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

Abstract

AbstractLetGbe a locally compact group and$\mathcal{U}G$be its largest semigroup compactification. Thensx≠xin$\mathcal{U}G$wheneversis an element inGother thane. This result was proved by Ellis in 1960 for the caseGdiscrete (and so$\mathcal{U}G$is the Stone–Čech compactification βGofG), and by Veech in 1977 for any locally compact group. We study this property in theWAP– compactification$\mathcal{W}G$ofG; and in$\mathcal{U}G$, we look at the situation whenxs≠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$\mathcal{W}G$due to Ruppert and Bouziad, an analogue in$\mathcal{W}G$of the “local structure theorem” proved by J. Pym in$\mathcal{U}G$, and an improvement of some earlier results proved by the author and J. Baker on$\mathcal{W}G$.