On Absolute Valued Algebras Containing a Central Algebraic Element

Abstract

Let  be an absolute valued algebra containing a nonzero central algebraic element. Then  is a pre-Hilbert algebra and is finite dimensional in the following cases:

1) A satisfies (x, x, x)=0.

2) A satisfies (x², x² , x² )=0.

3) A satisfies (x, x², x)=0.

In these cases  is isomorphic to  or . It may be conjectured that every absolute valued algebra containing a nonzero central element is pre-Hilbert algebra.