Generic measure preserving transformations and the closed groups they generate

Abstract

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.