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Isometries on extremely non-complex Banach spaces

Published online by Cambridge University Press:  14 July 2010

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)

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.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2010

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