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Non-secular, locally compact TRL groups

Published online by Cambridge University Press:  18 May 2009

R. H. Redfield
R. H. Redfield
Affiliation:
Mathematics Department, Monsh University, Clayton, Victoria Australia 3168
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In [12], Loy and Miller proved that a locally compact, eudoxian TR group is algebraically and order-theoretically (and hence, topologically) isomorphic to a finite product of copies of the real numbers. In [18], Wirth used their result to describe the subgroup of a locally compact TR group generated by the compact neighbourhoods of zero. The proof of Loy and Miller relied heavily on a result of Mackey (cf. [10], p. 390) and either the finite-dimensional case of the Choquet-Kendall Theorem (cf. [15], pp. 9–10) or the representation theory of Kakutani (cf. [11], Appendix). Below we use only elementary topological results and order-theoretic arguments and a theorem of Conrad [4] to characterize all non-secular, locally compact TRL groups (Theorem 3). Our proof of Theorem 3 allows us to deduce algebraically the theorems both of Loy and Miller and of Wirth, in both cases without appealing to the theorem of Conrad.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 21 , Issue 1 , January 1980 , pp. 1 - 7
Copyright
Copyright © Glasgow Mathematical Journal Trust 1980

References

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