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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
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