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Noetherian Banach Jordan pairs

Published online by Cambridge University Press:  01 March 2001

N. BOUDI ,
H. MARHNINE ,
C. ZARHOUTI ,
A. FERNANDEZ LOPEZ  and
E. GARCIA RUS
N. BOUDI
Affiliation:
Département de Mathématiques, Faculté des Sciences, B.P. 2121, Tétouan, Maroc
H. MARHNINE
Affiliation:
Département de Mathématiques, Faculté des Sciences, B.P. 2121, Tétouan, Maroc
C. ZARHOUTI
Affiliation:
Département de Mathématiques, Faculté des Sciences, B.P. 2121, Tétouan, Maroc
A. FERNANDEZ LOPEZ
Affiliation:
Departamento de Algebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
E. GARCIA RUS
Affiliation:
Departamento de Algebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
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Abstract

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.

For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.

Here we show the following results:

(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;

(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;

(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 130 , Issue 1 , January 2001 , pp. 25 - 36
Copyright
2001 Cambridge Philosophical Society

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Footnotes

This work is supported by ‘la Junta de Andalucía dentro del Convernio Marco de Cooperación Hispano-Marroquí’, by ‘l'Agence Espagnole de Coopération Internationale (Action Integrée 52/SEE/98)’ and by the DGICYT Grant PB97-1069-C02-01.