Uniform primeness of the Jordan algebra of Hermitian quaternion matrices.
type: Item
instance of: scholarly article
title: Uniform primeness of the Jordan algebra of Hermitian quaternion matrices.
zbMATH Open document ID: 1050.46054
DOI: 10.1016/S0024-3795(02)00635-3
author: Rok Strašek
published in: Linear Algebra and its Applications
publication date: 2003-05-27T00:00:00Z
review text: In [Math. Anal. Aust. Natl. Univ. 21, 297--317 (1989; Zbl 0701.46027)], \textit{M. Mathieu} introduced the concept of ultraprime algebra in the class of associative Banach algebras. According to his definition, an associative Banach algebra \(A\) is ultraprime if and only if there exists a positive constant \(k\) such that \(| M_{a,b} | \geq k \;| a| \;| b|\) for all \(a,b\in A\), where \(M_{a,b}\) is the mapping on \(A\) given by \(M_{a,b} (x) := axb\). In the settings of non-associative Jordan Banach algebras and real and complex JB*-triples, the concept of ultraprimeness was introduced and developed by Cabrera, Martínez and Rodríguez [\textit{M. Cabrera} and \textit{J. Martínez}, Arch. Math. 68, No. 3, 221--227 (1997; Zbl 0876.47022), Proc. Am. Math. Soc. 125, No. 7, 2033--2039 (1997; Zbl 0870.47024); \textit{M. Cabrera} and \textit{A. Rodríguez}, Extr. Math. 5, No. 3, 127--129 (1990; Zbl 0744.46038), Q. J. Math., Oxf. II. Ser. 43, No. 169, 1--7 (1992; Zbl 0758.46037), Proc. Lond. Math. Soc., III. Ser. 69, No. 3, 576--604 (1994; Zbl 0809.46044)] and Bunce, Chu, Galindo, Rodríguez, Stacho and Zalar [\textit{L. Bunce}, \textit{C.-H. H. Chu}, \textit{L. L. Stacho} and \textit{B. Zalar}, Q. J. Math., Oxf. II. Ser. 49, No. 195, 279--290 (1998; Zbl 1042.46514); \textit{C.-H. H. Chu}, \textit{A. M. Galindo} and \textit{A. Rodríguez}, Contemp. Math. 232, 105--109 (1999; Zbl 0935.46058)], respectively. If \(J\) is a real or complex JB*-triple then \(J\) is ultraprime iff there exists a positive constant \(k\) such that \(\|Q(a,b)\|\geq k \;\|a\| \;\|b\|\) for all \(a,b\in J\), where \(Q(a,b)\) is the mapping defined by \(Q(a,b) (x) = \{a, x, b\}\). When \(J\) is a Jordan Banach algebra we say that \(J\) is ultraprime iff it is ultraprime when it is considered as a JB*-triple, that is, \(Q(a,b)\) is replaced by the Jacobson-McCrimmon operator \(U_{a,b}\) defined by \(U_{a,b} (x) := (a\circ x)\circ b + (b\circ x)\circ a - (a\circ b)\circ x\). From the paper cited above it f…
reviewed by: Antonio M. Peralta
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 46K70
Mathematics Subject Classification ID: 17C65
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 1917815
zbMATH Keywords: ultraprimeness
zbMATH Keywords: Jordan Banach algebras
MaRDI profile type: Publication
cites work: Q4215651
cites work: Inner derivations on ultraprime normed real algebras
cites work: Inner derivations on ultraprime normed algebras
cites work: Q3985421
cites work: NONASSOCIATIVE ULTRAPRIME NORMED ALGEBRAS
cites work: Non-Degenerately Ultraprime Jordan-Banach Algebras: A Zel'Manovian Treatment
cites work: Q4252787
cites work: Q4321748
cites work: Q3479025
cites work: Uniform primeness of the Jordan algebra of symmetric operators
cites work: Quaternions and matrices of quaternions
Recommended article: Uniform primeness of the Jordan algebra of symmetric operators
Recommended article: Strong primeness of hermitian Jordan systems
Recommended article: The structure of strongly prime quadratic Jordan algebras
Recommended article: On uniform bounds of primeness in matrix rings
Recommended article: Primitivity in finitely generated special quadratic Jordan algebras
Recommended article: The structure of primitive quadratic Jordan algebras
Recommended article: Prime quotients of Jordan systems and Lie algebras
Recommended article: Unital quadratic quasi-Jordan algebras
description: scientific article; zbMATH DE number 1917815