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# The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact

Published:2011-04-27
Printed: Jun 2012
• Eli Glasner,
Department of Mathematics, Tel Aviv University, Ramat Aviv, Israel
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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 $G= \operatorname{Aut}(\mu)$ 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 $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal.
 Keywords: Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions
 MSC Classifications: 54H11 - Topological groups [See also 22A05] 22A05 - Structure of general topological groups 37B05 - Transformations and group actions with special properties (minimality, distality, proximality, etc.) 54H20 - Topological dynamics [See also 28Dxx, 37Bxx]