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

TL;DR

This paper proves that the automorphism group of an atomless standard Borel probability space is Roelcke precompact, identifies its compactification with Markov operators, and shows several function algebras coincide, extending Uspenskij's results.

Contribution

It establishes the Roelcke precompactness of Aut(μ), identifies its compactification with Markov operators, and proves total minimality, extending known results from unitary groups.

Findings

Aut(μ) is Roelcke precompact

Compactification identified as space of Markov operators

Algebras of various uniformly continuous functions coincide

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 $G= \Aut(\mu)$, 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.