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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.

Information

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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References

REFERENCES

[1]Becerra, J., Burgos, M., Kaidi, A. and Rodr'iguez, A.. Banach algebras with large groups of unitary elements. Quart. J. Math 58 (2007) 203–220.Google Scholar
[2]Becerra, J., Cowell, S., Rodr'iguez, A. and Wood, G. V.. Unitary Banach algebras. Studia Math 162 (2004), 25–51.Google Scholar
[3]Becerra, J., L'opez, G., Peralta, A. M. and Rodr'iguez, A.. Relatively weakly open sets in closed balls of Banach spaces, and real JB*-triples of finite rank. Math. Ann 330 (2004), 45–58.Google Scholar
[4]Becerra, J. and Martín, M.. The Daugavet property of C*-algebras, JB*-triples, and of their isometric preduals. J. Funct. Anal 224 (2005), 316–337.CrossRefGoogle Scholar
[5]Becerra, J. and Rodriguez, A.. The geometry of convex transitive Banach spaces. Bull. London Math. Soc. 31 (1999), 323–331.CrossRefGoogle Scholar
[6]Becerra, J. and Rodriguez, A.. Transistivity of the norm on Banach spaces having a Jordan structure. Manuscripta Math 102 (2000), 111–127.CrossRefGoogle Scholar
[7]Becerra, J., Rodriguez, A. and Wood, G.. Banach spaces whose algebras of operators are unitary: a holomorphic approach. Bull. London Math. Soc 35 (2003), 218–224.Google Scholar
[8]Bonsall, F. F. and Duncan, J.. Complete Normed Algebras (Springer-Verlag, 1973).CrossRefGoogle Scholar
[9]Cowie, E. R.. Isometries in Banach algebras. Ph. D. thesis. Swansea (1981).Google Scholar
[10]Cowie, E. R.. An analytic characterization of groups with no finite conjugacy classes. Proc. Amer. Math. Soc 87 (1983), 7–10.CrossRefGoogle Scholar
[11]Foias, C.. Sur certains theoremes de J. von Neumann concernant les ensembles spectraux. Acta Sci. Math 18 (1957), 15–20.Google Scholar
[12]Hansen, M. L. and Kadison, R. V.. Banach algebras with unitary norms. Pacific J. Math 175 (1996), 535–552.CrossRefGoogle Scholar
[13]Isidro, J. M., Kaup, W. and Rodríguez, A.. On real forms of JB*-triples. Manuscripta Math 86 (1995), 311–335.CrossRefGoogle Scholar
[14]Jacobson, N.. Structure of rings. Amer. Math. Soc. Coll. Publ. 37 (Providence, Rhode Island 1968).Google Scholar
[15]Kakutani, S. and Mackey, G. W.. Two characterizations of real Hilbert space. Ann. of Math 45 (1944), 50–58.CrossRefGoogle Scholar
[16]Kalton, N. J.. Spaces of compact operators. Math. Ann 208 (1974), 267–278.CrossRefGoogle Scholar
[17]Kaup, W.. Über die Automorphismen Grasmannerscher Mannigfaltigkeiten unendlicher Dimension. Math. Z 144 (1975), 75–96.CrossRefGoogle Scholar
[18]Kaup, W.. A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z 183 (1983), 503–529.CrossRefGoogle Scholar
[19]Martìnez, J. and Peralta, A. M.. Separate weak*-continuity of the triple product in dual real JB*-triples. Math. Z 234, 635–646.CrossRefGoogle Scholar
[20]Navarro, M. A.. Some characterizations of finite-dimensional Hilbert spaces. J. Math. Anal. Appl 223 (1998), 364–365.CrossRefGoogle Scholar
[21]Peralta, A. M. and Stach'o, L. L.. Atomic decomposition of real JBW*-triples. Quart. J. Math 52 (2001), 79–87.CrossRefGoogle Scholar
[22]Rickart, C. E.. General Theory of Banach Algebras (Kreiger, 1974).Google Scholar
[23]Rodrìguez, A.. Absolute valued algebras of degree two. In Nonassociative Algebra and its applications (Ed. S. González), 350–356 (Kluwer Academic Publishers, 1994).Google Scholar
[24]Rolewicz, S.. Metric Linear Spaces (Reidel, 1985).Google Scholar
[25]Wood, G. V.. Maximal algebra norms. Contemp. Math 321 (2003), 335–345.CrossRefGoogle Scholar