Absolute Valued Algebras with Involution

Abstract

We study absolute valued algebras with involution, as defined in Urbanik (1961). 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), 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.

Key Words:

• Absolute valued algebras
• Automorphisms
• Central idempotent
• Involution

2000 Mathematics Subject Classification:

• Primary 46E15, 46B04
• Secondary 46H70

Notes

*A la mémoire de notre regretté collègue le Professeur Mostafa Zaoui, qui nois a quitté prématurément le 10 Mars 2005.

Communicated by E. I. Zelmanov.