Generic measure preserving transformations and the closed groups they generate

TL;DR

This paper demonstrates that for a generic measure preserving transformation, the generated closed group is not isomorphic to a specific topological group of measurable functions, answering a question posed by Glasner and Weiss.

Contribution

It establishes a new spectral property of Koopman representations for generic transformations, distinguishing their generated groups from the group of all measurable functions.

Findings

The closed group generated by a generic measure preserving transformation is not isomorphic to L^0(λ, T).

Koopman representations of ergodic boolean actions have unique spectral properties.

The result answers a previously open question by Glasner and Weiss.

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.