Geodesic
Preduals of von Neumann Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 2, pp. 92-94.

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Proofs of two assertions are sketched. 1) If the Banach space of a von Neumann algebra $\mathfrak A$ is the third dual of some Banach space, then the space $\mathfrak A$ 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 (up to von Neumann algebra isomorphism) and is the unique second predual of $\mathfrak A$ (up to isometric isomorphism of Banach spaces). 2) An infinite-dimensional von Neumann algebra cannot have preduals of all orders.
Keywords: von Neumann algebra, Banach space, dual, predual. 
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A. I. Shtern. Preduals of von Neumann Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 2, pp. 92-94. http://geodesic.mathdoc.fr/item/FAA_2003_37_2_a10/

[1] Dixmier J., “Sur un théorème de Banach”, Duke Math. J., 15 (1948), 1057–1071 | DOI | MR | Zbl