Absolute Valued Algebras with Involution

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, x2, 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.