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Third-power associative absolute valued algebras with all its idempotents pairwise flexible. (English) Zbl 1405.17006
On the one hand, an algebra $$A$$ is called a third-power associative algebra if the identity $$x^2x=xx^2$$ holds for any $$x \in A$$.
On the other hand we recall that a real algebra $$A \neq 0$$ is said to be an absolute valued algebra if its vector space is a normed space whose norm $$| \cdot|$$ satisfies $$|xy|=|x|$$ $$|y|$$ for any $$x,y \in A$$.
The paper under review is devoted to study third power associative absolute valued algebras. In this framework it is known that if $$A$$ is finite-dimensional, then $$A$$ is (isometrically) isomorphic to $${\mathbb R}$$, $${\mathbb C}$$, $${\mathbb H}$$, $${\mathbb O}$$, $$\mathop{\mathbb C}\limits^*$$, $$\mathop{\mathbb H}\limits^*$$, $$\mathop{\mathbb O}\limits^*$$ or $${\mathbb P}$$. However, the classification in the infinite dimensional case is an open problem. In this line the author proves that in case either every two different nonzero idempotents of $$A$$ are pairwise flexible or each pair of different nonzero idempotents of $$A$$ generates a finite-dimensional subalgebra, then $$A$$ is finite dimensional and so (isometrically) isomorphic to one of the above ones.
##### MSC:
 17A80 Valued algebras
##### Keywords:
absolute valued algebras
Full Text:
##### References:
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