A Class of Distal Functions on Semitopological Semigroups

TL;DR

This paper generalizes the Weyl algebra of distal functions from the integers to broader semitopological semigroups, showing these functions are distal and exploring their properties in various algebraic structures.

Contribution

It extends the study of distal functions and the Weyl algebra to general semitopological semigroups, including countable rings and the bicyclic semigroup.

Findings

The algebra's elements are distal functions.

The algebra is characterized for the integers and countable rings.

Results include properties of the algebra on the bicyclic semigroup.

Abstract

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.