Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents

Abstract

This paper deals with the determination of the absolute valued algebras with a nonzero idempotent commuting with the remaining idempotents and satisfying $x^2 x = x x^2 $ for every $x$. We prove that, in addition to the absolute valued algebras $\mathbb R $, $\mathbb C $, $\mathbb H $, or $\mathbb O $ of the reals, complexes, division real quaternions or division real octonions, one such absolute valued algebra $A$ can also be isometrically isomorphic to some of the absolute valued algebras $\overset{\star}{\mathbb C}$, $\overset{\star}{\mathbb H}$, or $\overset{\star}{\mathbb O}$, obtained from $\mathbb C $, $\mathbb H$, and $\mathbb O $ by imposing a new product defined by multiplying the conjugates of the elements. In particular, every absolute valued algebra having the above properties is finite-dimensional. This generalizes some well known theorems of Albert, Urbanik and Wright, and El-Mallah.