Approximating the group algebra of the lamplighter by infinite matrix products (Q2155610)

The paper under review deals with the approximation of the group algebras of the lamplighter by infinite matrix products. Specifically, the authors introduced 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 they used for this investigation is a general approximation result for a class of crossed product algebras of special type. The connection between this class of algebras and a suitable class of group algebras is provided by the well-known Fourier transform often used in the functional analysis. Utilizing this machinery, the authors explored an explicit approximation for the lamplighter group algebras. The article under review is very well written and contains deep results in the subject like Theorem 1.2 (which summarizes Propositions 4.3, 4.6 and Theorem 4.11), Theorem 5.12 and Theorem 6.15, respectively. The provided reference list is complete and helpful. Taking into account all of this, the presented work will definitely be of some interest and importance for the experts in this topic.