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The uniform compactification of a locally compact abelian group

Published online by Cambridge University Press:  24 October 2008

M. Filali
M. Filali
Affiliation:
Department of Mathematics, University of Sheffield
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Extract

In recent years, the Stone-Čech compactification of certain semigroups (e.g. discrete semigroups) has been an interesting semigroup compactification (i.e. a compact right semitopological semigroup which contains a dense continuous homomorphic image of the given semigroup) to study, because an Arens-type product can be introduced. If G is a non-compact and non-discrete locally compact abelian group, then it is not possible to introduce such a product into the Stone-Čech compactification βG of G (see [1]). However, let UC(G) be the Banach algebra of bounded uniformly continuous complex functions on G, and let UG be the spectrum of UC(G) with the Gelfand topology. If f∈ UC(G), then the functions f and fy defined on G by

are also in UC(G).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 527 - 538
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Baker, J. W. and Butcher, R. J.. The Stone-Čech compactification of a topological semigroup. Math. Proc. Cambridge Philos. Soc. 80 (1976), 103–107.CrossRefGoogle Scholar
[2]Baker, J. W. and Milnes, P.. The ideal structure of the Stone-Čech compactification of a group. Math. Proc. Cambridge Philos. Soc. 82 (1977), 401–409.CrossRefGoogle Scholar
[3]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Compact Right Topological Semigroups and Generalisations of Almost Periodicity. Lecture Notes in Math. vol. 663 (Springer-Verlag, 1978).CrossRefGoogle Scholar
[4]Burckel, R. B.. Weakly Almost Periodic Functions on Semigroups (Gordon and Breach, 1970).Google Scholar
[5]Butcher, R. J.. The Stone-Cech compactification of a topological semigroup and its algebra of measures. Ph.D. thesis, University of Sheffield (1975).Google Scholar
[6]Filali, M.. An Arens-type product on the Banach dual of some C*-algebras of bounded continuous functions on a semitopological semigroup. Ph.D. thesis, University of Sheffield (1989).Google Scholar
[7]Gillman, L. and Jerison, M.. Rings of Continuous Functions (Van Nostrand, 1960).CrossRefGoogle Scholar
[8]Hamye, S. M.. Some compactifications of a semitopological semigroup and its algebra of charges. Ph.D. thesis, University of Sheffield (1986).CrossRefGoogle Scholar
[9]Hewitt, E. and Ross, K. A.. Abstract Harmonic Analysis, vol. 1 (Springer-Verlag, 1963).Google Scholar
[10]Hindman, N.. Minimal ideals and cancellation in βℕ. Semigroup Forum 25 (1982), 291–310.CrossRefGoogle Scholar
[11]Hindman, N. and Pym, J.. Free groups and semigroups in βℕ. Semigroup Forum 30 (1984), 177–193.CrossRefGoogle Scholar
[12]Parsons, D. J.. The Centre of the second dual of a commutative semigroup algebra. Math. Proc. Cambridge Philos. Soc. 95 (1984), 71–92.CrossRefGoogle Scholar
[13]Ruppert, W.. On semigroup compactifications of topological groups. Proc. Roy. Irish Acad. 79 (1979), 179–200.Google Scholar
[14]Simmons, G. F.. Introduction to Topology and Modern Analysis (McGraw-Hill, 1963).Google Scholar