Structure theory for real noncommutative Jordan \(H^*\)-algebras (Q1322061)

An \(H^*\)-algebra is an algebra defined on a real or complex Hilbert space, with inner product \((\cdot|\cdot)\), together with an involution \(*\) such that \((xy| z)= (y| x^* z)=(x| zy^*)\). This paper is devoted to the study of the real noncommutative Jordan \(H^*\)-algebras. The complex case was dealt with by \textit{J. A. Cuenca} and \textit{A. Rodríguez} [J. Algebra 106, 1-14 (1987; Zbl 0616.46047)]. The authors first reduce the structure theory to the classification of the topologically simple \(H^*\)-algebras. A careful study of the \(H^*\)-analogues of the finite dimensional simple noncommutative Jordan algebras is then carried out. The final result asserts that everything goes smoothly: the real topologically simple noncommutative Jordan \(H^*\)-algebras are more or less analogues of the simple finite dimensional noncommutative Jordan algebras.