Absolute Valued Algebras with Involution |
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Authors: | Abdellatif Rochdi Ángel Rodriguez Palacios |
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Institution: | 1. Department of Mathematics and Informatics, Faculty of Science , University Hassan II , Casablanca , Morocco abdellatifroc@hotmail.com;3. Departamento de Análisis Matemática , Universidad de Granada, Facultad de Ciencias , Granada , Spain |
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Abstract: | We study absolute valued algebras with involution, as defined in Urbanik (1961
Urbanik , K. ( 1961 ). Absolute valued algebras with an involution . Fundamenta Math. 49 : 247 – 258 . Google Scholar]). We prove that these algebras are finite-dimensional whenever they satisfy the identity (x, x 2, x) = 0, where (·, ·, ·) means associator. We show that, in dimension different from two, isomorphisms between absolute valued algebras with involution are in fact *-isomorphisms. Finally, we give a classification up to isomorphisms of all finite-dimensional absolute valued algebras with involution. As in the case of a parallel situation considered in Rochdi (2003
Rochdi , A. ( 2003 ). Eight-dimensional real absolute valued algebras with left unit whose automorphism group is trivial . Int. J. Math. Math. Sci. 70 : 4447 – 4454 .Crossref] , Google Scholar]), the triviality of the group of automorphisms of such an algebra can happen in dimension 8, and is equivalent to the nonexistence of 4-dimensional subalgebras. |
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Keywords: | Absolute valued algebras Automorphisms Central idempotent Involution |
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