Polish topometric groups
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- by Itaï Ben Yaacov, Alexander Berenstein and Julien Melleray PDF
- Trans. Amer. Math. Soc. 365 (2013), 3877-3897 Request permission
Abstract:
We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal. Our work is based on the concept of a Polish topometric group, defined in this article. Using Kechris and Rosendal’s work as a guide, we explore the consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group $\operatorname {Iso}(\mathbf {U}_1)$ of the bounded Urysohn space, the unitary group ${\mathcal U}(\ell _2)$ of a separable Hilbert space, and the automorphism group $\operatorname {Aut}([0,1],\lambda )$ of the Lebesgue measure algebra on $[0,1]$. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from $\operatorname {Aut}([0,1],\lambda )$ into a separable topological group is continuous.References
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Additional Information
- Itaï Ben Yaacov
- Affiliation: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
- MR Author ID: 699648
- Alexander Berenstein
- Affiliation: Departamento de Matemáticas, Universidad de los Andes, Carrera 1 # 18A-10, Bogotá, Colombia
- Julien Melleray
- Affiliation: Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
- MR Author ID: 781936
- Received by editor(s): July 8, 2011
- Received by editor(s) in revised form: December 8, 2011
- Published electronically: February 5, 2013
- Additional Notes: Work on this project was facilitated by ANR chaire d’excellence junior THEMODMET (ANR-06-CEXC-007), a visit of the third author at the Erwin Schrödinger Institute in Vienna, and the ECOS Nord program (action ECOS Nord C10M01). The first author was also supported by the Institut Universitaire de France
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3877-3897
- MSC (2010): Primary 03E15, 37B05
- DOI: https://doi.org/10.1090/S0002-9947-2013-05773-X
- MathSciNet review: 3042607