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.