Koshi, Shozo; Takahashi, Yuji Generating subsemigroups, orders, and a theorem of Glicksberg. (English) Zbl 0623.43005 Hokkaido Math. J. 16, 135-144 (1987). I. Glicksberg obtained the following theorem [Trans. Am. Math. Soc. 285, 235-240 (1984; Zbl 0548.43001), Theorem 1]: Let G be a locally compact abelian group with dual group \(\hat G,\) and let S be a proper closed generating subsemigroup of \(\hat G\) (i.e., S is a proper closed subsemigroup of \(\hat G\) such that \(S-S\) is dense in \(\hat G)\). Then the following holds. (i) There exists a nonzero \(\mu \in M_{S^ c}(G)\) which is singular with respect to the Haar measure of G unless \(G={\mathbb{R}}\times \Delta\) or \({\mathbb{T}}\times \Delta\) for a discrete group \(\Delta\); (ii) If \(G={\mathbb{R}}\times \Delta\) or \({\mathbb{T}}\times \Delta\) for a discrete group \(\Delta\) and if \(\mu \in M_{S^ c}(G)\), then \(\mu\) is absolutely continuous. The authors give another simple proof of the above Glicksberg’s theorem. Reviewer: H.Yamaguchi MSC: 43A05 Measures on groups and semigroups, etc. 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups Keywords:Riesz theorem; Fourier-Stieltjes transform; absolutely continuous measure; singular measure; locally compact abelian group; closed subsemigroup; Haar measure Citations:Zbl 0548.43001 PDFBibTeX XMLCite \textit{S. Koshi} and \textit{Y. Takahashi}, Hokkaido Math. J. 16, 135--144 (1987; Zbl 0623.43005) Full Text: DOI