Isomorphism of preduals implies isomorphism of the $W^*$-algebras or not??

Question

Let $M$ and $N$ are two von Neumann algebras such that their preduals $M_{*}$ and $N_{*}$ are isomorphic in the sense of Banach spaces, does it imply $M$ and $N$ are $*$-isomorphic or not??

Answer 1

No, but you can refine the data of a predual to get a positive answer: https://mathoverflow.net/questions/50302/can-we-recover-a-von-neumann-algebra-from-its-predual