A characterization of maximally almost periodic groups

It is proved that a topological group G is maximally almost periodic if and only if G admits an action $\pi$ on a compact Hausdorff space X such that the transformation group $(X,G,\pi )$ is equicontinuous and effective. Using this characterization, it is proved that if H is a closed uniform subgroup of a topological group G, then G is maximally almost periodic if and only if H is maximally almost periodic. The latter gives as corollaries the results of Kuranishi, Murakami, Grosser and Moskowitz concerning maximally almost periodic groups.