Generic measure preserving transformations and the closed groups they generate

• 成果类型: Article
• 署名作者: Solecki, Slawomir
• 署名单位: Cornell University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01154-5
• 发表日期: 2023
• 页码: 805-850
• 关键词: spectral properties

摘要

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, T) of all Lebesgue measurable functions from [0, 1] to 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, T) possess a non-trivial spectral property not shared by all unitary representations of L-0(lambda, T). The main tool underlying our arguments is a theorem on the form of unitary representations of L-0(lambda, T) from our earlier work.