The group \(\text{Aut}(\mu)\) is Roelcke precompact (Q2888782)
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scientific article; zbMATH DE number 6042600
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | The group \(\text{Aut}(\mu)\) is Roelcke precompact |
scientific article; zbMATH DE number 6042600 |
Statements
4 June 2012
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Roelcke precompact
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unitary group
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measure preserving transformations
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Markov operators
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weakly almost periodic functions
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0.82787454
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0.8261919
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0.8243197
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0.82327974
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0.82154167
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0.82048726
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0.81953955
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The group \(\text{Aut}(\mu)\) is Roelcke precompact (English)
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Let \((X,\mu)\) be a standard atomless Borel probability space and let \(Aut(\mu)\) be the group of measure preserving automorphisms of \((X,\mu)\) equipped with the weak topology.NEWLINENEWLINEThe author proves that \(Aut(\mu)\) is precompact in the Roelcke uniformity, the lower bound of left and right uniformities on \(Aut(\mu)\).NEWLINENEWLINETwo corollaries: the Roelcke and the weakly almost periodic compactification of \(G\) coincide; \(Aut(\mu)\) is totally minimal.NEWLINENEWLINESimilar results for the unitary group of a separable Hilbert space were obtained in [\textit{V. V. Uspenskij}, in: D. Dikranjan (ed.) et al., Abelian groups, module theory, and topology. Proceedings in honour of Adalberto Orsatti's 60th birthday, Padua, Italy, 1997. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 201, 411--419 (1998; Zbl 0914.22002)]
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