Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T13:05:52.996Z Has data issue: false hasContentIssue false
  • English
  • Français

Isometries on extremely non-complex Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  14 July 2010

Piotr Koszmider ,
Miguel Martín  and
Javier Merí
Piotr Koszmider
Affiliation:
Instytut Matematyki Politechniki Łódzkiej, ul. Wólczańska 215, 90-924 Łódź, Poland (pkoszmider.politechnika@gmail.com)
Miguel Martín
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (mmartins@ugr.es; jmeri@ugr.es)
Javier Merí
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (mmartins@ugr.es; jmeri@ugr.es)
Get access Rights & Permissions [Opens in a new window]

Abstract

Given a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that ‖ Id + T2 ‖ = 1 + ‖ T2 ‖ for every bounded linear operator T on it) whose dual contains E* as an L-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to ±Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

Keywords

few operatorscomplex structureDaugavet equationBanach spaces of continuous functionsextremely non-complexsurjective isometries

MSC classification

Secondary: 46B04: Isometric theory of Banach spaces 46B10: Duality and reflexivity
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 2 , April 2011 , pp. 325 - 348
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abramovich, Y. and Aliprantis, C., An invitation to operator theory, Graduate Texts in Mathematics, Volume 50 (American Mathematical Society, Providence, RI, 2002).Google Scholar
2.Abramovich, Y. and Aliprantis, C., Problems in operator theory, Graduate Texts in Mathematics, Volume 51 (American Mathematical Society, Providence, RI, 2002).Google Scholar
3.Albiac, F. and Kalton, N. J., Topics in Banach space theory, Graduate Texts in Mathematics, Volume 233 (Springer, 2006).Google Scholar
4.Daugavet, I. K., On a property of completely continuous operators in the space C, Usp. Mat. Nauk 18 (1963), 157–158 (in Russian).Google Scholar
5.Diestel, J., Sequences and series in Banach spaces, Graduate Texts in Mathematics, Volume 92 (Springer, 1984).Google Scholar
6.Dieudonné, J., Complex structures on real Banach spaces, Proc. Am. Math. Soc. 3 (1952), 162–164.CrossRefGoogle Scholar
7.Engel, K. J. and Nagel, R., One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, Volume 194 (Springer, 2000).Google Scholar
8.Engel, K. J. and Nagel, R., A short course on operator semigroups, Universitext (Springer, 2006).Google Scholar
9.Fajardo, R., Consistent constructions of Banach spaces C(K) with few operators, PhD Thesis, Universidade de São Paulo (2007).Google Scholar
10.Ferenczi, V., Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (2007), 462–488.CrossRefGoogle Scholar
11.Ferenczi, V. and Galego, E. Medina, Even infinite-dimensional real Banach spaces, J. Funct. Analysis 253 (2007), 534–549.CrossRefGoogle Scholar
12.Fleming, R. and Jamison, J., Isometries on Banach spaces: function spaces, Monographs and Surveys in Pure and Applied Mathematics, Volume 129 (>Chapman & Hall/CRC, Boca Raton, FL, 2003).Google Scholar
13.Fleming, R. and Jamison, J., Isometries on Banach spaces: vector-valued function spaces and operator spaces, Monographs and Surveys in Pure and Applied Mathematics, Volume 132 (Chapman & Hall/CRC, Boca Raton, FL, 2008).Google Scholar
14.Godefroy, G., Existence and uniqueness of isometric preduals: a survey, in Banach Space Theory, Iowa City, IA, 1987, Contemporary Mathematics, Volume 85, pp. 131–193 (American Mathematical Society, Providence, RI, 1989).Google Scholar
15.Harmand, P., Werner, D. and Werner, W., M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, Volume 1547 (Springer, 1993).Google Scholar
16.Kadets, V., Martín, M. and Merí, J., Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007), 2385–2411.Google Scholar
17.Kadets, V., Shvidkoy, R., Sirotkin, G. and Werner, D., Banach spaces with the Daugavet property, Trans. Am. Math. Soc. 352 (2000), 855–873.CrossRefGoogle Scholar
18.Koppelberg, S., Handbook of Boolean algebras, Volume 1 (North-Holland, Amsterdam, 1989).Google Scholar
19.Koszmider, P., Banach spaces of continuous functions with few operators, Math. Annalen 330 (2004), 151–183.CrossRefGoogle Scholar
20.Koszmider, P., Martín, M. and Merí, J., Extremely non-complex C(K) spaces, J. Math. Analysis Applic. 350 (2009), 584–598.CrossRefGoogle Scholar
21.Martín, M., The group of isometries of a Banach space and duality, J. Funct. Analysis 255 (2008), 2966–2976.CrossRefGoogle Scholar
22.Munroe, M. E., Measure and integration, 2nd edn (Addison-Wesley, Reading, MA, 1971).Google Scholar
23.Oikhberg, T., Some properties related to the Daugavet property, in Banach spaces and their applications in analysis, pp. 399–401 (Walter de Gruyter, Berlin, 2007).Google Scholar
24.Sakai, S., C*-algebras and W*-algebras (Springer, 1971).Google Scholar
25.Semadeni, Z., Banach spaces of Continuous functions Volume I, Monografie Matematyczne, Tom 55 (PWN/Polish Scientific Publishers, Warsaw, 1971).Google Scholar
26.Werner, D., An elementary approach to the Daugavet equation, in Interaction between functional analysis, harmonic analysis and probability (ed. Kalton, N., Saab, E. and Montgomery-Smith, S.), Lecture Notes in Pure and Applied Mathematics, Volume 175, pp. 449–454 (CRC Press, Boca Raton, FL, 1994).Google Scholar
27.Werner, D., Recent progress on the Daugavet property, Irish Math. Soc. Bull. 46 (2001), 77–97.CrossRefGoogle Scholar