Approximating the group algebra of the lamplighter by infinite matrix products

Abstract

AbstractIn this paper, we introduce a new technique in the study of the*{*}-regular closure of some specific group algebrasKGinside𝒰⁢(G){{\mathcal{U}}(G)}, the*{*}-algebra of unbounded operators affiliated to the group von Neumann algebra𝒩⁢(G){{\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 formCK⁢(X)⋊Tℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}, whereXis a totally disconnected compact metrizable space,Tis a homeomorphism ofX, andCK⁢(X){C_{K}(X)}stands for the algebra of locally constant functions onXwith values on an arbitrary fieldK. The connection between this class of algebras and a suitable class of group algebras is provided by the 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 ofℓ2{\ell^{2}}-Betti numbers arising from the lamplighter group, most of them transcendental.