Geodesic
The Group Aut (μ) is Roelcke Precompact
Canadian mathematical bulletin, Tome 55 (2012) no. 2, pp. 297-302

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2011-083-2
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
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[1] [1] Dikranjan, D., Prodanov, I. and Stoyanov, L., Topological groups: Characters, Dualities and Minimal Group Topologies. Monographs and Textbooks in Pure and Applied Mathematics 130. Marcel Dekker, New York, 1990. Google Scholar

[2] [2] Fathi, A., Le groupe de transformations de [0, 1] qui préservent la measure de Lebesgue est un groupe simple. Israel J. Math. 29(1978), no. 2-3, 302–308. Google Scholar | DOI

[3] [3] Giordano, T. and Pestov, V., Some extremely amenable groups. C. R. Acad. Sci. Paris 334(2002), no. 4, 273–278. Google Scholar

[4] [4] Giordano, T. and Pestov, V., Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu 6(2007), no. 2, 279–315. Google Scholar | DOI

[5] [5] Glasner, E., Ergodic Theory via Joinings. Math. Surveys and Monographs 101. American Mathematical Society, Providence, RI, 2003. Google Scholar

[6] [6] Glasner, E. and King, J., A zero-one law for dynamical properties. Contemporary Math. 215(1998), 231–242. Google Scholar

[7] [7] Glasner, E., Lemanczyk, M., and Weiss, B., A topological lens for a measure-preserving system. arXiv:0901.1247. Google Scholar

[8] [8] Glasner, E. and Megrelishvili, M., New algebras of functions on topological groups arising from G-spaces. Fund. Math. 201(2008), no. 1, 1–51. Google Scholar | DOI

[9] [9] Megrelishvili, M., Reflexively representable but not Hilbert representable compact flows and semitopological semigroups. Colloq. Math. 110(2008), no. 2, 383–407. Google Scholar | DOI

[10] [10] Roelcke, W. and Dierolf, S., Uniform Structures on Topological Groups and Their Quotients. McGraw-Hill, New York, 1981. Google Scholar

[11] [11] Stoyanov, L., Total minimality of the unitary groups. Math. Z. 187(1984), no. 2, 273–283. Google Scholar | DOI

[12] [12] Uspenskij, V. V., The Roelcke compactification of unitary groups. In: Abelian Groups,Module Theory, and Topology. Lecture Notes in Pure and Appl. Math, 201. Dekker, New York, 1998, pp 411–419. Google Scholar

[13] [13] Uspenskij, V. V., The Roelcke compactification of groups of homeomorphisms. Topology Appl. 111(2001), no. 1-2, 195–205. http://dx.doi.org/10.1016/S0166-8641(99)00185-6 Google Scholar

[14] [14] Uspenskij, V. V., Compactifications of topological groups. In: Proceedings of the Ninth Prague Topological Symposium. Topol. Atlas, North Bay, ON, 2002, pp. 331–346. Google Scholar

[15] [15] Uspenskij, V. V., On subgroups of minimal topological groups. Topology Appl. 155(2008), no. 14, 1580–1606. Google Scholar | DOI

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