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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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