2 7 N ov 2 00 5 Almost maximally almost - periodic group topologies determined by T - sequences ∗

A sequence {an} in a group G is a T -sequence if there is a Hausdorff group topology τ on G such that an τ −→ 0. In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a T -sequence, and investigate special sequences in the Prüfer groups Z(p∞). We show that for p 6= 2, there is a Hausdorff group topology τ on Z(p∞) that is determined by a T -sequence, which is close to being maximally almost-periodic—in other words, the von Neumann radical n(Z(p∞), τ) is a non-trivial finite subgroup. In particular, n(n(Z(p∞), τ)) ( n(Z(p∞), τ). We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a T -sequence with non-trivial finite von Neumann radical.