Structure theory for noncommutative Jordan \(H^ *\)-algebras (Q1821311)

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.