On the Structure of Third-Power Associative Absolute Valued Algebras

Abstract

This paper deals with pairs of nonzero idempotents e and f of a third-power associative absolute valued algebra A satisfying $(ef)e= e(fe)$ and $(fe)f=f(ef)$ (pairwise flexible idempotents), and the role that they play on the structure of A. We show that if g is a nonzero idempotent of A such that the nonzero idempotents commuting with g are pairwise flexible, then the subalgebra that they generate $B_g$ is isometrically isomorphic to ${\mathbb {R}}$, $\mathop {{\mathbb {C}}}\limits ^{\star }$, $\mathop {{\mathbb {H}}}\limits ^{\star }$, or $\mathop {{\mathbb {O}}}\limits ^{\star }$. Our main theorem proves the equivalence of the following assertions: (i) for every two different nonzero idempotents e and f, the nonzero idempotents of A that commute with $(e-f)^2$ are pairwise flexible; (ii) each pair of nonzero idempotents of A generates a finite-dimensional subalgebra; and (iii) either A is isometrically isomorphic to ${\mathbb {R}}$, ${\mathbb {C}}$, ${\mathbb {H}}$, ${\mathbb {O}}$, $\mathop {{\mathbb {C}}}\limits ^{\star }$, $\mathop {{\mathbb {H}}}\limits ^{\star }$ or $\mathop {{\mathbb {O}}}\limits ^{\star }$, or A contains a subalgebra B, which contains all idempotents of A and is isometrically isomorphic to the division absolute valued algebra ${\mathbb {P}}$ of the Okubo pseudo-octonions. More consequences on the structure of A related with the presence of pairwise flexible idempotents are given, among them several generalizations of some well-known theorems.