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Classification of injective JW-factors. (English. Russian original) Zbl 0561.46028
Funct. Anal. Appl. 18, 234-235 (1984); translation from Funkts. Anal. Prilozh. 18, No. 3, 67-68 (1984).
Let A be a JW-factor - a weakly closed Jordan algebra of self-adjoint operators on a Hilbert space with trivial center. And let $${\mathfrak A}(A)$$ be its enveloping von Neumann algebra which can be identified with the commutant A” of A. A is said to be injective if $${\mathfrak A}(A)$$ is an injective von Neumann algebra. The main result of the paper is based on the uniqueness of injective factors of type $$II_ 1,II_{\infty}$$, $$III_{\lambda}$$, $$0<\lambda \leq 1$$ (for the type $$III_ 1$$ factor see the recent result of U. Haagerup: Connes’ bicentralizer problem and uniqueness of the injective factor of type $$III_ 1$$. - Preprint No.10, Odense University (1984)] and the results of T. Giordano on classification of involutive anti-automorphisms of injective von Neumann factors [J. Funct. Anal. 51, 326-360 (1983; Zbl 0516.46045)]. The main theorem can be reformulated as follows:
Theorem. (i) (resp. (ii), (iii)) Up to isomorphism there are precisely two injective JW-factors of type $$II_ 1$$, (resp. $$II_{\infty}$$, $$III_ 1):$$ one is isomorphic to the self-adjoint part of a von Neumann algebra, the other is not.
(iv) Up to isomorphism there are precisely three injective JW-factors of type $$III_{\lambda}$$, $$0<\lambda <1:$$ one of them is isomorphic to the self-adjoint part of a von Neumann algebra, two others are not.
##### MSC:
 46L35 Classifications of $$C^*$$-algebras 17C65 Jordan structures on Banach spaces and algebras 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.)
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##### References:
 [1] A. Connes, Ann. Math.,104, 73-115 (1976). · Zbl 0343.46042 · doi:10.2307/1971057 [2] D. Topping, Mem. Am. Math. Soc., Vol. 53, Am. Math. Soc., Providence, R. I. (1965), pp. 1-48. [3] E. Stormer, Acta Math.,115, No. 3-4, 165-184 (1966). · Zbl 0139.30502 · doi:10.1007/BF02392206 [4] E. Stormer, Trans. Am. Math. Soc.,130, No. 1, 153-166 (1968). [5] Sh. A. Ayupov, Math. Z.,181, 253-268 (1982). · Zbl 0487.46045 · doi:10.1007/BF01215023 [6] Sh. A. Ayupov, Funkts. Anal. Prilozhen.,17, No. 1, 65-66 (1983). [7] Sh. A. Ayupov, Dokl. Akad. Nauk SSSR,267, No. 3, 521-524 (1982). [8] E. Stormer, Duke Math. J.,47, 145-153 (1980). · Zbl 0462.46044 · doi:10.1215/S0012-7094-80-04711-0 [9] T. Giordano and V. Jones, C. R. Acad. Sci. Paris,290A, 29-31 (1980). [10] T. Giordano, C. R. Acad. Sci. Paris,291A, 583-585 (1981).
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