On the continuity of left translations in the LUC-compactification
Let \(G\) be a non-compact locally compact topological group and \((\phi, G^{LUC})\) be the compactification of \(G\) that is universal with respect to the joint continuity property. (It turns out that \((\phi, G^{LUC})\) is in fact the universal semigroup compactification of \(G\).) Let \(G^*=G^{LUC}\setminus \phi(G)\). Certain properties concerning the continuity of the left translations of \(G^{LUC}\) have been intensively studied by several authors. The starting point for the present paper was a result proved by \textit{I. V. Protasov} and \textit{J. S. Pym} in a joint paper [Bull. Lond. Math. Soc. 33, 279--282 (2001; Zbl 1024.22002)]. They showed that in case \(G\) is \(\sigma\)-compact the subset of the semigroup \(G^*\) consisting of those points at which no left translation of \(G^*\) is continuous is dense in \(G^*\). This result is generalized in the main theorem of the present paper. For this, the author introduces, for an arbitrary non-compact locally compact topological group \(G\), a certain closed left ideal \(\mathcal{U}(G)\) of \(G^{LUC}\). (In case \(G\) is discrete \(\mathcal{U}(G)\) becomes the set of uniform ultrafilters on \(G\), and the \(\sigma\)-compactness of \(G\) implies that \(\mathcal{U}(G)\) coincides with \(G^*\).) Now the main theorem states that the subset of \(\mathcal{U}(G)\) consisting of those points at which no left translation of the semigroup \(\mathcal{U}(G)\) is continuous is dense in \(\mathcal{U}(G)\). As a consequence of this result the author derives a new proof for a result already obtained by \textit{Protasov} and \textit{Pym} in their paper mentioned above, namely that the topological center of \(G^*\) is empty.
scientific article; zbMATH DE number 5012982