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On the actions of a locally compact group on some of its semigroup compactifications

Published online by Cambridge University Press:  01 July 2007

M. FILALI
M. FILALI*
Affiliation:
Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland. e-mail: [email protected]
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Abstract

Let G be a locally compact group and be its largest semigroup compactification. Then sx≠x in whenever s is an element in G other than e. This result was proved by Ellis in 1960 for the case G discrete (and so is the Stone–Čech compactification βG of G), and by Veech in 1977 for any locally compact group. We study this property in the WAP – compactification of G; and in , we look at the situation when xs≠x. The points are separated by some weakly almost periodic functions which we are able to construct on a class of locally compact groups, which includes the so-called E-groups introduced by C. Chou and which is much larger than the class of SIN groups. The other consequences deduced with these functions are: a generalization of some theorems on the regularity of due to Ruppert and Bouziad, an analogue in of the “local structure theorem” proved by J. Pym in , and an improvement of some earlier results proved by the author and J. Baker on .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 1 , July 2007 , pp. 25 - 39
Copyright
Copyright © Cambridge Philosophical Society 2007

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References

REFERENCES

[1]Baker, J. W. and Filali, M.. On the analogue of Veech's theorem in the WAP-compactification of a locally compact group. Semigroup Forum 65 (2002), 107–112.CrossRefGoogle Scholar
[2]Berglund, J. F., Junghenn, H. D. and Milnes, P.. Analysis on Semigroups: Function Spaces, Compactifications, Representations (Wiley, 1989).Google Scholar
[3]Bouziad, A.. Contribution a la théorie des semigroupes semitopologiques. Thèse de doctorat, Université de Rouen (1989).Google Scholar
[4]Chou, C.. On the size of the set of left invariant means on a semigroup. Proc. Amer. Math. Soc. 23 (1969), 199–205.Google Scholar
[5]Chou, C.. Weakly almost periodic functions and almost convergent functions on a group. Trans. Amer. Math. Soc. 206 (1975), 175–200.CrossRefGoogle Scholar
[6]Ellis, R.. Lectures on Topological Dynamics (Benjamin, 1969).Google Scholar
[7]Ferri, S. and Strauss, D.. A note on the WAP-compactification and the LUC-compactification of a topological group. Semigroup Forum 69 (2004), 87–101.CrossRefGoogle Scholar
[8]Filali, M. and Pym, J. S.. Right cancellation in the LUC-compactification of a locally compact group Bulletin of London Math. Soc. 35 (2003), 128–134.CrossRefGoogle Scholar
[9]Junghenn, H. D. and Milnes, P.. Almost periodic compactifications of groups extensions Czechoslovak Math. J. 52 (127) (2002), 237–254.CrossRefGoogle Scholar
[10]Junghenn, H. D. and Milnes, P.. Distal compactifications of groups extensions. Rocky Mt. J. Math. 29 (1999), 209–227.CrossRefGoogle Scholar
[11]Lau, A. T., Milnes, P. and Pym, J. S.. Compactifications of locally compact groups and quotients. Math. Proc. Camb. Phil. Soc. 116 (1994), 451–463.CrossRefGoogle Scholar
[12]Lau, A. T., Milnes, P., Pym, J. S.. Flows on invariant subsets and compactifications of a locally compact group. Colloquiun Math. 78 (1998), 267–281.CrossRefGoogle Scholar
[13]Milnes, P.. Almost periodic compactifications of direct and semidirect products. Colloq. Math. 44 (1981), 125–136.CrossRefGoogle Scholar
[14]Palmer, T. W.. Classes of nonabelian, noncompact, locally compact groups. Rocky Mountain J. Math. 8 (1978), 683–741.CrossRefGoogle Scholar
[15]Pym, J.. A note on and Veech's theorem. Semigroup Forum 59 (1999), 171–174.CrossRefGoogle Scholar
[16]Ruppert, W.. On semigroup compactifications of topological groups. Proc. Roy. Irish Acad. Sect. A 79 (1979), 179–200.Google Scholar
[17]Ruppert, W.. Compact semitopological semigroups: an intrinsic theory. Lecture Notes in Math. 1079 (Springer-Verlag, 1984).CrossRefGoogle Scholar
[18]Veech, W. A.. Topological dynamics. Bull. Amer. Math. Soc. 83 (1977), 775–830.CrossRefGoogle Scholar