On Absolute Valued Algebras with a Central Algebraic Element and Satisfying Some Identities

Abstract

In [8], we have proven that if  is an absolute valued algebra containing a nonzero central algebraic element, then  is a pre-Hilbert algebra. Here we show that    is finite dimensional in the following cases:

1) A satisfies  (x², x, x) = 0 or (x, x, x²) = 0,

2) A satisfies  (x², x², x) = 0 or (x, x², x²) = 0, .

In these cases  A is isomorphic to R, C, H or O.