Preduals of von Neumann algebras (Q1428953)

In the paper under review, the author summarizes and sketches the proofs of the results given in the papers [Russ. J. Math. Phys. 10, 117--120 (2003; Zbl 1065.46039)] and [Adv. Stud. Contemp. Math., Kyungshang 7, 1--10 (2003; Zbl 1047.46042)]. The main results of the present paper are the following (from the abstract): (1) If the Banach space of a von Neumann algebra \({\mathcal A}\) is the third dual of some Banach space, then the space \({\mathcal A}\) is isometrically isomorphic to the second dual of some von Neumann algebra \(A\) and the von Neumann algebra \(A\) is uniquely determined by its enveloping von Neumann algebra and is the unique second preduals of \(\mathcal{A}\). (2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders. I strongly recommend to see [Zbl 1047.46042] for an assessment of these results.