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Banach spaces whose algebras of operators have a large group of unitary elements

Published online by Cambridge University Press:  01 January 2008

JULIO BECERRA GUERRERO ,
MARÍA BURGOS ,
EL AMIN KAIDI  and
ÁNGEL RODRÍGUEZ PALACIOS
JULIO BECERRA GUERRERO
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, Facultad de Ciencias, 18071-Granada, Spain. e-mail: juliobg@ugr.es
MARÍA BURGOS
Affiliation:
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Facultad de Ciencias Experimentales, 04120-Almería, Spain. e-mail: mburgos@ual.es; elamin@ual.es
EL AMIN KAIDI
Affiliation:
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Facultad de Ciencias Experimentales, 04120-Almería, Spain. e-mail: mburgos@ual.es; elamin@ual.es
ÁNGEL RODRÍGUEZ PALACIOS
Affiliation:
Departamento de Álgebra y Análisis Matemático, Universidad de Almería, Facultad de Ciencias Experimentales, 04120-Almería, Spain. e-mail: mburgos@ual.es; elamin@ual.es
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Abstract

We prove that a complex Banach space X is a Hilbert space if (and only if) the Banach algebra (of all bounded linear operator on X) is unitary and there exists a conjugate-linear algebra involution • on satisfying T• = T−1 for every surjective linear isometry T on X. Appropriate variants for real spaces of the result just quoted are also proven. Moreover, we show that a real Banach space X is a Hilbert space if and only if it is a real JB*-triple and is -unitary, where stands for the dual weak-operator topology.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 1 , January 2008 , pp. 97 - 108
Copyright
Copyright © Cambridge Philosophical Society 2008

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