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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: mfilali@cc.oulu.fi
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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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