Abstract:We give a construction of the Gurarij space analogous to Katětov’s construction of the Urysohn space. The adaptation of Katětov’s technique uses a generalisation of the Arens-Eells enveloping space to metric space with a distinguished normed subspace. This allows us to give a positive answer to a question of Uspenskij as to whether the linear isometry group of the Gurarij space is a universal Polish group.
- Itaï Ben Yaacov, Fraïssé limits of metric structures, submitted, arXiv:1203.4459.
- Itaï Ben Yaacov and C. Ward Henson, Generic orbits and type isolation in the Gurarij space, research notes.
- Richard J. Fleming and James E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 129, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1957004
- V. I. Gurariĭ, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Ž. 7 (1966), 1002–1013 (Russian). MR 0200697
- M. Katětov, On universal metric spaces, General topology and its relations to modern analysis and algebra, VI (Prague, 1986) Res. Exp. Math., vol. 16, Heldermann, Berlin, 1988, pp. 323–330. MR 952617
- Wiesław Kubiś and Sławomir Solecki, A proof of uniqueness of the Gurariĭ space, Israel J. Math. 195 (2013), no. 1, 449–456. MR 3101256, DOI 10.1007/s11856-012-0134-9
- Wolfgang Lusky, The Gurarij spaces are unique, Arch. Math. (Basel) 27 (1976), no. 6, 627–635. MR 433198, DOI 10.1007/BF01224730
- Vladimir Pestov, Topological groups: where to from here?, Proceedings of the 14th Summer Conference on General Topology and its Applications (Brookville, NY, 1999), 1999, pp. 421–502 (2001). MR 1876385
- V. V. Uspenskij, On the group of isometries of the Urysohn universal metric space, Comment. Math. Univ. Carolin. 31 (1990), no. 1, 181–182. MR 1056185
- Vladimir V. Uspenskij, On subgroups of minimal topological groups, Topology Appl. 155 (2008), no. 14, 1580–1606. MR 2435151, DOI 10.1016/j.topol.2008.03.001
- Nik Weaver, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1832645, DOI 10.1142/4100
- Itaï Ben Yaacov
- Affiliation: Université Claude Bernard – Lyon 1, Institut Camille Jordan, CNRS UMR 5208, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
- MR Author ID: 699648
- Received by editor(s): June 27, 2012
- Received by editor(s) in revised form: July 8, 2012, July 10, 2012, and July 31, 2012
- Published electronically: March 12, 2014
- Additional Notes: The author’s research was supported by the Institut Universitaire de France and ANR contract GruPoLoCo (ANR-11-JS01-008).
- Communicated by: Thomas Schlumprecht
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 142 (2014), 2459-2467
- MSC (2010): Primary 46B99, 22A05, 54D35
- DOI: https://doi.org/10.1090/S0002-9939-2014-11956-3
- MathSciNet review: 3195767