Absolute valued algebras containing a central idempotent
type: Item
instance of: scholarly article
title: Absolute valued algebras containing a central idempotent
zbMATH Open document ID: 0688.17001
DOI: 10.1016/0021-8693(90)90048-S
author: Mohamed Lamei El-Mallah
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: A nonassociative algebra over the reals is called an absolute valued algebra if it is a normed space satisfying \(\| xy\| =\| x\| \| y\|\). It is proved here that any absolute valued algebra containing a central idempotent is actually an inner product space, so that the norm comes from the inner product. This allows the author to classify the third-power associative finite dimensional absolute valued algebras. These are, up to isomorphism, the Hurwitz and para-Hurwitz division algebras and the pseudo-octonion algebra defined by \textit{S. Okubo} [Hadronic J. 5, 1564-1612 (1982; Zbl 0547.17001)].
Mathematics Subject Classification ID: 17A60
Mathematics Subject Classification ID: 17D99
Mathematics Subject Classification ID: 17A99
zbMATH DE Number: 4127459
zbMATH Keywords: absolute valued algebra
zbMATH Keywords: central idempotent
zbMATH Keywords: inner product space
zbMATH Keywords: third- power associative
reviewed by: Alberto Elduque
MaRDI profile type: Publication
cites work: Sur les dimensions des algèbres absolument valuees
cites work: Sur les algèbres absolument valuées qui vérifient l'identité \((x,x,x)=0\)
cites work: On finite dimensional absolute valued algebras satisfying \((x,x,x)=0\)
cites work: Q3207166
cites work: Algebras with nondegenerate associative symmetric bilinear forms permitting composition, II
cites work: Q3338405
cites work: A Remark on M. M. Day's Characterization of Inner-Product Spaces and a Conjecture of L. M. Blumenthal
cites work: Absolute-Valued Algebras
cites work: Absolute Valued Algebras
description: scientific article; zbMATH DE number 4127459