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 Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions

MSC Classifications:

54H11, 22A05, 37B05, 54H20 show english descriptions Topological groups [See also 22A05]
Structure of general topological groups
Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Topological dynamics [See also 28Dxx, 37Bxx] 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]

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