The topological center of the semigroup of free ultrafilters
It is well known [see, for example, \textit{N. Hindman}, Ultrafilters and Ramsey theory -- an update, Lect. Notes Math. 1401, 97-118 (1989; Zbl 0701.05060), p. 99] that the operation of a discrete semigroup \(S\) can be extended to the Stone-Čech compactification \(\beta S\) of the space \(S\) so that \(\beta S\) becomes a compact left topological semigroup. The topological center of a compact left topological semigroup \(S\) is the set of all elements \(s\in S\) such that the mapping \(\rho _{s}:S\rightarrow S,\rho _{s}(x)=xs\), is continuous. The author shows that for every infinite discrete group \(G\) there exists \(p\in\beta G\backslash G\) such that the mapping \(\rho _{p}:\beta G\backslash G\rightarrow\beta G\backslash G\) is discontinuous in every countable complete ultrafilter \(q\in\beta G\backslash G\) . It is also proved that for every group \(G\) the topological center of the semigroup \(\beta G\backslash G\) is empty.
scientific article; zbMATH DE number 1262568