Finite-dimensional absolute-valued algebras
type: Item
instance of: scholarly article
title: Finite-dimensional absolute-valued algebras
zbMATH Open document ID: 1290.17002
author: María Isabel Ramírez Alvarez
author: Antonio J. Calderón Martín
author: Amin Kaidi
author: Candido Martín González
author: A. Morales-Campoy
published in: Israel Journal of Mathematics
publication date: 2012-11-19T00:00:00Z
review text: A real (not necessarily associative) algebra \(A\neq 0\) is said to be an \textit{absolute-valued algebra} if \(A\) is a normed vector space such that \(\| xy\| = \| x\| \, \| y\| \) for all \(x, \, y \in A.\) The case that \(A\) has finite dimension has been considered by A. A. Albert. First he proves that a finite-dimensional absolute-valued algebra having a unit element is necessarily isomorphic to \({\mathbb R}\), \({\mathbb C}\), \({\mathbb H}\) or \({\mathbb O}\). Furthermore, in the finite-dimensional (but not necessarily unital) case, he takes a norm-one element \(a\in A\) and he introduces a new product \(\bullet \) in \(A\) defined by \(x\bullet y = R_a^{-1}(x)L_a^{-1}(y)\), where \(R_a\) (resp. \(L_a\)) is the linear operator of right (resp. left) multiplication \(R_a : x\mapsto xa\) (resp. \(L_a : x\mapsto ax\)). Now he observes that \((A, \bullet )\) is an absolute valued algebra with a unit element (namely, \(a^2\)). In consequence, he obtains that if \(C\) is a finite-dimensional absolute-valued algebra, with product denoted by \(\diamond \), then there are two isometric linear maps \(\varphi \) and \(\psi\) and a new product in the normed vector space \(C\), denoted by juxtaposition, which makes of this vector space a unitary absolute-valued algebra \(B\) such that \(x\diamond y = \varphi (x)\psi (y)\) for all \(x, \, y\in C\). On the other hand, the existence of nonzero idempotents in every finite-dimensional absolute-valued algebra implies easily that \(\varphi \) and \(\psi\) can be chosen satisfying, in addition, \(\varphi (1) = 1=\psi (1)\). This characterization of \(C\) in terms of the unital absolute-valued algebra \(B\) has many interesting consequences on the structure of \(C.\) For example, it makes possible an easy determination of the center \(Z(C)\) of \(C,\) and it allows to establish a criterion for isomorphism of two of these absolute-valued algebras \(C_1\) and \(C_2 .\) Well-known examples of no-unitary absolute valued algebras are…
reviewed by: José Antonio Cuenca Mira
Mathematics Subject Classification ID: 17A80
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 17A60
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 6108356
zbMATH Keywords: absolute-valued algebra
zbMATH Keywords: idempotent
zbMATH Keywords: division algebra
zbMATH Keywords: isometric linear map
MaRDI profile type: Publication
full work available at URL: s11856-011-0065-x
OpenAlex ID: W2072982803
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DOI: 10.1007/S11856-011-0065-X
description: scientific article; zbMATH DE number 6108356