A note on \(G^{\mathcal {LUC}}\) and Veech's theorem
The left uniformly continuous compactification \(G^{\mathcal{LUC}}\) of a locally compact group \(G\) is the compactification induced by the \(C^\ast\) algebra of bounded left uniformly continuous functions on \(G\) (and therefore coincides with the Stone-Čech compactification \(\beta G\) of \(G\) when \(G\) is discrete). \(G^{\mathcal{LUC}}\) carries a natural semigroup multiplication extending the group multiplication of \(G\). Veech's theorem, originally proven by \textit{W. A. Veech} [Bull. Am. Math. Soc. 83, 775-830 (1977; Zbl 0384.28018)] in the context of topological dynamics, asserts that, with this multiplication, \(s\mu \neq \mu\) whenever \(e\neq s\in G\) and \(\mu \in G^{\mathcal{LUC}}\). The present paper provides a new simpler proof of Veech's theorem based on a short proof of the usual partition lemma handling the discrete case and a structural approach to pass from the discrete to the general case.
left uniformly continuous compactification
scientific article; zbMATH DE number 1330705