zbMATH — the first resource for mathematics

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.
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
46L35 Classifications of \(C^*\)-algebras
17C65 Jordan structures on Banach spaces and algebras
Full Text: DOI
[1] D. Topping, ?Jordan algebras of self-adjoint operators,? Mem. Am. Math. Soc.,53, 1-48 (1965). · Zbl 0137.10203
[2] E. Stormer, ?Jordan algebras of type I,? Acta Math.,115, Nos. 3-4, 165-184 (1966). · Zbl 0139.30502 · doi:10.1007/BF02392206
[3] E. Stormer, ?Irreducible Jordan algebras of self-adjoint operators,? Trans. Am. Math. Soc.,130, 153-166 (1968).
[4] Sh. A. Ayupov, ?Extension of traces and type criterions for Jordan algebras of self-adjoint operators,? Math. Z.,181, 253-268 (1982). · Zbl 0487.46045 · doi:10.1007/BF01215023
[5] E. Stormer, ?On the Jordan structure of C*-algebras,? Trans. Am. Math. Soc.,120, 438-447 (1965).
[6] E. Stormer, ?Real structure in the hyperfinite factors,? Duke Math. J.,47, No. 1 145-153 (1980). · Zbl 0462.46044 · doi:10.1215/S0012-7094-80-04711-0
[7] E. M. Alfsen, H. Hanche-Olsen, and F. W. Shultz, ?State spaces of C*-algebras,? Acta Math.,144, Nos. 3-4, 267-305 (1980). · Zbl 0458.46047 · doi:10.1007/BF02392126
[8] M. Takesaki, Theory of Operator Algebras, I, Springer-Verlag, New York-Heidelberg-Berlin (1979). · Zbl 0436.46043
[9] E. M. Alfsen, F. W. Shultz, and E. Stormer, ?A Gelfand-Neumark theorem for Jordan algebras,? Adv. Math.28, No. 1, 11-56 (1978). · Zbl 0397.46065 · doi:10.1016/0001-8708(78)90044-0
[10] N. Jacobson, ?Structure and representations of Jordan algebras,? Am. Math. Soc. Colloq. Publ.,39, Providence R. I. (1968). · Zbl 0218.17010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.