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??



Isomorphism of preduals implies isomorphism of the $W^*$-algebras or not?
No. Take $M={\bf C}^{\oplus 4}$ (i.e. $\ell\_\infty$ on a 4-element set) and take $N=M\_2({\bf C})$.


Things should get more interesting if you require stricter bounds on the norm of the Banach space isomorphism between the preduals. For instance, isometric isomorphism of the preduals as Banach spaces immediately implies the two von Neumann algebras are isometrically isomorphic, and then based on Kadison's theorem (1951, Annals of Math.) this places various restrictions on how much the two von Neumann algebras can differ.


