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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*
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 SG is amenable; (b) given and a compact KG 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 SG is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every sS; (2) G is exponentially bounded if and only if for every nonempty open subset UG.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1994

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