The group $\Aut(μ)$ is Roelcke precompact

Details The group $\Aut(μ)$ is Roelcke precompact The group $\Aut(μ)$ is Roelcke precompact

2009-02-22

arxiv.org/abs/0902.3786v1

Mathematics

Dynamical Systems

Scientific paper

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 $G= \Aut(\mu)$, 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 $G$, all coincide. Again following Uspenskij we also conclude that $G$ is totally minimal.

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