Based on joint work with Matthias Neufang and Juris Steprans. By a variant of Foreman's 1994 construction, every tower ultrafilter on $\omega$ is the unique invariant mean for an amenable subgroup of $S_\infty$, the infinite symmetric group. From this we prove that in any model of ZFC with tower ultrafilters there is an element of $\ell_1(S_\infty)^{\ast\ast} \setminus \ell_1(S_\infty)$ whose action on $\ell_1(\omega)^{\ast\ast} $ is w* continuous. On the other hand, in ZFC there are always such elements whose action is not w* continuous.