  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.
