Quasiminimal distal function space and its semigroup compactification (Q1895276)

In an earlier paper [ibid. 14, 253-260 (1991; Zbl 0739.43009)] the author has introduced the notion of a quasidistal flow. A flow \((S,X, \pi)\) (where \(S\) is a semitopological semigroup acting via a homomorphism \(\pi : S \to X^X\) on a compact Hausdorff space \(X)\) is called quasidistal if the equation \(\lim_i s_i.x = \lim_i s_i.y\), for some net \((s_i)\) in \(S\) and \(x,y \in X\), always implies that \(s.x = s.y\) for all \(s \in S\). In other words, a flow \((S,X, \pi)\) is quasidistal if the associated point separating quotient flow \((S,X/ \sim, \pi_\sim)\) is distal. (Here \(\sim\) denotes the closed \(S\)-invariant equivalence relation \((x \sim y) \Leftrightarrow (s.x = s.y\) \(\forall s \in S)\) and \(\pi_\sim\) is naturally induced by \(\pi.)\) Similarly, the flow \((X,S, \pi)\) is called quasiminimal if the associated point separating quotient flow is minimal. A function \(f \in LMC (S)\) is said to be quasidistal [quasiminimal] if the associated flow \((S,X_f, \pi)\) on the orbit closure \(X_f\) of \(f\) (under the right translations \(f(.) \to f(s.))\) is quasidistal [quasiminimal]. The author observes that in \(LMC (S)\) the functions which are both quasiminimal and quasidistal form an admissible subalgebra and thus give rise to a right topological compactification \((Y, \beta)\) of \(S\). This compactification is maximal with respect to the property that \(\overline {Y^2}\) is left simple.