  We show that, for a generic measure preserving transformation $T$, the closed
group generated by $T$ is not isomorphic to the topological group $L^0(\lambda,
{\mathbb T})$ of all Lebesgue measurable functions from $[0,1]$ to $\mathbb T$
(taken with pointwise multiplication and the topology of convergence in
measure). This result answers a question of Glasner and Weiss. The main step in
the proof consists of showing that Koopman representations of ergodic boolean
actions of $L^0(\lambda, {\mathbb T})$ possess a non-trivial spectral property
not shared by all unitary representations of $L^0(\lambda, {\mathbb T})$. The
main tool underlying our arguments is a theorem on the form of unitary
representations of $L^0(\lambda, {\mathbb T})$ from our earlier work.
