Noetherian Jordan Banach algebras are finite-dimensional
type: Item
instance of: scholarly article
title: Noetherian Jordan Banach algebras are finite-dimensional
zbMATH Open document ID: 0923.46049
published in: Journal of Algebra
publication date: 1999-07-28T00:00:00Z
review text: An associative or alternative algebra is Noetherian if it satisfies the ascending chain condition on left ideals. \textit{A. M. Sinclair} and \textit{A. W. Tullo} [Math. Ann. 211, 151-153 (1974; Zbl 0275.46037)] showed that a complex Noetherian Banach associative algebra is finite-dimensional. This result was extended by the authors to the alternative case. For a Jordan algebra \(J\), the suitable Noetherian condition is the ascending chain condition on inner ideals. In the paper under review, the authors prove that a complex Noetherian Banach Jordan algebra is finite-dimensional. The proof uses fundamental results on structure theory of Jordan algebras such as nilpotence theorem, prime theorem and Goldie theorem for Jordan algebras, all of them due to E. Zelmanov, and powerful techniques of functional analysis. Reviewer's remark: As shown in a recent paper written by this reviewer jointly with other authors, a real Noetherian Banach Jordan algebra \(J\) is not necessarily finite-dimensional. However, if \(J\) is nondegenerate then it has finite capacity.
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 1270885
zbMATH Keywords: inner ideal
zbMATH Keywords: small radical
zbMATH Keywords: nilpotence theorem
zbMATH Keywords: prime theorem
zbMATH Keywords: Goldie theorem
zbMATH Keywords: Noetherian Banach associative algebra
zbMATH Keywords: Noetherian Banach Jordan algebra
author: Mohamed Benslimane
author: Nadia Boudi
reviewed by: Antonio Fernández López
MaRDI profile type: Publication
full work available at URL: jabr.1998.7650
OpenAlex ID: W2015069820
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DOI: 10.1006/JABR.1998.7650
description: scientific article; zbMATH DE number 1270885