Ultrafilters on semitopological semigroups (Q2571026)

It has been known since the sixties that the Stone-Čech compactification of a discrete semigroup can be given a natural structure of a compact right topological semigroup. In [\textit{N. Hindman} and \textit{D. Strauss}, Algebra in the Stone-Čech compactification: theory and applications (de Gruyter Expositions in Mathematics 27) (1998; Zbl 0918.22001)], one can find a self-contained presentation of this theory, and a survey of its many applications in infinite combinatorics and topological dynamics, among other fields. When considering semitopological semigroups instead of discrete ones, several natural compactifications arise, which can be realized as spectra of convenient subalgebras of the \(C^*\)-algebra of all continuous and bounded complex-valued functions defined on the semigroup. In the paper under review the authors take up an ultrafilter-based approach in the construction of such compactifications, and adapt some of the results included in the above mentioned reference to this setting. Such results deal with the notions of syndetic, piecewise syndetic and central sets. The paper is somewhat loosely written; it lacks further explanation of its motivation and general purposes, and some unnecessarily long proofs could very well have been removed after the referee's suggestions.