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Existence of Jordan algebras of selfadjoint operators of a given type. (English. Russian original) Zbl 0567.46035
Sib. Math. J. 25, 689-693 (1984); translation from Sib. Mat. Zh. 25, No. 5(147), 3-8 (1984).
Weakly closed Jordan algebras of bounded selfadjoint operators on a Hilbert space (JW-algebras) were considered by 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.
MSC:
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
46L35 Classifications of \(C^*\)-algebras
17C65 Jordan structures on Banach spaces and algebras
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