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Generating subsemigroups, orders, and a theorem of Glicksberg. (English) Zbl 0623.43005

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

Citations:

Zbl 0548.43001
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