Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T07:49:08.526Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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