Lipschitz groups and Lipschitz maps

Document Type : Research Paper

Author

University Paris 13, Paris Sorbonne Cité

Abstract

‎‎This contribution mainly focuses on some aspects of Lipschitz groups‎, ‎i.e.‎, ‎metrizable groups with Lipschitz multiplication and inversion map‎. ‎In the main result it is proved that metric groups‎, ‎with a translation-invariant metric‎, ‎may be characterized as particular group objects in the category of metric spaces and Lipschitz maps‎. ‎Moreover‎, ‎up to an adjustment of the metric‎, ‎any metrizable abelian group also is shown to be a Lipschitz group‎. ‎Finally we present a result similar to the fact that any topological nilpotent element $x$ in a Banach algebra gives rise to an invertible element $1-x$‎, ‎in the setting of complete Lipschitz groups‎.

Keywords

Main Subjects

References

[1] J. App ell, E. De Pascale and A. Vignoli, Nonlinear spectral theory, de Gruyter Series in Nonlinear Analysis and Applications, 10, Walter de Gruyter, 2004.
[2] A. Arkhangel'skii and M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, 1, World Scientic Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.
[3] P. Assouad, Plongements Lipschitziens dans Rn, Bul l. Soc. Math. France, 111 (1983) 429-448.
[4] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales (French), Fund. Math., 3 (1922) 133-181.
[5] B. Eckmann and P. J. Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann., 145 (1962) 227-255.
[6] M. Gromov, Geometric group theory - volume 2: Asymptotic invariants of innite groups, London Mathematical So ciety Lecture Note Series, 182, Cambridge University Press, 1991.
[7] V. L. Klee, Jr., Invariant metrics in groups (solution of a problem of Banach), Proc. Amer. Math. Soc., 3 (1952) 484-487.
[8] W. Kubs and M. Rubin, Extension and reconstruction theorems for the Urysohn universal metric space, Czechoslo-vak Math. J., 60 (2010) 1-29.
[9] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.
[10] P. Pansu, Metriques de Carnot-Caratheo dory et quasiisometries des espaces symetriques de rang un, Ann. of Math. (2), 129 (1989) 1-60.
[11] A. Papadop oulos, Metric spaces, convexity and nonpositive curvature, IRMA Lectures in Mathematics and Theo-retical Physics, 6, Europ ean Mathematical So ciety, 2005.
[12] B. Pavlovic, The category of compact metric spaces and its functional analytic duals, Publ. Inst. Math. (Beograd) (N.S.), 72 (2002) 29-38.
[13] C. E. Rickart, General theory of Banach algebras, The University series in higher mathematics, R. E. Krieger Pub. Co., 1974.
[14] D. R. Sherb ert, Banach algebras of Lipschitz functions, Pacic J. Math., 13 (1963) 1387-1399.
[15] N. Weaver, Lipschitz algebras, World Scientic Publishing Co., Inc., River Edge, NJ, 1999.

Files

History

  • Receive Date: 08 April 2013
  • Revise Date: 20 August 2015
  • Accept Date: 21 August 2015
  • Published Online: 01 March 2017

Share

How to cite

Statistics