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Uniform Boundedness for Groups

Published online by Cambridge University Press:  20 November 2018

Irving Glicksberg
Irving Glicksberg*
Affiliation:
University of Notre Dame
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Let G and H be locally compact abelian groups with character groups G*, H*, and let < . , . > denote the pairing between a group and its dual.

In 1952 Kaplansky proved the following result, using the structure of locally compact abelian groups and category arguments.

Theorem 1.1. Let τ: G → H be an algebraic homomorphism for which there is a dual τ* : H* → G* (so that < rg, h* > = < g, τ*h* > for all g in G, h* in H*). Then τ is continuous.

The result bears a striking similarity to a well-known fact about Banach spaces which is a consequence of uniform boundedness; the present note is devoted to an analogous “uniform boundedness” for groups, which yields a non-structural proof of Kaplansky's theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 269 - 276
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Grothendieck, A., Critères de compacité dans les espaces fonctionelles généraux, Amer. J. Math., 74 (1952), 168–186.Google Scholar
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3. Pettis, B. J., On continuity and openness of homomorphisms in topological groups, Ann. Math. (2), 52 (1950), 293–308.Google Scholar
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