Third-power associative absolute valued algebras with all its idempotents pairwise flexible
type: Item
instance of: scholarly article
title: Third-power associative absolute valued algebras with all its idempotents pairwise flexible
zbMATH Open document ID: 1405.17006
published in: Mediterranean Journal of Mathematics
publication date: 2017-11-09T00:00:00Z
review text: On the one hand, an algebra \(A\) is called a \textit{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 \textit{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.
Mathematics Subject Classification ID: 17A80
zbMATH DE Number: 6804292
zbMATH Keywords: absolute valued algebras
author: José Antonio Cuenca Mira
reviewed by: Antonio J. Calderón Martín
MaRDI profile type: Publication
full work available at URL: s00009-017-0996-5
OpenAlex ID: W2751445774
cites work: Absolute-valued algebraic algebras
cites work: Non-Associative Normed Algebras
cites work: On one-sided division infinite-dimensional normed real algebras
cites work: On composition and absolute-valued algebras
cites work: Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents
cites work: On the structure of third-power associative absolute valued algebras
cites work: Sur les algèbres absolument valuées qui vérifient l'identité \((x,x,x)=0\)
cites work: Absolute valued algebras containing a central idempotent
cites work: Sur les dimensions des algèbres absolument valuees
cites work: Q3207166
cites work: One-sided division absolute valued algebras
cites work: Absolute-Valued Algebras
DOI: 10.1007/S00009-017-0996-5
Recommended article: Third-power associative absolute valued algebras with a nonzero idempotent commuting with all idempotents
Recommended article: On the structure of third-power associative absolute valued algebras
Recommended article: Third power associative composition algebras
Recommended article: Absolute Valued Strongly Power-Associative Triple Systems
Recommended article: Q3815437
Recommended article: Q3342710
Recommended article: Third power associative rings satisfying a ( bc )= b ( ca )
Recommended article: Nonassociative Algebras Satisfying Identities of Degree Three
Recommended article: FLEXIBLE POWER-ASSOCIATIVE ALGEBRAS OF DEGREE 1
Recommended article: Third power associative, antiflexible rings satisfying (a, b, bc) = b(a, b, c)
description: scientific article; zbMATH DE number 6804292