Structure theory for noncommutative Jordan \(H^ *\)-algebras
type: Item
instance of: scholarly article
title: Structure theory for noncommutative Jordan \(H^ *\)-algebras
zbMATH Open document ID: 0616.46047
DOI: 10.1016/0021-8693(87)90018-4
author: José Antonio Cuenca Mira
author: Ángel Rodríguez Palacios
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The theory of associative Hilbert algebras as developed in [\textit{W. Ambrose}, Trans. Am. Math. Soc. 57, 364-386 (1945; Zbl 0060.269)] has been extended to various classes of nonassociative algebras, among them Jordan algebras, cf. [\textit{C. Viola Devapakkiam}, Math. Proc. Camb. Philos. Soc. 78, 293-300 (1975; Zbl 0357.17015) and with \textit{P. S. Rema}, ibid. 79, 307-319 (1976; Zbl 0357.17016)]. In this paper the authors study general nonassociative Hilbert algebras on three levels of abstraction. In {\S} 1, the most general setting, an H*-algebra is defined as a complex nonassociative algebra whose vector space is a Hilbert space satisfying (xy\(| z)=(x| zy*)=(y| x*z)\) for an involution \(x\to x*\). It is shown that a nonzero H*-algebra is semiprime iff it is the closure of an orthogonal sum of topologically simple \(H^*\)-algebras (namely its minimal closed ideals). In {\S} 2 noncommutative Jordan \(H^*\)-algebras are characterized. They are either anticommutative or commutative or quadratic or quasi- associative. In {\S} 3 the authors prove a coordinatization theorem for Jordan H*- algebras and use this to clarify topologically simple Jordan H*-algebras. The classification is a special case of {\S} 3 in [\textit{W. Kaup}, Math. Ann. 262, 57-75 (1983; Zbl 0482.32011)] since every Jordan H*-algebra is a JH*-triple which are classified in [W. Kaup (loc. cit.)]. The methods used in the two papers are different.
Mathematics Subject Classification ID: 46H70
Mathematics Subject Classification ID: 17C65
Mathematics Subject Classification ID: 46K15
Mathematics Subject Classification ID: 17C50
Mathematics Subject Classification ID: 17A15
zbMATH DE Number: 3998528
zbMATH Keywords: nonassociative Hilbert algebras
zbMATH Keywords: H*-algebra
zbMATH Keywords: noncommutative Jordan \(H^ *\)-algebras
zbMATH Keywords: coordinatization theorem
zbMATH Keywords: topologically simple Jordan H*- algebras
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90018-4
OpenAlex ID: W4213290886
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description: scientific article; zbMATH DE number 3998528