Weit, Yitzhak On the one-sided Wiener’s theorem for the motion group on \({\mathbb{R}}^ N\). (English) Zbl 0604.43002 Isr. J. Math. 55, 111-120 (1986). 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, 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. 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, V. Losert [J. Reine Angew. Math. 331, 47-57 (1982; Zbl 0472.43004)] in a remarkable way completely characterized groups with property (rWP). Reviewer: E.Kaniuth Cited in 1 Document MSC: 43A20 \(L^1\)-algebras on groups, semigroups, etc. 22D15 Group algebras of locally compact groups Keywords:proper one-sided ideals; Wiener Tauberian theorem; locally compact group; \(L^ 1\)-group algebra; motion groups Citations:Zbl 0399.22004; Zbl 0467.43003; Zbl 0472.43004 PDFBibTeX XMLCite \textit{Y. Weit}, Isr. J. Math. 55, 111--120 (1986; Zbl 0604.43002) Full Text: DOI References: [1] Ehrenpreis, L.; Mautner, F. I., Some properties of the Fourier transform on semisimple Lie groups III, Trans. Am. Math. Soc., 90, 431-484 (1959) · Zbl 0086.09904 [2] Gangolli, R., On the symmetry of L_1algebras of locally compact motion groups and the Wiener Tauberian Theorem, J. Funct. Anal., 25, 244-252 (1977) · Zbl 0347.43005 [3] Leptin, H., Ideal theory in group algebras of locally compact groups, Invent. Math., 31, 259-278 (1976) · Zbl 0328.22012 [4] Leptin, H., On one-sided harmonic analysis in non-commutative locally compact groups, J. Reine Angew. Math., 306, 122-153 (1979) · Zbl 0399.22004 [5] Stein, M.; Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces (1971), Princeton: Princeton University Press, Princeton · Zbl 0232.42007 [6] Varopoulos, N. T., Spectral synthesis on spheres, Proc. Camb. Phil. Soc., 62, 379-387 (1966) · Zbl 0154.39204 [7] Vilenkin, N. I., Special Functions and the Theory of Group Representations (1965), Moscow: Nauka, Moscow · Zbl 0144.38003 [8] Weit, Y., On the one-sided Wiener’s Theorem for the motion group, Ann. of Math., 111, 415-422 (1980) · Zbl 0407.43003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.