Generic measure preserving transformations and the closed groups they generate

DOI10.1007/S00222-022-01154-5zbMATH Open1514.37009arXiv2103.09429MaRDI QIDQ6363040

Sławomir Solecki

Publication date: 17 March 2021

Abstract: We show that, for a generic measure preserving transformation $T$, the closed group generated by $T$ is not isomorphic to the topological group $L^0(lambda,mathbbT)$ of all Lebesgue measurable functions from $[0,1]$ to $mathbbT$ (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 $L^0(lambda,mathbbT)$ possess a non-trivial spectral property not shared by all unitary representations of $L^0(lambda,mathbbT)$. The main tool underlying our arguments is a theorem on the form of unitary representations of $L^0(lambda,mathbbT)$ from our earlier work.

Mathematics Subject Classification ID

Representations of general topological groups and semigroups (22A25) Descriptive set theory (03E15) Ergodic theory on groups (22D40) General groups of measure-preserving transformations and dynamical systems (37A15) Measurable group actions (22F10)

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