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In this paper, we introduce a new technique in the study of the*{*}   -regular closure of some specific group algebras  KG  inside  ⁢   (  G  ){{\mathcal{U}}(G)}   , the*{*}   -algebra of unbounded operators affiliated to the group von Neumann algebra  ⁢   (  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 formC  K   ⁢   (  X  )⋊  T   ℤ{C_{K}(X)\rtimes_{T}{\mathbb{Z}}}   , where  X  is a totally disconnected compact metrizable space,  T  is a homeomorphism of  X , andC  K   ⁢   (  X  ){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 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 ofℓ  2{\ell^{2}}   -Betti numbers arising from the lamplighter group, most of them transcendental.

Approximating the group algebra of the lamplighter by infinite matrix products