Uniform primeness of the Jordan algebra of Hermitian quaternion matrices.
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 follows that every prime JB*-algebra and every prime JB*-triple (respectively, prime real JB*-triple) is ultraprime with constant \(k=\frac{1}{6}\) (respectively \(k=\frac{1}{12}\)). In the paper under review, the author improves the ultraprimeness constant in the case of the real JB*-triple of all hermitian operators on a finite-dimensional quaternionic Hilbert space by showing that, in this case, \(k=\frac{1}{12}\) can be replaced with \(\sqrt{2} -1\).
scientific article; zbMATH DE number 1917815