On the one-sided Wiener's theorem for the motion group on \({\mathbb{R}}^ N\)
Let G be a locally compact group and \(L^ 1(G)\) its \(L^ 1\)-group algebra. G satisfies the right Wiener's Tauberian theorem (rWT) if every proper closed right ideal in \(L^ 1(G)\) is contained in a maximal closed right ideal. The purpose of the paper under review is to show by quite explicit constructions that (rWT) fails to hold for the motion groups \(G_ n=SO(n)\ltimes {\mathbb{R}}^ n\), \(n\geq 2.\) It should be mentioned that the stronger right Wiener property (rWP), saying that every proper closed right ideal in \(L^ 1(G)\) is annihilated by some non-zero positive linear functional on \(L^ 1(G)\), has been studied previously by several authors. In fact, \textit{H. Leptin} [J. Reine Angew. Math. 306, 122-153 (1979; Zbl 0399.22004)] has shown that a connected group G satisfies (rWP) if and only if G is a direct product of a vector group and a compact group. \textit{R. W. Henrichs} [Proc. Am. Math. Soc. 80, 627-630 (1980; Zbl 0467.43003)] gave a much shorter proof of a slightly more general result. Finally, \textit{V. Losert} [J. Reine Angew. Math. 331, 47-57 (1982; Zbl 0472.43004)] in a remarkable way completely characterized groups with property (rWP).
scientific article; zbMATH DE number 3975584