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On Invariant Means which are Not Inverse Invariant

Published online by Cambridge University Press:  20 November 2018

M. Rajagopalan  and
K. G. Witz
M. Rajagopalan
Affiliation:
University of Illinois, Urbana, Illinois
K. G. Witz
Affiliation:
University of Illinois, Urbana, Illinois
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 222 - 224
Copyright
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
3. Rudin, W., Fourier analysis on groups (Interscience, 1962).Google Scholar