The Group Aut (μ) is Roelcke Precompact
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 297-302
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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 $G\,=\,\text{Aut(}\mu \text{)}$ 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.
Mots-clés :
54H11, 22A05, 37B05, 54H20, Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions
Glasner, Eli. The Group Aut (μ) is Roelcke Precompact. Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 297-302. doi: 10.4153/CMB-2011-083-2
@article{10_4153_CMB_2011_083_2,
author = {Glasner, Eli},
title = {The {Group} {Aut} (\ensuremath{\mu}) is {Roelcke} {Precompact}},
journal = {Canadian mathematical bulletin},
pages = {297--302},
year = {2012},
volume = {55},
number = {2},
doi = {10.4153/CMB-2011-083-2},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-083-2/}
}
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