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On Invariant Means which are Not Inverse Invariant
Published online by Cambridge University Press: 20 November 2018
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In (1) R. G. Douglas says: “For a finite abelian group there exists a unique invariant mean which must be inversion invariant. For an infinite torsion abelian group it is not clear what the situation is.” It is not hard to see that if every element of an abelian group G is of order 2, then every invariant mean on G is also inversion invariant (see 1, remark 4). In this note we prove the following theorem (Theorem 1 below): An abelian torsion group G has an invariant mean which is not inverse invariant if, and only if, 2G is infinite. This result, together with the theorems of Douglas, answers completely the question of the existence (on an arbitrary abelian group) of invariant means which are not inverse invariant.
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- Copyright © Canadian Mathematical Society 1968
References
1.
Douglas, R. G., On the inversion invariance of invariant means, Proc. Amer. Math. Soc, 00 1965), 642–645.Google Scholar
2.
Mitchell, T., Invariant means on semigroups and the constant functions, Thesis, Illinois Institute of Technology (1964).Google Scholar
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