The group aut(μ) is roelcke precompact

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Abstract

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 = Aut(μ) of automorphisms of an atomless standard Borel probability space (X, μ) is precompact. We identify the corresponding compactification as the space ofMarkov operators on L 2(μ) 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.

Original languageEnglish
Pages (from-to)297-302
Number of pages6
JournalCanadian Mathematical Bulletin
Volume55
Issue number2
DOIs
StatePublished - 2012

Keywords

  • Markov operators
  • Measure preserving transformations
  • Roelcke precompact
  • Unitary group
  • Weakly almost periodic functions

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