A note on the precompactness of weakly almost periodic groups

TL;DR

This paper establishes that a topological group has all its continuous actions on compact spaces be weakly almost periodic if and only if the group is precompact, extending known results to a broader class of groups.

Contribution

It proves the equivalence between precompactness of a topological group and the weakly almost periodicity of all its continuous actions on compact spaces.

Findings

For topological groups, all actions are weakly almost periodic iff the group is precompact.

The result generalizes previous findings for monothetic and locally compact groups.

Provides a characterization linking group topology with dynamical properties.

Abstract

An action of a group $G$ on a compact space $X$ is called weakly almost periodic if the orbit of every continuous function on $X$ is weakly relatively compact in $C(X)$. We observe that for a topological group $G$ the following are equivalent: (i) every continuous action of $G$ on a compact space is weakly almost periodic; (ii) $G$ is precompact. For monothetic groups the result was previously obtained by Akin and Glasner, while for locally compact groups it has been known for a long time.