Generic measure preserving transformations and the closed groups they generate

2021/03/17 by Solecki, Sławomir · 1 citation
#03E15 #22A25 #22F10 #37A15 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO) #Representation Theory (math.RT) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2103.09429

Abstract

We show that, for a generic measure preserving transformation T, the closed group generated by T is not isomorphic to the topological group L⁰(λ, \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 L⁰(λ, \mathbb T) possess a non-trivial spectral property not shared by all unitary representations of L⁰(λ, \mathbb T). The main tool underlying our arguments is a theorem on the form of unitary representations of L⁰(λ, \mathbb T) from our earlier work.

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• A generic transformation is invertible

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