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The Group Aut (μ) is Roelcke Precompact

Published online by Cambridge University Press:  20 November 2018

Eli Glasner
Eli Glasner*
Affiliation:
Department of Mathematics, Tel Aviv University, Ramat Aviv, Israele-mail: [email protected]
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Abstract

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Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G\,=\,\text{Aut(}\mu \text{)}$ of automorphisms of an atomless standard Borel probability space $(X,\,\mu )$ is precompact. We identify the corresponding compactification as the space of Markov operators on ${{L}_{2}}(\mu )$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal.

Keywords

54H1122A0537B0554H20Roelcke precompactunitary groupmeasure preserving transformationsMarkov operatorsweakly almost periodic functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 297 - 302
Copyright
Copyright © Canadian Mathematical Society 2012

References

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