Almost periodic function on a group

Remarks

It is a theorem that the left translates alone, or the right translates alone, already form a relatively compact set exactly when $f$ is almost periodic. For an abelian group, left and right translates agree, and the definition is the one of almost periodic function. The almost periodic functions form a closed subalgebra $AP(G)$ of $C_b(G)$ that contains the constants and is closed under complex conjugation.

Example: let $\pi$ be a continuous unitary representation of $G$ on $\C^n$, let $u, v \in \C^n$, and let $f(x) = \ip{\pi(x) u}{v}$. Then ${}_a f_b(x) = \ip{\pi(x) \pi(b) u}{\pi(a)^{*} v}$. The vectors $\pi(b) u$ and $\pi(a)^{*} v$ lie in spheres of $\C^n$, which are compact, and the map $(u', v') \mapsto (x \mapsto \ip{\pi(x) u'}{v'})$ is continuous into $C_b(G)$. So the translates of $f$ lie in a compact set, and $f$ is almost periodic.

Non-example: on $G = \R$, the function $f(x) = \arctan x$ is bounded and continuous but not almost periodic. The translates $f(x + n)$ converge pointwise to $\pi/2$ as $n \to \infty$, but $\abs{f(-n + n) - \pi/2} = \pi/2$ for all $n$, so no subsequence converges uniformly.

If $G$ is compact, every continuous function on $G$ is almost periodic.