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تسجيل مستخدم جديدApproximating the group algebra of the lamplighter by infinite matrix products
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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 $mathcal{U}(G)$, 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.
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In this paper we consider the algebraic crossed product $mathcal A := C_K(X) rtimes_T mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field and $C_K(X)$ denotes the $K$-algebra of locally constant $K$-valued
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