If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is injective. Equivalently, MAP groups are those groups the points of which are separated by the finite-dimensional unitary representations. My question is:
if $G$ is MAP, is $i$ a homeomorphism onto $i(G)$, or is it only injective?
(I would like to know if topological properties and regular measures on $i(G)$ can be pulled back on $G$.)
