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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

REFERENCES

1.Birkhoff, G., Lattice theory, 3rd. edition, Amer. Math. Soc. Colloquium Pub., Vol. 25 (Providence, R.I., 1976).Google Scholar
2.Bourbaki, N., Éléments de mathématique. I: Les structures fondamentales de I'analyse. III: Topologie générale. Actualités Sci. Indust., nos. 1029, 1045, 1084, 1142, 1143 (Hermann, 1947–1949).Google Scholar
3.Cameron, N. and Miller, J. B., Topology and axioms of interpolation in partially ordered spaces, J. Reine Angew. Math. 278/279 (1975), 1–13.Google Scholar
4.Conrad, P. F., The structure of a lattice-ordered group with a finite number of disjoint elements, Michigan Math. J. 7 (1960), 171–180.CrossRefGoogle Scholar
5.Conrad, P. F., Some Structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99 (1961), 212–240.CrossRefGoogle Scholar
6.Conrad, P. F., Lex-subgroups of lattice-ordered groups, Czechoslovak Math. J. 18 (1968), 86–103.CrossRefGoogle Scholar
7.Conrad, P. F., Lattice-ordered groups, Lecture notes (Tulane University, 1970).Google Scholar
8.Fuchs, L., Partially ordered algebraic systems, (Pergamon Press, 1963).Google Scholar
9.Fuchs, L., Riesz groups, Ann. Scuola Norm. Sup. Pisa Sci. Fis. Mat. 19 (1965), 1–34.Google Scholar
10.Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Vol. 1, (Springer, 1963).Google Scholar
11.Kelley, J. L., Namioka, I. et al. , Linear topological spaces, (Van Nostrand, 1963).CrossRefGoogle Scholar
12.Loy, R. T. and Miller, J. B., Tight Riesz groups, J. Austral. Math. Soc. Ser. A 13 (1972), 224–240.CrossRefGoogle Scholar
13.Miller, J. B., Subdirect representation of tight Riesz groups by hybrid products, J. Reine Angew. Math. 283/284 (1976), 110–124.Google Scholar
14.Miller, J. B., The order-dual of a TRL group I, J. Austral. Math. Soc. Ser. A 25 (1978), 129–141.CrossRefGoogle Scholar
15.Peressini, A. L., Ordered topological vector spaces, (Harper and Row, 1976).Google Scholar
16.Reilly, N. R., Representations of ordered groups with compatible tight Riesz orders, J. Austral. Math. Soc. Ser. A 20 (1975), 307–322.CrossRefGoogle Scholar
17.Wirth, A., Compatible tight Riesz orders, J. Austral. Math. Soc. Ser. A 15 (1973), 105–111.CrossRefGoogle Scholar
18.Wirth, A., Locally compact tight Riesz groups, J. Austral. Math. Soc. Ser. A 19 (1975), 247–251.CrossRefGoogle Scholar