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Exponential Boundedness and Amenability of Open Subsemigroupsof Locally Compact Groups

Published online by Cambridge University Press:  20 November 2018

Wojciech Jaworski
Wojciech Jaworski*
Affiliation:
Department of Mathematics, University of Ottawa, Ottawa, OntarioK1N6N5
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Abstract

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Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

Keywords

43A0722D0543A0560B15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1263 - 1274
Copyright
Copyright © Canadian Mathematical Society 1994

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