A Class of Distal Functions on Semitopological Semigroups

The norm closure of the algebra generated by the set ${n\mapsto {\lambda}^{n^k}:$ $\lambda \in \mathbb{T}$ and $k\in \mathbb{N}\}$ of functions on $(\mathbb{Z},+)$ was studied in \cite{S} (and was named as the Weyl algebra). In this paper, by a fruitful result of Namioka, this algebra is generalized for a general semitopological semigroup and, among other things, it is shown that the elements of the involved algebra are distal. In particular, we examine this algebra for $(\mathbb{Z},+)$ and (more generally) for the discrete (additive) group of any countable ring. Finally, our results are treated for a bicyclic semigroup.