On the ultrafilter semigroup of a topological group
Given a discrete semigroup \(S\), the operation can be extended to the Stone-Čech compactification \(\beta S\) of \(S\) by \(pq=\lim_{x\to p}\lim_{y\to q}xy\). Then \(\beta S\) is a semigroup with the right shift \(p\mapsto pq\) continuous, i.e., \(\beta S\) is a right topological semigroup. For a nondiscrete left topological group \(G\) let \(\text{Ult}(G)\) denote the set of all nonprincipal ultrafilters on \(G\) converging to the identity \(e\in G\). \(\text{Ult}(G)\) is called the ultrafilter semigroup of \(G\) and it is a closed subsemigroup of \(\beta G_d\) (the Stone-Čech compactification of \(G\) as a discrete semigroup). Therefore \(\text{Ult}(G)\) is either finite or has cardinality \(\geq2^{\mathfrak c}\). The existence of \(G\) with finite \(\text{Ult}(G)\) implies the existence of \(P\)-points in \(\omega^*\) [see \textit{Y. Zelenyuk}, Mat. Stud. 6, 41--52 (1996; Zbl 0927.22004)]. In the article under review the author proves this theorem: Assume that there are no \(P\)-points in \(\omega^*\). Then for every countable nondiscrete topological group \(G\), \(\text{Ult}(G)\) can be partitioned into closed right ideals each of which admits a continuous homomorphism onto the Bohr compactification of the integers.
scientific article; zbMATH DE number 5118903