Local topological structure in the LUC-compactification of a locally compact group and its relationship with Veech's theorem
If \(G\) is a topological group, \(G^{\text{LUC}}\) stands for the largest semigroup compactification for which the multiplication is jointly continuous. It can be obtained as the structure space of the \(C^\ast\)-algebra of all bounded left uniformly continuous functions on \(G\) (thence the term LUC in its name). In this paper a realization of \(G^{\text{LUC}}\) as a quotient of the semigroup \(\beta G_d\) (the Stone-Čech-compactification of \(G\) as a discrete space) is obtained. This quotient is the result of identifying two ultrafilters \(p,q\in \beta G_d\) whenever \(UP\cap UQ\neq \emptyset\) for every neighbourhood \(U\) of the identity of \(G\) and every \(P\in p\) and \(Q\in q\). This characterization of \(G^{\text{LUC}}\) is used to illustrate the close connection between Veech's theorem (one of the main results in the theory of LUC-compactifications stating that the map \(g\mapsto gs\) is injective on \(G\) for each \(s\in G^{\text{LUC}}\)) and local structure theorems that describe topologically neighbourhoods of points in \(G^{\text{LUC}}\). A local structure theorem was already used in \textit{J. Pym} [Semigroup Forum 59, No.~2, 171--174 (1999; Zbl 0980.22009)] to prove Veech's theorem. In this paper that local structure is improved to show that for every compact \(K\subset G\) and \(x\in G^{\text{LUC}}\) there is a neighbourhood \(V\) of \(K\) and a uniformly discrete \(P\subset G\) with \(x\in \overline{P}\) such that the multiplication map establishes a homeomorphism of \(V\times \beta P\) onto \(V\overline{P}\). Veech's theorem is an obvious direct consequence of the above local structure theorem for compact sets consisting of just two points. The latter can also be obtained as a direct consequence of Veech's theorem, suggesting a strong relationship between these two facts. Some extensions of the local structure theorem and of Veech's theorem are derived in the final part of the paper. The first makes use of the results of \textit{S. Ferri} and \textit{D. Strauss} [Semigroup Forum 69, No. ~1, 87--101 (2004; Zbl 1057.22002)] relating \(G^{\text{LUC}}\) to \(G^{\text{WAP}}\) to give a local structure theorem for \(G^{\text{WAP}}\) when \(G\) is SIN. The extension of Veech's theorem shows that there is an open set \(U\subset G^{\text{LUC}}\) with \(G\subset U\), \(G\neq U\), such that the multiplication map \(u\mapsto us\) is injective on \(U\) for every \(s\in G^{\text{LUC}}\).
scientific article; zbMATH DE number 5118909