$, where $X$ is a totally disconnected compact metrizable space, $T$ is a homeomorphism of $X$, and $C_K(X)$ stands for the algebra of locally constant functions on $X$ with values on an arbitrary field $K$. The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of $\ell^2$-Betti numbers arising from the lamplighter group, most of them transcendental.