Absolute Valued Algebras with Involution

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 .[Crossref] , [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.