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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Kaplansky theorem for JB*-triples


Authors: Francisco J. Fernández-Polo, Jorge J. Garcés and Antonio M. Peralta
Journal: Proc. Amer. Math. Soc. 140 (2012), 3179-3191
MSC (2010): Primary 46K70, 46L05, 46L10, 46L70; Secondary 17C65
Published electronically: January 18, 2012
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Abstract: Let $ T:E\rightarrow F$ be a not necessarily continuous triple homomorphism from a (complex) JB$ ^*$-triple (respectively, a (real) J$ ^*$B-triple) to a normed Jordan triple. The following statements hold:

  1. $ T$ has closed range whenever $ T$ is continuous.
  2. $ T$ is bounded below if and only if $ T$ is a triple monomorphism.
This result generalises classical theorems of I. Kaplansky and S.B. Cleveland in the setting of C$ ^*$-algebras and of A. Bensebah and J. Pérez, L. Rico and A. Rodríguez Palacios in the setting of JB$ ^*$-algebras.

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Additional Information

Francisco J. Fernández-Polo
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: pacopolo@ugr.es

Jorge J. Garcés
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: jgarces@correo.ugr.es

Antonio M. Peralta
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Email: aperalta@ugr.es

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11157-8
PII: S 0002-9939(2012)11157-8
Received by editor(s): February 23, 2010
Received by editor(s) in revised form: September 20, 2010, and March 28, 2011
Published electronically: January 18, 2012
Additional Notes: The authors were partially supported by D.G.I. project No. MTM2008-02186 and Junta de Andalucía grants FQM0199 and FQM3737.
Communicated by: Marius Junge
Article copyright: © Copyright 2012 American Mathematical Society