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On the structure of third-power associative absolute valued algebras. (English) Zbl 1369.17004
Summary: 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.

##### MSC:
 17A80 Valued algebras 17A75 Composition algebras 17A60 Structure theory for nonassociative algebras 17D99 Other nonassociative rings and algebras
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