Classification of injective JW-factors (Q2266415)

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 \textit{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 \textit{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.