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ZEL'MANOV'S THEOREM FOR PRIMITIVE JORDAN–BANACH ALGEBRAS

Published online by Cambridge University Press:  01 February 1998

M. CABRERA GARCÍA ,
A. MORENO GALINDO  and
A. RODRÍGUEZ PALACIOS
M. CABRERA GARCÍA
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain. E-mail: apalacio@goliat.ugr.es
A. MORENO GALINDO
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain. E-mail: apalacio@goliat.ugr.es
A. RODRÍGUEZ PALACIOS
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071-Granada, Spain. E-mail: apalacio@goliat.ugr.es
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Abstract

The following result is well known and easy to prove (see [14, Theorem 2.2.6]).

Theorem 0. If A is a primitive associative Banach algebra, then there exists a Banach space X such that A can be seen as a subalgebra of the Banach algebra BL(X) of all bounded linear operators on X in such a way that A acts irreducibly on X and the inclusion A[rarrhk ]BL(X) is continuous.

In fact, if X is any vector space on which the primitive Banach algebra A acts faithfully and irreducibly, then X can be converted in a Banach space in such a way that the requirements in Theorem 0 are satisfied and even the inclusion A[rarrhk ]BL(X) is contractive.

Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 231 - 244
Copyright
The London Mathematical Society 1998

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