Bounded subgroups as a von Neumann radical of an Abelian group

For an infinite Abelian group G, we give a complete description of those nonzero bounded subgroups of G which are the von Neumann radical for some Hausdorff group topology on G. If G is of infinite exponent, we prove that for every nonzero bounded subgroup H of G there exists a Hausdorff group topology τ on G such that H   is the von Neumann radical of (G,τ)(G,τ). If G has finite exponent, we show that the following are equivalent: (i) there exists a Hausdorff group topology τ on G such that H   is the von Neumann radical of (G,τ)(G,τ); (ii) G   contains a subgroup of the form Z(exp⁡(H))(ω)Z(exp⁡(H))(ω). In particular, an infinite Abelian group of finite exponent admits a Hausdorff minimally almost periodic group topology if and only if all its leading Ulm–Kaplansky invariants are infinite.