Existence of Jordan algebras of selfadjoint operators of a given type (Q1059823)

Weakly closed Jordan algebras of bounded selfadjoint operators on a Hilbert space (JW-algebras) were considered by \textit{D. Topping} as a real nonassociative counterpart of von Neumann algebras [Mem. Am. Math. Soc. 53, 48 p. (1965; Zbl 0137.102)]. He obtained the classification of JW- algebras by the types \(I_{fin}\), \(I_{\infty}\), \(II_ 1\), \(II_{\infty}\) and III and proved that if a JW-algebra is the selfadjoint part of a von Neumann algebra then this classification coincides with the usual one. The main result of the present paper shows the existence of JW-factors of any given type which are not isomorphic to the selfadjoint part of a von Neumann algebra.