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Zel'Manov's Theorem for Primitive Jordan–Banach Algebras
Author(s) -
Cabrera García M.,
Moreno Galindo A.,
Rodríguez Palacios A.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005882
Subject(s) - mathematics , subalgebra , banach algebra , banach space , pure mathematics , eberlein–šmulian theorem , bounded function , bounded inverse theorem , algebra over a field , discrete mathematics , finite rank operator , lp space , mathematical analysis
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 ↪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 ↪BL( X ) is contractive. Roughly speaking, the aim of this paper is to prove the appropriate Jordan variant of Theorem 0.