Preduals of von Neumann Algebras

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.